diff --git a/research/niemeier-lattices/CATALOG.txt b/research/niemeier-lattices/CATALOG.txt new file mode 100644 index 0000000000..fb551c9270 --- /dev/null +++ b/research/niemeier-lattices/CATALOG.txt @@ -0,0 +1,143 @@ +All 23 rooted Niemeier lattices: an exact low-shell census +======================================================= + +The machine-readable table is catalog.json, regenerated by code/catalog.py. +Norm means squared Euclidean norm. A line is an antipodal pair. H is the +common Coxeter number of the irreducible roots, not a binary frame dimension. +This table counts geometry; it does not certify a supplier or a kappa gain. + +Root system H norm 2 lines norm 4 lines in root lattice from glue +D24 46 552 85032 85032 0 +D16 E8 30 360 89640 73256 16384 +E8^3 30 360 89640 89640 0 +A24 25 300 91080 37950 53130 +D12^2 22 264 91944 42792 49152 +A17 E7 18 216 93096 28836 64260 +D10 E7^2 18 216 93096 33064 60032 +A15 D9 16 192 93672 23757 69915 +D8^3 14 168 94248 20520 73728 +A12^2 13 156 94536 16458 78078 +A11 D7 E6 12 144 94824 15227 79597 +E6^4 12 144 94824 16092 78732 +A9^2 D6 10 120 95400 10836 84564 +D6^4 10 120 95400 11304 84096 +A8^3 9 108 95688 8910 86778 +A7^2 D5^2 8 96 95976 7358 88618 +A6^4 7 84 96264 5712 90552 +A5^4 D4 6 72 96552 4332 92220 +D4^6 6 72 96552 4392 92160 +A4^6 5 60 96840 3090 93750 +A3^8 4 48 97128 2040 95088 +A2^12 3 36 97416 1188 96228 +A1^24 2 24 97704 552 97152 + +Derivation and independent checks +--------------------------------- +For q^{norm/2}, the primary-source identity is + + theta_L = E4^3 + (24H-720)Delta. + +Using E4=1+240 q+2160 q^2+... and Delta=q-24 q^2+... gives + + number of norm 2 vectors =24H, + number of norm 4 vectors =196560-576H, + number of norm 4 lines =98280-288H. + +For an irreducible root lattice, the norm 4 vector counts are + + A_n: 6*C(n+1,4), + D_n: 2 n+16*C(n,4), + E6:270, E7:756, E8:2160. + +The A_n vectors have two +1 and two -1 entries. The D_n vectors are ++/-2 e_i or have four +/-1 entries. The exceptional counts are explicitly +enumerated inside E8 by catalog.py, also recovering their 72,126,240 roots. +In a sum R=directsum R_i, a norm 4 vector is either a norm 4 vector in one +component or a pair of norm 2 roots in two components. Hence + + |R_4|=sum_i |(R_i)_4| + sum_{i^3-4]/48. + +Its diagonal is 1. Every nonzero off-diagonal has odd inner product, so +B(v,w)=1+ mod 2 is a fitting binary matrix. Unlike Leech, rooted +lattices may also have off-diagonal inner products +/-3; these are allowed +by this gate and must not be accidentally omitted in finite checks. +The generic real cubic-feature bound is 2600 and the generic binary bound +is 25. These are upper bounds for a particular kernel/fit, not lower +bounds for all constructions. + +Concrete structural cases +------------------------- +* D24+: the added half-integral coset has minimum norm 24/4=6. Thus the + norm 4 shell is exactly 24 axis lines and 8*C(24,4)=85008 four-support + lines. Sparse coordinate support makes this an explicit baseline. + +* E8^3: norm 4 vectors occupy one block with norm 4, or two blocks with + norms 2+2. There are 3*1080+3*(240*240/2)=89640 lines. Three-block cubic + features vanish, removing 8^3=512 dimensions from the 2600-dimensional + homogeneous cubic space. The surviving 360 one-block and 1728 two-block + features explain the 2088 real-rank construction. Two-block support also + gives a binary frame-width bound 17 for those factors. The exact cost + and dyadic factorization are treated separately in this package. + +* D16 E8: this is D16+ orthogonally summed with E8. The additional 16384 + norm 4 lines come from the D16 spinor coset, whose 32768 minimal vectors + have sixteen coordinates +/-1/2. There are only two coordinate blocks; + the E8^3 three-block-vanishing shortcut therefore does not apply. + +* D12^2: the two glue classes (s,v),(v,s) have norms 3+1, with 2048 spinor + vectors and 24 vector-coset vectors per component. Together they add + 2048*24=49152 lines. Their sum class (c,c) first appears at norm 6. + These low-rank block products deserve separate kernel/label treatment. + +* D8^3: a glue code generated by (s,v,v),(v,s,v),(v,v,s) has three + norm profiles 2+1+1 and three profiles 2+2+0 at norm 4. Here a spinor + coset has 128 minimal vectors, and a vector coset has 16. Consequently + the glue contributes [3*128*16^2+3*128^2]/2=73728 lines. The all-spinor + sum class first appears at norm 6. Both kinds of profile must be counted. + +* The remaining entries are candidates for a glue-aware shell treatment, + not rejected lattices. D4^6 offers a small 64-word hexacode glue group; + E6^4 has a 9-word group and triple-component norm 4 glue; A2^12 and A1^24 + have large coding-theoretic glue contributions. The census identifies + their available geometry but does not claim a competitive factorization. + +Natural root-only binary ranks +------------------------------ +For the norm 2 roots, B=1+ mod 2 is legal. After quotienting the +radical, its rank is 1+rank_F2(Cartan), namely + + A_n: n+1 for even n, n for odd n; + D_n: n-1 for even n, n for odd n; + E6: 7, E7: 7, E8: 9. + +These are ranks of this specified fitting matrix, not proved minimum +frame dimensions. They are recorded per component in catalog.json. +For D_n, the coordinate factor has n centers of width n-1, so its center +bill equals its n(n-1) root-line count; this particular unpunctured factor +has no rank deficit. Other factors, puncturing, and shared helper +structures require their own accounting. + +See SOURCES.txt for exact primary-source locations. No conclusion here +rules out all suppliers, all kernels, or all schedules of any lattice. diff --git a/research/niemeier-lattices/CONTINUATION.txt b/research/niemeier-lattices/CONTINUATION.txt new file mode 100644 index 0000000000..89af76df34 --- /dev/null +++ b/research/niemeier-lattices/CONTINUATION.txt @@ -0,0 +1,172 @@ +Further construction search, 2026-10-12 +===================================== + +No new kappa is certified. This pass attempts actual routing constructions +and alternative fitting kernels. This package is submitted as separate +research following PR #598; it does not change the selected exponent. + +Pinned comparison context +---------------- +Fetched origin/main at 5996f9043c27bafa90b23e1c937de92a738f1a76. +The initial open numerical comparison inspected was PR #600, Jeremy Kothe's +dfb7b827990d7b586b3cd778e3d750c050dd4778, reporting +kappa = 0.001612722852614221. Its helper profiles are pinned in +workbench/continuation/endpoint-costs/sources/. These are inherited inputs, +not a fresh independent audit of that PR's multiplication theorem. + +At the time of the research, a freshness check found draft PR #604 at +af8c23b24482386aeed314b379b92482c2fca993, reporting +kappa = 0.001620147045734211. Its then-current description stated that heavy fresh-copy +verification and CI were still running. This is DreamingOfClouds' work with +Claude Opus 5.5 via Claude Code, building on Jeremy Kothe's PR #601. +It ends fourteen cold factory helpers at four-dimensional frames. The +metadata/body is retained in workbench/continuation/PR604.json. The tensor +invoice here uses its explicitly pinned PR #600 profiles, not PR #604. +The all-stage hybrid obstruction also fails the higher reported bound. + +Native tensor lift: concrete, but not competitive +------------------------------------------------ +Remove an orthogonal eight-root frame from each E8 factor. The remaining +112 x 112 relative-sign pairs give 25,088 mixed ports in binary dimension +17. A same-label Walsh change separates two sectors. Each sector is a +disjoint collection of 112 native E8 blocks. + +The construction has 224 native invocations, center charge 14,336 and +112,896 inherited helpers. Exact checks verify the signed transport to the +inherited Hamming-coordinate prototype, all 224 isometric frame embeddings, +the eight-dimensional spectator complements, and rank-eight center descents. +Virtual entrances and hyperplane endpoints retain their native residual +widths. The native scalar word is inherited, not re-emitted and replayed here. + +Pricing the actual PR #600 profiles and paid spectator banks gives best +nominal kappa about 0.000872182222 over odd stage counts 3 through 101 +(seven stages, ATM3 variant). CK is slightly worse. This is about 46% below +the comparison, not a gain. It does not exclude a different shared circuit +or prove a global bound over all stage counts for this tensor construction. + +An exact edge partition provides a possible shared-circuit ingredient: +135 orthogonal eight-root cliques cover every E8 orthogonality edge once, +each root occurring nine times. If G is the root Gram matrix and H is +(G entrywise-squared minus J)/4, then + + H = 3 I - (1/4) sum_C 1_C 1_C^T. + +The decomposition is dyadic. Applying its clique words independently +duplicates source inputs across incomparable frames; the identity alone +does not make those repeated gathers free. + +Spinor routing: actual reusable subwords +--------------------------------------- +A fixed pairing of the 16 coordinates partitions the spinor ports into +128 complementary rational eight-dimensional packets, each with 128 ports. +At a common packet frame, a reversible sum of existing source registers +can feed the opposite packet's targets and then be undone. This single +matching layer needs no extra helper roles. Different pairings generally +require incompatible frame histories, so their costs cannot be combined +by simply reusing the same data registers. + +A complete orthogonal 16-port clique-adjacency word uses 34 dirty helpers. +The three helper classes have paths [2,1,13] eight times, [4,3,9] twelve +times, and [8,7,1] fourteen times. Exact integer replay checks every input +column, nested/common frames, target endpoints, and restoration of all +source and helper values. Across 1,024 Hadamard cliques this is 34,816 +helpers. This is a compensation subword; a compatible center phase is +still required for a full supplier. + +One matching layer can be fused at the clique word's eight-dimensional +stage. Sixteen source pivots at a packet frame hold its clique-half totals. +The invertible binary mixer I+J on those 16 pivots changes each total into +the sum of the other 15. The existing fanout then supplies clique adjacency +XOR matching adjacency, with no additional paid helper paths. The mixer +must be reversed during cleanup; this fusion is over F2. + +The final fused variant cleans each copied helper immediately at the +parent span, before its source pivot joins the sibling total. Its helper +paths become [2,1,1] eight times, [4,3,1] twelve times, and [8,7,1] fourteen +times per clique. The checked rank mass falls from 1,024 to 832 per clique +(18.75%). This includes source and target paths, which remain unchanged. +Across the shell, helper endpoint widths are 4 for 8,192 roles, 8 for +12,288 roles, and 16 for 14,336 roles. The final fused replay checks all +1,056 input columns of a representative 256-port block and rejects an +omitted cleanup operation. All 64 blocks have the same enumerated geometry. +Native admission still needs the exact endpoint wrappers and their paid +spectator banks; this is not an exponent certificate. + +The associated alternative decoder C' = J_H + M has binary rank 1,024. +Within each complementary packet pair, its quotient is + + [[I, A], [A, I]], A = I+J_16, A^2 = I. + +Its image consists of (z,Az). Every nonzero vector has support on both +packet halves, each of which spans its full eight-dimensional space. +Therefore every nonzero image column spans all 16 rational dimensions. +Every rank-minimal center factorization costs 1,024 x 16 = 16,384, equal +to the spinor port count. This fused route has no positive rank surplus +under that center model, despite its cheaper compensation circuit. + +Changing the fitting kernel +---------------------------- +The full-shell hybrid retains the original sparse/sparse and sparse/spinor +orthogonality entries, but replaces spinor/spinor entries by Hadamard-clique +blocks. Exact binary rank is 1,851, versus 815 for the original full kernel. +Its full-width center bill is 29,616 on 30,960 ports. An exact active-pressure +bound gives kappa < 0.000448646990529330 at every odd stage count, even +omitting all helper, center-pressure and idle-pressure costs. + +Further exact section searches retain all quadratic Boolean functions on +four coordinates and k independent cubic terms: + + k 0 1 2 3 4 + spinor ports 1024 2048 4096 8192 16384 + binary decoder rank 112 180 272 392 576 + +Pruning the full-shell spinor interactions to the corresponding disjoint +sections, while keeping all sparse and cross entries, gives full-shell +ranks 2,407, 2,033, 1,675 and 1,359 for k=0,1,2,3 respectively. These are +valid fitting matrices but no cheap complete routing words were found. + +The sparse/spinor cross block has exact rank 470. A tested natural-row +factor has every decoder support spanning the full rational 16-space: +low scalar rank again fails to guarantee a cheap physical decoder. + +The earlier residual D = C + J_H has a stronger obstruction than the +natural-column check: every nonzero vector in its image spans at least +nine rational address dimensions. The proof uses the degree-six Boolean +polynomial for C, invariance of J_H under H, and the minimum-weight bound +for degree-five Boolean polynomials. A support in at most eight rational +dimensions would have at most 128 spinor lines. Its H-derivatives would +have at most 256 entries, below the 512 minimum for a nonzero derivative, +forcing a whole orthogonal 16-port H-clique into that small subspace. + +Consequently any rank-minimal 1,268-column factorization implemented by +full-frame aggregates and private decoder copies pays at least 11,412 +descent ranks. This exceeds the spinor surplus of 7,168. The result covers +every minimal basis of that residual, but not redundant factorizations, +ordinary helper computations or fused circuits with different endpoints. + +Reproduction and scope +----------------------- +workbench/continuation/ contains the scripts and exact receipts under +tensor-routes/, endpoint-costs/, spinor-routes/ and section-search/. +Portable drivers replay the published receipts with outputs outside the +checkout; use the top-level verify.py --workbench command in README.txt. +The native tensor interface is inherited rather than independently +re-emitted. Optional search scripts and partial exploratory receipts are +identified in the subtree READMEs and are not elevated into global bounds. + +The experiments do not establish a new integer-multiplication exponent, +a universal obstruction to Niemeier lattices, or an O(1) indexing result. + +Attribution +------------ +New experiments: DominikScholz with OpenAI GPT-6 Astra in T3 Code. +The tensor control transports the existing E8/Hamming native word and +PR #600 helper profiles; it does not claim those circuits as new. Preserve +the contributor lineage in NOTICE and in the pinned upstream sources. + +Related primary sources consulted for orthogonal-basis and Pauli structure: +M. Waegell and P. K. Aravind, https://arxiv.org/abs/1502.04350; +Simon Schmidt, https://arxiv.org/abs/2209.14906; +K. Abdukhalikov, E. Bannai and S. Suda, https://arxiv.org/abs/0802.1425. +The finite partitions, rank receipts and circuit deductions above are +checked locally, not attributed to those papers. diff --git a/research/niemeier-lattices/COSTS.txt b/research/niemeier-lattices/COSTS.txt new file mode 100644 index 0000000000..576e4e0e11 --- /dev/null +++ b/research/niemeier-lattices/COSTS.txt @@ -0,0 +1,183 @@ +Niemeier supplier budgets: scope and exact comparisons +==================================================== + +These are necessary one-supplier moment screens in the inherited complex +binary-frame model. They are not complete words or overall exponent gains. +The scalar factorizations, binary fits, and center supports are checked +separately by the geometry modules. A native circuit, restored dirty +helpers, physical hosting, a bit-side construction, and outer transfer +checks are still missing. + +The retained bit supplier is already a separate bottleneck. PR592 at +9092320f7932dbe61a78f15b6c5fd6a0cbd41307 cites PR543's bit exponent + + c = 413456747036699736067 / 250000000000000000000000 + = 0.001653826988146798944268. + +Its assembly takes the minimum of the bit and complex exponents, and +reports rejection of the next bit endpoint c+10^-24. The corresponding +nominal outer cap c/(1+c) is about 0.001651096360425896: only 2.37964% +above PR592's declared kappa 0.001612719427075359. A larger overall gain +therefore also needs a better bit supplier. The retained bit input and +assembly source are pinned here as provenance, not freshly replayed. + +Exact scalar inputs and the useful comparison +-------------------------------------------- +The Pauli construction transforms the traceless quadratic factors before +combining them with the linear factor in the other E8 block. This changes +the real center basis and reduces binary support widths on both encoder +and decoder sides. It is not merely a reduction in real matrix rank. + +The main comparisons below all use the same hosting-aware idle model and +the actual heterogeneous center widths. Each optimum is certified over +every odd stage count L >= 3, including the infinite tail. + + candidate N h center bill nominal kappa L + E8 root reference 120 9 72 .004515128332 5 + E8^2 unary-correction factor 30960 17 14144 .005728684107 3 + E8^2 Fano/Hadamard factor 30960 17 13648 .006163602190 3 + E8^2 full Pauli factor 30960 17 12752 .006945535353 3 + E8^3 unary-correction factor 89640 25 38976 .004266157982 3 + E8^3 full Pauli factor 89640 25 35400 .004996800857 3 + D24 norm-four factor 85032 23 55224 .001265452501 5 + D24 with isolated axes omitted 85008 23 55200 .001266293539 5 + +The E8^2 Pauli factor has 816 centers: 256 of width 17 and 560 of width +15. The E8^3 Pauli factor has 2,088 centers: 408 of width 25 and 1,680 +of width 15. Their bills are respectively 12,752 and 35,400. + +The cost verifier binds these parameters directly to E8-SECTIONS.json, +E8-CUBED.json, and D24.json. The older unary two-block comparisons restrict +the certified unary identity and charge every center at the full ambient +width. They are conservative comparisons rather than additional minimal +support claims. + +The old E8 reference already has large zero-helper headroom over the +published frontier. This is why comparing only against the frontier can +mislead. E8^2's new Pauli value is about 53.83% above that same-model E8 +reference, which is useful structural headroom but still not a gain from +a complete construction. All percentages based on these roots retain +the unbuilt helper and bit-side obligations. + +The E8^3 all-full-width fallback, with 2,112 centers each charged at 25, +has nominal kappa only 0.001628114676. Thus center support matters greatly. +The smaller 2,088-center literal cubic factor without its necessary +linear correction is not used as a certified input. The improved Pauli +factor supplies a different exact dyadic rank-minimal factorization. + +Moment model +------------ +Let n_w count centers of support width w, C=sum_w w*n_w, and d=h-1. +For odd L, the inherited paired schedule has + + m=dL+1, W=(L-1)N, D=W-LC. + +Required active data mass is 2NLd at child rank d. Center mass is L*w*n_w +at rank w. For K=(L-1)/2, the idle slots are the first pair's tail +2(K-1)d, and, for p=1,...,K-1, the pre-stretch 2pd, bridge d, and any +positive tail 2(K-p-1)d. An idle slot of rank r carries mass 2Nr and is +optimistically allowed child rank max(r,h), as in PR595's slot relaxation. +The code checks total rank mass = mW-D exactly. No actual owner packing +or helper inventory is inferred from those fractional idle capacities. + +With alpha=kappa/(1-kappa), a child of rank r and mass M has pressure + + g(M,r,alpha)=M*((m/r)^alpha-1). + +The nominal root solves sum g=D. A second, weaker necessary screen +relaxes every center's child rank to h while retaining its actual bill C. +This is separately named relaxed_center_cap in the receipt; it does not +silently replace every center's support width by h in the primary model. + +All bounds use exact rational log and exponential series with explicit +tails. Outward rounding to denominator 10^40 controls intermediate sizes +without sacrificing rigor. Floating arithmetic only nominates a root. +The selected stage's lower root bracket exceeds every other candidate's +upper endpoint in the finite search. To cover all later L, use only the +active climbs and the (L-3)/2 bridge slots: + + pressure/(NL) >= d*alpha*[2 log L + +(1-3/L) log((dL+1)/h)]. + +The right side increases for L>=3. Once it exceeds 1-C/N, it exceeds +D/(NL) at every subsequent stage. This proves the infinite-tail exclusion. +When C>=N, D=L(N-C)-N is negative for every stage immediately. + +Helper and extra-operation budget +-------------------------------- +All 70 distinct width-15 Pauli support frames in E8^2 have restricted +standard-dot rank 13 and radical dimension 2. This does not force padding +to width 17 in the complex framework. The pinned inherited note +notes/general-clifford-frames.tex constructs exact Gaussian-dyadic +representatives for every binary subspace U, including degenerate U. +It proves Fourier rank(T_U)=dim U and nested transition rank +rank(T_V T_U^-1)=dim V-dim U when U is contained in V. Thus a private +center descending from U to zero costs dim U=15. The invertible binary +map used in the construction need not preserve the standard dot product; +its Clifford lift includes the required quadratic phases. No irrational +extension or additional recursive padding is necessary. + +Consequently 12,752 is a justified sum of individual complex center-child +ranks, not merely an unsupported label-dimension estimate. It is not a +realizable total invoice yet: a complete native scalar word, source and +target incidence, adapters, dirty-helper routing and restoration still +need to be supplied and charged. The frame note credits the construction +to icekylinx with OpenAI GPT-6 Astra assistance and its exposition and +integration to Codex. + +For the strongest E8^2 case at L=3, m=49, W=61920 and D=23664. At the +frontier, +20%, and +100% one-supplier targets, unused pressure is about +18,215.1281, 17,122.0550, and 12,738.7135 respectively. This schedule +has no idle slots; there is no free hosting capacity to assign to helpers. + +If an invoice adds one paid rank-r child in every local call, adds no +stock, and removes no previous child, it consumes + + L*r*(m/r)^alpha + +of that unused pressure, including the loss of deficit. At the +20% +target, E8^2 allows at most 5,664 additional rank-one children per call, +or 177 pairs of rank-16 children per call. The latter is far fewer than +816 centers. Thus a factory needing a full enter/exit pair for every +center fails this particular budget. These are exact invoice scenarios, +not universal lower bounds on all possible helper factories. + +More generally, if D changes to D-e and the net pressure changes by +DeltaP after crediting removed children, the necessary condition is +e+DeltaP < the recorded slack. A banked helper, changed controller, or +shared helper must be priced with its actual stock and residual changes. +The receipt also gives L=5,7,9 budgets, where idle slots are available. + +There are exactly 70 distinct width-15 frames, each shared by eight +centers, and all 256 width-17 centers share the full frame. This makes +batching by 71 frame classes a concrete next experiment. For scale only, +71 paid rank-16 enter/exit pairs would fit the 177-pair budget above; +there is no construction showing that one pair per frame class suffices. +Equal center frames allow common-frame scalar additions and can share +encoder work. They do not supply a complete local word automatically. +The scalar shadow must still implement the identity shear and restore +all dirt; this includes the K-I compensation network and its legal target +chains. A target ending at p_t^perp cannot visit the full h-dimensional +center frame. Private center copies descend and scatter, while helpers +must obey their own frame and restoration rules. Sequential reuse is +useful only if those transitions and the intermediate dirt response are +actually restored or charged. Native batching and low-width compensation +are therefore the next concrete circuit tasks. + +Catalog baseline and root-only units +----------------------------------- +As a uniform comparison only, assign h=25 and 2,600 full-width centers +to each norm-four shell. No such rank or factorization is asserted for +the other lattices. Every one of these 23 hypothetical models fails the +frontier, because its C/N exceeds 0.591018161737. This is the globally +optimized necessary center-bill threshold at h=25; the best stage is 5. +For a +20% target that threshold is 0.561851627124, attained at stage 3. +At h=23 the corresponding frontier threshold is 0.608211423703. + +Root shells split orthogonally by irreducible component, so repeated +isomorphic components merely repeat an existing local unit. For the +direct D_n root-coordinate construction, N=n(n-1), each of n coordinate +encoders has width n-1, and C=N. Its deficit is always -N. This excludes +that construction, not all alternate encoders or controllers. The useful +new behavior here comes from norm-four mixed-component ports and their +quadratic operator basis, not from the Niemeier shell count alone. diff --git a/research/niemeier-lattices/CURRENT.txt b/research/niemeier-lattices/CURRENT.txt new file mode 100644 index 0000000000..318699992b --- /dev/null +++ b/research/niemeier-lattices/CURRENT.txt @@ -0,0 +1,100 @@ +Initial construction experiments, 2026-10-12 +======================================== + +Follow-up construction pass: see CONTINUATION.txt and workbench/continuation/. +That pass adds native tensor lifts, a fused spinor compensation subword, +and alternative fitting-kernel searches. No new kappa is certified. + +This research is submitted separately from the earlier Leech/Turyn PR #598. +There is no new certified kappa. These experiments seek a complete paid +construction. CONTINUATION.txt records later results, including a stronger +residual obstruction and the fused proper-ending compensation subword. + +Positive local results +---------------------- +1. E8^2 has a matching small rational bit geometry: use its original + norm-four vectors divided by two as norm-one address vectors in Q^16, + and the F2 scalar decoder C(v,w)=[=0 mod 4]. It has diagonal one + and its off-diagonal support is exactly rational orthogonality. Exact + elimination on all 30,960 rows gives binary rank 815, so full centers cost + 13040. D16+ has the same 30,960-port/rank 815 pair. This supersedes the + expensive h816 dual as the useful bit candidate in INTERFACES.txt. + +2. A legal Pauli encoder PARTIAL word can use 43,056 roles. Fixed-root input + groups first aggregate in 9-dimensional frames, then fan out their eight + totals to the 15-dimensional Pauli frames. The selected root minor has + determinant 256, so the mixer and inverse are dyadic. Exact frame + inclusions and affine dirty-state restoration are checked. It implements + the K contribution, not the missing I-K compensation. + +3. Financially, keeping all 816 complex centers at full width 17 is better: + C = 13,872 avoids 42,240 encoder roles. The bit analogue uses 815 full-width 16 + centers. At a +20% one-supplier moment target, a specified five-stage + banked invoice leaves room for 48,780 complex compensation helpers of + path [1,7,8,1], or 64,651 bit helpers of path [1,7,7,1]. These depend on the + stated source/target chains. They include restored stock and padding; + they are budgets awaiting actual words, not supplier certificates. + +4. On the 16,384 spinor ports of D16+, a Boolean group-basis change turns + the decoder into multiplication by e8 in 14 square-zero variables. + An explicit Darboux transformation gives e8=e4(x1*y1,...,x7*y7). + The correction therefore factors into 35 commuting degree-eight + creation shears, 64 additions each: 2,240 binary additions in that basis. + Exact graded ranks [1,14,91,364,91,14,1] sum 576. The basis changes have + physical costs and are not included in that addition count. + +5. Orthogonal 16-port Hadamard blocks reduce the spinor correction to + clique adjacency plus a residual of rank 1,268. This is exact, but its + literal natural-column decoding remains too expensive. + +Tested constructions that fail +----------------------------- +* E8^2's complex I-K has full rank 30,960. The exact spectrum rules out a + low-rank scalar correction, not a reusable arithmetic circuit. +* The direct 14-layer Mobius transform forces BOTH operands of each gate + into their common frame. Ultimately every wire has visited FULL16. + Output caps have dimension 15, so at least 16,384 extra descent ranks are + needed. The spinor-only rank deficit is negative; the full-shell bit + version has kappa<0.000541880688 at every odd stage count, even with + omitted costs made free. The analogous complex upper bound is + 0.000186502122. Partial and fused transforms are outside this result. +* The specified independent rectangle factory requires at least 1,640,925 + roles for spinor orthogonality. Its ordinary per-role rank 16 accounting + fails even if all center costs are removed. This assumes a+b-1 separate + input/output carriers per rectangle; shared outputs, reusable DAGs, + fused children and different endpoints are not covered by that bound. +* Each natural column of the rank 1,268 residual has 6,420 targets spanning + a 15-dimensional cap. A full-frame aggregate can scatter to that column + at a common line only after paying 15 ranks. Thus the literal column- + basis construction needs 19,020 descent ranks, before other charges. + Different bases and fused routing remain open. + +Reproduction and limitations +---------------------------- +workbench/ retains scripts, exact receipts and derivations from the initial +experiments. Portable replay drivers now regenerate their outputs outside +the checkout and compare them with the frozen receipts. The top-level +README gives the complete normal and optimized commands. Initial sampled +search receipts remain explicitly distinguished from final certificates. +No formal review or end-to-end Niemeier word is claimed. + +Pinned historical frontier context +---------------------- +The earlier cost package deliberately pins PR #592's reported value and +retained PR #543 bit supplier. Its 2.38% cap applies only when that bit input +is kept. A later check found PR #600 reporting kappa 0.001612722852614221 +at dfb7b827990d7b586b3cd778e3d750c050dd4778. PR #589 at +a5afc8c6169ea61ba7c51dfd3cade02cc7b13b46 reports the stronger bit exponent +0.001733427039589844234749; its nominal c/(1+c) is 0.001730427469823603, +about 7.30% above PR #600. These are read upstream claims, not fresh supplier +replays or proof that they compose with a new Niemeier word. PR #589 credits +DreamingOfClouds with Claude Opus 5.5 via Claude Code and its predecessor +authors. Its exact source metadata is retained in workbench/bit-routes. + +Remaining substantive task +------------------------- +A cheaper joint routing/restoration circuit is needed. Neither a small +kernel rank, the 35-shear transformed computation, nor the rank 1,268 +residual suffices if moving into and out of their frames consumes the +saving. A useful next construction must avoid the tested full-frame +round trip and emit its complete scalar word with all helper paths. diff --git a/research/niemeier-lattices/INTERFACES.txt b/research/niemeier-lattices/INTERFACES.txt new file mode 100644 index 0000000000..ae54649ead --- /dev/null +++ b/research/niemeier-lattices/INTERFACES.txt @@ -0,0 +1,132 @@ +E8 section kernels and the two compiler interfaces +================================================== + +This note distinguishes the local certificate in code/e8_sections.py from +an end-to-end exponent claim. Counts and center bills are read from that +certificate; this note does not replace the complete cost receipt. + +1. What the new complex-side pair supplies + +For squared-norm-four lattice representatives v,w, put + + K(v,w) = ((v.w)^3 - 4(v.w))/48, + B(v,w) = 1 + (v.w mod 2). + +Both diagonal entries equal one. At distinct lines, K can be nonzero only +when v.w is odd, so B is then zero. The certified E8^2 section has 30,960 +ports, rank_Q(K)=816 and rank_F2(B)=17. A binary Gram realization of B gives +the norm-one source lines and target caps used by the complex Clifford +frame interface. The symmetric dyadic factorizations in e8_sections.py +supply an exact scalar center and certified binary source-support widths. +Those widths are dimensions over F2, not over Q. + +The use of rational dyadic coefficients is compatible with the retained +complex scalar primitives. The general Clifford frame lemma also admits +degenerate binary intermediate subspaces; their dimensions alone do not +impose an orthonormal-basis condition. It still requires exact frame +representatives, common labels on every scalar gate, and paid residual +recipes. In particular, a matrix factorization by itself is not a local +transparent word, a helper allocation, or a complete recursive instance. + +The relative-sign pair (u,v),(u,-v) has equal binary labels. Consequently +its rational Walsh change H2 on one side and H2/2 on the inverse side can +be performed within an equal-label group. It preserves the local common +frame. This observation removes no helper, exit, restoration or lifetime +obligation unless an emitted word actually uses that change of basis. + +2. Why the bit dimension is not automatically 17 + +The bit interchange interface uses rational address projectors. Its scalar +payload can be F2, or any fixed finite alphabet, but this is independent of +the address-label field. For example, the existing E8 bit word has rational +address dimension eight, while the E8 complex word has binary address +label dimension nine. Its bit checker requires exact rational +orthogonality of each source line to the target cap. + +An F2 scalar product being zero says that an integer lift has even scalar +product; it does not say that its rational scalar product is zero. Neither +a dyadic scalar factorization nor reduction of its coefficients modulo an +odd prime converts a binary orthogonality certificate into a rational one. +Thus the h=17 complex center widths cannot be inserted into the bit cost +model without an additional rational address construction. + +The fixed finite-alphabet transfer does permit the scalar dyadic word to +be reduced modulo an odd prime. That permits finite pointwise gates on +bit inputs encoded as the symbols zero and one, with all scratch restored +at the boundary. The address prime is chosen separately to preserve the +finite rational matrix and minor tables. This transfer changes neither +the rational address geometry nor its charged ranks. + +3. A concrete dual bit construction, with a larger frame + +The same pair can exchange its characteristic-zero and characteristic-two +roles. Take the rational cubic feature space of K as the address space, +and use B as the scalar matrix over F2. The exact symmetric diagonal +factorization gives a nondegenerate rational form with signature (800,16) +in dimension 816. Each port has norm one in this form. Whenever a distinct +pair has B(v,w)=1, its integer scalar product is even, hence K(v,w)=0. +Therefore all entries of the off-diagonal matrix A=B+I can be delivered +along rationally orthogonal source and target lines. + +A finite, deliberately uncompressed local implementation is available: +use the rank-17 factorization of B for 17 full-frame center totals, and a +separate helper for every ordered nonzero entry of A. Copy the associated +source through that helper's norm-one line and inject it into its target +cap. A source line is contained in that cap. Both are nondegenerate, even +though the ambient form is indefinite; nested orthogonal-projector +transitions are therefore legitimate. Apply the standard old-value/new- +value transparent wrapper to restore arbitrary helper and center data. +The target contribution is B+A=I over F2. + +This proves the existence of a local bit scalar/frame construction with +rational frame dimension 816 and a full-frame center bill at most +17*816=13,872. It does not prove a useful bit saving: the direct helper +count can be quadratic in the number of ports. A useful instance needs a +much smaller reversible aggregation circuit, exact nondegeneracy for all +of its rational intermediate frames, and its own complete paid profile. +No minimality assertion is made about the rational frame dimension or +the center bill. + +The complex free Walsh step does not transfer unchanged. For roots u,v +of squared norm two, (u,v).(u,-v)=0, so these two ports have K-pairing zero +and norm one individually. They are distinct orthogonal rational address +lines in this dual construction. A Walsh gate touching both cannot be +placed at their separate one-dimensional source frames without a new +common-frame schedule and its rank charges. + +4. What remains before an exponent claim + +A useful local word must specify both orientations, every scalar operation +and common frame, all scratch and source cleanup on arbitrary input +columns, and actual center-copy lifetimes. Its entire physical recursive +list must then include helper entry and exit residuals, all-role endpoints, +data corrections, adapters, remainder children, and any hosting owners. +The finite scalar/precision, tape, row and stopping inequalities and the +appropriate exact moment engines must be rerun on that list. Finally the +complex and bit receipts must be composed through the retained outer +inequalities. Improving only one rail can leave the other rail as the +limiting quantity. No numerical kappa follows from the kernel ranks or +center bill alone. + +Repository sources for these distinctions + +* notes/general-clifford-frames.tex: arbitrary binary frame subspaces, + exact representatives and nested paid children. +* research/e8-complex-gauge-kernel/PROOF.md, Sections 2.1-2.4: the common- + frame conjugation identity, transparent scalar word, cleanup and copied + centers, including arbitrary dirty inputs and both orientations. +* research/e8-bit-carrier-skeleton/code/bit/independent/checkword.py: + rational line/cap legality and exact scalar F2 replay; this specialized + E8 checker itself is not a checker for the proposed 816-dimensional word. +* notes/partial-swap-construction.tex and notes/structured-bulk-bit.tex: + rational projectors, nondegenerate nested subspaces, partial swaps and + paid ordered-affine wrappers. +* notes/incidence-construction.tex: transparent B+A=I-style scalar + restoration with rational source/target orthogonality, including an + indefinite ambient form with nondegenerate intermediate spans. +* notes/finite-alphabet-transfer.tex and notes/prime-field28-construction.tex: + finite payload alphabet versus rational address geometry. +* notes/stopped-product-interface.tex: adapters, remainders, all-width + reduction and the stopped recurrence. +* certificates/selected-result.json: a selected complete conditional + result; not a record of every more recent open research PR. diff --git a/research/niemeier-lattices/MANIFEST.json b/research/niemeier-lattices/MANIFEST.json new file mode 100644 index 0000000000..f2c4f41cdb --- /dev/null +++ b/research/niemeier-lattices/MANIFEST.json @@ -0,0 +1,139 @@ +{ + "schema": 1, + "scope": "Immutable package inventory; semantic receipt equality is verified separately.", + "sha256": { + "CATALOG.txt": "f1bf81f22a1a1fdea39b897f19cc0eb8f3842cfce77ef2309c1da3c73f04edef", + "CONTINUATION.txt": "f295eb71b9f01b9aab683c487f1843fe9c64be8aebafed10cba36facdea93fdd", + "COSTS.txt": 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Apache-2.0, following the repository license. + +User guidance: examine the other 23 rooted Niemeier lattices, following +the suggestion attributed to David Ryan (@davidryan59) in the supplied +image. Earlier user guidance on Leech, Turyn and code-space transforms +is preserved in the three predecessor packages. This is research with +no new kappa or O(1) indexing claim. + +New work: the reproducible low-shell census; D24+ and E8^3 scalar factors; +E8^2 sections; dyadic Fano elimination; the symmetric Pauli operator-space +factorization and exhaustive support/frame certificates; scoped exact +moment screens; and the complex/bit interface analysis. + +The continuation adds rational h=16 bit labels, D16+ spinor operator +transforms, exact residual/cross ranks, the scoped support obstruction, +tensor frame embeddings and paid invoices, and the clique/matching +compensation subword with earlier cleanup. OpenAI GPT-6 Astra, running +in T3 Code, developed the research, certificates, code and write-up under +DominikScholz's direction. These are research contributions, not a claim +to a new multiplication exponent or to priority for classical identities. + +Lattice classification and theta identities credit J. H. Conway, +N. J. A. Sloane, Gabriele Nebe, Shoyu Nagaoka and Sho Takemori. The E8 +geometry, Fano plane, Walsh/Hadamard transforms and Pauli matrices are +classical constructions; no priority is claimed for them. SOURCES.txt +gives primary sources. Their use and the supplier deductions here are +distinguished from claims in those sources. + +The general binary-subspace Clifford-frame construction is inherited from +icekylinx with the OpenAI GPT-6 Astra assistance credited in +notes/general-clifford-frames.tex. The common-frame scalar interface, +transparent cleanup and copied-center lineage includes Zhihao Chen / +jacklightChen, icekylinx, Jacob Sussman and the other contributors retained +in the repository and predecessor NOTICE files. Cost-source pins are in +sources/COST-SOURCES.json; INTERFACES.txt identifies the inherited +construction sources. + +The exact interval engine comes from research/leech-lattice in PR598. +The inherited hosting relaxation credits DreamingOfClouds and the +disclosed Claude Opus 5.5 assistance via Claude Code for PR595. PR592's +reported comparison and pinned assembly source credit Jeremy Kothe / +jeremykothe2 and the disclosed Claude Opus 5.5 assistance in the omp +harness. The retained bit input credits eumemic with substantial +Anthropic Claude assistance via Claude Code for PR543, and that PR's +predecessor contributors. The cited upstream circuits and endpoints +are not represented as freshly replayed here. + +The tensor transport inherits the native E8/Hamming prototype from +research/frame-sa504, credited to icekylinx with OpenAI GPT-6 Astra +assistance and its preserved Hamming-input lineage. Its exact source hash and inherited +interface are recorded beside verify_tensor.py. Its native circuit is +not freshly replayed by this contribution. The full original attribution +is retained in that predecessor package's NOTICE. + +The continuation's PR600 profiles credit Jeremy Kothe / jeremykothe2 +with the disclosed Claude Opus 5.5 assistance in the omp harness. +Its archived source files and NOTICE.md are retained without modification +in workbench/continuation/endpoint-costs/sources, with exact hashes. +PR604's contemporary proper-endpoint result credits DreamingOfClouds +with Claude Opus 5.5 via Claude Code, building on Jeremy Kothe's PR601. +This package does not claim priority over that work for ending cold +helpers in proper subspaces. PR589's stronger bit-side comparison +likewise credits DreamingOfClouds and its retained predecessor lineage. + +Related orthogonal-basis and Pauli references credit Mordecai Waegell, +P. K. Aravind, Simon Schmidt, K. Abdukhalikov, E. Bannai and S. Suda. +The binary-code background includes A. R. Calderbank, P. J. Cameron, +W. M. Kantor and J. J. Seidel. References are listed in SOURCES.txt; +the supplier deductions and finite experiments are those made here. + +The conditional supplier framework is inherited from original OpenAI #109 +and the community research maintained by Douglas Colkitt / CrocSwap. +The predecessor NOTICE files retain their full supplier lineage. Imported +source files remain unchanged; they are not presented as newly authored. diff --git a/research/niemeier-lattices/PROOF.txt b/research/niemeier-lattices/PROOF.txt new file mode 100644 index 0000000000..dcbf296f20 --- /dev/null +++ b/research/niemeier-lattices/PROOF.txt @@ -0,0 +1,204 @@ +Exact local constructions and scope +================================== + +1. Shells and the common parity kernel + +Take one canonical representative v from each antipodal pair of squared +norm four. For integral lattice pairings t=, define + + K(v,w)=(t^3-4t)/48, B(v,w)=1+(t mod 2). + +Both diagonal entries are one. Distinct canonical lines have |t|<=3. +The even off-diagonal values t=-2,0,2 vanish in K, while odd values have +B=0. Thus K and B form a fitting pair. The values +/-3, present in rooted +lattices, are retained. This observation is an exact universal argument, +not an inference from sampled pairs. + +CATALOG.txt derives the 23 shell counts from the cited theta identity. +The two fully constructed 24-dimensional shells are D24+ and E8^3. +E8^2 denotes the norm-four section with the third E8 block zero; it is +16-dimensional, not another 24-dimensional Niemeier lattice. + +2. D24+ + +The half-integral glue coset has minimum squared norm six, so the norm-four +lines are 24 axes 2e_i and 8*C(24,4)=85008 signed four-coordinate lines. +Modulo two their supports lie in the even-weight subspace of F2^24. +Its radical is the all-ones vector. Quotienting gives an alternating form +of dimension 22, and adjoining the constant direction realizes B in the +standard dot product on F2^23. code/d24.py checks this isometry on a basis +of the whole even-weight space, and checks the full label span is 23. + +Let A_j(v)=v_j/2 on the axes and zero otherwise; let Y_j(v)=v_j off the +axes and zero on them. Use M_ij(v)=v_i^2*v_j for i!=j and +T_ijk(v)=v_i*v_j*v_k for i= 3.11, NumPy and SymPy. Z3 is needed only for the +optional clique discovery script, not for the frozen certificate replay. +The exact interval engine and native Hamming prototype are inherited from +other research directories in this repository and checked by source hash. +From the repository root: + + python3 -B research/niemeier-lattices/verify.py --workbench + python3 -B -I -O research/niemeier-lattices/verify.py --workbench + +Omit --workbench for only the census, geometric factors and initial exact +cost screens. Optional --output /tmp/niemeier-replay writes their combined +receipt outside the package. The workbench drivers use isolated temporary +outputs and compare their regenerated receipts to the committed inputs. +They regenerate coordinate NPZ inputs; the saved cross-factor NPZ is +compared by arrays, not ZIP bytes. Only the MOD4-RANK.json elapsed-seconds +field is excluded from semantic comparisons. Checks use exceptions and +remain active under optimized Python. + +MANIFEST.json hashes the complete package except itself and Python caches. +This binds immutable published inputs; receipt replay checks mathematical +contents. The workbench READMEs identify optional discovery/history files +that are retained for context but are not independent certified results. +Finite certificates do not verify the full multiplication theorem or +freshly replay the cited upstream suppliers and their reported endpoints. + +Reading order and claim boundaries +---------------------------------- +CURRENT.txt records the initial construction attempts; CONTINUATION.txt +records the tensor, fused-word and alternative-kernel follow-up. Both are +research reports with explicitly pinned historical comparisons. +PROOF.txt, CATALOG.txt and catalog.json cover core lattice algebra. +COSTS.txt contains conditional zero-helper screens and moment budgets. +INTERFACES.txt distinguishes complex and bit address fields; its expensive +h=816 bit dual is superseded by the rational h=16 construction in CURRENT. +workbench/continuation/spinor-routes/PROOF.txt states the residual support +obstruction and its implementation assumptions. +SOURCES.txt, NOTICE and the subtree source metadata preserve references, +source pins, contributor lineage and substantial AI disclosure. + +The outstanding route is a joint routing/restoration circuit whose paid +frame histories retain a positive center surplus. Neither a low scalar +rank nor a cheap transformed computation alone establishes that saving. diff --git a/research/niemeier-lattices/SOURCES.txt b/research/niemeier-lattices/SOURCES.txt new file mode 100644 index 0000000000..1fcd3308db --- /dev/null +++ b/research/niemeier-lattices/SOURCES.txt @@ -0,0 +1,63 @@ +Primary sources for the Niemeier census +====================================== + +1. Gabriele Nebe and Neil J. A. Sloane, A Catalogue of Lattices. + https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/index.html + The Niemeier section lists all 23 rooted systems and links their lattice + entries. Used to cross-check the classification and component labels. + +2. Shoyu Nagaoka and Sho Takemori, Notes on theta series for Niemeier lattices, + arXiv:1504.06715. + https://arxiv.org/pdf/1504.06715 + Equation (4), printed page 5, gives the genus-one theta identity + theta_L=E4^3+(24H-720)Delta. Table 1, printed page 9, independently lists + the 24 systems, including Leech, and their common Coxeter numbers. + The coefficient extraction and antipodal line counts in catalog.py are + calculations made here, rather than numerical performance claims in + that paper. Our H denotes the Coxeter number, not the binary frame size. + +3. J. H. Conway and N. J. A. Sloane, On the Enumeration of Lattices of + Determinant One, Journal of Number Theory 15 (1982), 83-94. + https://neilsloane.com/doc/Me81.pdf + Table V on printed page 91 lists the Niemeier root systems, glue generators, + and root counts. Section 4 and Table VI give the completeness check. + The paper identifies the D4^6 glue with the length-six hexacode over F4. + Root-system and discriminant-coset descriptions appear in Section 2. + +4. Nebe–Sloane entries for the exceptional root lattices: + https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/E6.html + https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/E7.html + https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/E8.html + The E8 page supplies a standard integral/half-integral coordinate basis. + The E7 entry identifies the sum-zero hyperplane inside E8; the E6 entry + adds x_1+x_8=0. catalog.py enumerates their norm-two and norm-four shells + exactly in integer coordinates x=2 v, independently checking all counts. + +5. Individual catalogue entries for the first explicit constructions: + https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/NIEM_D24.html + https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/NIEM_D16_E8.html + https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/NIEM_3E8.html + https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/NIEM_2D12.html + https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/NIEM_3D8.html + +All geometric kernels, factor counts, support-width estimates, and algorithmic +screens in this research package are our deductions/computations. The sources +above establish lattice data; they do not establish a matrix-multiplication +exponent or prove optimality of the screened supplier families. + +Related primary literature consulted during the continuation +----------------------------------------------------------- +M. Waegell and P. K. Aravind, https://arxiv.org/abs/1502.04350. +Simon Schmidt, https://arxiv.org/abs/2209.14906. +K. Abdukhalikov, E. Bannai and S. Suda, +https://arxiv.org/abs/0802.1425. +A. R. Calderbank, P. J. Cameron, W. M. Kantor and J. J. Seidel, +Z4-Kerdock codes, orthogonal spreads, and extremal Euclidean line-sets, +https://doi.org/10.1112/S0024611597000403. + +These provide context for orthogonal bases, Pauli structure and binary +code constructions. The specific finite partitions, rank receipts, frame +obstructions and paid circuit deductions in this package are checked here +and are not attributed to those papers. Pinned computational dependencies +are separately recorded in sources/COST-SOURCES.json and the continuation +subtrees; reading a source or a PR is not a fresh supplier replay. diff --git a/research/niemeier-lattices/catalog.json b/research/niemeier-lattices/catalog.json new file mode 100644 index 0000000000..52c9c8af91 --- /dev/null +++ b/research/niemeier-lattices/catalog.json @@ -0,0 +1,1239 @@ +{ + "scope": "Exact shell counts, not a supplier or exponent certificate", + "norm_convention": "Squared Euclidean norm; antipodal pairs count once", + "theta_convention": "q exponent is squared norm / 2", + "theta_identity": "theta_L = E4^3 + (24H-720) Delta", + "norm4_line_formula": "98280 - 288H", + "exceptional_shell_vector_counts": { + "6": { + "2": 72, + "4": 270 + }, + "7": { + "2": 126, + "4": 756 + }, + "8": { + "2": 240, + "4": 2160 + } + }, + "lattices": [ + { + "root_system": "D24", + "coxeter_number": 46, + "root_lines": 552, + "norm4_lines": 85032, + "root_sublattice_norm4_lines": 85032, + "glue_only_norm4_lines": 0, + "glue_group_order": 2, + "components": [ + { + "type": "D", + "rank": 24, + "coxeter_number": 46, + "determinant": 4, + "root_vectors": 1104, + "norm4_vectors": 170064, + "natural_root_fitting_rank": 23 + } + ] + }, + { + "root_system": "D16 E8", + "coxeter_number": 30, + "root_lines": 360, + "norm4_lines": 89640, + "root_sublattice_norm4_lines": 73256, + "glue_only_norm4_lines": 16384, + "glue_group_order": 2, + "components": [ + { + "type": "D", + "rank": 16, + "coxeter_number": 30, + "determinant": 4, + "root_vectors": 480, + "norm4_vectors": 29152, + "natural_root_fitting_rank": 15 + }, + { + "type": "E", + "rank": 8, + "coxeter_number": 30, + "determinant": 1, + "root_vectors": 240, + "norm4_vectors": 2160, + "natural_root_fitting_rank": 9 + } + ] + }, + { + "root_system": "E8^3", + "coxeter_number": 30, + "root_lines": 360, + "norm4_lines": 89640, + "root_sublattice_norm4_lines": 89640, + 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"root_vectors": 2, + "norm4_vectors": 0, + "natural_root_fitting_rank": 1 + }, + { + "type": "A", + "rank": 1, + "coxeter_number": 2, + "determinant": 2, + "root_vectors": 2, + "norm4_vectors": 0, + "natural_root_fitting_rank": 1 + }, + { + "type": "A", + "rank": 1, + "coxeter_number": 2, + "determinant": 2, + "root_vectors": 2, + "norm4_vectors": 0, + "natural_root_fitting_rank": 1 + }, + { + "type": "A", + "rank": 1, + "coxeter_number": 2, + "determinant": 2, + "root_vectors": 2, + "norm4_vectors": 0, + "natural_root_fitting_rank": 1 + }, + { + "type": "A", + "rank": 1, + "coxeter_number": 2, + "determinant": 2, + "root_vectors": 2, + "norm4_vectors": 0, + "natural_root_fitting_rank": 1 + }, + { + "type": "A", + "rank": 1, + "coxeter_number": 2, + "determinant": 2, + "root_vectors": 2, + "norm4_vectors": 0, + "natural_root_fitting_rank": 1 + }, + { + "type": "A", + "rank": 1, + "coxeter_number": 2, + "determinant": 2, + "root_vectors": 2, + "norm4_vectors": 0, + "natural_root_fitting_rank": 1 + }, + { + "type": "A", + "rank": 1, + "coxeter_number": 2, + "determinant": 2, + "root_vectors": 2, + "norm4_vectors": 0, + "natural_root_fitting_rank": 1 + } + ] + } + ] +} diff --git a/research/niemeier-lattices/code/catalog.py b/research/niemeier-lattices/code/catalog.py new file mode 100644 index 0000000000..a5c9263ffb --- /dev/null +++ b/research/niemeier-lattices/code/catalog.py @@ -0,0 +1,159 @@ +"""Exact low-shell census of the 23 rooted Niemeier lattices. + +The classification and theta identity are cited in SOURCES.txt. Root-lattice +norm-four counts are elementary coordinate counts; exceptional counts are +independently enumerated inside E8 here. This is a census, not a cost certificate. +""" +from itertools import combinations, product +from math import comb, isqrt, prod +from pathlib import Path +import json + +ROOT_SYSTEMS = [ + ("D24", [("D", 24, 1)]), + ("D16 E8", [("D", 16, 1), ("E", 8, 1)]), + ("E8^3", [("E", 8, 3)]), + ("A24", [("A", 24, 1)]), + ("D12^2", [("D", 12, 2)]), + ("A17 E7", [("A", 17, 1), ("E", 7, 1)]), + ("D10 E7^2", [("D", 10, 1), ("E", 7, 2)]), + ("A15 D9", [("A", 15, 1), ("D", 9, 1)]), + ("D8^3", [("D", 8, 3)]), + ("A12^2", [("A", 12, 2)]), + ("A11 D7 E6", [("A", 11, 1), ("D", 7, 1), ("E", 6, 1)]), + ("E6^4", [("E", 6, 4)]), + ("A9^2 D6", [("A", 9, 2), ("D", 6, 1)]), + ("D6^4", [("D", 6, 4)]), + ("A8^3", [("A", 8, 3)]), + ("A7^2 D5^2", [("A", 7, 2), ("D", 5, 2)]), + ("A6^4", [("A", 6, 4)]), + ("A5^4 D4", [("A", 5, 4), ("D", 4, 1)]), + ("D4^6", [("D", 4, 6)]), + ("A4^6", [("A", 4, 6)]), + ("A3^8", [("A", 3, 8)]), + ("A2^12", [("A", 2, 12)]), + ("A1^24", [("A", 1, 24)]), +] + + +def require(condition, message): + if not condition: + raise ValueError(message) + + +def exceptional_shell_counts(): + """Enumerate x=2v in the E8 norm-two and norm-four shells.""" + shells = {2: set(), 4: set()} + for support in combinations(range(8), 2): + for signs in product((-2, 2), repeat=2): + x = [0] * 8 + for i, a in zip(support, signs): + x[i] = a + shells[2].add(tuple(x)) + for x in product((-1, 1), repeat=8): + if sum(x) % 4 == 0: + shells[2].add(x) + for i in range(8): + for a in (-4, 4): + x = [0] * 8 + x[i] = a + shells[4].add(tuple(x)) + for support in combinations(range(8), 4): + for signs in product((-2, 2), repeat=4): + x = [0] * 8 + for i, a in zip(support, signs): + x[i] = a + shells[4].add(tuple(x)) + for i in range(8): + for signs in product((-1, 1), repeat=8): + x = list(signs) + x[i] *= 3 + if sum(x) % 4 == 0: + shells[4].add(tuple(x)) + answer = {} + for n in (6, 7, 8): + answer[n] = {} + for norm, shell in shells.items(): + selected = [x for x in shell if (n == 8 or sum(x) == 0) + and (n != 6 or x[0] + x[7] == 0)] + require(all(sum(a * a for a in x) == 4 * norm for x in selected), + f"E{n} shell {norm}: every vector has the required norm") + require(all(tuple(-a for a in x) in shell for x in selected), + f"E{n} shell {norm}: every antipode is present") + answer[n][norm] = len(selected) + require(answer == {6: {2: 72, 4: 270}, 7: {2: 126, 4: 756}, + 8: {2: 240, 4: 2160}}, + "Exact exceptional root and norm-four shell counts") + return answer + + +def component_data(kind, n, exceptional): + if kind == "A": + h, determinant, roots = n + 1, n + 1, n * (n + 1) + norm4 = 6 * comb(n + 1, 4) + natural_binary_rank = n + 1 if n % 2 == 0 else n + elif kind == "D": + h, determinant, roots = 2 * n - 2, 4, 2 * n * (n - 1) + norm4 = 2 * n + 16 * comb(n, 4) + natural_binary_rank = n - 1 if n % 2 == 0 else n + else: + h = {6: 12, 7: 18, 8: 30}[n] + determinant = {6: 3, 7: 2, 8: 1}[n] + roots, norm4 = exceptional[n][2], exceptional[n][4] + natural_binary_rank = {6: 7, 7: 7, 8: 9}[n] + return {"type": kind, "rank": n, "coxeter_number": h, + "determinant": determinant, "root_vectors": roots, + "norm4_vectors": norm4, + "natural_root_fitting_rank": natural_binary_rank} + + +def census(): + exceptional = exceptional_shell_counts() + rows = [] + for name, factors in ROOT_SYSTEMS: + components = [component_data(kind, n, exceptional) + for kind, n, count in factors for _ in range(count)] + require(sum(c["rank"] for c in components) == 24, + f"{name}: component ranks sum to 24") + hs = {c["coxeter_number"] for c in components} + require(len(hs) == 1, f"{name}: components share a Coxeter number") + h = hs.pop() + roots = sum(c["root_vectors"] for c in components) + require(roots == 24 * h, f"{name}: root count equals 24H") + determinant = prod(c["determinant"] for c in components) + glue_order = isqrt(determinant) + require(glue_order * glue_order == determinant, + f"{name}: root determinant is a perfect square") + root_norm4_vectors = sum(c["norm4_vectors"] for c in components) + root_norm4_vectors += sum(a["root_vectors"] * b["root_vectors"] + for a, b in combinations(components, 2)) + require(root_norm4_vectors % 2 == 0, + f"{name}: norm-four vectors form antipodal pairs") + shell4_lines = 98280 - 288 * h + root_norm4_lines = root_norm4_vectors // 2 + require(0 <= root_norm4_lines <= shell4_lines, + f"{name}: root norm-four shell is contained in the full shell") + rows.append({"root_system": name, "coxeter_number": h, + "root_lines": roots // 2, "norm4_lines": shell4_lines, + "root_sublattice_norm4_lines": root_norm4_lines, + "glue_only_norm4_lines": shell4_lines - root_norm4_lines, + "glue_group_order": glue_order, "components": components}) + require(len(rows) == 23, "The census contains all 23 rooted systems") + return {"scope": "Exact shell counts, not a supplier or exponent certificate", + "norm_convention": "Squared Euclidean norm; antipodal pairs count once", + "theta_convention": "q exponent is squared norm / 2", + "theta_identity": "theta_L = E4^3 + (24H-720) Delta", + "norm4_line_formula": "98280 - 288H", + "exceptional_shell_vector_counts": exceptional, "lattices": rows} + + +if __name__ == "__main__": + result = census() + destination = Path(__file__).resolve().parents[1] / "catalog.json" + destination.write_text(json.dumps(result, indent=2) + "\n") + for row in result["lattices"]: + print(f'{row["root_system"]:14} H={row["coxeter_number"]:2}' + f' roots={row["root_lines"]:3} N4={row["norm4_lines"]:5}' + f' root-part={row["root_sublattice_norm4_lines"]:5}' + f' glue-part={row["glue_only_norm4_lines"]:5}' + f' glue-order={row["glue_group_order"]}') diff --git a/research/niemeier-lattices/code/costs.py b/research/niemeier-lattices/code/costs.py new file mode 100644 index 0000000000..e6ed87c2e8 --- /dev/null +++ b/research/niemeier-lattices/code/costs.py @@ -0,0 +1,314 @@ +"""Exact necessary moment screens; no scalar word or exponent claim.""" +from fractions import Fraction +from functools import lru_cache +import hashlib +import importlib.util +import json +from math import expm1, floor, log +from pathlib import Path + +ROOT = Path(__file__).resolve().parents[1] +RESEARCH = ROOT.parent +FRONTIER = Fraction(1612719427075359, 10**18) +FACTORS = (Fraction(1), Fraction(11, 10), Fraction(6, 5), Fraction(3, 2), Fraction(2)) +INTERVAL_SCALE = 10**40 + + +def require(value, message): + if not value: + raise ValueError(message) + + +def source_check(): + specification = json.loads((ROOT/'sources/COST-SOURCES.json').read_text()) + for name, digest in specification['inherited_files'].items(): + require(hashlib.sha256((RESEARCH/name).read_bytes()).hexdigest() == digest, f'Cost source hash: {name}') + for name, digest in specification['repository_files'].items(): + require(hashlib.sha256((RESEARCH.parent/name).read_bytes()).hexdigest() == digest, + f'Frame lemma source hash: {name}') + for name, source in specification['bit_source_files'].items(): + require(hashlib.sha256((ROOT/'sources'/name).read_bytes()).hexdigest() == source['sha256'], + f'Bit comparison source hash: {name}') + bit_data = (ROOT/'sources/PR543-EXPECTED.json').read_bytes() + blob = b'blob '+str(len(bit_data)).encode()+b'\0'+bit_data + require(hashlib.sha1(blob).hexdigest() == specification['bit_citation']['git_blob_sha1'], + 'Original cited bit receipt blob') + require(FRONTIER == Fraction(specification['frontier']['declared_kappa']), 'Pinned comparison value') + return specification + + +def verify_input_links(specification, receipts=None): + """Bind cost parameters to independently replayed geometry or stated bounds.""" + names = ('E8-CUBED.json', 'E8-SECTIONS.json', 'D24.json') + if receipts is None: + receipts = {name: json.loads((ROOT/'evidence'/name).read_text()) for name in names} + cube, sections, d24 = (receipts[name] for name in names) + section = {case['blocks']: case for case in sections['cases']} + expected = {} + + def add(name, ports, h, histogram, source): + expected[name] = dict(ports=ports, binary_dimension=h, + center_width_histogram=histogram, source=source) + + add('E8_cubed_verified_factor', cube['norm_four_lines'], cube['fitting']['dimension'], + cube['factorization']['center_width_histogram'], 'E8-CUBED.json: factorization') + add('E8_cubed_all_full_width_fallback', cube['norm_four_lines'], cube['fitting']['dimension'], + {'25': cube['factorization']['center_count']}, 'E8-CUBED.json: all centers charged at ambient width') + for blocks, word in ((2, 'squared'), (3, 'cubed')): + case = section[blocks] + for suffix, field in (('Fano_factor', 'coordinate_factorization'), + ('Pauli_factor', 'operator_factorization')): + add(f'E8_{word}_{suffix}', case['port_count'], case['fitting']['dimension'], + case[field]['center_width_histogram'], f'E8-SECTIONS.json: blocks={blocks}, {field}') + add('D24_verified_factor', d24['port_count'], d24['binary_dimension'], + d24['center_width_histogram'], 'D24.json: full factor') + omitted = d24['omitted_axis_variant'] + add('D24_omit_axes', omitted['port_count'], d24['binary_dimension'], + omitted['center_width_histogram'], 'D24.json: omitted_axis_variant') + # Restrict the certified unary identity to two blocks. Each block has + # 56 repeated cubics, 56 distinct cubics and 16 unary centers; each + # ordered block pair has 8*36 mixed centers. Full ambient width is safe. + unary_count = 2*(56+56+16)+2*8*36 + add('E8_squared_unary_factor', section[2]['port_count'], section[2]['fitting']['dimension'], + {'17': unary_count}, 'E8-CUBED.json: two-block restriction, ambient-width upper bound') + add('E8_squared_pair_only', section[2]['port_count']-2*cube['e8_norm_four_lines'], + section[2]['fitting']['dimension'], {'17': unary_count}, + 'Same two-block factor restricted to pair ports, ambient-width upper bound') + add('E8_root_reference', cube['e8_root_lines'], 9, {'9': 8}, + 'Inherited eight-coordinate E8 root unit, ambient-width bound') + add('D24_root_coordinate_unit', 24*23, 23, {'23': 24}, + 'D_n root-coordinate formula at n=24') + require(len(specification['cases']) == len(expected), 'Complete cost input coverage') + require({case['name'] for case in specification['cases']} == set(expected), 'Unique cost case names') + for case in specification['cases']: + link = expected[case['name']] + require(all(case[key] == link[key] for key in + ('ports', 'binary_dimension', 'center_width_histogram')), + f'Cost input disagrees with geometry: {case["name"]}') + return expected + + +path = RESEARCH/'leech-lattice/code/screen.py' +spec = importlib.util.spec_from_file_location('frozen_niemeier_moment_bounds', path) +engine = importlib.util.module_from_spec(spec) +spec.loader.exec_module(engine) + + +@lru_cache(maxsize=None) +def logarithm(base): + low, high = engine.ln_bounds(base, terms=24) + return rounded_bounds(low, high) + + +def rounded_bounds(low, high): + # Outward rounding keeps all later sums on one rational denominator. + # This preserves the proof intervals and avoids enormous denominator lcms. + return (Fraction(low.numerator*INTERVAL_SCALE//low.denominator, INTERVAL_SCALE), + Fraction(-((-high.numerator*INTERVAL_SCALE)//high.denominator), INTERVAL_SCALE)) + + +@lru_cache(maxsize=None) +def power_bounds(base, alpha): + low, high = logarithm(base) + lower, upper = engine.exp_bounds(alpha*low, alpha*high, terms=12) + return rounded_bounds(lower, upper) + + +def idle_slots(stages, d): + pairs = (stages-1)//2 + slots = [2*(pairs-1)*d] if pairs > 1 else [] + for pair in range(1, pairs): + slots.extend((2*pair*d, d)) + if pairs-pair-1: + slots.append(2*(pairs-pair-1)*d) + return slots + + +def instance(n, h, histogram, stages, relaxed=False): + require(n > 0 and h >= 2 and stages >= 3 and stages % 2 == 1, 'Moment parameters') + require(all(1 <= width <= h and count >= 0 for width, count in histogram.items()), 'Center widths') + d, m = h-1, (h-1)*stages+1 + bill = sum(width*count for width, count in histogram.items()) + deficit = (stages-1)*n-stages*bill + terms = [(2*n*stages*d, Fraction(m, d))] + terms.extend((stages*width*count, Fraction(m, h if relaxed else width)) + for width, count in sorted(histogram.items()) if count) + terms.extend((2*n*rank, Fraction(m, max(rank, h))) for rank in idle_slots(stages, d)) + require(sum(mass for mass, _ in terms) == m*(stages-1)*n-deficit, 'Exact forced rank-mass identity') + return dict(ports=n, binary_dimension=h, center_bill=bill, stages=stages, m=m, + stock=(stages-1)*n, deficit=deficit, terms=terms) + + +def pressure_bounds(row, alpha): + low = high = Fraction(0) + for mass, base in row['terms']: + lower, upper = power_bounds(base, alpha) + low += mass*(lower-1) + high += mass*(upper-1) + return low, high + + +def nominate(row): + if row['deficit'] <= 0: + return 0. + lo, hi = 0., .02 + for _ in range(60): + a = (lo+hi)/2 + p = sum(mass*expm1(a*log(float(base))) for mass, base in row['terms']) + if p < row['deficit']: + lo = a + else: + hi = a + return lo + + +def scaled(low, high, scale=10**18): + return [low.numerator*scale//low.denominator, -((-high.numerator*scale)//high.denominator)] + + +def bracket(row): + integer = floor(nominate(row)*10**15) + lo, hi = Fraction(integer-2, 10**15), Fraction(integer+3, 10**15) + require(pressure_bounds(row, lo)[1] < row['deficit'] < pressure_bounds(row, hi)[0], 'Certified root bracket') + return lo, hi + + +def tail_bound(n, h, bill, alpha, start): + # Active climbs and the (L-3)/2 bridge slots alone suffice. Each term + # of this divided-by-L bound increases for real L >= start >= 3. + d = h-1 + log_l = logarithm(Fraction(start))[0] + log_bridge = logarithm(Fraction(d*start+1, h))[0] + lower = d*alpha*(2*log_l+(1-Fraction(3, start))*log_bridge) + return lower, Fraction(n-bill, n) + + +def dominates(row, alpha): + linear = alpha*sum(mass*logarithm(base)[0] for mass, base in row['terms']) + return linear > row['deficit'] or pressure_bounds(row, alpha)[0] > row['deficit'] + + +def compare(row, target): + alpha = target/(1-target) + low, high = pressure_bounds(row, alpha) + slack = row['deficit']-high, row['deficit']-low + return dict(target_kappa=str(target), slack_interval_scaled_1e12=scaled(*slack, 10**12), + verdict='NOMINAL_HEADROOM' if slack[0] > 0 else 'FAILS_NECESSARY_SCREEN' if slack[1] < 0 else 'UNDECIDED') + + +def optimize(n, h, histogram, relaxed=False): + bill = sum(width*count for width, count in histogram.items()) + if bill >= n: + return dict(status='NO_POSITIVE_DEFICIT', center_bill=bill, + proof='D=L*(N-C)-N is negative for every positive stage count') + limit = 101 + while True: + rows = [instance(n, h, histogram, stages, relaxed) for stages in range(3, limit+1, 2)] + best = max(rows, key=nominate) + low, high = bracket(best) + tail_low, tail_right = tail_bound(n, h, bill, low, limit) + if tail_low > tail_right: + break + limit = 2*limit-1 + require(limit <= 3201, 'Finite tail search resource bound') + require(all(row is best or row['deficit'] <= 0 or dominates(row, low) for row in rows), + 'Selected stage strictly dominates every other finite candidate') + return dict(status='CERTIFIED_GLOBAL_ODD_STAGE_MAXIMUM', best_stages=best['stages'], m=best['m'], + center_bill=bill, stock=best['stock'], deficit=best['deficit'], + kappa_interval_scaled_1e18=scaled(low/(1+low), high/(1+high)), + finite_odd_stage_range=[3, limit], tail_start=limit, + tail_pressure_lower_bound=str(tail_low), tail_deficit_upper_bound=str(tail_right), + comparisons=[compare(best, FRONTIER*factor) for factor in FACTORS]) + + +def extra_children_budget(row, target, child_rank, copies_per_call=1): + alpha = target/(1-target) + low, high = pressure_bounds(row, alpha) + slack = row['deficit']-high, row['deficit']-low + power_low, power_high = power_bounds(Fraction(row['m'], child_rank), alpha) + cost_low = row['stages']*copies_per_call*child_rank*power_low + cost_high = row['stages']*copies_per_call*child_rank*power_high + if slack[0] < 0: + return dict(child_rank=child_rank, copies_per_call=copies_per_call, admissible_count=0) + lower, upper = slack[0]/cost_high, slack[1]/cost_low + lo, hi = lower.numerator//lower.denominator, upper.numerator//upper.denominator + require(lo == hi, 'Exact integer additional-child budget') + require(lo*cost_high < slack[0] and (lo+1)*cost_low > slack[1], + 'Strict admissibility and rejection of the next child count') + return dict(child_rank=child_rank, copies_per_call=copies_per_call, admissible_count=lo, + scope='Additional paid children in every local call, no stock change and no hosting credit') + + +def center_threshold(h, target): + alpha = target/(1-target) + candidates = [] + for stages in range(3, 102, 2): + row = instance(1, h, {}, stages, True) + low, high = pressure_bounds(row, alpha) + pl, ph = power_bounds(Fraction(row['m'], h), alpha) + numerator_low, numerator_high = stages-1-high, stages-1-low + # Once the numerator is negative, C >= 0 cannot pass at this stage. + if numerator_high <= 0: + continue + require(numerator_low > 0, 'Resolved center-threshold numerator sign') + candidates.append((numerator_low/(stages*ph), numerator_high/(stages*pl), stages)) + best = max(candidates, key=lambda item: item[0]) + require(all(item is best or item[1] < best[0] for item in candidates), 'Exact optimal center threshold stage') + tl, tr = tail_bound(1, h, best[0], alpha, 101) + require(tl > tr, 'All later stages have smaller center threshold') + return dict(binary_dimension=h, target_kappa=str(target), best_stages=best[2], + maximum_center_bill_per_port_interval_scaled_1e12=scaled(best[0], best[1], 10**12), + scope='Relaxed center cap h and inherited hosting-aware slots; a necessary threshold') + + +def run(): + sources = source_check() + specification = json.loads((ROOT/'evidence/COST-INPUTS.json').read_text()) + links = verify_input_links(specification) + results = [] + for case in specification['cases']: + histogram = {int(width): count for width, count in case['center_width_histogram'].items()} + results.append(dict(name=case['name'], input_scope=case['scope'], + fixed_center_widths=optimize(case['ports'], case['binary_dimension'], histogram), + relaxed_center_cap=optimize(case['ports'], case['binary_dimension'], histogram, True))) + budgets = [] + for name in specification['budget_cases']: + candidate = next(case for case in specification['cases'] if case['name'] == name) + histogram = {int(width): count for width, count in candidate['center_width_histogram'].items()} + h = candidate['binary_dimension'] + for stages in (3, 5, 7, 9): + row = instance(candidate['ports'], h, histogram, stages) + entries = [] + for factor in FACTORS: + target = FRONTIER*factor + entries.append(dict(**compare(row, target), additional_children=[ + extra_children_budget(row, target, rank, copies) + for rank, copies in ((1, 1), (h-1, 2))])) + budgets.append(dict(candidate=name, stages=stages, m=row['m'], stock=row['stock'], deficit=row['deficit'], + idle_rank_mass=sum(2*candidate['ports']*rank for rank in idle_slots(stages, h-1)), targets=entries)) + catalog = json.loads((ROOT/'catalog.json').read_text()) + uniform = [] + cap = center_threshold(25, FRONTIER) + cap_upper = Fraction(cap['maximum_center_bill_per_port_interval_scaled_1e12'][1], 10**12) + for lattice in catalog['lattices']: + ratio = Fraction(65000, lattice['norm4_lines']) + require(ratio > cap_upper, 'Uniform catalog baseline fails for all odd stages') + uniform.append(dict(root_system=lattice['root_system'], ports=lattice['norm4_lines'], + hypothetical_center_bill_per_port=str(ratio), + result='FAILS_FRONTIER_FOR_EVERY_ODD_STAGE')) + bit = json.loads((ROOT/'sources/PR543-EXPECTED.json').read_text())['nine'] + bit_low = Fraction(bit['bit_lower']) + bit_high = bit_low+Fraction(1, 10**24) + return dict(status='PASS_SCOPED_NIEMEIER_MOMENT_SCREENS', sources=sources, + geometry_input_links=links, cases=results, + additional_child_budgets=budgets, + cited_bit_limitation=dict(moment_lower=str(bit_low), claimed_rejected_next_endpoint=str(bit_high), + nominal_outer_cap_interval_scaled_1e18=scaled(bit_low/(1+bit_low), bit_high/(1+bit_high)), + maximum_gain_over_frontier_percent_interval_scaled_1e12=scaled( + 100*(bit_low/(1+bit_low)/FRONTIER-1), 100*(bit_high/(1+bit_high)/FRONTIER-1), 10**12), + scope='Retained PR543 bit supplier as cited and repriced by PR592. These upstream moment endpoints were read, not freshly replayed here. Larger overall gains require improving the bit supplier as well.'), + necessary_center_thresholds=[center_threshold(h, FRONTIER*factor) + for h in (23, 25) for factor in (1, Fraction(6, 5))], + uniform_catalog_baseline=uniform, + uniform_catalog_scope='Hypothetical h=25 and 2600 centers of width 25 for each norm-four shell; no rank, labeling, support or factorization claim for the other lattices', + scope='Necessary one-supplier moment screens only. No native scalar word, helper inventory, physical hosting, bit supplier or outer transfer is supplied; no overall kappa improvement is claimed.') diff --git a/research/niemeier-lattices/code/d24.py b/research/niemeier-lattices/code/d24.py new file mode 100644 index 0000000000..aa6010dfbd --- /dev/null +++ b/research/niemeier-lattices/code/d24.py @@ -0,0 +1,116 @@ +"""Exact D24+ norm-four shell and a support-priced dyadic factorization.""" +from collections import Counter, defaultdict +from itertools import combinations, product +import numpy as np + + +def require(condition, message): + if not condition: + raise ValueError(message) + + +def rank_binary(values): + pivots = {} + for value in values: + value = int(value) + while value: + pivot = value.bit_length()-1 + if pivot in pivots: + value ^= pivots[pivot] + else: + pivots[pivot] = value + break + return len(pivots) + + +def label(support): + parity = sum(1 << j for j in support) + q = (parity & ((1 << 22)-1)) ^ (((parity >> 22) & 1)*((1 << 22)-1)) + even = q | ((q.bit_count() & 1) << 22) + return ((1 << 23)-1) ^ even + + +def construct(): + rows = [] + for j in range(24): + row = np.zeros(24, dtype=np.int16) + row[j] = 2 + rows.append(row) + for support in combinations(range(24), 4): + for signs in product((-1, 1), repeat=3): + row = np.zeros(24, dtype=np.int16) + row[list(support)] = (1,)+signs + rows.append(row) + vectors = np.asarray(rows) + require(vectors.shape == (85032, 24), 'D24 norm-four line count') + require(np.all(np.sum(vectors*vectors, axis=1) == 4), 'D24 shell norm') + return vectors + + +def certify_d24(): + vectors = construct() + supports = list(combinations(range(24), 4)) + labels = [label(support) for support in supports] + require(rank_binary([label(())]+labels) == 23, 'D24 binary dimension') + require(all(value.bit_count() % 2 for value in labels), 'Odd standard-dot labels') + # The quotient removes the radical of the even-weight space in F2^24. + # On its basis e_i+e_23, the quotient is q_i=x_i+x_22, i<22. + even_basis = [(i, 23) for i in range(23)] + for first in even_basis: + for second in even_basis: + actual = (label(first) & label(second)).bit_count() % 2 + require(actual == (1 ^ (len(set(first) & set(second)) % 2)), + 'Exact binary pairing on the even-weight basis') + require(label(()) == (1 << 23)-1, 'Constant odd direction') + unary, mixed, triple = defaultdict(list), defaultdict(list), defaultdict(list) + for support, value in zip(supports, labels): + for j in support: + unary[j].append(value) + for pair in combinations(support, 2): + mixed[pair].append(value) + for indices in combinations(support, 3): + triple[indices].append(value) + unary_hist = Counter(rank_binary(unary[j]) for j in range(24)) + mixed_hist = Counter(rank_binary(mixed[pair]) for pair in combinations(range(24), 2)) + triple_hist = Counter(rank_binary(triple[indices]) for indices in combinations(range(24), 3)) + require(unary_hist == {23:24}, 'Unary support dimensions') + require(mixed_hist == {22:276}, 'Mixed support dimensions') + require(triple_hist == {21:2024}, 'Triple support dimensions') + # The diagonal factorization is exact: each center uses the same support + # at source and target, with dyadic signed coefficients. + indices = np.unique(np.r_[np.arange(24), np.linspace(24, len(vectors)-1, 128, dtype=int)]) + sample = vectors[indices].astype(np.int64) + axis = np.zeros_like(sample) + axis[np.abs(sample) == 2] = sample[np.abs(sample) == 2]//2 + y = sample.copy() + y[np.abs(sample) == 2] = 0 + pair_features = np.stack([sample[:, i]**2*sample[:, j] + for i in range(24) for j in range(24) if i != j], axis=1) + triple_features = np.stack([sample[:, i]*sample[:, j]*sample[:, k] + for i,j,k in combinations(range(24),3)], axis=1) + gram_times_16 = 16*(axis @ axis.T) - y @ y.T + pair_features @ pair_features.T + 2*(triple_features @ triple_features.T) + dots = sample @ sample.T + require(np.array_equal(3*gram_times_16, dots**3-4*dots), 'Dyadic cubic identity sample') + sample_labels = [label(np.flatnonzero(row % 2)) for row in sample] + binary_gram = np.array([[(a & b).bit_count() % 2 for b in sample_labels] for a in sample_labels]) + require(np.array_equal(binary_gram, 1 ^ (dots & 1)), 'Standard-dot fitting identity sample') + active = gram_times_16 != 0 + np.fill_diagonal(active, False) + require(not binary_gram[active].any() and np.all(np.diag(gram_times_16) == 16), + 'Normalized fitting sample') + full_hist = {'1':24, '23':24, '22':552, '21':2024} + reduced_hist = {'23':24, '22':552, '21':2024} + return dict(status='PASS_D24_NORM4_DYADIC_FACTORIZATION', + lattice='D24+', port_count=85032, binary_dimension=23, + real_rank=2600, center_count=2624, center_width_histogram=full_hist, + center_bill=sum(int(width)*count for width,count in full_hist.items()), + omitted_axis_variant=dict(port_count=85008, real_rank=2576, + center_count=2600, center_width_histogram=reduced_hist, + center_bill=sum(int(width)*count for width,count in reduced_hist.items())), + shell_types={'axis_lines':24, 'four_coordinate_lines':85008}, + factorization='K=AA^T+(MM^T-YY^T)/16+TT^T/8; A_j=x_j/2 on axes only, Y_j=x_j off axes, M_ij=x_i^2*x_j (i!=j), T_ijk=x_i*x_j*x_k', + universal_identity='Expand (x.y)^3-4(x.y). Unary terms equal 48 A_j A_j -3 Y_j Y_j; mixed coefficients are3 and triple coefficients6. Division by48 gives the displayed dyadic factorization for every shell pair.', + rank_argument='Independent sign characters separate2024 triple functions and24 unary sectors. In unary sectorj, the23 inclusion functions on3-subsets ofthe other23coordinates are independent overQ; the coefficient is(I-J/9)/16 with nonzero eigenvalues1/16 and-14/144. Hence off-axis rank2024+24*23=2576. The24 isolated axis ports add24.', + exact_support_checks=dict(unary=24, unordered_mixed=276, triple=2024), + sample_pairs=len(indices)**2, + scope='Exact scalar factorization, binary fitting, and source/target support bill. No controller, dirty-helper word, hosting map, or kappa certificate is supplied.') diff --git a/research/niemeier-lattices/code/e8_cubed.py b/research/niemeier-lattices/code/e8_cubed.py new file mode 100644 index 0000000000..d4444d142f --- /dev/null +++ b/research/niemeier-lattices/code/e8_cubed.py @@ -0,0 +1,302 @@ +"""Exact E8^3 norm-four shell, binary fit, and dyadic scalar factorization. + +Coordinates are doubled: roots have squared norm 8 and the norm-four shell +has squared norm 16. All rank calculations are over finite fields; no +floating-point rank or sampled all-pairs assertion is used. +""" +from collections import Counter +from fractions import Fraction +from itertools import combinations, combinations_with_replacement, product +import hashlib +import json +import numpy as np + + +def require(condition, message): + if not condition: + raise ValueError(message) + + +def insert_binary(basis, value): + value = int(value) + while value: + pivot = value.bit_length() - 1 + if pivot in basis: + value ^= basis[pivot] + else: + basis[pivot] = value + return True + return False + + +def inverse_binary(matrix): + size = len(matrix) + work = np.concatenate((matrix.copy(), np.eye(size, dtype=np.uint8)), axis=1) + for col in range(size): + choices = np.flatnonzero(work[col:, col]) + require(len(choices) > 0, 'Invertible binary Gram matrix') + pivot = col + int(choices[0]) + work[[col, pivot]] = work[[pivot, col]] + for row in range(size): + if row != col and work[row, col]: + work[row] ^= work[col] + return work[:, size:] + + +def symplectic_transform(matrix): + size = len(matrix) + require(size % 2 == 0 and np.array_equal(matrix, matrix.T) + and not np.diag(matrix).any(), 'Alternating binary form') + transform = np.eye(size, dtype=np.uint8) + work = matrix.copy() + for first in range(0, size, 2): + choices = np.argwhere(work[first:, first:]) + require(len(choices) > 0, 'Nondegenerate alternating form') + a, b = (int(value) + first for value in choices[0]) + for current, target in ((a, first), (b if b != first else a, first + 1)): + work[[current, target]] = work[[target, current]] + work[:, [current, target]] = work[:, [target, current]] + transform[[current, target]] = transform[[target, current]] + for row in range(first + 2, size): + left, right = int(work[row, first + 1]), int(work[row, first]) + if left: + transform[row] ^= transform[first] + if right: + transform[row] ^= transform[first + 1] + work = (transform @ matrix @ transform.T) & 1 + target = np.zeros_like(matrix) + for first in range(0, size, 2): + target[first, first + 1] = target[first + 1, first] = 1 + require(np.array_equal(work, target), 'Symplectic basis identity') + return transform + + +def canonical_lines(vectors): + array = np.asarray(vectors, dtype=np.int16) + first = np.argmax(array != 0, axis=1) + signs = np.where(array[np.arange(len(array)), first] > 0, 1, -1) + return np.unique(array * signs[:, None], axis=0).astype(np.int16) + + +def e8_shells(): + roots, fourth = [], [] + for positions in combinations(range(8), 2): + for signs in product((-2, 2), repeat=2): + row = np.zeros(8, dtype=np.int16) + row[list(positions)] = signs + roots.append(row) + for signs in product((-1, 1), repeat=8): + if sum(signs) % 4 == 0: + roots.append(np.array(signs, dtype=np.int16)) + for coordinate in range(8): + for sign in (-4, 4): + row = np.zeros(8, dtype=np.int16) + row[coordinate] = sign + fourth.append(row) + for positions in combinations(range(8), 4): + for signs in product((-2, 2), repeat=4): + row = np.zeros(8, dtype=np.int16) + row[list(positions)] = signs + fourth.append(row) + for coordinate in range(8): + for signs in product((-1, 1), repeat=8): + row = np.array(signs, dtype=np.int16) + row[coordinate] *= 3 + if int(row.sum()) % 4 == 0: + fourth.append(row) + require(len(roots) == 240 and len(fourth) == 2160, 'E8 shell vector counts') + roots, fourth = canonical_lines(roots), canonical_lines(fourth) + require(roots.shape == (120, 8) and fourth.shape == (1080, 8), 'E8 line counts') + for rows, norm in ((roots, 8), (fourth, 16)): + require(np.all((rows * rows).sum(axis=1) == norm), 'Exact E8 shell norms') + require(np.all(rows.sum(axis=1) % 4 == 0), 'Doubled E8 parity condition') + require(np.all((rows % 2) == (rows[:, :1] % 2)), 'Doubled E8 coordinate parity') + return roots, fourth + + +def construct_shell(): + roots, fourth = e8_shells() + pieces, profiles = [], [] + for block in range(3): + rows = np.zeros((1080, 24), dtype=np.int16) + rows[:, 8 * block:8 * (block + 1)] = fourth + pieces.append(rows) + profiles.extend([1 << block] * len(rows)) + for first, second in combinations(range(3), 2): + rows = np.zeros((28800, 24), dtype=np.int16) + rows[:, 8 * first:8 * (first + 1)] = np.repeat(roots, 240, axis=0) + rows[:, 8 * second:8 * (second + 1)] = ( + np.tile(np.repeat(roots, 2, axis=0), (120, 1)) + * np.tile([1, -1], 14400)[:, None]) + pieces.append(rows) + profiles.extend([(1 << first) | (1 << second)] * len(rows)) + rows = np.concatenate(pieces) + require(rows.shape == (89640, 24), 'E8 cubed norm-four line count') + require(np.all((rows * rows).sum(axis=1) == 16), 'E8 cubed shell norms') + require(len({row.tobytes() for row in rows}) == len(rows), 'Distinct canonical lines') + first = np.argmax(rows != 0, axis=1) + require(np.all(rows[np.arange(len(rows)), first] > 0), 'Canonical antipodal orientation') + return roots, fourth, rows, np.array(profiles, dtype=np.uint8) + + +def binary_labels(roots, ports): + blocks = ports.shape[1] // 8 + dimension = 8 * blocks + 1 + integer_gram = roots @ roots.T + require(not (integer_gram % 4).any(), 'Integral E8 lattice pairings') + gram = ((integer_gram // 4) & 1).astype(np.uint8) + work, available, selected = gram.copy(), list(range(120)), [] + while work[np.ix_(available, available)].any(): + a, b = np.argwhere(work[np.ix_(available, available)])[0] + a, b = available[int(a)], available[int(b)] + selected.extend((a, b)) + left, right = work[a].copy(), work[b].copy() + work ^= np.outer(left, right) ^ np.outer(right, left) + available.remove(a) + available.remove(b) + require(len(selected) == 8, 'E8 mod-two rank eight') + basis = roots[selected] + inverse = inverse_binary(gram[np.ix_(selected, selected)]) + coordinates = np.zeros((len(ports), dimension - 1), dtype=np.uint8) + form = np.zeros((dimension - 1, dimension - 1), dtype=np.uint8) + for block in range(blocks): + pairings = ports[:, 8 * block:8 * (block + 1)] @ basis.T + require(not (pairings % 4).any(), 'Integral shell-basis pairings') + coordinates[:, 8 * block:8 * (block + 1)] = (pairings // 4) & 1 + form[8 * block:8 * (block + 1), 8 * block:8 * (block + 1)] = inverse + even_gram = np.ones((dimension - 1, dimension - 1), dtype=np.uint8) ^ np.eye(dimension - 1, dtype=np.uint8) + change = (inverse_binary(symplectic_transform(form)) + @ symplectic_transform(even_gram)) & 1 + require(np.array_equal((change @ even_gram @ change.T) & 1, form), + 'Standard-dot even-space isometry') + even_basis = np.concatenate((np.eye(dimension - 1, dtype=np.uint8), + np.ones((dimension - 1, 1), dtype=np.uint8)), axis=1) + labels = ((coordinates @ change @ even_basis) & 1) ^ 1 + require(np.all(labels.sum(axis=1) % 2 == 1), 'Odd standard-dot labels') + packed = labels.astype(np.uint32) @ (1 << np.arange(dimension, dtype=np.uint32)) + spanning, witnesses = {}, [] + for index, value in enumerate(packed): + if insert_binary(spanning, value): + witnesses.append(index) + require(len(spanning) == dimension, 'Full binary label span') + samples = np.unique(np.concatenate((witnesses, np.linspace(0, len(ports) - 1, 32, dtype=int)))) + for index in samples: + dots = ports @ ports[index] + require(not (dots % 4).any(), 'Integral shell pairings') + require(np.array_equal((labels @ labels[index]) & 1, 1 ^ ((dots // 4) & 1)), + 'Exact label identity on additional diagnostic rows') + return labels, packed, dict(dimension=dimension, root_basis_indices=selected, + root_basis_vectors=basis.tolist(), alternating_gram=form.tolist(), + standard_dot_coordinate_map=((change @ even_basis) & 1).tolist(), + span_witnesses=witnesses, diagnostic_rows=samples.tolist(), + identity='p(u) dot p(v) = 1 + (u dot v / 4) mod 2', + universal_proof=f'The eight E8 basis roots have an odd integral Gram determinant. In lattice pairing coordinates its rational inverse reduces to the displayed binary inverse. {"Three" if blocks == 3 else "Two"} orthogonal blocks give the alternating rank-{dimension - 1} form; the verified even-space isometry and the orthogonal all-ones vector realize it by the standard dot product in dimension {dimension}.') + + +def modular_rank_witnesses(rows, prime=101): + basis, selected = {}, [] + for index, raw in enumerate(rows): + row = np.asarray(raw, dtype=np.int64).copy() % prime + for pivot in sorted(basis): + row = (row - row[pivot] * basis[pivot]) % prime + nonzero = np.flatnonzero(row) + if not len(nonzero): + continue + pivot = int(nonzero[0]) + basis[pivot] = row * pow(int(row[pivot]), -1, prime) % prime + selected.append(index) + if len(basis) == rows.shape[1]: + break + return selected + + +def real_rank(roots, fourth): + witnesses = {} + for name, rows, degree, expected in ( + ('pure_cubics', fourth, 3, 120), + ('root_quadratics', roots, 2, 36), ('root_linears', roots, 1, 8)): + monomials = list(combinations_with_replacement(range(8), degree)) + features = np.stack([np.prod(rows[:, monomial].astype(np.int64), axis=1) + for monomial in monomials], axis=1) + selected = modular_rank_witnesses(features) + require(len(selected) == expected, f'Exact rational feature rank: {name}') + witnesses[name] = dict(prime=101, rank=expected, independent_rows=selected) + trace_norm = (Fraction(1) + Fraction(23, 3)) / 256 + correction_eigenvalue = 1 - 64 * trace_norm + require(trace_norm == Fraction(13, 384) and correction_eigenvalue == Fraction(-7, 6), + 'Nonzero trace-correction eigenvalue') + return dict(real_kernel_rank=2088, pure_cubic_features=360, mixed_cubic_features=1728, + omitted_three_block_features=512, finite_rank_witnesses=witnesses, + trace_correction_eigenvalue=str(correction_eigenvalue), signature=[2064, 24], + proof='The pure norm-four sectors make all 3 times 120 pure cubics independent. On each root-pair sector, changing one block sign separates the two mixed parities; ranks 36 and 8 give 288 independent features per ordered block pair. Thus the 2088 retained cubic columns are independent. Let D contain cubic multinomial coefficients and represent u_i as (sum_j u_i*u_j^2)/16. The coefficient matrix is (D-64 L L^T)/3072, with L^T D^-1 L=(13/384)I_24. Its rank-24 correction has nonzero eigenvalue -7/6, proving rank 2088 and the displayed signature.') + + +def support_orbit(ports, packed, name, coordinates=None): + if coordinates is not None: + values = np.prod(ports[:, coordinates].astype(np.int32), axis=1) + else: + value = ports[:, 0].astype(np.int32) + quotient = (value ** 3 - value) // 3 + values = 5 * value - quotient if name == 'unary_f' else quotient - value + support = np.flatnonzero(values) + basis, witnesses = {}, [] + for index in support: + if insert_binary(basis, packed[index]): + witnesses.append(int(index)) + return dict(name=name, representative_coordinates=list(coordinates) if coordinates else [0], + support_size=len(support), binary_span=len(basis), independent_ports=witnesses) + + +def factorization(ports, packed): + orbits = [support_orbit(ports, packed, name, coordinates) for name, coordinates in ( + ('pure_cube', (0, 0, 0)), ('pure_repeated', (0, 0, 1)), + ('pure_distinct', (0, 1, 2)), ('mixed_repeated', (0, 0, 8)), + ('mixed_distinct', (0, 1, 8)), ('unary_f', None), ('unary_g', None))] + require([orbit['binary_span'] for orbit in orbits] == [25, 25, 25, 17, 17, 25, 25], + 'Complete binary support spans for each center orbit') + for left, right in product(range(-4, 5), repeat=2): + q_left, q_right = (left ** 3 - left) // 3, (right ** 3 - right) // 3 + require(left ** 3 * right ** 3 - 64 * left * right == + 3 * (-(5 * left - q_left) * (5 * right - q_right) + + 4 * (q_left - left) * (q_right - right)), + 'Dyadic unary factorization identity') + monomials = [indices for indices in combinations_with_replacement(range(24), 3) + if len({index // 8 for index in indices}) <= 2] + counts = Counter() + for indices in monomials: + blocks = len({index // 8 for index in indices}) + distinct = len(set(indices)) + counts[(blocks, distinct)] += 1 + require(dict(counts) == {(1, 1): 24, (1, 2): 168, (1, 3): 168, + (2, 2): 384, (2, 3): 1344}, 'Cubic feature orbit counts') + return dict(center_count=2112, center_width_histogram={'25': 384, '17': 1728}, + center_bill=38976, literal_cubic_only_bill=38376, + literal_cubic_only_bill_scope='Omits the linear correction; not a certified factorization bill.', + support_orbits=orbits, + dyadic_unary_identity='q(u)=(u^3-u)/3; (u^3*v^3-64*u*v)/3072 = (-(5*u-q(u))*(5*v-q(v))+4*(q(u)-u)*(q(v)-v))/1024', + diagonal_coefficients={'unary_f': '-1/1024', 'unary_g': '1/256', + 'repeated_cubic': '1/1024', 'distinct_cubic': '1/512'}, + proof='Expand (u dot v)^3. All three-block monomials vanish on this shell. Keep the 168 pure repeated, 168 pure distinct, 384 mixed repeated, and 1344 mixed distinct monomials; replace each of the 24 pure-cube plus linear terms by the two displayed unary centers. All coefficients are dyadic. The same feature supports occur on encoder and decoder sides because the factorization is symmetric and diagonal. Block and coordinate permutations preserve E8^3 and induce invertible binary label changes, so the seven full-support orbit checks establish all 2112 center widths.', + scope='Exact scalar fitting kernel and symmetric dyadic factorization with binary support bill. A finite recursive word, helper exits, scheduling, and a new kappa are not supplied by this certificate.') + + +def certify_e8_cubed(): + roots, fourth, ports, profiles = construct_shell() + _, packed, fitting = binary_labels(roots, ports) + values = {str(t): str(Fraction(t ** 3 - 4 * t, 48)) for t in range(-4, 5)} + require(values['4'] == '1' and all(values[str(t)] == '0' for t in (-2, 0, 2)), + 'Scalar kernel diagonal and even off-diagonal zeros') + return dict(status='PASS_EXACT_E8_CUBED_SCALAR_FACTORIZATION', + lattice='E8 cubed', doubled_coordinate_scale=2, norm_four_lines=89640, + norm_four_vectors=179280, e8_root_lines=120, e8_norm_four_lines=1080, + profile_counts={str(key): value for key, value in sorted(Counter(map(int, profiles)).items())}, + ports_sha256=hashlib.sha256(ports.astype(' pivot and (basis[higher] >> pivot) & 1: + basis[higher] ^= basis[pivot] + canonical = tuple(basis[pivot] for pivot in sorted(basis)) + require(len(canonical) == 15, 'Every Pauli partner support has dimension15') + partners.append(canonical) + frames[canonical] += 1 + require(len(set(partners)) == 1, 'All eight coordinate partners share one binary frame') + gram_basis = {} + for left in partners[0]: + gram_row = sum((((left & right).bit_count() & 1) << index) + for index, right in enumerate(partners[0])) + e8.insert_binary(gram_basis, gram_row) + require(len(gram_basis) == 13, 'Pauli support frame has a two-dimensional radical') + groups.append(dict(linear_block=first, quadratic_block=second, + pauli_shift_character=[shift, character], binary_frame=list(partners[0]), + restricted_dot_rank=13, radical_dimension=2)) + require(len(frames) == 70 and set(frames.values()) == {8}, 'Exactly 70 shared Pauli frames') + return dict(distinct_width15_frames=70, centers_per_width15_frame=8, + full_width17_frame_centers=256, restricted_width15_dot_rank=13, + width15_radical_dimension=2, groups=groups, + scope='Exact binary support subspaces of the two-block Pauli factorization. The width15 subspaces are degenerate (restricted dot rank13), so any helper theorem must handle those subspaces. Sharing a frame nominates batches; no helper schedule or routing cost is inferred.') + + +def section_rank(roots, fourth, blocks): + baseline = e8.real_rank(roots, fourth) + dimension = 8 * blocks + trace_norm = Fraction(dimension + 2, 768) + eigenvalue = 1 - 64 * trace_norm + expected = 120 * blocks + 288 * blocks * (blocks - 1) + require(eigenvalue != 0, 'Nondegenerate real coefficient form') + return dict(real_kernel_rank=expected, pure_cubic_features=120 * blocks, + mixed_cubic_features=288 * blocks * (blocks - 1), + finite_rank_witnesses=baseline['finite_rank_witnesses'], + trace_correction_eigenvalue=str(eigenvalue), + signature=[expected - dimension, dimension], + proof='Pure sectors and opposite relative signs separate independent cubic spaces, using the displayed finite rank witnesses 120,36,8. For the retained cubic feature matrix F, K=F(D-64 L L^T)F^T/3072. Since the shell norm is16 in doubled coordinates, u_i=sum_j u_i*u_j^2/16, and L^T D^-1 L=((8*blocks+2)/768)I. The nonzero displayed correction eigenvalue proves the exact real rank.') + + +def walsh_sections(): + hadamard = np.array([[1, 1], [1, -1]], dtype=np.int32) + require(np.array_equal(hadamard @ hadamard, 2 * np.eye(2, dtype=np.int32)), + 'Relative-sign Walsh inverse is H/2') + return dict(matrix=hadamard.tolist(), inverse_denominator=2, + same_label_identity='p(a,b)=p(a,-b), since all lattice pairings agree modulo two.', + three_block_sectors=3, ports_per_three_block_sector=29880, + real_rank_per_three_block_sector=696, + two_block_sectors=2, ports_per_two_block_sector=15480, + real_rank_per_two_block_sector=408, + pair_kernel_after_transform='For two AB ports in the odd-A sector, s_A*(s_A^2+3*s_B^2-4)/24, where s_A and s_B are ordinary root inner products.', + proof='Every retained cubic or linear feature is odd in exactly one block. Transform each present relative-sign pair (a,b),(a,-b) by H on inputs and H/2 on outputs. This same-label similarity preserves the binary labels and unit diagonal, and separates the odd-block characters. Pure ports already have one character. At a fixed root a, the same-sector pair kernel is s_B^2/4, whose real rank is36 by the exact root-quadratic witness; this is not an eight-dimensional E8 frame replication.', + scope='An exact free change of basis within equal-label ports. It does not by itself supply improved helper exits or a complete recursive word.') + + +def certify_sections(): + roots, fourth, full, profiles = e8.construct_shell() + results = [] + for blocks in (2, 3): + ids = np.flatnonzero((profiles & (7 ^ ((1 << blocks) - 1))) == 0) + ports = full[ids, :8 * blocks] + expected = 1080 * blocks + 28800 * (blocks * (blocks - 1) // 2) + require(len(ports) == expected, 'Exact E8 section shell count') + _, packed, fitting = e8.binary_labels(roots, ports) + coordinate_factor = fano_factorization(ports, packed, blocks) + frames = support_frames(ports, packed) if blocks == 2 else None + results.append(dict(blocks=blocks, port_count=len(ports), fitting=fitting, + real_rank=section_rank(roots, fourth, blocks), + coordinate_factorization=coordinate_factor, + operator_factorization=operator_factorization(ports, packed, blocks, coordinate_factor), + shared_support_frames=frames)) + return dict(status='PASS_EXACT_E8_SECTION_FACTORIZATIONS', cases=results, + relative_sign_walsh=walsh_sections()) + + +if __name__ == '__main__': + print(json.dumps(certify_sections(), indent=2, sort_keys=True)) diff --git a/research/niemeier-lattices/evidence/COST-INPUTS.json b/research/niemeier-lattices/evidence/COST-INPUTS.json new file mode 100644 index 0000000000..1ad055baee --- /dev/null +++ b/research/niemeier-lattices/evidence/COST-INPUTS.json @@ -0,0 +1,127 @@ +{ + "cases": [ + { + "name": "E8_cubed_verified_factor", + "ports": 89640, + "binary_dimension": 25, + "center_width_histogram": { + "25": 384, + "17": 1728 + }, + "scope": "Exact scalar dyadic factor and source/target support widths; controller and helpers unbuilt" + }, + { + "name": "E8_cubed_all_full_width_fallback", + "ports": 89640, + "binary_dimension": 25, + "center_width_histogram": { + "25": 2112 + }, + "scope": "Same center count charged entirely at width 25; conservative comparison, not an alternative factor" + }, + { + "name": "D24_verified_factor", + "ports": 85032, + "binary_dimension": 23, + "center_width_histogram": { + "1": 24, + "23": 24, + "22": 552, + "21": 2024 + }, + "scope": "Exact scalar dyadic factor including 24 isolated axis ports; controller and helpers unbuilt" + }, + { + "name": "D24_omit_axes", + "ports": 85008, + "binary_dimension": 23, + "center_width_histogram": { + "23": 24, + "22": 552, + "21": 2024 + }, + "scope": "Exact axis-omitted scalar factor; controller and helpers unbuilt" + }, + { + "name": "E8_root_reference", + "ports": 120, + "binary_dimension": 9, + "center_width_histogram": { + "9": 8 + }, + "scope": "Inherited E8 root unit reference with eight full-width real encoders; same zero-helper relaxation" + }, + { + "name": "D24_root_coordinate_unit", + "ports": 552, + "binary_dimension": 23, + "center_width_histogram": { + "23": 24 + }, + "scope": "Root-only D24 coordinate encoding has center bill equal to its port count" + }, + { + "name": "E8_cubed_Fano_factor", + "ports": 89640, + "binary_dimension": 25, + "center_width_histogram": { + "25": 408, + "17": 1344, + "15": 336 + }, + "scope": "Exact rank-minimal Fano/Hadamard dyadic factor; controller and helpers unbuilt" + }, + { + "name": "E8_squared_unary_factor", + "ports": 30960, + "binary_dimension": 17, + "center_width_histogram": { + "17": 832 + }, + "scope": "Two-block restriction of the certified unary dyadic recipe" + }, + { + "name": "E8_squared_Fano_factor", + "ports": 30960, + "binary_dimension": 17, + "center_width_histogram": { + "17": 704, + "15": 112 + }, + "scope": "Exact rank-minimal two-block Fano/Hadamard factor; controller and helpers unbuilt" + }, + { + "name": "E8_squared_pair_only", + "ports": 28800, + "binary_dimension": 17, + "center_width_histogram": { + "17": 832 + }, + "scope": "Pair-only restriction with the conservative 832-center recipe; no rank-minimal factor claimed" + }, + { + "name": "E8_cubed_Pauli_factor", + "ports": 89640, + "binary_dimension": 25, + "center_width_histogram": { + "25": 408, + "15": 1680 + }, + "scope": "Exact rank-minimal symmetric dyadic factor after a Pauli basis change in the traceless quadratic operator space; controller and helpers unbuilt" + }, + { + "name": "E8_squared_Pauli_factor", + "ports": 30960, + "binary_dimension": 17, + "center_width_histogram": { + "17": 256, + "15": 560 + }, + "scope": "Exact rank-minimal symmetric dyadic factor after a Pauli basis change in the traceless quadratic operator space; controller and helpers unbuilt" + } + ], + "budget_cases": [ + "E8_cubed_Pauli_factor", + "E8_squared_Pauli_factor" + ] +} diff --git a/research/niemeier-lattices/evidence/COSTS.json b/research/niemeier-lattices/evidence/COSTS.json new file mode 100644 index 0000000000..c400e02a0f --- /dev/null +++ b/research/niemeier-lattices/evidence/COSTS.json @@ -0,0 +1,2708 @@ +{ + "additional_child_budgets": [ + { + "candidate": "E8_cubed_Pauli_factor", + "deficit": 73080, + "idle_rank_mass": 0, + "m": 73, + "stages": 3, + "stock": 179280, + "targets": [ + { + 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Larger overall gains require improving the bit supplier as well." + }, + "geometry_input_links": { + "D24_omit_axes": { + "binary_dimension": 23, + "center_width_histogram": { + "21": 2024, + "22": 552, + "23": 24 + }, + "ports": 85008, + "source": "D24.json: omitted_axis_variant" + }, + "D24_root_coordinate_unit": { + "binary_dimension": 23, + "center_width_histogram": { + "23": 24 + }, + "ports": 552, + "source": "D_n root-coordinate formula at n=24" + }, + "D24_verified_factor": { + "binary_dimension": 23, + "center_width_histogram": { + "1": 24, + "21": 2024, + "22": 552, + "23": 24 + }, + "ports": 85032, + "source": "D24.json: full factor" + }, + "E8_cubed_Fano_factor": { + "binary_dimension": 25, + "center_width_histogram": { + "15": 336, + "17": 1344, + "25": 408 + }, + "ports": 89640, + "source": "E8-SECTIONS.json: blocks=3, coordinate_factorization" + }, + "E8_cubed_Pauli_factor": { + "binary_dimension": 25, + "center_width_histogram": { + "15": 1680, + "25": 408 + }, + "ports": 89640, + "source": "E8-SECTIONS.json: blocks=3, operator_factorization" + }, + "E8_cubed_all_full_width_fallback": { + "binary_dimension": 25, + "center_width_histogram": { + "25": 2112 + }, + "ports": 89640, + "source": "E8-CUBED.json: all centers charged at ambient width" + }, + "E8_cubed_verified_factor": { + "binary_dimension": 25, + "center_width_histogram": { + "17": 1728, + "25": 384 + }, + "ports": 89640, + "source": "E8-CUBED.json: factorization" + }, + "E8_root_reference": { + "binary_dimension": 9, + "center_width_histogram": { + "9": 8 + }, + "ports": 120, + "source": "Inherited eight-coordinate E8 root unit, ambient-width bound" + }, + "E8_squared_Fano_factor": { + "binary_dimension": 17, + "center_width_histogram": { + "15": 112, + "17": 704 + }, + "ports": 30960, + "source": "E8-SECTIONS.json: blocks=2, coordinate_factorization" + }, + "E8_squared_Pauli_factor": { + "binary_dimension": 17, + "center_width_histogram": { + "15": 560, + "17": 256 + }, + "ports": 30960, + "source": "E8-SECTIONS.json: blocks=2, operator_factorization" + }, + "E8_squared_pair_only": { + "binary_dimension": 17, + "center_width_histogram": { + "17": 832 + }, + "ports": 28800, + "source": "Same two-block factor restricted to pair ports, ambient-width upper bound" + }, + "E8_squared_unary_factor": { + "binary_dimension": 17, + "center_width_histogram": { + "17": 832 + }, + "ports": 30960, + "source": "E8-CUBED.json: two-block restriction, ambient-width upper bound" + } + }, + "necessary_center_thresholds": [ + { + "best_stages": 5, + "binary_dimension": 23, + "maximum_center_bill_per_port_interval_scaled_1e12": [ + 608211423702, + 608211423703 + ], + "scope": "Relaxed center cap h and inherited hosting-aware slots; a necessary threshold", + "target_kappa": "1612719427075359/1000000000000000000" + }, + { + "best_stages": 3, + "binary_dimension": 23, + "maximum_center_bill_per_port_interval_scaled_1e12": [ + 570367347333, + 570367347334 + ], + "scope": "Relaxed center cap h and inherited hosting-aware slots; a necessary threshold", + "target_kappa": "4838158281226077/2500000000000000000" + }, + { + "best_stages": 5, + "binary_dimension": 25, + "maximum_center_bill_per_port_interval_scaled_1e12": [ + 591018161736, + 591018161737 + ], + "scope": "Relaxed center cap h and inherited hosting-aware slots; a necessary threshold", + "target_kappa": "1612719427075359/1000000000000000000" + }, + { + "best_stages": 3, + "binary_dimension": 25, + "maximum_center_bill_per_port_interval_scaled_1e12": [ + 561851627123, + 561851627124 + ], + "scope": "Relaxed center cap h and inherited hosting-aware slots; a necessary threshold", + "target_kappa": "4838158281226077/2500000000000000000" + } + ], + "scope": "Necessary one-supplier moment screens only. No native scalar word, helper inventory, physical hosting, bit supplier or outer transfer is supplied; no overall kappa improvement is claimed.", + "sources": { + "accounting": { + "head": "39112c2577452fa278489fb9fd77a9aecb2cdc24", + "pr": 595, + "scope": "Inherited B_K compressed-window, star-copy and whole-residual hosting assumptions; complete PR analysis not replayed here" + }, + "bit_citation": { + "fresh_circuit_replay": false, + "git_blob_sha1": "b61d859826246e0c3dabbe096e57ad7015eadfd5", + "original_commit": "2065fe4f72a914d907f1046603f5a36f17018535", + "original_path": "research/e8-bit-r420/EXPECTED.json", + "original_pr": 543, + "original_repository": "eumemic/integer-mult-bounds" + }, + "bit_source_files": { + "PR543-EXPECTED.json": { + "sha256": "06eccee9f6b77e140833df83407e7996dc948c17d49b0779ab171fbe463fd28d", + "url": "https://raw.githubusercontent.com/CrocSwap/integer-mult-bounds/71c99ac935a33bb7b59a1b4a7af8725a939b0b90/research/complex-multicold/inputs/PR543-EXPECTED.json" + }, + "PR592-ASSEMBLE.py": { + "sha256": "71fca807f2e70ed7439e369be4bb3be21b487e942b92f0e7629d7521e9398a65", + "url": "https://raw.githubusercontent.com/CrocSwap/integer-mult-bounds/9092320f7932dbe61a78f15b6c5fd6a0cbd41307/research/complex-571-hpeq4/code/assemble.py" + } + }, + "complex_frame_lemma": { + "attribution": "Frame construction credited by the inherited note to icekylinx with OpenAI GPT-6 Astra assistance; exposition and integration use Codex.", + "path": "notes/general-clifford-frames.tex", + "scope": "Exact Gaussian-dyadic representatives for every binary subspace, including degenerate subspaces. Nested U to 0 transitions have Fourier rank dim U; this justifies individual center widths, not an entire helper schedule." + }, + "frontier": { + "declared_kappa": "0.001612719427075359", + "freshly_replayed": false, + "head": "9092320f7932dbe61a78f15b6c5fd6a0cbd41307", + "pr": 592 + }, + "inherited_files": { + "leech-lattice/code/screen.py": "6118abd81e8860bdb551526029d8b36958835de52adb8f50c1a43bb247005385", + "turyn-port-labels/sources/PR592.json": "08c02ef05272279e1a518c77e5661ff9996b00c1b08159af9a8887d2611d2795", + "turyn-port-labels/sources/PR595-SKELETON.py": "9cfbc1316389a46922765cdae85fdde2042f7681986d1c97a19174c4922ea29c", + "turyn-port-labels/sources/PR595.json": "2cc7c3c343a94b880bcb38d5bcc2e37abec6884740e6176806b41cdc4a24f5b3" + }, + "repository_files": { + "notes/general-clifford-frames.tex": "c9bb9655b594583c6b4bab8cb485233f1d1e588f83858cf1e4c3530cee8fed54" + } + }, + "status": "PASS_SCOPED_NIEMEIER_MOMENT_SCREENS", + "uniform_catalog_baseline": [ + { + "hypothetical_center_bill_per_port": "8125/10629", + "ports": 85032, + "result": "FAILS_FRONTIER_FOR_EVERY_ODD_STAGE", + "root_system": "D24" + }, + { + "hypothetical_center_bill_per_port": "1625/2241", + "ports": 89640, + "result": "FAILS_FRONTIER_FOR_EVERY_ODD_STAGE", + "root_system": "D16 E8" + }, + { + "hypothetical_center_bill_per_port": "1625/2241", + "ports": 89640, + "result": "FAILS_FRONTIER_FOR_EVERY_ODD_STAGE", + "root_system": "E8^3" + }, + { + "hypothetical_center_bill_per_port": "1625/2277", + "ports": 91080, + "result": "FAILS_FRONTIER_FOR_EVERY_ODD_STAGE", + "root_system": "A24" + }, + { + "hypothetical_center_bill_per_port": "8125/11493", + "ports": 91944, + "result": "FAILS_FRONTIER_FOR_EVERY_ODD_STAGE", + "root_system": "D12^2" + }, + { + "hypothetical_center_bill_per_port": "8125/11637", + "ports": 93096, + "result": "FAILS_FRONTIER_FOR_EVERY_ODD_STAGE", + "root_system": "A17 E7" + }, + { + "hypothetical_center_bill_per_port": "8125/11637", + "ports": 93096, + "result": "FAILS_FRONTIER_FOR_EVERY_ODD_STAGE", + "root_system": "D10 E7^2" + }, + { + "hypothetical_center_bill_per_port": "8125/11709", + "ports": 93672, + "result": "FAILS_FRONTIER_FOR_EVERY_ODD_STAGE", + "root_system": "A15 D9" + }, + { + "hypothetical_center_bill_per_port": "8125/11781", + "ports": 94248, + "result": "FAILS_FRONTIER_FOR_EVERY_ODD_STAGE", + "root_system": "D8^3" + }, + { + "hypothetical_center_bill_per_port": "625/909", + "ports": 94536, + "result": "FAILS_FRONTIER_FOR_EVERY_ODD_STAGE", + "root_system": "A12^2" + }, + { + "hypothetical_center_bill_per_port": "8125/11853", + "ports": 94824, + "result": "FAILS_FRONTIER_FOR_EVERY_ODD_STAGE", + "root_system": "A11 D7 E6" + }, + { + "hypothetical_center_bill_per_port": "8125/11853", + "ports": 94824, + "result": "FAILS_FRONTIER_FOR_EVERY_ODD_STAGE", + "root_system": "E6^4" + }, + { + "hypothetical_center_bill_per_port": "325/477", + "ports": 95400, + "result": "FAILS_FRONTIER_FOR_EVERY_ODD_STAGE", + "root_system": "A9^2 D6" + }, + { + "hypothetical_center_bill_per_port": "325/477", + "ports": 95400, + "result": "FAILS_FRONTIER_FOR_EVERY_ODD_STAGE", + "root_system": "D6^4" + }, + { + "hypothetical_center_bill_per_port": "8125/11961", + "ports": 95688, + "result": "FAILS_FRONTIER_FOR_EVERY_ODD_STAGE", + "root_system": "A8^3" + }, + { + "hypothetical_center_bill_per_port": "8125/11997", + "ports": 95976, + "result": "FAILS_FRONTIER_FOR_EVERY_ODD_STAGE", + "root_system": "A7^2 D5^2" + }, + { + "hypothetical_center_bill_per_port": "8125/12033", + "ports": 96264, + "result": "FAILS_FRONTIER_FOR_EVERY_ODD_STAGE", + "root_system": "A6^4" + }, + { + "hypothetical_center_bill_per_port": "8125/12069", + "ports": 96552, + "result": "FAILS_FRONTIER_FOR_EVERY_ODD_STAGE", + "root_system": "A5^4 D4" + }, + { + "hypothetical_center_bill_per_port": "8125/12069", + "ports": 96552, + "result": "FAILS_FRONTIER_FOR_EVERY_ODD_STAGE", + "root_system": "D4^6" + }, + { + "hypothetical_center_bill_per_port": "1625/2421", + "ports": 96840, + "result": "FAILS_FRONTIER_FOR_EVERY_ODD_STAGE", + "root_system": "A4^6" + }, + { + "hypothetical_center_bill_per_port": "8125/12141", + "ports": 97128, + "result": "FAILS_FRONTIER_FOR_EVERY_ODD_STAGE", + "root_system": "A3^8" + }, + { + "hypothetical_center_bill_per_port": "8125/12177", + "ports": 97416, + "result": "FAILS_FRONTIER_FOR_EVERY_ODD_STAGE", + "root_system": "A2^12" + }, + { + "hypothetical_center_bill_per_port": "8125/12213", + "ports": 97704, + "result": "FAILS_FRONTIER_FOR_EVERY_ODD_STAGE", + "root_system": "A1^24" + } + ], + "uniform_catalog_scope": "Hypothetical h=25 and 2600 centers of width 25 for each norm-four shell; no rank, labeling, support or factorization claim for the other lattices" +} diff --git a/research/niemeier-lattices/evidence/D24.json b/research/niemeier-lattices/evidence/D24.json new file mode 100644 index 0000000000..94dc9fb252 --- /dev/null +++ b/research/niemeier-lattices/evidence/D24.json @@ -0,0 +1,40 @@ +{ + "binary_dimension": 23, + "center_bill": 55224, + "center_count": 2624, + "center_width_histogram": { + "1": 24, + "21": 2024, + "22": 552, + "23": 24 + }, + "exact_support_checks": { + "triple": 2024, + "unary": 24, + "unordered_mixed": 276 + }, + "factorization": "K=AA^T+(MM^T-YY^T)/16+TT^T/8; A_j=x_j/2 on axes only, Y_j=x_j off axes, M_ij=x_i^2*x_j (i!=j), T_ijk=x_i*x_j*x_k", + "lattice": "D24+", + "omitted_axis_variant": { + "center_bill": 55200, + "center_count": 2600, + "center_width_histogram": { + "21": 2024, + "22": 552, + "23": 24 + }, + "port_count": 85008, + "real_rank": 2576 + }, + "port_count": 85032, + "rank_argument": "Independent sign characters separate2024 triple functions and24 unary sectors. In unary sectorj, the23 inclusion functions on3-subsets ofthe other23coordinates are independent overQ; the coefficient is(I-J/9)/16 with nonzero eigenvalues1/16 and-14/144. Hence off-axis rank2024+24*23=2576. The24 isolated axis ports add24.", + "real_rank": 2600, + "sample_pairs": 23104, + "scope": "Exact scalar factorization, binary fitting, and source/target support bill. No controller, dirty-helper word, hosting map, or kappa certificate is supplied.", + "shell_types": { + "axis_lines": 24, + "four_coordinate_lines": 85008 + }, + "status": "PASS_D24_NORM4_DYADIC_FACTORIZATION", + "universal_identity": "Expand (x.y)^3-4(x.y). Unary terms equal 48 A_j A_j -3 Y_j Y_j; mixed coefficients are3 and triple coefficients6. Division by48 gives the displayed dyadic factorization for every shell pair." +} diff --git a/research/niemeier-lattices/evidence/E8-CUBED.json b/research/niemeier-lattices/evidence/E8-CUBED.json new file mode 100644 index 0000000000..e378f898f6 --- /dev/null +++ b/research/niemeier-lattices/evidence/E8-CUBED.json @@ -0,0 +1,1948 @@ +{ + "doubled_coordinate_scale": 2, + "e8_norm_four_lines": 1080, + "e8_root_lines": 120, + "enumeration_proof": "Every nonzero E8 vector has even norm at least two. Norm four in E8 cubed is therefore either one norm-four block, giving 3*1080 lines, or two root blocks, giving 3*120*120*2 lines. Relative signs are counted as separate scalar ports.", + "factorization": { + "center_bill": 38976, + "center_count": 2112, + "center_width_histogram": { + "17": 1728, + "25": 384 + }, + "diagonal_coefficients": { + "distinct_cubic": "1/512", + "repeated_cubic": "1/1024", + "unary_f": "-1/1024", + "unary_g": "1/256" + }, + "dyadic_unary_identity": "q(u)=(u^3-u)/3; (u^3*v^3-64*u*v)/3072 = (-(5*u-q(u))*(5*v-q(v))+4*(q(u)-u)*(q(v)-v))/1024", + "literal_cubic_only_bill": 38376, + "literal_cubic_only_bill_scope": "Omits the linear correction; not a certified factorization bill.", + "proof": "Expand (u dot v)^3. All three-block monomials vanish on this shell. Keep the 168 pure repeated, 168 pure distinct, 384 mixed repeated, and 1344 mixed distinct monomials; replace each of the 24 pure-cube plus linear terms by the two displayed unary centers. All coefficients are dyadic. The same feature supports occur on encoder and decoder sides because the factorization is symmetric and diagonal. Block and coordinate permutations preserve E8^3 and induce invertible binary label changes, so the seven full-support orbit checks establish all 2112 center widths.", + "scope": "Exact scalar fitting kernel and symmetric dyadic factorization with binary support bill. A finite recursive word, helper exits, scheduling, and a new kappa are not supplied by this certificate.", + "support_orbits": [ + { + "binary_span": 25, + "independent_ports": [ + 287, + 288, + 289, + 290, + 291, + 295, + 303, + 319, + 735, + 13320, + 13322, + 13324, + 13332, + 13344, + 13360, + 13380, + 13404, + 42120, + 42122, + 42124, + 42132, + 42144, + 42160, + 42180, + 42204 + ], + "name": "pure_cube", + "representative_coordinates": [ + 0, + 0, + 0 + ], + "support_size": 38233 + }, + { + "binary_span": 25, + "independent_ports": [ + 287, + 288, + 289, + 290, + 291, + 295, + 303, + 319, + 735, + 13320, + 13322, + 13324, + 13332, + 13344, + 13360, + 13380, + 13404, + 42120, + 42122, + 42124, + 42132, + 42144, + 42160, + 42180, + 42204 + ], + "name": "pure_repeated", + "representative_coordinates": [ + 0, + 0, + 1 + ], + "support_size": 32312 + }, + { + "binary_span": 25, + "independent_ports": [ + 287, + 288, + 289, + 290, + 291, + 295, + 303, + 319, + 735, + 13320, + 13322, + 13324, + 13332, + 13344, + 13360, + 13380, + 13404, + 42120, + 42122, + 42124, + 42132, + 42144, + 42160, + 42180, + 42204 + ], + "name": "pure_distinct", + "representative_coordinates": [ + 0, + 1, + 2 + ], + "support_size": 31272 + }, + { + "binary_span": 17, + "independent_ports": [ + 13404, + 13406, + 13408, + 13410, + 13412, + 13420, + 13436, + 13468, + 13532, + 13644, + 13884, + 14124, + 14364, + 15324, + 17244, + 21084, + 28764 + ], + "name": "mixed_repeated", + "representative_coordinates": [ + 0, + 0, + 8 + ], + "support_size": 12168 + }, + { + "binary_span": 17, + "independent_ports": [ + 13404, + 13406, + 13408, + 13410, + 13412, + 13420, + 13436, + 13468, + 13532, + 13644, + 13884, + 14124, + 14364, + 15324, + 17244, + 21084, + 28764 + ], + "name": "mixed_distinct", + "representative_coordinates": [ + 0, + 1, + 8 + ], + "support_size": 10296 + }, + { + "binary_span": 25, + "independent_ports": [ + 287, + 288, + 289, + 290, + 291, + 295, + 303, + 319, + 735, + 13320, + 13322, + 13324, + 13332, + 13344, + 13360, + 13380, + 13404, + 42120, + 42122, + 42124, + 42132, + 42144, + 42160, + 42180, + 42204 + ], + "name": "unary_f", + "representative_coordinates": [ + 0 + ], + "support_size": 38232 + }, + { + "binary_span": 25, + "independent_ports": [ + 287, + 288, + 289, + 290, + 291, + 295, + 303, + 319, + 1079, + 13320, + 13322, + 13324, + 13332, + 13344, + 13360, + 13380, + 13404, + 42120, + 42122, + 42124, + 42132, + 42144, + 42160, + 42180, + 42204 + ], + "name": "unary_g", + "representative_coordinates": [ + 0 + ], + "support_size": 31233 + } + ] + }, + "fitting": { + "alternating_gram": [ + [ + 0, + 0, + 0, + 1, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0 + ], + [ + 0, + 0, + 1, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0 + ], + [ + 0, + 1, + 0, + 1, + 0, + 1, + 0, + 1, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 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In lattice pairing coordinates its rational inverse reduces to the displayed binary inverse. Three orthogonal blocks give the alternating rank-24 form; the verified even-space isometry and the orthogonal all-ones vector realize it by the standard dot product in dimension 25." + }, + "kernel": { + "fitting_proof": "Distinct canonical norm-four lines have integral inner product t between -3 and 3. Even t gives zero kernel; odd t gives zero binary pairing under the proved label identity. At t=4 the real and binary diagonal entries are both one.", + "formula": "K(u,v)=(t^3-4*t)/48, t=(u dot v)/4", + "values": { + "-1": "1/16", + "-2": "0", + "-3": "-5/16", + "-4": "-1", + "0": "0", + "1": "-1/16", + "2": "0", + "3": "5/16", + "4": "1" + } + }, + "lattice": "E8 cubed", + "norm_four_lines": 89640, + "norm_four_vectors": 179280, + "ports_sha256": "fa9888cd554c640fc70cd61a7ca3ee3e43346d78524dd8e540359d4a5519e4b3", + "profile_counts": { + "1": 1080, + "2": 1080, + "3": 28800, + "4": 1080, + "5": 28800, + "6": 28800 + }, + "real_rank": { + "finite_rank_witnesses": { + "pure_cubics": { + "independent_rows": [ + 0, + 1, + 2, + 3, + 4, + 5, + 6, + 7, + 8, + 9, + 10, + 11, + 12, + 13, + 14, + 15, + 16, + 17, + 18, + 19, + 20, + 21, + 22, + 23, + 24, + 25, + 26, + 28, + 32, + 33, + 34, + 35, + 36, + 40, + 44, + 45, + 46, + 47, + 48, + 49, + 50, + 51, + 52, + 53, + 55, + 56, + 57, + 58, + 63, + 69, + 70, + 71, + 73, + 81, + 101, + 125, + 126, + 127, + 128, + 129, + 130, + 131, + 132, + 133, + 134, + 135, + 136, + 137, + 140, + 141, + 142, + 144, + 145, + 146, + 147, + 158, + 166, + 167, + 168, + 169, + 172, + 190, + 246, + 286, + 287, + 288, + 289, + 290, + 291, + 292, + 293, + 294, + 295, + 296, + 297, + 298, + 299, + 300, + 301, + 302, + 303, + 304, + 305, + 306, + 307, + 311, + 319, + 320, + 321, + 322, + 323, + 327, + 335, + 343, + 347, + 349, + 350, + 415, + 511, + 735 + ], + "prime": 101, + "rank": 120 + }, + "root_linears": { + "independent_rows": [ + 0, + 1, + 2, + 6, + 12, + 20, + 30, + 42 + ], + "prime": 101, + "rank": 8 + }, + "root_quadratics": { + "independent_rows": [ + 0, + 1, + 2, + 3, + 4, + 5, + 6, + 7, + 8, + 9, + 12, + 13, + 14, + 15, + 16, + 20, + 21, + 22, + 23, + 24, + 25, + 30, + 31, + 32, + 33, + 34, + 35, + 36, + 42, + 43, + 44, + 45, + 46, + 50, + 58, + 74 + ], + "prime": 101, + "rank": 36 + } + }, + "mixed_cubic_features": 1728, + "omitted_three_block_features": 512, + "proof": "The pure norm-four sectors make all 3 times 120 pure cubics independent. On each root-pair sector, changing one block sign separates the two mixed parities; ranks 36 and 8 give 288 independent features per ordered block pair. Thus the 2088 retained cubic columns are independent. Let D contain cubic multinomial coefficients and represent u_i as (sum_j u_i*u_j^2)/16. The coefficient matrix is (D-64 L L^T)/3072, with L^T D^-1 L=(13/384)I_24. Its rank-24 correction has nonzero eigenvalue -7/6, proving rank 2088 and the displayed signature.", + "pure_cubic_features": 360, + "real_kernel_rank": 2088, + "signature": [ + 2064, + 24 + ], + "trace_correction_eigenvalue": "-7/6" + }, + "status": "PASS_EXACT_E8_CUBED_SCALAR_FACTORIZATION" +} diff --git a/research/niemeier-lattices/evidence/E8-SECTIONS.json b/research/niemeier-lattices/evidence/E8-SECTIONS.json new file mode 100644 index 0000000000..d950242a5d --- /dev/null +++ b/research/niemeier-lattices/evidence/E8-SECTIONS.json @@ -0,0 +1,7712 @@ +{ + "cases": [ + { + "blocks": 2, + "coordinate_factorization": { + "center_bill": 13648, + "center_count": 816, + "center_width_histogram": { + "15": 112, + "17": 704 + }, + "definitions": "For coordinate i, m_j=u_i*u_j^2 for the seven other coordinates in its E8 block; f=(16*u_i-u_i^3)/3; g=(u_i^3-4*u_i)/3. Let h_T=sum_{j in T}m_j-f for the seven Fano triples. In each other block let b_k=sum_j H[k,j]*u_i*u_j^2, k=0,...,7. Retain all distinct-coordinate cubic monomials supported on at most two blocks. Add g, the seven h_T, the seven b_k with k>0 per other block, and the listed trace center(s) b_0 or b_0+c_0 and b_0-c_0.", + "diagonal_coefficients": { + "fano_h": "1/2048", + "mixed_distinct": "1/512", + "mixed_hadamard": "1/8192", + "pure_distinct": "1/512", + "trace": "-3/8192", + "unary_g": "1/256" + }, + "fano_triples": [ + [ + 0, + 1, + 2 + ], + [ + 0, + 3, + 4 + ], + [ + 0, + 5, + 6 + ], + [ + 1, + 3, + 5 + ], + [ + 1, + 4, + 6 + ], + [ + 2, + 3, + 6 + ], + [ + 2, + 4, + 5 + ] + ], + "hadamard_matrix": [ + [ + 1, + 1, + 1, + 1, + 1, + 1, + 1, + 1 + ], + [ + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1 + ], + [ + 1, + 1, + -1, + -1, + 1, + 1, + -1, + -1 + ], + [ + 1, + -1, + -1, + 1, + 1, + -1, + -1, + 1 + ], + [ + 1, + 1, + 1, + 1, + -1, + -1, + -1, + -1 + ], + [ + 1, + -1, + 1, + -1, + -1, + 1, + -1, + 1 + ], + [ + 1, + 1, + -1, + -1, + -1, + -1, + 1, + 1 + ], + [ + 1, + -1, + -1, + 1, + -1, + 1, + 1, + -1 + ] + ], + "identity": "3*f=sum_j m_j+sum_other_blocks b_0. The Fano identity is sum_j m_j*m_j_prime-f*f_prime = (sum_T h_T*h_T_prime-(sum_other_blocks b_0)*(sum_other_blocks b_0_prime))/2. Hadamard transforms give each other-block contribution sum_k b_k*b_k_prime/8. The displayed trace coefficients diagonalize the remaining one- or two-variable form.", + "scope": "An exact scalar factorization and support bill, not a complete recursive circuit or a new kappa.", + "support_orbits": [ + { + "binary_span": 17, + "independent_ports": [ + 287, + 288, + 289, + 290, + 291, + 295, + 303, + 319, + 1079, + 12240, + 12242, + 12244, + 12252, + 12264, + 12280, + 12300, + 12324 + ], + "multiplicity": 16, + "name": "unary_g", + "support_size": 15873 + }, + { + "binary_span": 17, + "independent_ports": [ + 287, + 288, + 289, + 290, + 291, + 295, + 303, + 319, + 735, + 12240, + 12242, + 12244, + 12252, + 12264, + 12280, + 12300, + 12324 + ], + "multiplicity": 112, + "name": "fano_h", + "support_size": 17928 + }, + { + "binary_span": 15, + "independent_ports": [ + 12246, + 12248, + 12254, + 12266, + 12270, + 12282, + 12302, + 12486, + 12726, + 12966, + 13206, + 14166, + 16086, + 19926, + 27606 + ], + "multiplicity": 112, + "name": "mixed_hadamard", + "support_size": 3744 + }, + { + "binary_span": 17, + "independent_ports": [ + 287, + 288, + 289, + 290, + 291, + 295, + 303, + 319, + 735, + 12240, + 12242, + 12244, + 12252, + 12264, + 12280, + 12300, + 12324 + ], + "multiplicity": 112, + "name": "pure_distinct", + "support_size": 15912 + }, + { + "binary_span": 17, + "independent_ports": [ + 12324, + 12326, + 12328, + 12330, + 12332, + 12340, + 12356, + 12388, + 12452, + 12564, + 12804, + 13044, + 13284, + 14244, + 16164, + 20004, + 27684 + ], + "multiplicity": 448, + "name": "mixed_distinct", + "support_size": 10296 + }, + { + "binary_span": 17, + "independent_ports": [ + 12240, + 12242, + 12244, + 12246, + 12252, + 12264, + 12280, + 12300, + 12324, + 12480, + 12720, + 12960, + 13200, + 14160, + 16080, + 19920, + 27600 + ], + "multiplicity": 16, + "name": "trace", + "support_size": 18720 + } + ], + "support_proof": "All support rows, including zero tests, are exhausted for one representative of each listed orbit. Block permutations and within-block coordinate permutations preserve the lattice and induce invertible binary label maps. They act transitively on the listed center supports: a Fano center chooses a three-subset among seven coordinates, and a nonconstant Hadamard row chooses a balanced four/four partition. Thus the representative support dimensions apply to every center. Encoder and decoder supports agree because this factorization is symmetric and diagonal." + }, + "fitting": { + "alternating_gram": [ + [ + 0, + 0, + 0, + 1, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0 + ], + [ + 0, + 0, + 1, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0 + ], + [ + 0, + 1, + 0, + 1, + 0, + 1, + 0, + 1, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0 + ], + [ + 1, + 0, + 1, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0 + ], + [ + 0, + 0, + 0, + 0, + 0, + 1, + 0, + 1, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0 + ], + [ + 0, + 0, + 1, + 0, + 1, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0 + ], + [ + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 1, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0 + ], + [ + 0, + 0, + 1, + 0, + 1, + 0, + 1, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0 + ], + [ + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 1, + 0, + 0, + 0, + 0 + ], + [ + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 1, + 0, + 0, + 0, + 0, + 0 + ], + [ + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 1, + 0, + 1, + 0, + 1, + 0, + 1 + ], + [ + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 1, + 0, + 1, + 0, + 0, + 0, + 0, + 0 + ], + [ + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 1, + 0, + 1 + ], + [ + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 1, + 0, + 1, + 0, + 0, + 0 + ], + [ + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 1 + ], + [ + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 1, + 0, + 1, + 0, + 1, + 0 + ] + ], + "diagnostic_rows": [ + 0, + 3, + 4, + 12, + 13, + 16, + 45, + 126, + 287, + 998, + 1080, + 1083, + 1092, + 1093, + 1096, + 1125, + 1206, + 1367, + 1997, + 2996, + 3994, + 4993, + 5992, + 6990, + 7989, + 8988, + 9986, + 10985, + 11984, + 12982, + 13981, + 14980, + 15978, + 16977, + 17976, + 18974, + 19973, + 20972, + 21970, + 22969, + 23968, + 24966, + 25965, + 26964, + 27962, + 28961, + 29960, + 30959 + ], + "dimension": 17, + "identity": "p(u) dot p(v) = 1 + (u dot v / 4) mod 2", + "root_basis_indices": [ + 0, + 2, + 1, + 42, + 6, + 12, + 20, + 30 + ], + "root_basis_vectors": [ + [ + 0, + 0, + 0, + 0, + 0, + 0, + 2, + -2 + ], + [ + 0, + 0, + 0, + 0, + 0, + 2, + -2, + 0 + ], + [ + 0, + 0, + 0, + 0, + 0, + 0, + 2, + 2 + ], + [ + 1, + -1, + -1, + -1, + -1, + -1, + -1, + 1 + ], + [ + 0, + 0, + 0, + 0, + 2, + -2, + 0, + 0 + ], + [ + 0, + 0, + 0, + 2, + -2, + 0, + 0, + 0 + ], + [ + 0, + 0, + 2, + -2, + 0, + 0, + 0, + 0 + ], + [ + 0, + 2, + -2, + 0, + 0, + 0, + 0, + 0 + ] + ], + "span_witnesses": [ + 0, + 3, + 4, + 12, + 13, + 16, + 45, + 126, + 287, + 1080, + 1083, + 1092, + 1093, + 1096, + 1125, + 1206, + 1367 + ], + "standard_dot_coordinate_map": [ + [ + 1, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 1 + ], + [ + 1, + 1, + 0, + 1, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 1 + ], + [ + 0, + 1, + 1, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0 + ], + [ + 0, + 1, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 1 + ], + [ + 1, + 1, + 1, + 1, + 1, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 1 + ], + [ + 0, + 0, + 1, + 0, + 0, + 1, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0 + ], + [ + 1, + 1, + 1, + 1, + 1, + 1, + 1, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 1 + ], + [ + 1, + 1, + 0, + 1, + 1, + 0, + 0, + 1, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 1 + ], + [ + 1, + 1, + 1, + 1, + 1, + 1, + 1, + 1, + 1, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 1 + ], + [ + 1, + 1, + 1, + 1, + 1, + 1, + 1, + 1, + 1, + 1, + 0, + 1, + 0, + 0, + 0, + 0, + 1 + ], + [ + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 1, + 1, + 0, + 0, + 0, + 0, + 0, + 0 + ], + [ + 1, + 1, + 1, + 1, + 1, + 1, + 1, + 1, + 0, + 1, + 0, + 0, + 0, + 0, + 0, + 0, + 1 + ], + [ + 1, + 1, + 1, + 1, + 1, + 1, + 1, + 1, + 1, + 1, + 1, + 1, + 1, + 0, + 0, + 0, + 1 + ], + [ + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 1, + 0, + 0, + 1, + 0, + 0, + 0 + ], + [ + 1, + 1, + 1, + 1, + 1, + 1, + 1, + 1, + 1, + 1, + 1, + 1, + 1, + 1, + 1, + 0, + 1 + ], + [ + 1, + 1, + 1, + 1, + 1, + 1, + 1, + 1, + 1, + 1, + 0, + 1, + 1, + 0, + 0, + 1, + 1 + ] + ], + "universal_proof": "The eight E8 basis roots have an odd integral Gram determinant. In lattice pairing coordinates its rational inverse reduces to the displayed binary inverse. Two orthogonal blocks give the alternating rank-16 form; the verified even-space isometry and the orthogonal all-ones vector realize it by the standard dot product in dimension 17." + }, + "operator_factorization": { + "center_bill": 12752, + "center_count": 816, + "center_width_histogram": { + "15": 560, + "17": 256 + }, + "full_factorization_diagnostic_rows": [ + 0, + 1, + 2, + 3, + 4, + 5, + 6, + 7, + 8, + 9, + 10, + 11, + 12, + 13, + 14, + 15, + 16, + 17, + 18, + 19, + 20, + 21, + 22, + 23, + 24, + 25, + 26, + 27, + 28, + 29, + 30, + 31, + 32, + 33, + 34, + 35, + 36, + 37, + 38, + 39, + 40, + 41, + 42, + 43, + 44, + 45, + 46, + 47, + 48, + 49, + 50, + 51, + 52, + 53, + 54, + 55, + 56, + 57, + 58, + 59, + 60, + 61, + 62, + 63, + 243, + 487, + 731, + 975, + 1218, + 1462, + 1706, + 1950, + 2193, + 2437, + 2681, + 2925, + 3169, + 3412, + 3656, + 3900, + 4144, + 4387, + 4631, + 4875, + 5119, + 5362, + 5606, + 5850, + 6094, + 6338, + 6581, + 6825, + 7069, + 7313, + 7556, + 7800, + 8044, + 8288, + 8532, + 8775, + 9019, + 9263, + 9507, + 9750, + 9994, + 10238, + 10482, + 10725, + 10969, + 11213, + 11457, + 11701, + 11944, + 12188, + 12432, + 12676, + 12919, + 13163, + 13407, + 13651, + 13894, + 14138, + 14382, + 14626, + 14870, + 15113, + 15357, + 15601, + 15845, + 16088, + 16332, + 16576, + 16820, + 17064, + 17307, + 17551, + 17795, + 18039, + 18282, + 18526, + 18770, + 19014, + 19257, + 19501, + 19745, + 19989, + 20233, + 20476, + 20720, + 20964, + 21208, + 21451, + 21695, + 21939, + 22183, + 22426, + 22670, + 22914, + 23158, + 23402, + 23645, + 23889, + 24133, + 24377, + 24620, + 24864, + 25108, + 25352, + 25596, + 25839, + 26083, + 26327, + 26571, + 26814, + 27058, + 27302, + 27546, + 27789, + 28033, + 28277, + 28521, + 28765, + 29008, + 29252, + 29496, + 29740, + 29983, + 30227, + 30471, + 30715, + 30959 + ], + "mixed_center_coefficient": "1/8192", + "mixed_center_rule": "For each ordered pair of distinct E8 blocks A,B, each coordinate i in A, and each of the 35 operators P, use u_A,i * (u_B^T P u_B).", + "operator_rule": "P_(a,b)e_x=(-1)^(b dot x)e_(x+a), a,b,x in F2^3, a dot b=0. 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The identity replaces the seven nonconstant diagonal Hadamard centers and 28 mixed off-diagonal cubic centers per ordered block/coordinate by35 Pauli centers, all with coefficient1/8192. The trace direction is unchanged, so the prior Fano trace diagonalization and the full kernel identity remain exact. All35 support sets are exhausted on actual ports; block permutations and permutations within the linear block cover all coordinate choices. Encoder and decoder supports agree in this symmetric dyadic diagonal factorization.", + "quadratic_identity": "sum_{P nonidentity} (u^T P u)*(v^T P v) = 8*(u dot v)^2 - (u dot u)*(v dot v)", + "retained_diagonal_coefficients": { + "fano_h": "1/2048", + "pure_distinct": "1/512", + "trace": "-3/8192", + "unary_g": "1/256" + }, + "retained_support_orbits": [ + { + "binary_span": 17, + "independent_ports": [ + 287, + 288, + 289, + 290, + 291, + 295, + 303, + 319, + 1079, + 12240, + 12242, + 12244, + 12252, + 12264, + 12280, + 12300, + 12324 + ], + "multiplicity": 16, + "name": "unary_g", + "support_size": 15873 + }, + { + "binary_span": 17, + "independent_ports": [ + 287, + 288, + 289, + 290, + 291, + 295, + 303, + 319, + 735, + 12240, + 12242, + 12244, + 12252, + 12264, + 12280, + 12300, + 12324 + ], + "multiplicity": 112, + "name": "fano_h", + "support_size": 17928 + }, + { + "binary_span": 17, + "independent_ports": [ + 287, + 288, + 289, + 290, + 291, + 295, + 303, + 319, + 735, + 12240, + 12242, + 12244, + 12252, + 12264, + 12280, + 12300, + 12324 + ], + "multiplicity": 112, + "name": "pure_distinct", + "support_size": 15912 + }, + { + "binary_span": 17, + "independent_ports": [ + 12240, + 12242, + 12244, + 12246, + 12252, + 12264, + 12280, + 12300, + 12324, + 12480, + 12720, + 12960, + 13200, + 14160, + 16080, + 19920, + 27600 + ], + "multiplicity": 16, + "name": "trace", + "support_size": 18720 + } + ], + "scope": "Exact rank-minimal scalar factorization, binary support bill, and a genuine reduction versus the coordinate basis. Helper exits, a finite recursive word, and a new kappa remain separate obligations." + }, + "port_count": 30960, + "real_rank": { + "finite_rank_witnesses": { + "pure_cubics": { + "independent_rows": [ + 0, + 1, + 2, + 3, + 4, + 5, + 6, + 7, + 8, + 9, + 10, + 11, + 12, + 13, + 14, + 15, + 16, + 17, + 18, + 19, + 20, + 21, + 22, + 23, + 24, + 25, + 26, + 28, + 32, + 33, + 34, + 35, + 36, + 40, + 44, + 45, + 46, + 47, + 48, + 49, + 50, + 51, + 52, + 53, + 55, + 56, + 57, + 58, + 63, + 69, + 70, + 71, + 73, + 81, + 101, + 125, + 126, + 127, + 128, + 129, + 130, + 131, + 132, + 133, + 134, + 135, + 136, + 137, + 140, + 141, + 142, + 144, + 145, + 146, + 147, + 158, + 166, + 167, + 168, + 169, + 172, + 190, + 246, + 286, + 287, + 288, + 289, + 290, + 291, + 292, + 293, + 294, + 295, + 296, + 297, + 298, + 299, + 300, + 301, + 302, + 303, + 304, + 305, + 306, + 307, + 311, + 319, + 320, + 321, + 322, + 323, + 327, + 335, + 343, + 347, + 349, + 350, + 415, + 511, + 735 + ], + "prime": 101, + "rank": 120 + }, + "root_linears": { + "independent_rows": [ + 0, + 1, + 2, + 6, + 12, + 20, + 30, + 42 + ], + "prime": 101, + "rank": 8 + }, + "root_quadratics": { + "independent_rows": [ + 0, + 1, + 2, + 3, + 4, + 5, + 6, + 7, + 8, + 9, + 12, + 13, + 14, + 15, + 16, + 20, + 21, + 22, + 23, + 24, + 25, + 30, + 31, + 32, + 33, + 34, + 35, + 36, + 42, + 43, + 44, + 45, + 46, + 50, + 58, + 74 + ], + "prime": 101, + "rank": 36 + } + }, + "mixed_cubic_features": 576, + "proof": "Pure sectors and opposite relative signs separate independent cubic spaces, using the displayed finite rank witnesses 120,36,8. For the retained cubic feature matrix F, K=F(D-64 L L^T)F^T/3072. Since the shell norm is16 in doubled coordinates, u_i=sum_j u_i*u_j^2/16, and L^T D^-1 L=((8*blocks+2)/768)I. 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The width15 subspaces are degenerate (restricted dot rank13), so any helper theorem must handle those subspaces. Sharing a frame nominates batches; no helper schedule or routing cost is inferred.", + "width15_radical_dimension": 2 + } + }, + { + "blocks": 3, + "coordinate_factorization": { + "center_bill": 38088, + "center_count": 2088, + "center_width_histogram": { + "15": 336, + "17": 1344, + "25": 408 + }, + "definitions": "For coordinate i, m_j=u_i*u_j^2 for the seven other coordinates in its E8 block; f=(16*u_i-u_i^3)/3; g=(u_i^3-4*u_i)/3. Let h_T=sum_{j in T}m_j-f for the seven Fano triples. In each other block let b_k=sum_j H[k,j]*u_i*u_j^2, k=0,...,7. Retain all distinct-coordinate cubic monomials supported on at most two blocks. Add g, the seven h_T, the seven b_k with k>0 per other block, and the listed trace center(s) b_0 or b_0+c_0 and b_0-c_0.", + "diagonal_coefficients": { + "fano_h": "1/2048", + "mixed_distinct": "1/512", + "mixed_hadamard": "1/8192", + "pure_distinct": "1/512", + "trace_minus": "1/16384", + "trace_plus": "-7/16384", + "unary_g": "1/256" + }, + "fano_triples": [ + [ + 0, + 1, + 2 + ], + [ + 0, + 3, + 4 + ], + [ + 0, + 5, + 6 + ], + [ + 1, + 3, + 5 + ], + [ + 1, + 4, + 6 + ], + [ + 2, + 3, + 6 + ], + [ + 2, + 4, + 5 + ] + ], + "hadamard_matrix": [ + [ + 1, + 1, + 1, + 1, + 1, + 1, + 1, + 1 + ], + [ + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1 + ], + [ + 1, + 1, + -1, + -1, + 1, + 1, + -1, + -1 + ], + [ + 1, + -1, + -1, + 1, + 1, + -1, + -1, + 1 + ], + [ + 1, + 1, + 1, + 1, + -1, + -1, + -1, + -1 + ], + [ + 1, + -1, + 1, + -1, + -1, + 1, + -1, + 1 + ], + [ + 1, + 1, + -1, + -1, + -1, + -1, + 1, + 1 + ], + [ + 1, + -1, + -1, + 1, + -1, + 1, + 1, + -1 + ] + ], + "identity": "3*f=sum_j m_j+sum_other_blocks b_0. The Fano identity is sum_j m_j*m_j_prime-f*f_prime = (sum_T h_T*h_T_prime-(sum_other_blocks b_0)*(sum_other_blocks b_0_prime))/2. Hadamard transforms give each other-block contribution sum_k b_k*b_k_prime/8. The displayed trace coefficients diagonalize the remaining one- or two-variable form.", + "scope": "An exact scalar factorization and support bill, not a complete recursive circuit or a new kappa.", + "support_orbits": [ + { + "binary_span": 25, + "independent_ports": [ + 287, + 288, + 289, + 290, + 291, + 295, + 303, + 319, + 1079, + 13320, + 13322, + 13324, + 13332, + 13344, + 13360, + 13380, + 13404, + 42120, + 42122, + 42124, + 42132, + 42144, + 42160, + 42180, + 42204 + ], + "multiplicity": 24, + "name": "unary_g", + "support_size": 31233 + }, + { + "binary_span": 25, + "independent_ports": [ + 287, + 288, + 289, + 290, + 291, + 295, + 303, + 319, + 735, + 13320, + 13322, + 13324, + 13332, + 13344, + 13360, + 13380, + 13404, + 42120, + 42122, + 42124, + 42132, + 42144, + 42160, + 42180, + 42204 + ], + "multiplicity": 168, + "name": "fano_h", + "support_size": 35208 + }, + { + "binary_span": 15, + "independent_ports": [ + 13326, + 13328, + 13334, + 13346, + 13350, + 13362, + 13382, + 13566, + 13806, + 14046, + 14286, + 15246, + 17166, + 21006, + 28686 + ], + "multiplicity": 336, + "name": "mixed_hadamard", + "support_size": 3744 + }, + { + "binary_span": 25, + "independent_ports": [ + 287, + 288, + 289, + 290, + 291, + 295, + 303, + 319, + 735, + 13320, + 13322, + 13324, + 13332, + 13344, + 13360, + 13380, + 13404, + 42120, + 42122, + 42124, + 42132, + 42144, + 42160, + 42180, + 42204 + ], + "multiplicity": 168, + "name": "pure_distinct", + "support_size": 31272 + }, + { + "binary_span": 17, + "independent_ports": [ + 13404, + 13406, + 13408, + 13410, + 13412, + 13420, + 13436, + 13468, + 13532, + 13644, + 13884, + 14124, + 14364, + 15324, + 17244, + 21084, + 28764 + ], + "multiplicity": 1344, + "name": "mixed_distinct", + "support_size": 10296 + }, + { + "binary_span": 25, + "independent_ports": [ + 13320, + 13322, + 13324, + 13326, + 13332, + 13344, + 13360, + 13380, + 13404, + 13560, + 13800, + 14040, + 14280, + 15240, + 17160, + 21000, + 28680, + 42120, + 42122, + 42124, + 42132, + 42144, + 42160, + 42180, + 42204 + ], + "multiplicity": 24, + "name": "trace_plus", + "support_size": 37440 + }, + { + "binary_span": 25, + "independent_ports": [ + 13320, + 13322, + 13324, + 13326, + 13332, + 13344, + 13360, + 13380, + 13404, + 13560, + 13800, + 14040, + 14280, + 15240, + 17160, + 21000, + 28680, + 42120, + 42122, + 42124, + 42132, + 42144, + 42160, + 42180, + 42204 + ], + "multiplicity": 24, + "name": "trace_minus", + "support_size": 37440 + } + ], + "support_proof": "All support rows, including zero tests, are exhausted for one representative of each listed orbit. Block permutations and within-block coordinate permutations preserve the lattice and induce invertible binary label maps. They act transitively on the listed center supports: a Fano center chooses a three-subset among seven coordinates, and a nonconstant Hadamard row chooses a balanced four/four partition. Thus the representative support dimensions apply to every center. Encoder and decoder supports agree because this factorization is symmetric and diagonal." + }, + "fitting": { + "alternating_gram": [ + [ + 0, + 0, + 0, + 1, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0 + ], + [ + 0, + 0, + 1, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0 + ], + [ + 0, + 1, + 0, + 1, + 0, + 1, + 0, + 1, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0 + ], + [ + 1, + 0, + 1, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0 + ], + [ + 0, + 0, + 0, + 0, + 0, + 1, + 0, + 1, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0 + ], + [ + 0, + 0, + 1, + 0, + 1, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0 + ], + [ + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 1, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, + 0, 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In lattice pairing coordinates its rational inverse reduces to the displayed binary inverse. Three orthogonal blocks give the alternating rank-24 form; the verified even-space isometry and the orthogonal all-ones vector realize it by the standard dot product in dimension 25." + }, + "operator_factorization": { + "center_bill": 35400, + "center_count": 2088, + "center_width_histogram": { + "15": 1680, + "25": 408 + }, + "full_factorization_diagnostic_rows": [ + 0, + 1, + 2, + 3, + 4, + 5, + 6, + 7, + 8, + 9, + 10, + 11, + 12, + 13, + 14, + 15, + 16, + 17, + 18, + 19, + 20, + 21, + 22, + 23, + 24, + 25, + 26, + 27, + 28, + 29, + 30, + 31, + 32, + 33, + 34, + 35, + 36, + 37, + 38, + 39, + 40, + 41, + 42, + 43, + 44, + 45, + 46, + 47, + 48, + 49, + 50, + 51, + 52, + 53, + 54, + 55, + 56, + 57, + 58, + 59, + 60, + 61, + 62, + 63, + 705, + 1411, + 2117, + 2823, + 3529, + 4234, + 4940, + 5646, + 6352, + 7058, + 7764, + 8469, + 9175, + 9881, + 10587, + 11293, + 11998, + 12704, + 13410, + 14116, + 14822, + 15528, + 16233, + 16939, + 17645, + 18351, + 19057, + 19762, + 20468, + 21174, + 21880, + 22586, + 23292, + 23997, + 24703, + 25409, + 26115, + 26821, + 27526, + 28232, + 28938, + 29644, + 30350, + 31056, + 31761, + 32467, + 33173, + 33879, + 34585, + 35290, + 35996, + 36702, + 37408, + 38114, + 38820, + 39525, + 40231, + 40937, + 41643, + 42349, + 43054, + 43760, + 44466, + 45172, + 45878, + 46584, + 47289, + 47995, + 48701, + 49407, + 50113, + 50818, + 51524, + 52230, + 52936, + 53642, + 54348, + 55053, + 55759, + 56465, + 57171, + 57877, + 58582, + 59288, + 59994, + 60700, + 61406, + 62112, + 62817, + 63523, + 64229, + 64935, + 65641, + 66346, + 67052, + 67758, + 68464, + 69170, + 69876, + 70581, + 71287, + 71993, + 72699, + 73405, + 74110, + 74816, + 75522, + 76228, + 76934, + 77640, + 78345, + 79051, + 79757, + 80463, + 81169, + 81874, + 82580, + 83286, + 83992, + 84698, + 85404, + 86109, + 86815, + 87521, + 88227, + 88933, + 89639 + ], + "mixed_center_coefficient": "1/8192", + "mixed_center_rule": "For each ordered pair of distinct E8 blocks A,B, each coordinate i in A, and each of the 35 operators P, use u_A,i * (u_B^T P u_B).", + "operator_rule": "P_(a,b)e_x=(-1)^(b dot x)e_(x+a), a,b,x in F2^3, a dot b=0. 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"support_size": 3744 + } + ], + "proof": "The 36 explicitly checked symmetric Pauli matrices are an orthogonal basis of Sym_8 with squared Frobenius norm8; this proves the quadratic identity for all real u,v. The identity replaces the seven nonconstant diagonal Hadamard centers and 28 mixed off-diagonal cubic centers per ordered block/coordinate by35 Pauli centers, all with coefficient1/8192. The trace direction is unchanged, so the prior Fano trace diagonalization and the full kernel identity remain exact. All35 support sets are exhausted on actual ports; block permutations and permutations within the linear block cover all coordinate choices. Encoder and decoder supports agree in this symmetric dyadic diagonal factorization.", + "quadratic_identity": "sum_{P nonidentity} (u^T P u)*(v^T P v) = 8*(u dot v)^2 - (u dot u)*(v dot v)", + "retained_diagonal_coefficients": { + "fano_h": "1/2048", + "pure_distinct": "1/512", + "trace_minus": "1/16384", + "trace_plus": "-7/16384", + "unary_g": "1/256" + }, + "retained_support_orbits": [ + { + "binary_span": 25, + "independent_ports": [ + 287, + 288, + 289, + 290, + 291, + 295, + 303, + 319, + 1079, + 13320, + 13322, + 13324, + 13332, + 13344, + 13360, + 13380, + 13404, + 42120, + 42122, + 42124, + 42132, + 42144, + 42160, + 42180, + 42204 + ], + "multiplicity": 24, + "name": "unary_g", + "support_size": 31233 + }, + { + "binary_span": 25, + "independent_ports": [ + 287, + 288, + 289, + 290, + 291, + 295, + 303, + 319, + 735, + 13320, + 13322, + 13324, + 13332, + 13344, + 13360, + 13380, + 13404, + 42120, + 42122, + 42124, + 42132, + 42144, + 42160, + 42180, + 42204 + ], + "multiplicity": 168, + "name": "fano_h", + "support_size": 35208 + }, + { + "binary_span": 25, + "independent_ports": [ + 287, + 288, + 289, + 290, + 291, + 295, + 303, + 319, + 735, + 13320, + 13322, + 13324, + 13332, + 13344, + 13360, + 13380, + 13404, + 42120, + 42122, + 42124, + 42132, + 42144, + 42160, + 42180, + 42204 + ], + "multiplicity": 168, + "name": "pure_distinct", + "support_size": 31272 + }, + { + "binary_span": 25, + "independent_ports": [ + 13320, + 13322, + 13324, + 13326, + 13332, + 13344, + 13360, + 13380, + 13404, + 13560, + 13800, + 14040, + 14280, + 15240, + 17160, + 21000, + 28680, + 42120, + 42122, + 42124, + 42132, + 42144, + 42160, + 42180, + 42204 + ], + "multiplicity": 24, + "name": "trace_plus", + "support_size": 37440 + }, + { + "binary_span": 25, + "independent_ports": [ + 13320, + 13322, + 13324, + 13326, + 13332, + 13344, + 13360, + 13380, + 13404, + 13560, + 13800, + 14040, + 14280, + 15240, + 17160, + 21000, + 28680, + 42120, + 42122, + 42124, + 42132, + 42144, + 42160, + 42180, + 42204 + ], + "multiplicity": 24, + "name": "trace_minus", + "support_size": 37440 + } + ], + "scope": "Exact rank-minimal scalar factorization, binary support bill, and a genuine reduction versus the coordinate basis. Helper exits, a finite recursive word, and a new kappa remain separate obligations." + }, + "port_count": 89640, + "real_rank": { + "finite_rank_witnesses": { + "pure_cubics": { + "independent_rows": [ + 0, + 1, + 2, + 3, + 4, + 5, + 6, + 7, + 8, + 9, + 10, + 11, + 12, + 13, + 14, + 15, + 16, + 17, + 18, + 19, + 20, + 21, + 22, + 23, + 24, + 25, + 26, + 28, + 32, + 33, + 34, + 35, + 36, + 40, + 44, + 45, + 46, + 47, + 48, + 49, + 50, + 51, + 52, + 53, + 55, + 56, + 57, + 58, + 63, + 69, + 70, + 71, + 73, + 81, + 101, + 125, + 126, + 127, + 128, + 129, + 130, + 131, + 132, + 133, + 134, + 135, + 136, + 137, + 140, + 141, + 142, + 144, + 145, + 146, + 147, + 158, + 166, + 167, + 168, + 169, + 172, + 190, + 246, + 286, + 287, + 288, + 289, + 290, + 291, + 292, + 293, + 294, + 295, + 296, + 297, + 298, + 299, + 300, + 301, + 302, + 303, + 304, + 305, + 306, + 307, + 311, + 319, + 320, + 321, + 322, + 323, + 327, + 335, + 343, + 347, + 349, + 350, + 415, + 511, + 735 + ], + "prime": 101, + "rank": 120 + }, + "root_linears": { + "independent_rows": [ + 0, + 1, + 2, + 6, + 12, + 20, + 30, + 42 + ], + "prime": 101, + "rank": 8 + }, + "root_quadratics": { + "independent_rows": [ + 0, + 1, + 2, + 3, + 4, + 5, + 6, + 7, + 8, + 9, + 12, + 13, + 14, + 15, + 16, + 20, + 21, + 22, + 23, + 24, + 25, + 30, + 31, + 32, + 33, + 34, + 35, + 36, + 42, + 43, + 44, + 45, + 46, + 50, + 58, + 74 + ], + "prime": 101, + "rank": 36 + } + }, + "mixed_cubic_features": 1728, + "proof": "Pure sectors and opposite relative signs separate independent cubic spaces, using the displayed finite rank witnesses 120,36,8. For the retained cubic feature matrix F, K=F(D-64 L L^T)F^T/3072. Since the shell norm is16 in doubled coordinates, u_i=sum_j u_i*u_j^2/16, and L^T D^-1 L=((8*blocks+2)/768)I. The nonzero displayed correction eigenvalue proves the exact real rank.", + "pure_cubic_features": 360, + "real_kernel_rank": 2088, + "signature": [ + 2064, + 24 + ], + "trace_correction_eigenvalue": "-7/6" + }, + "shared_support_frames": null + } + ], + "relative_sign_walsh": { + "inverse_denominator": 2, + "matrix": [ + [ + 1, + 1 + ], + [ + 1, + -1 + ] + ], + "pair_kernel_after_transform": "For two AB ports in the odd-A sector, s_A*(s_A^2+3*s_B^2-4)/24, where s_A and s_B are ordinary root inner products.", + "ports_per_three_block_sector": 29880, + "ports_per_two_block_sector": 15480, + "proof": "Every retained cubic or linear feature is odd in exactly one block. Transform each present relative-sign pair (a,b),(a,-b) by H on inputs and H/2 on outputs. This same-label similarity preserves the binary labels and unit diagonal, and separates the odd-block characters. Pure ports already have one character. At a fixed root a, the same-sector pair kernel is s_B^2/4, whose real rank is36 by the exact root-quadratic witness; this is not an eight-dimensional E8 frame replication.", + "real_rank_per_three_block_sector": 696, + "real_rank_per_two_block_sector": 408, + "same_label_identity": "p(a,b)=p(a,-b), since all lattice pairings agree modulo two.", + "scope": "An exact free change of basis within equal-label ports. It does not by itself supply improved helper exits or a complete recursive word.", + "three_block_sectors": 3, + "two_block_sectors": 2 + }, + "status": "PASS_EXACT_E8_SECTION_FACTORIZATIONS" +} diff --git a/research/niemeier-lattices/sources/COST-SOURCES.json b/research/niemeier-lattices/sources/COST-SOURCES.json new file mode 100644 index 0000000000..8e4b452fa7 --- /dev/null +++ b/research/niemeier-lattices/sources/COST-SOURCES.json @@ -0,0 +1,45 @@ +{ + "inherited_files": { + "leech-lattice/code/screen.py": "6118abd81e8860bdb551526029d8b36958835de52adb8f50c1a43bb247005385", + "turyn-port-labels/sources/PR592.json": "08c02ef05272279e1a518c77e5661ff9996b00c1b08159af9a8887d2611d2795", + "turyn-port-labels/sources/PR595.json": "2cc7c3c343a94b880bcb38d5bcc2e37abec6884740e6176806b41cdc4a24f5b3", + "turyn-port-labels/sources/PR595-SKELETON.py": "9cfbc1316389a46922765cdae85fdde2042f7681986d1c97a19174c4922ea29c" + }, + "frontier": { + "pr": 592, + "head": "9092320f7932dbe61a78f15b6c5fd6a0cbd41307", + "declared_kappa": "0.001612719427075359", + "freshly_replayed": false + }, + "accounting": { + "pr": 595, + "head": "39112c2577452fa278489fb9fd77a9aecb2cdc24", + "scope": "Inherited B_K compressed-window, star-copy and whole-residual hosting assumptions; complete PR analysis not replayed here" + }, + "bit_source_files": { + "PR592-ASSEMBLE.py": { + "url": "https://raw.githubusercontent.com/CrocSwap/integer-mult-bounds/9092320f7932dbe61a78f15b6c5fd6a0cbd41307/research/complex-571-hpeq4/code/assemble.py", + "sha256": "71fca807f2e70ed7439e369be4bb3be21b487e942b92f0e7629d7521e9398a65" + }, + "PR543-EXPECTED.json": { + "url": "https://raw.githubusercontent.com/CrocSwap/integer-mult-bounds/71c99ac935a33bb7b59a1b4a7af8725a939b0b90/research/complex-multicold/inputs/PR543-EXPECTED.json", + "sha256": "06eccee9f6b77e140833df83407e7996dc948c17d49b0779ab171fbe463fd28d" + } + }, + "bit_citation": { + "original_pr": 543, + "original_repository": "eumemic/integer-mult-bounds", + "original_commit": "2065fe4f72a914d907f1046603f5a36f17018535", + "original_path": "research/e8-bit-r420/EXPECTED.json", + "git_blob_sha1": "b61d859826246e0c3dabbe096e57ad7015eadfd5", + "fresh_circuit_replay": false + }, + "repository_files": { + "notes/general-clifford-frames.tex": "c9bb9655b594583c6b4bab8cb485233f1d1e588f83858cf1e4c3530cee8fed54" + }, + "complex_frame_lemma": { + "path": "notes/general-clifford-frames.tex", + "scope": "Exact Gaussian-dyadic representatives for every binary subspace, including degenerate subspaces. Nested U to 0 transitions have Fourier rank dim U; this justifies individual center widths, not an entire helper schedule.", + "attribution": "Frame construction credited by the inherited note to icekylinx with OpenAI GPT-6 Astra assistance; exposition and integration use Codex." + } +} diff --git a/research/niemeier-lattices/sources/PR543-EXPECTED.json b/research/niemeier-lattices/sources/PR543-EXPECTED.json new file mode 100644 index 0000000000..b61d859826 --- /dev/null +++ b/research/niemeier-lattices/sources/PR543-EXPECTED.json @@ -0,0 +1,239 @@ +{ + "word": { + "file": "data/bitE8mut-r420.json.gz", + "file_sha256": "9721a3819efd68307f66f00d84a952eb28eb7c3ad2ce79653ce295c98a764784", + "content_sha256": "e597e3885c0a8daa262afa5985ee2930443329126aece76d33159d81882bdcc2", + "plan_sha256": "d5c2494b4baae801143049e172db303b3c8e25a002e9d9715a1d79dcb2d74c58", + "R": 420, + "retained_totals": 9, + "gates": { + "D": 120, + "A": 2064, + "B": 837, + "C": 2661 + }, + "local_histogram": { + "1": 1320, + "2": 1440, + "4": 60, + "5": 120, + "8": 9 + }, + "legality": { + "records": 29874, + "common_frame_ADDs": 26916, + "rank_mass": 5112 + }, + "five_stage_columns": 900, + "five_stage_payload_bits": 100, + "nine_stage_columns": 1380, + "nine_stage_payload_bits": 100, + "eleven_stage_columns": 1620, + "eleven_stage_payload_bits": 100 + }, + "interface": { + "hypotheses": { + "H1_no_target_feeds_z": true, + "H2_D_frame0_targets_from_helpers": true, + "H3_A_z_only_scatter_targets_B_split": true, + "H4_C_literal_reverse_at_full": true, + "H5_V_identity": true + }, + "conclusion": { + "U_block_form": true, + "J_identity": true, + "K_equals_D": true, + "C_is_U_inverse": true + }, + "interleaving": { + "z_gates_A": 2064, + "source_writes_A": 0, + "z_gates_B": 597, + "source_writes_B": 597, + "helper_writes_B": 0, + "sources_rewritten_B": 120, + "B_z_write_frame_dims": { + "4": 477, + "6": 120 + }, + "B_z_operand_kinds": { + "source<-helper": 120, + "source<-source": 477 + }, + 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{ + "kappa": "1584931111785787/1000000000000000000", + "kappa_decimal": "0.001584931111785787", + "complex_lower": "1589991533606044174031/1000000000000000000000000", + "binding_component": "bit" + }, + "pr531": { + "kappa": "1517760977820331/1000000000000000000", + "kappa_decimal": "0.001517760977820331", + "complex_lower": "304013615566499291631/200000000000000000000000", + "binding_component": "complex" + }, + "pr484": { + "kappa": "662515748707749/500000000000000000", + "kappa_decimal": "0.001325031497415498", + "complex_lower": "663394767670133/500000000000000000", + "binding_component": "complex" + }, + "pr471": { + "kappa": "260706019542937/200000000000000000", + "kappa_decimal": "0.001303530097714685", + "complex_lower": "1305231506267591/1000000000000000000", + "binding_component": "complex" + }, + "pr458": { + "kappa": "323880830669511/250000000000000000", + "kappa_decimal": "0.001295523322678044", + "complex_lower": "1297203880559577927527/1000000000000000000000000", + "binding_component": "complex" + } + } + }, + "pr493_r489_eleven": { + "word": "../e8-bit-carrier-skeleton/data/bit/bitE8mut-r489.json.gz", + "bit_lower": "391360506745414227149/250000000000000000000000", + "bit_lower_decimal": "0.001565442026981656908596", + "emitted_program_sha256": "e37b52e4861c39154b5891460fb45df8651c92cd68961078f8f06596f057c822", + "kappa": { + "pr535": { + "kappa": "97687203033227/62500000000000000", + "kappa_decimal": "0.001562995248531632", + "complex_lower": "1589991533606044174031/1000000000000000000000000", + "binding_component": "bit" + }, + "pr531": { + "kappa": "1517760977820331/1000000000000000000", + "kappa_decimal": "0.001517760977820331", + "complex_lower": "304013615566499291631/200000000000000000000000", + "binding_component": "complex" + }, + "pr484": { + "kappa": "662515748707749/500000000000000000", + "kappa_decimal": "0.001325031497415498", + "complex_lower": "663394767670133/500000000000000000", + "binding_component": "complex" + }, + "pr471": { + "kappa": "260706019542937/200000000000000000", + "kappa_decimal": "0.001303530097714685", + "complex_lower": "1305231506267591/1000000000000000000", + "binding_component": "complex" + }, + "pr458": { + "kappa": "323880830669511/250000000000000000", + "kappa_decimal": "0.001295523322678044", + "complex_lower": "1297203880559577927527/1000000000000000000000000", + "binding_component": "complex" + } + } + } + } +} diff --git a/research/niemeier-lattices/sources/PR592-ASSEMBLE.py b/research/niemeier-lattices/sources/PR592-ASSEMBLE.py new file mode 100644 index 0000000000..1261127a60 --- /dev/null +++ b/research/niemeier-lattices/sources/PR592-ASSEMBLE.py @@ -0,0 +1,213 @@ +"""Hardened 63 width-seven (#571 prefix word) and 9 plain eight-cold (#568 E8) provider assembly with the native exact-complex moment. + +Prepared with OpenAI Codex assistance. The rational engines and outer +constraints are retained from PR547/PR548 and their preserved ancestry. +The bit operator is the cited unchanged PR543 supplier. The complex owner +and invoice inputs must be freshly emitted by the packaged physical replay. +All conservative admission guards and conservative moment/outer witnesses run +before the PR559 native-complex correction. The bit surcharge is retained. +""" +from pathlib import Path +from fractions import Fraction as Q +from decimal import Decimal, localcontext +import argparse,json,hashlib,importlib.util,os,sys +from collections import Counter +assert __debug__, 'Assertions required' +P=Path(sys.argv[1] if __name__=='__main__' else os.environ['GAIN_PR564_SOURCE']).resolve() +read=lambda p:json.loads(p.read_text()) +sha=lambda p:hashlib.sha256(p.read_bytes()).hexdigest() +hash_ok=lambda x:isinstance(x,str) and len(x)==64 and all(t in '0123456789abcdef'for t in x) +digest=lambda x:hashlib.sha256(json.dumps(x,separators=(',',':')).encode()).hexdigest() +def load(n): + s=importlib.util.spec_from_file_location(n,P/'engines'/(n+'.py'));m=importlib.util.module_from_spec(s);s.loader.exec_module(m);return m +M,B,A=[load(n)for n in ('moment','base_two_moment','outer')] +def dec(x): + with localcontext()as ctx:ctx.prec=70;return str(Decimal(x.numerator)/Decimal(x.denominator)) +from witness import DISPATCH,CALL_ORDER,COUNTS,PIDS,X,DISCOVER,pin + + +def assemble(owner,invoice,owner_sha,invoice_sha): + # These are admission receipts, not optional collections of successful rows. + # Require both orientations and every finite check before pricing anything. + assert owner['status']=='PASS_INDEPENDENT_ASYNC_OWNERS_AND_PROFILE' + assert invoice['status']=='PASS_INDEPENDENT_MIXED_ASYNC_SCALAR_PREFIX_AND_FINITE_INVOICE' + assert all(hash_ok(x)for x in (owner_sha,invoice_sha,owner['template_sha256'],owner['controller_audit_sha256'],invoice['parent_template_sha256'])) + checks={'local_primitive_bound','logical','router','literal','mass','precision','tape','row','halving'} + assert set(invoice['strict_checks'])==checks + assert all(invoice['strict_checks'][name] is True for name in checks) + assert set(invoice['providers'])==set(PIDS) + assert invoice['provider_counts']==COUNTS + assert owner['all_live_owners_per_call_distinct'] is True + assert owner['owners_per_call']==728 and owner['actual_helper_owner_charts']==owner['hosted_residuals']==pin('expected.hosted_residuals',owner['hosted_residuals']) + pin('expected.rank_mass',owner['rank_mass']) + pin('expected.actual_73column_sigmas',owner['actual_73column_sigmas']);pin('expected.cross_replica_helper_roles',owner['cross_replica_helper_roles']) + pin('expected.idle_remainders',owner['explicit_idle_remainder_children']) + assert invoice['complete_invocation_sweeps']==invoice['bridge_sweeps']==72 + assert invoice['live']==owner['live'] and invoice['physical']==owner['physical'] + assert invoice['helpers']==504 and invoice['centres']==8 + assert invoice['physical_stock_identity']==dict(data_and_owner_families=7168,local_helpers=504,private_centres=8,coefficient_work=2) + assert sum(invoice['physical_stock_identity'].values())==invoice['physical'] + assert invoice['controller_audit_sha256']==owner['controller_audit_sha256'] + for pid,p in invoice['providers'].items(): + assert set(p['local'])=={'forward','backward'} + assert p['helpers']==504 and p['centres']==8 and p['n']==728 and p['v']==112 and p['h']==9 + K='providers.'+pid+'.';pin(K+'entering',p['entering_helpers']);pin(K+'entrance',p['entrance_rank']) + pin(K+'inventory',p['width_inventory']) + assert all(type(w) is int and 0=0 for n in idle.values()) + idle=+idle;assert sum(idle.values())==owner['explicit_idle_remainder_children'] + pin('expected.idle_remainder_profile',{str(r):n for r,n in sorted(idle.items())}) + paid.update(idle) + assert dict(paid)=={int(r):n for r,n in owner['paid_profile'].items()} + assert invoice['compiler']==64*74**3 + assert invoice['local_adapters']==max(p['local_adapters']for p in invoice['providers'].values()) + assert invoice['local_primitive_bound']==max(p['local_primitive_bound']for p in invoice['providers'].values()) + # Recount the retained dedicated-bank envelope independently. Its geometry + # stays fixed, while every local primitive and wrapper comes from the actual + # selected provider/orientation sums above. + parent_live,parent_physical,parent_idle=11704,12218,16128 + assert owner['source_idle_children_charged']==parent_idle + bridges,shifts=2*112*72,4*112*8;moves=2*shifts + nonlocal_items=(4*112*8*9+2*112*8*9+2*72*504+parent_idle+9*504+bridges+shifts+moves+parent_live+parent_physical+invoice['wrapper_elementary_factors_charged']) + outside=8*nonlocal_items+64*73 + pervertex=(invoice['selected_real_component_primitives']+bridges+shifts+moves+parent_live+(72*invoice['local_adapters']+outside)*invoice['compiler']+64*parent_physical) + rebased=pervertex+65*(invoice['physical']-parent_physical)+(invoice['live']-parent_live) + assert invoice['unhosted_pervertex']==pervertex and invoice['hosted_rebased_pervertex']==rebased + used=max(pervertex,rebased) + prior=invoice['retained_prior_overcharge'] + if prior is not None: + assert hash_ok(prior['sha256']) and type(prior['previous_pervertex_allowance']) is int and prior['previous_pervertex_allowance']>0 + assert prior['added_complete_new_pervertex']==pervertex + assert prior['policy']=='Retain old allowance and add entire fresh mixed recount.' + used=max(used,prior['previous_pervertex_allowance']+pervertex) + assert invoice['conservative_pervertex_used']==used + finite=invoice['finite'] + assert 02*48*48) + assert actual_checks==invoice['strict_checks'] and all(actual_checks.values()) + bit_path=P/'inputs/PR543-EXPECTED.json';bit_data=bit_path.read_bytes() + assert hashlib.sha1(b'blob '+str(len(bit_data)).encode()+b'\0'+bit_data).hexdigest()=='b61d859826246e0c3dabbe096e57ad7015eadfd5' + bit=json.loads(bit_data)['nine'] + c=Q(bit['bit_lower']) + H={int(r):n for r,n in owner['paid_profile'].items()};pin('expected.global_profile',owner['paid_profile']) + # Claimed lower endpoints come from the spec; discovery derives them with the same certified moment engine. + # Either way they are verified below (accepted at b, rejected at b+1e-24). + if DISCOVER:b=Q(M.certify(H,73,owner['live'],False)['lower']);conservative_b=Q(M.certify(H,73,owner['live'],True)['lower']) + else:b=Q(X['complex_lower']);conservative_b=Q(X['conservative_complex_lower']) + assert owner['template_sha256']==invoice['template_sha256'] and owner_sha==invoice['owner_audit_sha256'] + pin('expected.live',owner['live']);pin('expected.physical',owner['physical']);assert owner['deficit']==2560 + assert owner['banked_residuals']==0 + def check_moment(side,h,m,W,lower,deficit,bad): + assert m*W-sum(r*n for r,n in h.items())==deficit and m*m>2*max(h)**2 + rows=[] + for z,accept in [(lower,True),(lower+Q(1,10**24),False)]: + l,u=M.moment(h,m,W,z,bad);ll,uu=B.moment(m,W,list(h.items()),z) + if bad: + fl,fu=B.moment(m,W,[(1,32*m*m*sum(h.values()))],z);ll+=fl/10**16;uu+=fu/10**16 + assert (u<1 and uu<1)if accept else(l>1 and ll>1) + rows.append(dict(side=side,surcharge=bad,endpoint=str(z),accept=accept,mercator=[str(l-1),str(u-1)],base_two=[str(ll-1),str(uu-1)])) + return rows + bitH={int(r):n for r,n in bit['profile'].items()} + bit_moments=check_moment('bit',bitH,64,bit['bit']['live'],c,9984,True) + conservative_moments=check_moment('conservative_complex',H,73,owner['live'],conservative_b,2560,True) + assert bit['bit']['two_level_coefficient_bits']0 + ceil=lambda x:-(-x.numerator//x.denominator);cut=max(ceil(36/gap**2),ceil(2*(4+512)/gap));assert cut*gap**2>=36 and cut*gap>=2*(4+512) + cuts.append(dict(exponent=str(a),gap=str(gap),cutoff_log2=str(cut)));chain.append(a) + q=chain[-1]*(1-2*eta);margin=(1-eta)*q/(1+q);ticks=margin*10**18;k=Q((ticks.numerator-1)//ticks.denominator,10**18) + bridge={'semantic':{'C0':{'base':2,'exponent':210000},'strict_literal_gap':1},'rows':{'degree_gap':Q(10**6)-Q(51*20161,25)}} + outer=A.assembly(chain[-1],bb,bridge,k,eta=eta,beta=beta);assert len(outer['strict_constraints'])==47 and all(v>0 for v in outer['strict_constraints'].values()) + controls=[] + for name,kk,ee,be in [('next_grid',k+Q(1,10**18),eta,beta),('zero_eta',k,Q(0),beta),('zero_beta',k,eta,Q(0)),('unit_beta',k,eta,Q(1))]: + try:A.assembly(chain[-1],bb,bridge,kk,eta=ee,beta=be) + except AssertionError:controls.append(name) + else:raise AssertionError(name) + return k,cuts,outer,controls + conservative_k,conservative_cuts,conservative_outer,conservative_controls=outer_witness(conservative_b) + pin('expected.conservative_kappa_decimal',dec(conservative_k)) + # Only after every conservative supplier/admission/outer check succeeds is + # the reviewed native exact-complex recurrence selected. Bit charge stays. + native_moments=check_moment('complex',H,73,owner['live'],b,2560,False) + k,cuts,outer,controls=outer_witness(b) + pin('expected.kappa_decimal',dec(k)) + assert M.moment(H,73,owner['live'],b,True)[0]>1 + assert M.moment(bitH,64,bit['bit']['live'],c+Q(1,10**24),False)[1]<1 + model_controls=['native_complex_endpoint_rejected_with_old_surcharge','omitting_bit_surcharge_would_fail_next_endpoint_rejection'] + moments=bit_moments+native_moments + input_hashes={'owner_audit':owner_sha,'fresh_invoice':invoice_sha,'inputs/PR543-EXPECTED.json':sha(bit_path)} + input_hashes.update({'engines/'+name+'.py':sha(P/'engines'/(name+'.py'))for name in ('moment','base_two_moment','outer')}) + return dict(status='PASS_FRESH_MIXED_048_ALLCOLD_NATIVE_EXACT_ASSEMBLY',kappa=str(k),kappa_decimal=dec(k),complex_lower=str(b),complex_decimal=dec(b),bit_lower=str(c),bit_decimal=dec(c),binding='complex',gain_over_pinned569_percent=dec(100*(k/Q(X['gain_baseline_kappa'])-1)),complex_profile=H,provider_counts=invoice['provider_counts'],surcharge_model={'complex':False,'bit':True},conservative_complex_lower=str(conservative_b),conservative_complex_decimal=dec(conservative_b),conservative_kappa=str(conservative_k),conservative_kappa_decimal=dec(conservative_k),conservative_moments=conservative_moments,conservative_bootstrap=conservative_cuts,conservative_outer={n:str(v)for n,v in conservative_outer['strict_constraints'].items()},conservative_controls=conservative_controls,model_controls=model_controls,native_correction_gain=dec(k-conservative_k),template_sha256=owner['template_sha256'],input_sha256=input_hashes,bit_supplier=dict(pr=543,repository='eumemic/integer-mult-bounds',commit='2065fe4f72a914d907f1046603f5a36f17018535',path='research/e8-bit-r420/EXPECTED.json',git_blob_sha1='b61d859826246e0c3dabbe096e57ad7015eadfd5',replay_scope='Cited unchanged supplier; this assembly reprices its full profile and does not rebuild its bit circuit.'),moments=moments,bootstrap=cuts,outer={n:str(v)for n,v in outer['strict_constraints'].items()},controls=controls,scope='Conditional native exact-complex witness after complete conservative admission, moment and outer checks; PR559 exact-cover recurrence correction only, bit bad-residue surcharge and H1-H5 retained. The frozen PR582 plain-joint-prefix, added-kernel and entrance-promoted provider dispatch with the new frame coalescences composed with cited unchanged PR543 bit supplier. All-size compiler, common-label/mutable-source realization, whole-call hosting, precision/tape and analytic transfer remain assumptions. No unconditional theorem or full Lean certification.') +if __name__=='__main__': + import sys + package,C,out=map(Path,sys.argv[1:4]);assert package.resolve()==P and not out.exists() + owner_path=C/'audit/OWNERS-AND-PROFILE.json';invoice_path=C/'audit/PREFIX-AND-INVOICE.json' + owner,invoice=read(owner_path),read(invoice_path) + assert owner['template_sha256']==read(C/'hosted/HOSTED.json')['template_sha256'] + r=assemble(owner,invoice,sha(owner_path),sha(invoice_path));out.write_text(json.dumps(r,indent=2)+'\n');print(r['status'],r['kappa_decimal']) diff --git a/research/niemeier-lattices/verify.py b/research/niemeier-lattices/verify.py new file mode 100644 index 0000000000..c243096b1a --- /dev/null +++ b/research/niemeier-lattices/verify.py @@ -0,0 +1,102 @@ +#!/usr/bin/env python3 +"""Replay the Niemeier geometry, scoped costs, and optional construction workbench.""" +import argparse +import hashlib +import importlib.util +import json +from pathlib import Path +import sys +import subprocess +import tempfile + +sys.dont_write_bytecode = True +ROOT = Path(__file__).resolve().parent + + +def require(condition, message): + if not condition: + raise ValueError(message) + + +def module(name): + spec = importlib.util.spec_from_file_location(name, ROOT/'code'/f'{name}.py') + result = importlib.util.module_from_spec(spec) + spec.loader.exec_module(result) + return result + + +def canonical(value): + return json.loads(json.dumps(value)) + + +def verify_manifest(): + manifest = json.loads((ROOT/'MANIFEST.json').read_text()) + actual = {str(path.relative_to(ROOT)): hashlib.sha256(path.read_bytes()).hexdigest() + for path in ROOT.rglob('*') if path.is_file() + and '__pycache__' not in path.parts and path.suffix != '.pyc' + and path.name != 'MANIFEST.json'} + require(actual == manifest['sha256'], 'Package file inventory and SHA256 manifest') + + +def replay_workbench(): + flags = ['-B', '-I'] + if sys.flags.optimize: + flags.append('-' + 'O' * sys.flags.optimize) + with tempfile.TemporaryDirectory(prefix='niemeier-workbench-') as temporary: + output = Path(temporary) + cases = [ + ('workbench/bit-routes/verify_portable.py', []), + ('workbench/helper-routes/replay.py', [str(output/'helpers')]), + ('workbench/continuation/tensor-routes/replay.py', [str(output/'tensor')]), + ('workbench/paid-controller/verify_portable.py', + ['--workspace', str(output/'paid')]), + ('workbench/continuation/endpoint-costs/verify_portable.py', + ['--workspace', str(output/'endpoints')]), + ('workbench/root-probes/rectangles.py', ['--output', str(output/'rectangles.json')]), + ] + for relative, arguments in cases: + if relative.endswith('verify_portable.py') and 'bit-routes' not in relative: + if not sys.flags.optimize: + arguments.append('--normal') + print(f'Replaying {relative}', flush=True) + subprocess.run([sys.executable, *flags, str(ROOT/relative), *arguments], check=True) + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument('--output', type=Path) + parser.add_argument('--workbench', action='store_true', + help='Also replay the construction and scoped-obstruction receipts') + args = parser.parse_args() + verify_manifest() + census = canonical(module('catalog').census()) + require(census == json.loads((ROOT/'catalog.json').read_text()), 'Exact catalogue receipt') + receipts = { + 'D24.json': module('d24').certify_d24(), + 'E8-CUBED.json': module('e8_cubed').certify_e8_cubed(), + 'E8-SECTIONS.json': module('e8_sections').certify_sections(), + } + for name, result in receipts.items(): + require(canonical(result) == json.loads((ROOT/'evidence'/name).read_text()), + f'Exact geometric receipt: {name}') + costs = module('costs') + inputs = json.loads((ROOT/'evidence/COST-INPUTS.json').read_text()) + costs.verify_input_links(inputs, receipts) + result = costs.run() + require(canonical(result) == json.loads((ROOT/'evidence/COSTS.json').read_text()), + 'Exact scoped cost receipt') + if args.output: + output = args.output.resolve() + require(not output.is_relative_to(ROOT), 'Replay output stays outside the research package') + output.mkdir(parents=True, exist_ok=True) + (output/'REPLAY.json').write_text(json.dumps(dict(catalog=census, geometry=receipts, + costs=result), indent=2, sort_keys=True)+'\n') + print('PASS_NIEMEIER_GEOMETRY_AND_SCOPED_COSTS', flush=True) + if args.workbench: + replay_workbench() + print('PASS_NIEMEIER_WORKBENCH') + print('No complete supplier or new kappa is certified.') + + +if __name__ == '__main__': + main() diff --git a/research/niemeier-lattices/workbench/bit-routes/CLIQUE-RESIDUAL-SUPPORT.json b/research/niemeier-lattices/workbench/bit-routes/CLIQUE-RESIDUAL-SUPPORT.json new file mode 100644 index 0000000000..6ff86c9b74 --- /dev/null +++ b/research/niemeier-lattices/workbench/bit-routes/CLIQUE-RESIDUAL-SUPPORT.json @@ -0,0 +1,306 @@ +{ + "status": "PASS_EXACT_RESIDUAL_COLUMN_CAP_SPAN", + "column_port": 0, + "support_count": 6420, + "rational_support_rank": 15, + "support_witness_ports": [ + 127, + 191, + 223, + 239, + 247, + 251, + 253, + 254, + 255, + 319, + 575, + 1087, + 2111, + 4159, + 8255 + ], + "support_witness_vectors": [ + [ + 1, + -1, + -1, + -1, + -1, + -1, + -1, + -1, + 1, + 1, + 1, + 1, + 1, + 1, + 1, + -1 + ], + [ + 1, + -1, + -1, + -1, + -1, + -1, + -1, + 1, + -1, + 1, + 1, + 1, + 1, + 1, + 1, + -1 + ], + [ + 1, + -1, + -1, + -1, + -1, + -1, + 1, + -1, + -1, + 1, + 1, + 1, + 1, + 1, + 1, + -1 + ], + [ + 1, + -1, + -1, + -1, + -1, + 1, + -1, + -1, + -1, + 1, + 1, + 1, + 1, + 1, + 1, + -1 + ], + [ + 1, + -1, + -1, + -1, + 1, + -1, + -1, + -1, + -1, + 1, + 1, + 1, + 1, + 1, + 1, + -1 + ], + [ + 1, + -1, + -1, + 1, + -1, + -1, + -1, + -1, + -1, + 1, + 1, + 1, + 1, + 1, + 1, + -1 + ], + [ + 1, + -1, + 1, + -1, + -1, + -1, + -1, + -1, + -1, + 1, + 1, + 1, + 1, + 1, + 1, + -1 + ], + [ + 1, + 1, + -1, + -1, + -1, + -1, + -1, + -1, + -1, + 1, + 1, + 1, + 1, + 1, + 1, + -1 + ], + [ + 1, + -1, + -1, + -1, + -1, + -1, + -1, + -1, + -1, + 1, + 1, + 1, + 1, + 1, + 1, + 1 + ], + [ + 1, + -1, + -1, + -1, + -1, + -1, + -1, + 1, + 1, + -1, + 1, + 1, + 1, + 1, + 1, + -1 + ], + [ + 1, + -1, + -1, + -1, + -1, + -1, + -1, + 1, + 1, + 1, + -1, + 1, + 1, + 1, + 1, + -1 + ], + [ + 1, + -1, + -1, + -1, + -1, + -1, + -1, + 1, + 1, + 1, + 1, + -1, + 1, + 1, + 1, + -1 + ], + [ + 1, + -1, + -1, + -1, + -1, + -1, + -1, + 1, + 1, + 1, + 1, + 1, + -1, + 1, + 1, + -1 + ], + [ + 1, + -1, + -1, + -1, + -1, + -1, + -1, + 1, + 1, + 1, + 1, + 1, + 1, + -1, + 1, + -1 + ], + [ + 1, + -1, + -1, + -1, + -1, + -1, + -1, + 1, + 1, + 1, + 1, + 1, + 1, + 1, + -1, + -1 + ] + ], + "removed_coordinate": 0, + "minor_determinant": -16384, + "modular_prime": 101, + "common_target_cap_dimension": 1, + "natural_column_basis_size": 1268, + "full_frame_to_common_cap_rank": 15, + "literal_private_copy_rank_mass": 19020, + "spinor_port_count": 16384, + "excess_over_port_count": 2636, + "translation_proof": "XOR translation by j multiplies the sixteen real coordinates by the signs of the doubled spinor port j; this is an invertible diagonal orthogonal transformation and preserves the subgroup-removed support relation.", + "scope": "The literal natural-column rank-basis implementation with full-frame aggregates and separately lowered private decoder copies. Not a lower bound for arbitrary rank factorizations, alternate bases, or fused shared DAGs." +} diff --git a/research/niemeier-lattices/workbench/bit-routes/CLIQUE-RESIDUAL.json b/research/niemeier-lattices/workbench/bit-routes/CLIQUE-RESIDUAL.json new file mode 100644 index 0000000000..e0315de030 --- /dev/null +++ b/research/niemeier-lattices/workbench/bit-routes/CLIQUE-RESIDUAL.json @@ -0,0 +1,67 @@ +{ + "dimension": 14, + "port_count": 16384, + "H": [ + 0, + 5461, + 9830, + 13107, + 14456, + 11565, + 7710, + 2891, + 16256, + 10965, + 6630, + 3251, + 2040, + 4781, + 8606, + 13515 + ], + "subgroup_generators": [ + 5461, + 9830, + 14456, + 16256 + ], + "clique_size": 16, + "clique_count": 1024, + "J_H_rank": 1024, + "C_rank": 576, + "residual_rank": 1268, + "residual_degree_counts": { + "0": 0, + "1": 0, + "2": 0, + "3": 0, + "4": 616, + "5": 560, + "6": 252, + "7": 64, + "8": 2996, + "9": 0, + "10": 0, + "11": 0, + "12": 0, + "13": 0, + "14": 0 + }, + "J_H_degree_counts": { + "0": 0, 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additions of #555's punctured R439 bit word (v = 111) are moved out of their D4 mini-node gates and rerun at other frames. Nothing else changes: the same registers, scatter, retained totals (nine COPY ranks of 7, decoder constant 63), frozen hosting selections and pipelines.\n\n| | #555 | this PR | change |\n|---|---|---|---|\n| bit c, nine stages | 0.001733052823330162099157 | **0.001733427039589844234749** | +0.0216% |\n| bit c, eleven stages | 0.001710608806098012673516 | **0.001710951900777583975622** | +0.0201% |\n| κ with #548's complex b | 0.001590984155135328 | 0.001590984155135328 | none |\n\nκ stays complex-bound for every cited complex supplier (#548, #544, #535, #531), so this is bit headroom only.\n\n### What changed\n- **hoist:** in A[1993] (frame 509, rank 4), the summand `helper 222 += helper 248` becomes its own gate at rank-1 frame 79, placed just before that gate.\n- **sink:** in A[1994] (frame 509), the summand `helper 191 += helper 247` becomes its own gate at the rank-7 pseudo-total frame 517, placed after A[2028].\n\nBoth summands commute with every addition they cross, and the dirty-register endpoint is unchanged on every column. The local paid profile moves by `+2e1 − e3 − 3e4 + e6 + e7`, and rank mass stays at 5,129.\n\nThe mechanisms come from #541 and #548 (Dugongue) and #542 (rohanarun) on the R480 word. Their move lists are R480 gate ids and don't apply to #555's word, which has no two-x-source additions and no zero-response gates. `tune/tune.py` therefore re-runs the mechanisms as searches. It reproduces #541's and #548's R480 profiles exactly. On #555's word it finds exactly these two moves, and pruning, Steinberg 3→2 and least-frame steps find nothing further.\n\n### Checks\n```sh\nbash research/e8-bit-p111/tune/verify.sh /absolute/new/output # tuned word\nbash research/e8-bit-p111/verify.sh /absolute/new/output # #555's own check, unchanged\n```\n- **Byte-for-byte rebuild.** `tune/verify.sh` rebuilds #555's word from its frozen plan, applies `moves.json` and compares both words byte for byte.\n- **Checkers.** It runs the independent replay and interface lemma, the word-level and record-level mutation controls, the official legality and formal-column checkers (5, 9 and 11 stages) and the Gram witnesses.\n- **Price.** It runs both price pipelines and asserts the nine-stage c exactly against `tune/EXPECTED.json`.\n- **#555 untouched.** #555's `verify.sh`, `code/` and `data/` are byte-identical, and `MANIFEST.sha256` lists `tune/`.\n- **Receipts.** Both scripts pass on a fresh `git archive` of this PR's head (a5afc8c): `tune/verify.sh` exits 0 in about 6 min and #555's `verify.sh` in about 10 min.\n- **Independent audit.** The audit re-priced an independently built word with #555's pipeline and matched every field of `EXPECTED.json`. Pricing #555's word flattened with no moves gives exactly #555's c, so the whole gain comes from the two moves.\n\n### Notes\n- **Stacked on #555** (unmerged): the diff shows #555's commit until it merges. This PR adds one commit, touching only `research/e8-bit-p111/{README.md, MANIFEST.sha256, tune/}`.\n- **Retained assumptions:** all of #555's retained assumptions apply unchanged. This is a conditional finite construction, not a Lean proof.\n\n**Credit:** #555 (eumemic) is the base, built on #543 (eumemic), #545 and #510 (DreamingOfClouds). The mechanisms are #541 and #548 (Dugongue) and #542 (rohanarun).\n\nDreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code.\n\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\n","headRefOid":"a5afc8c6169ea61ba7c51dfd3cade02cc7b13b46","number":589,"title":"Bit c = 0.001733427039589844: two A-section transports on #555's punctured R439 word (+0.0216%, κ unchanged)","url":"https://github.com/CrocSwap/integer-mult-bounds/pull/589"} diff --git a/research/niemeier-lattices/workbench/bit-routes/README.md b/research/niemeier-lattices/workbench/bit-routes/README.md new file mode 100644 index 0000000000..214c031ce1 --- /dev/null +++ b/research/niemeier-lattices/workbench/bit-routes/README.md @@ -0,0 +1,20 @@ +# Bit routes and spinor continuation + +These finite experiments investigate Niemeier and D16 spinor suppliers. They include exact kernel ranks, scoped implementation obstructions, and unsuccessful bounded searches. They do **not** certify an improved κ. + +From `research/niemeier-lattices`, with NumPy and SymPy installed: + +```sh +python3 workbench/bit-routes/verify_portable.py +python3 -I -O workbench/bit-routes/verify_portable.py +``` + +The verifier copies the package into a temporary directory, regenerates the E8² and D16 coordinate arrays, runs 18 generators, and compares 19 saved receipts. JSON comparisons exclude only the historical `seconds` timing field in `MOD4-RANK.json`; the factor NPZ is compared by array contents. Saved checkout receipts are never regeneration targets. + +The same verifier covers this directory and `../continuation/section-search` and `../continuation/spinor-routes`. A top-level verifier can reuse its own isolated copy with `replay(isolated_package, frozen_package, optimized=True)`. + +[REPLAY.txt](REPLAY.txt) gives the dependency and result scopes. In particular, `NEIGHBOR-CUBICS.json` preserves an early sampled lower bound; its mixed rank 678 is superseded by the final exact rank 679 in `CUBIC-RELATIONS.json`. The small clique searches retain their bounded scope. [The spinor proofs](../continuation/spinor-routes/PROOF.txt) concern specified implementations, not all possible factorizations or circuits. + +Direct execution of an individual generator is a regeneration operation that writes beside the script. Use a copy for those commands. `PR589.json` is a historical context snapshot, not a regenerated mathematical receipt. + +Both commands passed on 2026-10-12 with Python 3.14.8, NumPy 2.5.4, and SymPy 1.14.0. The normal run took approximately 83 seconds of generator time; the `-I -O` run reported 83.47 seconds including the temporary copy and comparisons. Runtime depends on the machine. diff --git a/research/niemeier-lattices/workbench/bit-routes/REPLAY.txt b/research/niemeier-lattices/workbench/bit-routes/REPLAY.txt new file mode 100644 index 0000000000..5d457211d2 --- /dev/null +++ b/research/niemeier-lattices/workbench/bit-routes/REPLAY.txt @@ -0,0 +1,53 @@ +Portable replay for the bit routes and spinor continuation +======================================================== + +From any working directory, run this file's sibling verifier using a Python +interpreter with NumPy and SymPy installed: + + python3 workbench/bit-routes/verify_portable.py + python3 -I -O workbench/bit-routes/verify_portable.py + +The command paths above are relative to research/niemeier-lattices. The +verifier itself resolves all inputs relative to its file, copies the package +to a temporary directory, regenerates both coordinate NPZ files, and runs +all18 generators in dependency order. It never writes checkout receipts. +Direct generator execution is a regeneration operation: generators write +beside themselves, so run them in a copy if preserving frozen receipts. + +The verifier compares all JSON data, excluding only the measured `seconds` +field in MOD4-RANK.json. CROSS-FACTOR.npz is compared by named array content, +not archive bytes. ports.npz and d16-ports.npz are generated intermediate +inputs and need not be committed. PR589.json and the historical *.log files +are context snapshots, not regenerated mathematical receipts. + +Exact final computations +------------------------ +MOD4-RANK, MIXED-ORBIT-RANKS, D16-RANK, D16-SECTIONS, SPINOR-OPERATOR, +CLIQUE-RESIDUAL, CLIQUE-RESIDUAL-SUPPORT, and CUBIC-RELATIONS are the final +finite computations from bit-routes. CUBIC-RELATIONS uses a modular rank +lower bound on a specified deterministic subset, then checks the complete +integer kernel relation on every orthogonal neighbor. Its resulting rank679 +supersedes the earlier mixed-neighbor sample rank678. + +SECTIONS, HYBRID, HYBRID-SECTIONS, FRAME-BOUNDS, CROSS-RANK, and CROSS-FACTOR +are the final continuation receipts. FRAME-BOUNDS checks the finite premises +of the written mathematical proofs; its rank1268 is verified separately by +CLIQUE-RESIDUAL in the same run. A rank or center bill is not a complete +supplier or an improved exponent certificate. + +Exploratory computations preserved and replayed +----------------------------------------------- +NEIGHBOR-CUBICS records the original fixed-seed sample only. Its ranks674 +and678 are sampled lower bounds, not exact full-neighbor ranks. The later +CUBIC-RELATIONS computation establishes679 for the selected mixed port. + +CLIQUE-LEADING is an explicit failed candidate: an odd433-clique relation +cancels its leading4-form, but its exact convolution rank2942 is too high. +CLIQUE-SMALL and CLIQUE-HALF record bounded permutation-orbit searches. +Their negative outcomes do not exclude other clique collections. All of +these deterministic experiments are replayed without being promoted to +universal mathematical claims. + +The helper/schedule and rank-minimal-factorization scopes are stated in +continuation/spinor-routes/PROOF.txt. Nonminimal factorizations and other +fused schedules remain open. diff --git a/research/niemeier-lattices/workbench/bit-routes/RESULTS.txt b/research/niemeier-lattices/workbench/bit-routes/RESULTS.txt new file mode 100644 index 0000000000..b15eed4c78 --- /dev/null +++ b/research/niemeier-lattices/workbench/bit-routes/RESULTS.txt @@ -0,0 +1,154 @@ +Niemeier bit routes: exact local results, no new kappa +==================================================== +All work here is scratch. No PR or published package is changed. + +A genuine small-dimensional bit pair +----------------------------------- +For norm-four lattice lines use rational address Gram K(v,w)=v.w/4. +Define the binary scalar central matrix C(v,w)=[v.w=0 mod4]. Since distinct +lines have inner product in {-3,...,3}, C=I+A, where A is the ordinary +orthogonality adjacency. This is a legal bit pair with rational frame +size equal to the lattice dimension. It does not use the complex F2 frame +as a rational address frame. + +The integer-valued polynomial binom(t-1,3) modulo two equals [t=0 mod4]. +Its degree-three multivariate binomial basis gives rank at most +binom(d+3,3), independent of a floating-point computation. + +Exact exhaustive integer dot products and packed GF2 elimination give: + + geometry ports rational h binary rank full center bill + E8^2 norm-four shell 30960 16 815 13040 + E8^2 mixed block ports 28800 16 782 12512 + D16+ norm-four shell 30960 16 815 13040 + D16+ spinor lines 16384 16 576 9216 + D16 norm-four sparse lines 14576 16 575 9200 + +The full-center bill is h times the displayed rank, not a complete local +rank histogram. The graph compensation still needs an efficient legal +transparent word with arbitrary dirt restored. + +On the E8^2 mixed ports, there are three invariant orthogonality types: +block pairings (0,0), opposite unit pairings, and opposite norm-two +pairings (the relative-sign partner). Including each type or omitting it +independently gives binary central ranks, in three-bit mask order: + + [28800,28800,28800,28800,14400,1998,14400,782]. + +Thus retaining all three types is best within this eight-member family. +This is not a global minrank claim. + +Files: mod4_rank.py/MOD4-RANK.json; mixed_rank.py/MIXED-ORBIT-RANKS.json; +d16_rank.py/D16-RANK.json; section_ranks.py/D16-SECTIONS.json. + +D16+ spinor operator transformation +---------------------------------- +Index the canonical spinors by x in F2^14: + + v(x)=(1,(-1)^x_1,...,(-1)^x_14,(-1)^(sum x))/2. + +Distinct ports are orthogonal precisely when wt(x+y) is 7 or 8. Their +central decoder is group convolution by delta_0 + [weight7] + [weight8]. +Write g_i=1+z_i in the F2 group algebra, so z_i^2=0. The exact upper +Boolean zeta transform turns this kernel into e8(z_1,...,z_14). + +For zero-based i=0,...,6, use the invertible Darboux coordinates + + x_i=z_(2i)+z_(2i+1), + y_i=z_(2i+1)+sum_(j>i)(z_(2j)+z_(2j+1)). + +Then e2(z)=sum x_i*y_i and e8(z)=e4(x_0*y_0,...,x_6*y_6). +The latter has 35 monomials. Each creates eight variables and has rank64 +on the full 16384-dimensional exterior algebra. Every two such monomials +have product zero, since their four-element pair subsets intersect. +Therefore I+e8 factors into 35 commuting shears, totaling 2240 scalar +CNOT operations IN THE TRANSFORMED BASIS. Exact graded ranks are +1,14,91,364,91,14,1, whose sum is576. + +The two basis changes are not free common-frame operations. An unfused +14-layer in-place Mobius transform forces both operands of every addition +to the same union frame, ultimately FULL16 for every role. Lowering each +output to a target cap requires at least16384 additional rank-one copies, +which the spinor-only margin16384-9216=7168 cannot pay. This blocks that +direct schedule, not every possible fused transform. + +Files: spinor_operator.py/SPINOR-OPERATOR.json. The exact square-zero +polynomial identities and graded matrix ranks pass with Python -I -O. +Independent physical-frame receipt is in +workbench/helper-routes/MOBIUS-FRAMES.json (owned by the routes agent). + +A compatible Hadamard-clique subtraction +--------------------------------------- +For a in F2^4 define the14-bit mask H(a) by + + H(a)_(j-1)=parity(a & j), j=1,...,14. + +The four subgroup generators are5461,9830,14456,16256. Each nonzero shift +has canonical weight7 or8. Thus the1024 cosets of H are sixteen-port +orthogonal cliques in rational dimension16. + +If J_H is convolution by the whole subgroup, its rank is1024. The legal +clique adjacency is P=I+J_H. The residual from the full spinor adjacency +is A+P=C+J_H and has EXACT binary rank1268 (not merely the upper bound1600). +This may be useful for a fused schedule, but low algebraic residual rank +does not itself give compatible encoder and decoder frames. + +Files: clique_residual.py/CLIQUE-RESIDUAL.json; checked with -I -O. + +A scoped E8^2 packet-cover obstruction +------------------------------------ +For one mixed port (u,v), its11831 orthogonal neighbors lie in the +15-dimensional cap a.u+b.v=0. Their homogeneous cubic evaluation rank is +679, out of680 possible cubic monomials. An exact integer vanishing +relation supplies the matching upper bound, so this rank is over Q too: + + ell*(||a||^2-||b||^2)=0, where ell=a.u=-b.v. + +Indeed, mixed neighbors have equal block norms and pure neighbors have +ell=0. This spans the entire vanishing-cubic space. The quadratic factor +on the cap has rank14 and signature(7,7), with radical spanned by(u,-v). +It cannot be a product of two linear forms. Therefore these neighbors +cannot be covered by three proper rational linear hyperplanes. Block +Weyl symmetry gives the same conclusion for every mixed port. + +This is an obstruction to such a three-hyperplane packet-cover plan, +not to every mutable-source algorithm or every supplier. + +File: cubic_relations.py/CUBIC-RELATIONS.json. A modular rank679 sample +includes the unique orthogonal sign-partner; omitting that rare point +can give the misleading sample rank678. The integer relation is checked +on ALL11831 neighbors. The earlier NEIGHBOR-CUBICS.json records that +initial bounded sample and is not the final certificate. + +Pricing interpretation +---------------------- +The budget agent's separate ordinary banked screen gives meaningful +headroom for the h16/rank815 pair, conditional on a small enough actual +word. Its conservative five-stage helper profile allows approximately +43879 helpers for a +20% target, or60062 to cross the retained frontier. +No such word was constructed here. Independent rectangle factories and +unfused transforms do not meet this requirement. No new bit saving or +integer-multiplication exponent is established by these rank receipts. + +Literal natural-column implementation of the rank1268 residual +--------------------------------------------------------------- +A column of D=C+J_H is supported on6420 spinor targets: the6435 +orthogonal neighbors with the15 nonzero H clique shifts removed. +For the all-positive column, every support vector lies in the sum-zero +rational cap. Fifteen listed support vectors have a nonzero exact integer +15-by-15 minor after deleting the first coordinate. Consequently the +support spans the entire15-dimensional cap. XOR translations are diagonal +real sign isometries, so every column has the same exact span property. + +Therefore the intersection of all target caps for one such decoder +column is precisely the one-dimensional line of that column's port. +If a rank1268 column-basis factorization uses full-frame aggregates and +one private decoder copy per retained natural column, each copy must +lower from dimension16 to at most1. Its rank mass is at least +1268*15=19020 per invocation, already larger than all16384 spinor ports, +even before charging the original576 full-frame centers or the clique +subroutine. This rejects that literal implementation of the1268 residual. +It does not reject different encoder/decoder bases or fused shared DAGs. + +Files: clique_residual_support.py/CLIQUE-RESIDUAL-SUPPORT.json. +The finite witness and scope check pass with Python -I -O. diff --git a/research/niemeier-lattices/workbench/bit-routes/SPINOR-OPERATOR.json b/research/niemeier-lattices/workbench/bit-routes/SPINOR-OPERATOR.json new file mode 100644 index 0000000000..3351be0ef2 --- /dev/null +++ b/research/niemeier-lattices/workbench/bit-routes/SPINOR-OPERATOR.json @@ -0,0 +1,65 @@ +{ + "status": "PASS_EXACT_SPINOR_OPERATOR_IDENTITY", + "dimension": 14, + "port_count": 16384, + "rational_frame_dimension": 16, + "group_kernel_weights": [ + 0, + 7, + 8 + ], + "z_polynomial_degree": 8, + "z_polynomial_monomials": 3003, + "darboux_pairs": 7, + "creation_monomials": [ + 255, + 831, + 3135, + 12351, + 975, + 3279, + 12495, + 3855, + 13071, + 15375, + 1011, + 3315, + 12531, + 3891, + 13107, + 15411, + 4035, + 13251, + 15555, + 16131, + 1020, + 3324, + 12540, + 3900, + 13116, + 15420, + 4044, + 13260, + 15564, + 16140, + 4080, + 13296, + 15600, + 16176, + 16320 + ], + "creation_operator_rank": 64, + "creation_shear_scalar_additions": 2240, + "graded_ranks": [ + 1, + 14, + 91, + 364, + 91, + 14, + 1 + ], + "central_rank": 576, + "darboux_rule": "x_i=z_(2i)+z_(2i+1); y_i=z_(2i+1)+sum_(j>i)(z_(2j)+z_(2j+1)), zero-based i=0,...,6.", + "scope": "Exact scalar basis-change identity. The two conjugating basis changes are not free common-frame operations, and this is not a complete bit supplier." +} diff --git a/research/niemeier-lattices/workbench/bit-routes/clique_residual.py b/research/niemeier-lattices/workbench/bit-routes/clique_residual.py new file mode 100644 index 0000000000..836fd528a9 --- /dev/null +++ b/research/niemeier-lattices/workbench/bit-routes/clique_residual.py @@ -0,0 +1,49 @@ +from pathlib import Path +WORKBENCH = Path(__file__).resolve().parents[1] +PACKAGE = WORKBENCH.parent +import json +D=14;N=1<>bit&1:row=((row&mask)<>shift)&mask) + v=row + while v: + p=v.bit_length()-1 + if p in basis:v^=basis[p] + else:basis[p]=v;break + +def zeta(c): + c=c.copy() + for bit in range(D): + for i in range(N): + if not i>>bit&1:c[i]^=c[i|(1<>i)&1)for i in range(D)]+[1-2*(m.bit_count()%2)] +def require(b,s): + if not b:raise ValueError(s) +require(len(support)==6420,'Exact natural-column support count') +require(all(sum(port(m))==0 for m in support),'All targets lie in the all-positive port cap') +basis={};witness=[] +for m in support: + row=[a%101 for a in port(m)[1:]] + for pivot,old in sorted(basis.items()): + c=row[pivot];row=[(a-c*b)%101 for a,b in zip(row,old)] + nonzero=[i for i,c in enumerate(row)if c] + if nonzero: + pivot=nonzero[0];scale=pow(row[pivot],-1,101);basis[pivot]=[a*scale%101 for a in row];witness.append(m) + if len(basis)==15:break +require(len(basis)==15,'Full rank15 inside the rational cap') +minor=sp.Matrix([port(m)[1:]for m in witness]);det=int(minor.det()) +require(det!=0 and det%101!=0,'Nonzero exact integer fifteen-by-fifteen minor') +result={'status':'PASS_EXACT_RESIDUAL_COLUMN_CAP_SPAN','column_port':0, + 'support_count':len(support),'rational_support_rank':15, + 'support_witness_ports':witness,'support_witness_vectors':[port(m)for m in witness], + 'removed_coordinate':0,'minor_determinant':det,'modular_prime':101, + 'common_target_cap_dimension':1,'natural_column_basis_size':1268, + 'full_frame_to_common_cap_rank':15,'literal_private_copy_rank_mass':1268*15, + 'spinor_port_count':16384,'excess_over_port_count':1268*15-16384, + 'translation_proof':'XOR translation by j multiplies the sixteen real coordinates by the signs of the doubled spinor port j; this is an invertible diagonal orthogonal transformation and preserves the subgroup-removed support relation.', + 'scope':'The literal natural-column rank-basis implementation with full-frame aggregates and separately lowered private decoder copies. Not a lower bound for arbitrary rank factorizations, alternate bases, or fused shared DAGs.'} +(WORKBENCH / 'bit-routes/CLIQUE-RESIDUAL-SUPPORT.json').write_text(json.dumps(result,indent=2)+'\n') +print(json.dumps({k:v for k,v in result.items()if k!='support_witness_vectors'},indent=2)) diff --git a/research/niemeier-lattices/workbench/bit-routes/cubic_relations.py b/research/niemeier-lattices/workbench/bit-routes/cubic_relations.py new file mode 100644 index 0000000000..5e14daf04f --- /dev/null +++ b/research/niemeier-lattices/workbench/bit-routes/cubic_relations.py @@ -0,0 +1,25 @@ +from pathlib import Path +WORKBENCH = Path(__file__).resolve().parents[1] +PACKAGE = WORKBENCH.parent +from itertools import combinations_with_replacement +import json,time +import numpy as np +import sympy as sp +rows=np.load((WORKBENCH / 'bit-routes/ports.npz'))['ports'];v=rows[2160];ids=np.flatnonzero(rows@v==0);rem=int(np.flatnonzero(v)[0]);coords=np.delete(rows[ids],rem,axis=1).astype(np.int64);mons=np.array(list(combinations_with_replacement(range(15),3))) +F0=np.prod(coords[:,mons],axis=2);rng=np.random.default_rng(123);sample=np.unique(np.concatenate((rng.permutation(len(coords))[:1700], np.flatnonzero(ids==2161))));F=F0[sample]%101;pivs=[];rank=0 +for col in range(F.shape[1]): + nz=np.flatnonzero(F[rank:,col]) + if not len(nz):continue + j=rank+int(nz[0]);F[[rank,j]]=F[[j,rank]];F[rank,col:]=F[rank,col:]*pow(int(F[rank,col]),-1,101)%101 + F[rank+1:,col:]=(F[rank+1:,col:]-F[rank+1:,col,None]*F[rank,col:])%101 + pivs.append(col);rank+=1 +free=[i for i in range(680) if i not in pivs];null=[];expressions=[];names=[f'a{i}'for i in range(8)]+[f'b{i}'for i in range(8)];names.pop(rem);sym=sp.symbols(' '.join(names)) +for f in free: + c=np.zeros(680,dtype=np.int64);c[f]=1 + for j in range(rank-1,-1,-1):c[pivs[j]]=-int(F[j]@c)%101 + for den in range(1,65): + z=(c*den+50)%101-50 + if not np.any(F0@z):break + else:raise ValueError('no integer lift') + null.append(z.tolist());expr=sum(int(k)*sp.prod(sym[int(i)]for i in mon)for k,mon in zip(z,mons)if k);expressions.append(str(sp.factor(expr)));print(den,sp.factor(expr),flush=True) +res={'target_port':2160,'prime':101,'sample_port_ids':ids[sample].tolist(),'all_neighbor_count':len(ids),'target_vector':v.tolist(),'removed_coordinate':rem,'coordinate_names':names,'rank':rank,'free_columns':free,'monomials':mons.tolist(),'integer_kernel_basis':null,'factorizations':expressions};(WORKBENCH / 'bit-routes/CUBIC-RELATIONS.json').write_text(json.dumps(res,indent=2)+'\n') diff --git a/research/niemeier-lattices/workbench/bit-routes/d16_rank.py b/research/niemeier-lattices/workbench/bit-routes/d16_rank.py new file mode 100644 index 0000000000..a957d5e212 --- /dev/null +++ b/research/niemeier-lattices/workbench/bit-routes/d16_rank.py @@ -0,0 +1,30 @@ +from pathlib import Path +WORKBENCH = Path(__file__).resolve().parents[1] +PACKAGE = WORKBENCH.parent +from itertools import combinations, product +import importlib.util,json,time +import numpy as np +p=(PACKAGE / 'code/e8_cubed.py') +spec=importlib.util.spec_from_file_location('e8',p);e8=importlib.util.module_from_spec(spec);spec.loader.exec_module(e8) +rows=[] +for i in range(16): + a=[0]*16;a[i]=4;rows.append(a) +for inds in combinations(range(16),4): + for sg in product((-2,2),repeat=3): + a=[0]*16;a[inds[0]]=2 + for i,s in zip(inds[1:],sg):a[i]=s + rows.append(a) +for sg in product((-1,1),repeat=14): + a=[1]+list(sg);a.append(1 if sum(s==-1 for s in a)%2==0 else -1);rows.append(a) +rows=np.array(rows,dtype=np.int16);N=len(rows) +if N!=30960 or np.any(np.sum(rows*rows,axis=1)!=16):raise ValueError('shell') +np.savez_compressed((WORKBENCH / 'bit-routes/d16-ports.npz'),ports=rows) +basis={};witness=[];t0=time.time() +for start in range(0,N,128): + ids=np.arange(start,min(start+128,N));gram=rows[ids]@rows.T + if np.any(gram%4):raise ValueError('Pairing integrality') + fit=gram%16==0 + for i,row in zip(ids,np.packbits(fit,axis=1,bitorder='little')): + if e8.insert_binary(basis,int.from_bytes(row.tobytes(),'little')):witness.append(int(i)) + if start%2048==0:print(start,len(basis),round(time.time()-t0,1),flush=True) +res={'N':N,'h':16,'rank':len(basis),'C':len(basis)*16,'witness':witness};(WORKBENCH / 'bit-routes/D16-RANK.json').write_text(json.dumps(res,indent=2)+'\n');print({k:v for k,v in res.items()if k!='witness'}) diff --git a/research/niemeier-lattices/workbench/bit-routes/mixed_rank.py b/research/niemeier-lattices/workbench/bit-routes/mixed_rank.py new file mode 100644 index 0000000000..dd8f61ed6b --- /dev/null +++ b/research/niemeier-lattices/workbench/bit-routes/mixed_rank.py @@ -0,0 +1,22 @@ +from pathlib import Path +WORKBENCH = Path(__file__).resolve().parents[1] +PACKAGE = WORKBENCH.parent +import importlib.util,json,time +import numpy as np +p=(PACKAGE / 'code/e8_cubed.py') +spec=importlib.util.spec_from_file_location('e8',p);e8=importlib.util.module_from_spec(spec);spec.loader.exec_module(e8) +data=np.load((WORKBENCH / 'bit-routes/ports.npz'));rows=data['ports'][2160:];N=len(rows) +bases=[{} for _ in range(8)];t0=time.time() +for start in range(0,N,96): + ids=np.arange(start,min(start+96,N));a=rows[ids,:8]@rows[:,:8].T;b=rows[ids,8:]@rows[:,8:].T + types=[(a==0)&(b==0),((a==4)&(b==-4))|((a==-4)&(b==4)),((a==8)&(b==-8))|((a==-8)&(b==8))] + diag=ids[:,None]==np.arange(N)[None,:] + for mask in range(8): + fit=diag.copy() + for k in range(3): + if mask>>k&1:fit^=types[k] + for row in np.packbits(fit,axis=1,bitorder='little'): + e8.insert_binary(bases[mask],int.from_bytes(row.tobytes(),'little')) + if start%1920==0:print(start,[len(x) for x in bases],round(time.time()-t0,1),flush=True) +res={'N':N,'h':16,'ranks':{str(mask):len(bases[mask])for mask in range(8)},'orbit_bits':['both_block_inner0','opposite_block_inner1','opposite_block_inner2']} +(WORKBENCH / 'bit-routes/MIXED-ORBIT-RANKS.json').write_text(json.dumps(res,indent=2)+'\n');print(res,flush=True) diff --git a/research/niemeier-lattices/workbench/bit-routes/mod4_rank.py b/research/niemeier-lattices/workbench/bit-routes/mod4_rank.py new file mode 100644 index 0000000000..bd4746573c --- /dev/null +++ b/research/niemeier-lattices/workbench/bit-routes/mod4_rank.py @@ -0,0 +1,27 @@ +from pathlib import Path +WORKBENCH = Path(__file__).resolve().parents[1] +PACKAGE = WORKBENCH.parent +import importlib.util, json, time +import numpy as np +p=(PACKAGE / 'code/e8_cubed.py') +spec=importlib.util.spec_from_file_location('e8',p);e8=importlib.util.module_from_spec(spec);spec.loader.exec_module(e8) +roots,fourth,full,prof=e8.construct_shell() +rows=full[(prof&4)==0,:16] +N=len(rows) +np.savez_compressed((WORKBENCH / 'bit-routes/ports.npz'),ports=rows,roots=roots,fourth=fourth) +basis={};witness=[];t0=time.time() +# Check pure blocks first, then interleave mixed-port rows across both root factors. +order=list(range(2160))+list(range(2160,N)) +for start in range(0,N,128): + ids=order[start:start+128] + gram=rows[ids]@rows.T + if np.any(gram%4): raise ValueError('Nonintegral pairing') + fit=(gram%16==0) + if not np.all(fit[np.arange(len(ids)),ids]):raise ValueError('Diagonal') + for i,a in zip(ids,np.packbits(fit,axis=1,bitorder='little')): + v=int.from_bytes(a.tobytes(),'little') + if e8.insert_binary(basis,v):witness.append(i) + if start%2048==0:print(start,len(basis),round(time.time()-t0,2),flush=True) +res={'N':N,'rational_frame_dimension':16,'binary_central_rank':len(basis),'center_bill_upper':16*len(basis),'independent_rows':witness,'seconds':time.time()-t0} +(WORKBENCH / 'bit-routes/MOD4-RANK.json').write_text(json.dumps(res,indent=2)+'\n') +print(res,flush=True) diff --git a/research/niemeier-lattices/workbench/bit-routes/neighbor_cubics.py b/research/niemeier-lattices/workbench/bit-routes/neighbor_cubics.py new file mode 100644 index 0000000000..8df1e7c5db --- /dev/null +++ b/research/niemeier-lattices/workbench/bit-routes/neighbor_cubics.py @@ -0,0 +1,24 @@ +from pathlib import Path +WORKBENCH = Path(__file__).resolve().parents[1] +PACKAGE = WORKBENCH.parent +from itertools import combinations_with_replacement +import json,time +import numpy as np +rows=np.load((WORKBENCH / 'bit-routes/ports.npz'))['ports'] +results=[] +for tag,index in [('pure',0),('mixed',2160)]: + v=rows[index];neighbors=np.flatnonzero(rows@v==0);removed=int(np.flatnonzero(v)[0]);coords=np.delete(rows[neighbors],removed,axis=1).astype(np.int64) + rng=np.random.default_rng(123);sample=rng.permutation(len(coords))[:min(len(coords),1700)] + mons=np.array(list(combinations_with_replacement(range(15),3))) + F=np.prod(coords[sample][:,mons],axis=2)%101;ids=neighbors[sample].copy();rank=0;piv=[];st=time.time() + for col in range(F.shape[1]): + nz=np.flatnonzero(F[rank:,col]) + if not len(nz):continue + j=rank+int(nz[0]);F[[rank,j]]=F[[j,rank]];ids[[rank,j]]=ids[[j,rank]] + F[rank,col:]=F[rank,col:]*pow(int(F[rank,col]),-1,101)%101 + F[rank+1:,col:]=(F[rank+1:,col:]-F[rank+1:,col,None]*F[rank,col:])%101 + piv.append(int(ids[rank]));rank+=1 + if rank==F.shape[1]:break + res=dict(tag=tag,port=index,neighbor_count=len(neighbors),quadratic_rank=None,cubic_rank=rank,cubic_dimension=680,prime=101,removed_coordinate=removed,independent_ports=piv) + results.append(res);print(tag,len(neighbors),rank,round(time.time()-st,2),flush=True) +(WORKBENCH / 'bit-routes/NEIGHBOR-CUBICS.json').write_text(json.dumps(results,indent=2)+'\n') diff --git a/research/niemeier-lattices/workbench/bit-routes/section_ranks.py b/research/niemeier-lattices/workbench/bit-routes/section_ranks.py new file mode 100644 index 0000000000..39b44e56e3 --- /dev/null +++ b/research/niemeier-lattices/workbench/bit-routes/section_ranks.py @@ -0,0 +1,15 @@ +from pathlib import Path +WORKBENCH = Path(__file__).resolve().parents[1] +PACKAGE = WORKBENCH.parent +import importlib.util,json,time +import numpy as np +p=(PACKAGE / 'code/e8_cubed.py');s=importlib.util.spec_from_file_location('e8',p);m=importlib.util.module_from_spec(s);s.loader.exec_module(m) +allrows=np.load((WORKBENCH / 'bit-routes/d16-ports.npz'))['ports'];out=[] +for name,rows in [('D16spinor',allrows[14576:]),('D16sparse',allrows[:14576])]: + B={};start=time.time() + for i in range(0,len(rows),128): + G=rows[i:i+128]@rows.T;F=(G%16==0) + for a in np.packbits(F,axis=1,bitorder='little'):m.insert_binary(B,int.from_bytes(a.tobytes(),'little')) + out.append(dict(name=name,N=len(rows),h=16,rank=len(B),C=16*len(B))) + print(out[-1],time.time()-start,flush=True) +(WORKBENCH / 'bit-routes/D16-SECTIONS.json').write_text(json.dumps(out,indent=2)+'\n') diff --git a/research/niemeier-lattices/workbench/bit-routes/spinor_operator.py b/research/niemeier-lattices/workbench/bit-routes/spinor_operator.py new file mode 100644 index 0000000000..99f006b577 --- /dev/null +++ b/research/niemeier-lattices/workbench/bit-routes/spinor_operator.py @@ -0,0 +1,97 @@ +"""Exact finite group-algebra transform for the D16+ spinor bit decoder.""" +from itertools import combinations +from math import comb +from pathlib import Path +WORKBENCH = Path(__file__).resolve().parents[1] +PACKAGE = WORKBENCH.parent +import json + +D=14 +ALL=(1<>i&1:coeff[mask]^=coeff[mask|(1<i}q_j. Their linear transformation is triangular +# after the seven elementary coordinate-pair changes, hence invertible. +pairs=[];forms=[] +for i in range(7): + x={1<<(2*i),1<<(2*i+1)} + y={1<<(2*i+1)}|{1<7, so every cross-product operator is zero. +monomials=[sum(3<<(2*i)for i in inds)for inds in combinations(range(7),4)] +require(len(monomials)==35 and all(m.bit_count()==8 for m in monomials), + '35 creation monomials of degree eight') +require(all(a&b for a,b in combinations(monomials,2)), + 'Every pair of creation operators has zero product') +require(all(2**(D-m.bit_count())==64 for m in monomials),'Rank64 per creation operator') + +# Exact ranks of the seven graded inclusion matrices (not a floating rank). +def binary_rank(rows): + basis={} + for v in rows: + while v: + k=v.bit_length()-1 + if k in basis:v^=basis[k] + else:basis[k]=v;break + return len(basis) + +grades=[] +for j in range(7): + source=list(combinations(range(D),j));target=list(combinations(range(D),j+8)) + src=[sum(1<i)(z_(2j)+z_(2j+1)), zero-based i=0,...,6.', + 'scope':'Exact scalar basis-change identity. The two conjugating basis changes are not free common-frame operations, and this is not a complete bit supplier.'} +(WORKBENCH / 'bit-routes/SPINOR-OPERATOR.json').write_text(json.dumps(result,indent=2)+'\n') +print(json.dumps(result,indent=2)) diff --git a/research/niemeier-lattices/workbench/bit-routes/verify_portable.py b/research/niemeier-lattices/workbench/bit-routes/verify_portable.py new file mode 100644 index 0000000000..3c56517fea --- /dev/null +++ b/research/niemeier-lattices/workbench/bit-routes/verify_portable.py @@ -0,0 +1,113 @@ +"""Replay these three research subtrees without modifying checkout receipts. + +All generators run in an isolated copy of the package. Coordinate NPZ inputs +are regenerated, JSON is compared structurally, and the saved factor NPZ is +compared by array contents. Timing metadata is deliberately not certified. +""" +from pathlib import Path +import argparse +import json +import shutil +import subprocess +import sys +import tempfile +import time + +import numpy as np + +PACKAGE = Path(__file__).resolve().parents[2] + +# Dependency order is intentional. hybrid_sections imports hybrid and therefore +# regenerates HYBRID.json too. clique_rank completes the leading-search receipt. +REPLAYS = ( + ('bit-routes/mod4_rank.py', ('MOD4-RANK.json',)), + ('bit-routes/d16_rank.py', ('D16-RANK.json',)), + ('bit-routes/mixed_rank.py', ('MIXED-ORBIT-RANKS.json',)), + ('bit-routes/section_ranks.py', ('D16-SECTIONS.json',)), + ('bit-routes/neighbor_cubics.py', ('NEIGHBOR-CUBICS.json',)), + ('bit-routes/cubic_relations.py', ('CUBIC-RELATIONS.json',)), + ('bit-routes/spinor_operator.py', ('SPINOR-OPERATOR.json',)), + ('bit-routes/clique_residual.py', ('CLIQUE-RESIDUAL.json',)), + ('bit-routes/clique_residual_support.py', ('CLIQUE-RESIDUAL-SUPPORT.json',)), + ('continuation/section-search/sections.py', ('SECTIONS.json',)), + ('continuation/section-search/hybrid_sections.py', ('HYBRID.json', 'HYBRID-SECTIONS.json')), + ('continuation/spinor-routes/frame_bounds.py', ('FRAME-BOUNDS.json',)), + ('continuation/spinor-routes/cross_rank.py', ('CROSS-RANK.json',)), + ('continuation/spinor-routes/cross_factor.py', ('CROSS-FACTOR.json', 'CROSS-FACTOR.npz')), + ('continuation/spinor-routes/clique_leading.py', ()), + ('continuation/spinor-routes/clique_rank.py', ('CLIQUE-LEADING.json',)), + ('continuation/spinor-routes/clique_small.py', ('CLIQUE-SMALL.json',)), + ('continuation/spinor-routes/clique_half.py', ('CLIQUE-HALF.json',)), +) + + +def require(condition, message): + if not condition: + raise ValueError(message) + + +def compare_receipt(expected, actual): + require(actual.is_file(), f'Missing regenerated receipt: {actual.name}') + if expected.suffix == '.npz': + with np.load(expected, allow_pickle=False) as left, np.load(actual, allow_pickle=False) as right: + require(set(left.files) == set(right.files), f'NPZ keys differ: {expected.name}') + for key in left.files: + require(np.array_equal(left[key], right[key]), f'NPZ array differs: {expected.name}:{key}') + return + left = json.loads(expected.read_text()) + right = json.loads(actual.read_text()) + if expected.name == 'MOD4-RANK.json': + left.pop('seconds', None) + right.pop('seconds', None) + require(left == right, f'Regenerated receipt differs: {expected}') + + +def replay(isolated_package, frozen_package=PACKAGE, optimized=False): + """Replay in an already isolated package copy and return checked paths. + + The caller must supply distinct directories. This hook permits a top-level + package verifier to reuse its own temporary workspace. + """ + isolated_package = Path(isolated_package).resolve() + frozen_package = Path(frozen_package).resolve() + require(isolated_package != frozen_package, 'Replay must use an isolated package copy') + checked = [] + for relative, outputs in REPLAYS: + script = isolated_package / 'workbench' / relative + command = [sys.executable, '-I'] + if optimized: + command.append('-O') + command.append(str(script)) + start = time.monotonic() + completed = subprocess.run(command, cwd=isolated_package, text=True, + stdout=subprocess.PIPE, stderr=subprocess.STDOUT) + require(completed.returncode == 0, + f'{relative} failed:\n{completed.stdout[-6000:]}') + directory = Path(relative).parent + for name in outputs: + path = Path('workbench') / directory / name + compare_receipt(frozen_package / path, isolated_package / path) + checked.append(str(path)) + print(f'PASS {relative} ({time.monotonic()-start:.1f}s)', flush=True) + return checked + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument('--optimized', action='store_true', + help='Run generators with -I -O (also inherited from this interpreter)') + args = parser.parse_args() + started = time.monotonic() + with tempfile.TemporaryDirectory(prefix='niemeier-bit-replay-') as temporary: + isolated = Path(temporary) / 'niemeier-lattices' + shutil.copytree(PACKAGE, isolated, + ignore=shutil.ignore_patterns('__pycache__', '*.pyc')) + checked = replay(isolated, optimized=args.optimized or bool(sys.flags.optimize)) + print(json.dumps({'status': 'PASS', 'compared_receipts': len(checked), + 'coordinate_inputs': 'regenerated', + 'elapsed_seconds': round(time.monotonic()-started, 2), + 'frozen_checkout_modified': False}, indent=2)) + + +if __name__ == '__main__': + main() diff --git a/research/niemeier-lattices/workbench/continuation/PR604.json b/research/niemeier-lattices/workbench/continuation/PR604.json new file mode 100644 index 0000000000..533b60e78c --- /dev/null +++ b/research/niemeier-lattices/workbench/continuation/PR604.json @@ -0,0 +1 @@ +{"author":{"id":"U_kgDOClpS9w","is_bot":false,"login":"DreamingOfClouds","name":""},"body":"> ⚠️ **Draft:** the heavy verification (full fresh-copy replays, CI) is still running. Build on this at your own risk.\n\n**Conditional κ = 0.001620147045734211, +0.368% over #601** (0.001614202339036103). The conservative model gives 0.001620147042858282. The dispatch is ck ×44, atm3 ×22, jp8 ×6. The value comes from `ASSEMBLY.json` of the strict `verify.py` run on this package's literal owner map.\n\nThe complex side binds. The bit supplier value (#543, 0.0016538) is above the complex bound, so κ is the same with or without the bit cap.\n\n### What changes relative to #601\nIn every call, **14 cold factory-scratch helpers** (`S b.10`, one per factory block) end at a **4-dimensional frame** instead of FULL, so their residual width drops from 9 to 4. One typed step, `cold_end`, is appended to each provider chain (ck, atm3, jp8) after `hyperplane_end`. It has three parts:\n1. **Premix:** 142 rank-0 gates at frame `[]` at the head of section A, after the entry adapters and paid wrappers. They swap `S b.0`/`S b.10` dirt and fan the candidate's initial value out over `S b.1…S b.7`. The compensation reads no cold-ended helper.\n2. **Low-frame cleanup:** four multi-gate tail gates per candidate at its frame U_b (`h += −½·P_a`, `h += P_b`, `P_c += h`, `P_d += h`). After them, h's row and column of the state are unit vectors.\n3. **Exact FULL remainder:** a dyadic Gauss–Jordan refactorisation of the rest of the inverse, touching no helper that ends below FULL.\n\nThe local profile changes by {1:+28, 5:+14, 6:−28} per call per provider. The dispatch and the literal owner map are re-solved under every inherited window, family and capacity constraint.\n\n**Necessary condition used throughout:** h can end at U only if, at the end of B, the supports of its row in the inverse state map and of its dirt column both lie in the registers whose end-of-B frames lie in U. The premix collapses each candidate's dirt column to three low-frame registers.\n\n### Why the cold ends are legitimate\n- **Lean:** #599 (`hpe_labels_multi`, `hpe_word_multi`, `exact_cold`) covers multi-gate tails and an unrestricted entrance (E = ⊥), building on #593. It is now merged.\n- **(N) for every (0, U_b) pair** is checked exactly, global phase included (`hpe_adapters_local.py`), with wrapper mutants rejected.\n- **The refactored cleanup is not trusted.** It is accepted only for words marked `refactored_cleanup`: non-target registers only, dyadic additions with signed binary scales, and the arbitrary-dirt endpoint replay of the full program exact in both orientations. The transport also checks the row and column cone conditions.\n- **Wrapper order** is enforced twice: by the transport's zero-frame assertion, and by an event-level check in `hpe_audit_local.py` with a negative control (a premix gate hoisted before the pre-wrapper).\n- **The verifier changes are inert** for words without cold ends.\n\n### Checks\n```sh\nresearch/complex-571-cold14/verify.sh /tmp/complex-571-cold14 .. # kappa 0.001620147045734211\n```\n- **Full rebuild:** `verify.py` rebuilds the three provider words through their typed chains, with pinned hashes and paired controls.\n- **Local replay:** it runs exact scalar replay with the cone pre-check, both reflected ledgers, the independent local replay with mutants, exact wrappers and all phase edges.\n- **Global:** it runs binding with the physical owner audit, a fresh finite invoice with both moment engines, and the assembly, phase-timing and generated-identity controls.\n- **Independent audit:** the three cold words replay exactly with independent code (forward, dual, inverse-reverse and random-dirt). There are 0 nesting violations, and the cone fits are tight at 60/60 per word. The compensation reads no wrapped helper, and every word-level mutant is rejected.\n\n### Future optimization\n| quantity | value |\n|---|---|\n| LP ceiling, #601's dispatch, all three providers cold | ≈ +0.49% |\n| LP ceiling, free dispatch over {ck, atm3, jp8} cold (jp8-heavy) | ≈ +0.71% |\n| this package's literal map | +0.368% |\n\n- **What limits a jp8-heavy dispatch:** time-refinability, i.e. one helper per port per call within each owner group. The windows don't limit it. An exact, edge-colourable owner model (per-time Hall conditions) is the remaining owner-side lever.\n- **End dimension:** d = 4 is the best end dimension for these units. Raising it lifts one partner and adds 2ψ(d−4) + ψ(9−d) − ψ(5) to the local cost (ψ(r) = r·ln(73/r)), which exceeds the remaining hosting headroom.\n- **Unit supply:** the 14 units per word are not certified optimal here.\n\n**Credits:**\n- **jeremykothe2:** #591, #592, #593, #596, #600 and #601 (the hyperplane-end mechanism, verifier, Lean and the base packages).\n- **icekylinx:** #531 and #535.\n- **DreamingOfClouds:** #509, #516 and #599 (Lean).\n- The pinned #568, #569, #571, #575 and #583 packages and their authors are listed in NOTICE.md.\n\nThe approach borrows from the uncomputation literature: Bennett 1989; Gidney 2015; Häner–Roetteler–Svore 2017.\n\nDreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code.\n\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\n","headRefOid":"c825e501c5525a08bb98f51034e85cc21e7e7373","isDraft":true,"number":604,"updatedAt":"2026-10-12T17:33:58Z","url":"https://github.com/CrocSwap/integer-mult-bounds/pull/604"} diff --git a/research/niemeier-lattices/workbench/continuation/endpoint-costs/CLIQUE-WORD.json b/research/niemeier-lattices/workbench/continuation/endpoint-costs/CLIQUE-WORD.json new file mode 100644 index 0000000000..55ed8a38fb --- /dev/null +++ b/research/niemeier-lattices/workbench/continuation/endpoint-costs/CLIQUE-WORD.json @@ -0,0 +1,223 @@ +{ + "scope": "One complete orthogonal16-clique adjacency subword. Coordinates denote its orthogonal basis; arbitrary old helpers restored. No fitting kernel, complete lattice word, or exponent claim.", + "ports": 16, + "helpers": 34, + "helper_paths": { + "(2, 1, 13)": 8, + "(4, 3, 9)": 12, + "(8, 7, 1)": 14 + }, + "source_paths": [ + [ + 1, + 1, + 1, + 3, + 1, + 7, + 1 + ], + [ + 1, + 14 + ], + [ + 1, + 1, + 1, + 12 + ], + [ + 1, + 14 + ], + [ + 1, + 1, + 1, + 3, + 1, + 8 + ], + [ + 1, + 14 + ], + [ + 1, + 1, + 1, + 12 + ], + [ + 1, + 14 + ], + [ + 1, + 1, + 1, + 3, + 1, + 7, + 1 + ], + [ + 1, + 14 + ], + [ + 1, + 1, + 1, + 12 + ], + [ + 1, + 14 + ], + [ + 1, + 1, + 1, + 3, + 1, + 8 + ], + [ + 1, + 14 + ], + [ + 1, + 1, + 1, + 12 + ], + [ + 1, + 14 + ] + ], + "target_paths": [ + [ + 1, + 2, + 4, + 8 + ], + [ + 1, + 2, + 4, + 8 + ], + [ + 1, + 2, + 4, + 8 + ], + [ + 1, + 2, + 4, + 8 + ], + [ + 1, + 2, + 4, + 8 + ], + [ + 1, + 2, + 4, + 8 + ], + [ + 1, + 2, + 4, + 8 + ], + [ + 1, + 2, + 4, + 8 + ], + [ + 1, + 2, + 4, + 8 + ], + [ + 1, + 2, + 4, + 8 + ], + [ + 1, + 2, + 4, + 8 + ], + [ + 1, + 2, + 4, + 8 + ], + [ + 1, + 2, + 4, + 8 + ], + [ + 1, + 2, + 4, + 8 + ], + [ + 1, + 2, + 4, + 8 + ], + [ + 1, + 2, + 4, + 8 + ] + ], + "paid_profile": { + "1": 76, + "2": 24, + "3": 16, + "4": 28, + "7": 16, + "8": 32, + "9": 12, + "12": 4, + "13": 8, + "14": 8 + }, + "paid_mass": 1024, + "expected_mass": 1024, + "scalar_adds": 194, + "checks": [ + "All scalar columns over exact integers", + "Every ADD has identical current frames", + "Every frame chain is nested", + "All source and helper columns restored", + "Every output is exactly all other source inputs", + "Every target ends in its exact orthogonal hyperplane" + ] +} diff --git a/research/niemeier-lattices/workbench/continuation/endpoint-costs/FUSED-CLIQUE-WORD.json b/research/niemeier-lattices/workbench/continuation/endpoint-costs/FUSED-CLIQUE-WORD.json new file mode 100644 index 0000000000..99e4eb74fd --- /dev/null +++ b/research/niemeier-lattices/workbench/continuation/endpoint-costs/FUSED-CLIQUE-WORD.json @@ -0,0 +1,124 @@ +{ + "status": "PASS_EXACT_FUSED_SUBWORD", + "scope": "All64 pairing-frame pairs have identical16-clique incidence and the same scalar coefficient word. One representative combinatorial block replays all1056 input columns; full16384-port geometry alignment is enumerated. This proves a compensation subword, not an admitted supplier or an exponent.", + "geometry": { + "spinor_ports": 16384, + "hadamard_cliques": 1024, + "pairing_frames": 128, + "orthogonal_frame_pairs": 64, + "halves_per_frame": 16, + "ports_per_half": 8, + "ports_per_frame": 128, + "coordinate_pairs": [ + [ + 0, + 1 + ], + [ + 2, + 3 + ], + [ + 4, + 5 + ], + [ + 6, + 7 + ], + [ + 8, + 9 + ], + [ + 10, + 11 + ], + [ + 12, + 13 + ], + [ + 14, + 15 + ] + ], + "canonical_spinor_bitstrings_sha256": "0828152d9e2eb1be6c3550fea196b36a1387f2854410969387d3fa0b9a0381d3" + }, + "ordinary_clique_block": { + "cliques": 16, + "ports": 256, + "helpers": 544, + "helper_endpoint_widths": { + "4": 128, + "8": 192, + "16": 224 + }, + "paid_profile": { + "1": 1536, + "2": 384, + "3": 256, + "4": 448, + "7": 256, + "8": 512, + "12": 64, + "14": 128 + }, + "scalar_CNOTs": 3104, + "mixer_CNOTs_per_orientation": 0, + "all_input_columns_checked": 1056, + "output_row_degree": 15 + }, + "fused_block": { + "cliques": 16, + "ports": 256, + "helpers": 544, + "helper_endpoint_widths": { + "4": 128, + "8": 192, + "16": 224 + }, + "paid_profile": { + "1": 1536, + "2": 384, + "3": 256, + "4": 448, + "7": 256, + "8": 512, + "12": 64, + "14": 128 + }, + "scalar_CNOTs": 4096, + "mixer_CNOTs_per_orientation": 248, + "all_input_columns_checked": 1056, + "output_row_degree": 127 + }, + "full_endpoint_fused_control": { + "cliques": 16, + "ports": 256, + "helpers": 544, + "helper_endpoint_widths": { + "16": 544 + }, + "paid_profile": { + "1": 1216, + "2": 384, + "3": 256, + "4": 448, + "7": 256, + "8": 512, + "9": 192, + "12": 64, + "13": 128, + "14": 128 + }, + "scalar_CNOTs": 4096, + "mixer_CNOTs_per_orientation": 248, + "all_input_columns_checked": 1056, + "output_row_degree": 127 + }, + "total_helpers": 34816, + "negative_control": "Omitting one inverse CNOT fails arbitrary-column restoration", + "endpoint_scope": "Proper-ending helpers need exact endpoint wrappers and width-aware banks with16-width spectators. The frame-level word restores their scalar dirt at the stated terminal frame.", + "center_warning": "J_H plus matching has no established center saving here; this file makes no gain claim." +} diff --git a/research/niemeier-lattices/workbench/continuation/endpoint-costs/HANDOFF.txt b/research/niemeier-lattices/workbench/continuation/endpoint-costs/HANDOFF.txt new file mode 100644 index 0000000000..fd03fc23ac --- /dev/null +++ b/research/niemeier-lattices/workbench/continuation/endpoint-costs/HANDOFF.txt @@ -0,0 +1,83 @@ +Local constructive endpoint continuation +====================================== +Prepared for DominikScholz with GPT-6 Astra assistance, following the request +for an actual kappa gain. This packaged research makes no improvement claim. + +The new positive result is a fully replayed compensation subword. An +orthogonal clique of 16 ports admits clique adjacency with 34 dirty helpers. +The input data itself holds partial sums. At a binary-tree cut with child +size s, a data pivot holds its child's total; s-1 copied helpers serve all +but one opposite-child target, and that pivot serves the final target. +Only then do the two pivots grow to the parent span and aggregate. + +The proper-ending implementation cleans copied helpers at the parent +span before that aggregation. Their exact per-clique paths are: + eight helpers: [2,1,1], terminal width 4; + twelve helpers:[4,3,1], terminal width 8; + fourteen helpers:[8,7,1], terminal width 16. +Their initial dirt is precompensated on the intended output at frame 0. +This is a concrete cleanup schedule, not deletion of an assumed tail. +All target paths are [1,2,4,8]. The full per-clique paid histogram is +{1:96,2:24,3:16,4:28,7:16,8:32,12:4,14:8}, rank mass 832. +The helper mass is 352; source and target mass is 480. +Proper endings still require exact native endpoint adapters and banks: +a helper of width w carries 16-w spectators; at ambient m a safe private +owner bank has q=floor((m-(16-w))/w) copies and pad m-qw. +The frame-level construction does not itself certify that native wrapper +implementation, controller ownership, or bit bad-branch envelope. + +One fixed coordinate-pairing matching layer can be fused for no additional +paid paths. Full enumeration of 16,384 even-parity sign lines verifies 1,024 +Hadamard cliques, each split 8+8 between orthogonal pairing frames. Each of +128 pairing frames contains 16 clique halves and 128 distinct ports. +Opposite frames have exactly the same 16-clique incidence. +At that shared 8D frame, use the invertible characteristic-two matrix I+J +on the 16 data pivots holding half totals. Since 16 is even, (I+J)^2=I. +The ordinary top-cut fanout now computes clique adjacency XOR matching. +The emitted matrix uses 248 CNOTs per orientation; its inverse is included +in final source cleanup. All 1,056 input columns in a representative 256-port +block replay exactly, including arbitrary helper dirt. Every block has the +same verified incidence, so the 64 blocks cover the full spinor shell. +There are 34,816 helpers in total, with endpoint inventory +{4:8192,8:12288,16:14336}. An omitted cleanup CNOT is rejected. + +Run: + python3 -I -O research/niemeier-lattices/workbench/continuation/endpoint-costs/verify_portable.py +Run from the checkout root; generated outputs use an external workspace. +Receipt: FUSED-CLIQUE-WORD.json. The ordinary full-ending word in +clique_word.py/CLIQUE-WORD.json is retained as a simpler control. +This is a compensation subword, not a full supplier or exponent gain. +The fitting kernel J_H+M still needs sufficient center savings; the separate +rank/image work finds no such saving for this construction. + +Two other current routes are quantified: +1. The complete punctured native tensor lift has N = 25,088, C = 14,336, +224 native v112 invocations and 112,896 helpers. Exact PR600 paid profiles, +8D spectator preclimbs, and width-aware banks give best nominal kappa +0.000872182222 for ATM3 and 0.000867937701 for CK, both at L = 7 among odd L from 3 through 101. +Those values are about 46% below PR600's declared 0.001612722852614221. +The native tensor embedding is certified separately by the routes agent. +This invoice inherits the PR600 receipt; it is not a fresh local native-word +replay. See native_tensor_price.py and NATIVE-TENSOR-PRICE.json. + +2. Full-shell hybrid rank 1851, N = 30,960, h = 16, C = 29,616 fails every odd stage, +even ignoring helpers, center pressure, idle pressure, and bank padding. +The positive deficit first occurs at L = 25. Active pressure/(LN) at the +frontier is above 0.156522 while D/(LN) is always below 1344/30960=0.0434109. +Hence its necessary kappa is less than0.000448646990529330 for all stages. +See hybrid_bound.py/HYBRID-BOUND.json. Both normal and -I -O runs pass; the +cheaper rank 815 control is not rejected by this sufficient obstruction. + +Endpoint source context: current PR600 head + dfb7b827990d7b586b3cd778e3d750c050dd4778 +and main 5996f9043c27bafa90b23e1c937de92a738f1a76. +Exact fetched URLs and hashes are in sources/SOURCES.json. +The upstream hyperplane-end mechanism requires compatible cleanup operands +and endpoint-specific residual wrappers; an arbitrary favorable first or +last frame cannot simply be declared a free endpoint. + +The most concrete next direction is to retain this shared-data, early- +cleanup compensation architecture while finding a lower-center fitting +kernel compatible with its one shared 8D cut. Adding more matching layers +alone is constrained by the separate augmentation/rank result; unrelated +cuts also demand a new monotone frame schedule. No further gain is claimed. diff --git a/research/niemeier-lattices/workbench/continuation/endpoint-costs/HYBRID-BOUND.json b/research/niemeier-lattices/workbench/continuation/endpoint-costs/HYBRID-BOUND.json new file mode 100644 index 0000000000..9a5ccaf885 --- /dev/null +++ b/research/niemeier-lattices/workbench/continuation/endpoint-costs/HYBRID-BOUND.json @@ -0,0 +1,32 @@ +{ + "status": "PASS_EXACT_ALL_ODD_STAGES", + "result": { + "N": 30960, + "h": 16, + "center_roles": 1851, + "center_histogram": { + "16": 1851, + "1": 0 + }, + "center_bill": 29616, + "center_bill_per_port": "617/645", + "gap": 1344, + "first_odd_stage_with_positive_deficit": 25, + "deficit_at_previous_odd_stage": -48, + "frontier": "1612722852614221/1000000000000000000", + "normalized_active_pressure_lower_bound_at_frontier": "78261079943499913732025124276131833977/500000000000000000000000000000000000000", + "normalized_deficit_strict_upper_bound": "28/645", + "global_necessary_kappa_upper_bound_interval_scaled_1e18": [ + 448646990529329, + 448646990529330 + ], + "fraction_of_frontier_upper_bound": "14459824870215685357335259553179057214000000000000000000/51977814418152943696654367100285811822963156208224506517", + "status": "FAILS_FRONTIER_FOR_EVERY_ODD_STAGE", + "scope": "Necessary bound using active data climbs alone, with no helper costs, center pressure, idle pressure or physical-hosting penalties. Not the optimal root of the full necessary model." + }, + "negative_control": "Rank815 without added center cost is not rejected by this sufficient obstruction", + "source": { + "path": "workbench/paid-controller/unfused/verify.py", + "sha256": "62fd386bd7950dc1edfd83c22c412af0d59165197f2b9cd68ef54434a9e68382" + } +} diff --git a/research/niemeier-lattices/workbench/continuation/endpoint-costs/NATIVE-TENSOR-PRICE.json b/research/niemeier-lattices/workbench/continuation/endpoint-costs/NATIVE-TENSOR-PRICE.json new file mode 100644 index 0000000000..3958c4ed3f --- /dev/null +++ b/research/niemeier-lattices/workbench/continuation/endpoint-costs/NATIVE-TENSOR-PRICE.json @@ -0,0 +1,4514 @@ +{ + "scope": "Exact rational moment invoice conditional on the inherited PR600 native contract and complement-lift interface. No new native scalar-word replay or claimed exponent.", + "source": { + "pr": 600, + "head": "dfb7b827990d7b586b3cd778e3d750c050dd4778", + "expected_sha256": "55e972cdc22fe3b8503af590abde3860fcc8a35d7d0f1a8564238727a40c8b5c", + "frontier": "1612722852614221/1000000000000000000" + }, + "parameters": { + "ports": 25088, + "ambient": 17, + "native_invocations": 224, + "helpers": 112896, + "center_bill": 14336 + }, + "models": { + "atm3": { + "best": { + "stages": 7, + "m": 113, + "stock": "763802592/3575", + "deficit": 50176, + "kappa_scaled_1e18": [ + 872182221530663, + 872182221535655 + ], + "pressure_at_frontier": "20771462728146138623524463109050177359495571767/223437500000000000000000000000000000000000", + "pressure_exceeds_deficit": true, + "banks": { + "4": { + "roles_per_invocation": 48, + "spectators": 13, + "per_bank": 25, + "pad": 13 + }, + "5": { + "roles_per_invocation": 8, + "spectators": 12, + "per_bank": 20, + "pad": 13 + }, + "7": { + "roles_per_invocation": 70, + 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improvement. It realizes orthogonal clique adjacency with dirty +helpers, and fuses a compatible matching layer over characteristic two. +Early cleanup permits proper helper endings at widths 4, 8 and 16. +Native endpoint wrappers, spectator banks and assembly admission remain +separate obligations. HANDOFF.txt gives the construction and its scope. + +From the checkout root, run: + python3 -I -O research/niemeier-lattices/workbench/continuation/endpoint-costs/verify_portable.py --workspace /path/outside-checkout/endpoint-replay + python3 -I research/niemeier-lattices/workbench/continuation/endpoint-costs/verify_portable.py --normal --workspace /path/outside-checkout/endpoint-normal + +The workspace is optional and defaults to a fresh temporary directory. +Only the standard library is required. Generated receipts and logs go to +that external workspace. The verifier compares all parsed JSON fields, +including the original full-ending clique control and the later proper- +ending fused receipt. It checks all pinned upstream bytes before use. +The five local replay entry points use exception-based checks under -O. +The omitted-cleanup negative control remains active in the fused replay. + +clique_word.py checks the ordinary full-ending construction over integers. +fused_clique_word.py checks the characteristic-two word, full spinor/clique +alignment, the proper-ending version and its full-ending control. +hybrid_bound.py proves the stated all-stage obstruction using active +pressure alone. native_tensor_price.py uses PR600's pinned local receipt +and the stated tensor lift; it does not replay PR600's native pipeline. +tensor_price.py is an earlier conditional equal-factor invoice, retained +for comparison; it is not the later complete tensor construction. + +The numerical comparison remains the historical PR600 declaration +0.001612722852614221 at head dfb7b827990d7b586b3cd778e3d750c050dd4778. +It is not silently updated to subsequent PRs. The enclosing continuation +records PR604 separately; none of these saved invoices uses its value or +claims to replay its cold-end construction. + +Prepared for DominikScholz with OpenAI GPT-6 Astra assistance in T3 Code. +The upstream hyperplane-end mechanism, code, receipts and their inherited +lineage credit Jeremy Kothe / jeremykothe2 and the disclosed Claude Opus 5.5 +assistance. sources/NOTICE.md is preserved byte-for-byte from PR600, as are +its source files. sources/SOURCES.json records exact source URLs and hashes. +The general Clifford-frame note is also preserved and pinned. + +Archival upstream Python files under sources/ are quoted implementation +context and are NOT executed by this portable verifier. Their original +assertions and upstream no-O requirements are unchanged. This is the +explicit exception to the local replay's optimized-mode support; no +fresh upstream supplier replay is claimed. Original upstream paths in +sources/README.md are likewise retained as part of that pinned document. diff --git a/research/niemeier-lattices/workbench/continuation/endpoint-costs/TENSOR-PRICE.json b/research/niemeier-lattices/workbench/continuation/endpoint-costs/TENSOR-PRICE.json new file mode 100644 index 0000000000..153c7d8038 --- /dev/null +++ b/research/niemeier-lattices/workbench/continuation/endpoint-costs/TENSOR-PRICE.json @@ -0,0 +1,737 @@ +[ + { + "L": 3, + "profile": [ + 9 + ], + "m": 49, + "R": 120960, + "max_R_at_frontier": 120220, + "kappa_interval_scaled_1e18": [ + 1607182354045570, + 1607182354050555 + ] + }, + { + "L": 3, + "profile": [ + 1, + 7, + 1 + ], + "m": 49, + "R": 120960, + "max_R_at_frontier": 91393, + "kappa_interval_scaled_1e18": [ + 1365302819370816, + 1365302819375803 + ] + }, + { + "L": 3, + "profile": [ + 1, + 1, + 1, + 1, + 1, + 1, + 1, + 1, + 1 + ], + "m": 49, + "R": 120960, + "max_R_at_frontier": 59690, + "kappa_interval_scaled_1e18": [ + 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"max_R_at_frontier": 0, + "kappa_interval_scaled_1e18": [ + 652864071050709, + 652864071055704 + ] + } +] diff --git a/research/niemeier-lattices/workbench/continuation/endpoint-costs/clique_word.py b/research/niemeier-lattices/workbench/continuation/endpoint-costs/clique_word.py new file mode 100644 index 0000000000..97e2cc6ac1 --- /dev/null +++ b/research/niemeier-lattices/workbench/continuation/endpoint-costs/clique_word.py @@ -0,0 +1,64 @@ +from pathlib import Path +import sys +sys.dont_write_bytecode = True +sys.path.insert(0, str(Path(__file__).resolve().parents[2] / "paid-controller")) +from portable import WORKBENCH, output_path, source_name +"""Complete paid adjacency word for an orthogonal basis; exact integer replay.""" +from pathlib import Path +from collections import Counter +import json +N=16;FULL=(1<>(2*i))^(w>>(2*i+1)))&1)<>(2*i))^(w>>(2*i+1)))&1)==((sigma>>i)&1)for i in range(8))for w in half),'Raw sign vectors satisfy all8 frame equations') + return dict(spinor_ports=len(words),hadamard_cliques=len(cliques),pairing_frames=len(groups),orthogonal_frame_pairs=len(groups)//2,halves_per_frame=16,ports_per_half=8,ports_per_frame=128,coordinate_pairs=[[2*i,2*i+1]for i in range(8)],canonical_spinor_bitstrings_sha256=hashlib.sha256(b''.join(w.to_bytes(2,'little')for w in words)).hexdigest()) + +def mixer_gates(n=16): + rows=[((1<>col&1) + if pivot!=col: + for a,b in ((pivot,col),(col,pivot),(pivot,col)):rows[b]^=rows[a];gates.append((a,b)) + for i in range(n): + if i!=col and rows[i]>>col&1:rows[i]^=rows[col];gates.append((col,i)) + require(rows==[1<r['deficit'],banks=banks)) + best=max(rows,key=lambda r:r['kappa_scaled_1e18'][0]) + result[provider]=dict(best=best,rows=rows) + print(provider,'best L',best['stages'],'kappa',best['kappa_scaled_1e18'],'relative',float(F(best['kappa_scaled_1e18'][0],10**18)/FRONTIER-1)) + out=dict(scope='Exact rational moment invoice conditional on the inherited PR600 native contract and complement-lift interface. No new native scalar-word replay or claimed exponent.',source=dict(pr=600,head='dfb7b827990d7b586b3cd778e3d750c050dd4778',expected_sha256=hashlib.sha256(raw).hexdigest(),frontier=str(FRONTIER)),parameters=dict(ports=N,ambient=H,native_invocations=BLOCKS,helpers=BLOCKS*504,center_bill=C),models=result) + output_path('continuation/endpoint-costs/NATIVE-TENSOR-PRICE.json').write_text(json.dumps(out,indent=2)+'\n') +if __name__=='__main__':main() diff --git a/research/niemeier-lattices/workbench/continuation/endpoint-costs/sources/EXPECTED.json b/research/niemeier-lattices/workbench/continuation/endpoint-costs/sources/EXPECTED.json new file mode 100644 index 0000000000..2f4d3df109 --- /dev/null +++ b/research/niemeier-lattices/workbench/continuation/endpoint-costs/sources/EXPECTED.json @@ -0,0 +1,1388 @@ +{ + "schema": "witness-expected/1", + "witness_sha256": "67dfa7d7eb44927fa883e18f36fe9e64c6188540578c8837f3520560ab494f9c", + "build": { + "atm3": { + "steps": [ + { + 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7682, + "deficit": 2560, + "entering_helper_roles": 16416, + "idle_remainder_profile": { + "1": 2, + "3": 279, + "5": 1, + "12": 1, + "48": 1969 + }, + "complex_lower": "1615327928879591249231/1000000000000000000000000", + "conservative_complex_lower": "323065585203866292309/200000000000000000000000", + "kappa": "1612722852614221/1000000000000000000", + "kappa_decimal": "0.001612722852614221", + "conservative_kappa_decimal": "0.001612722849763179", + "mutations_rejected": 23, + "assembly_controls_rejected": 114 + } +} diff --git a/research/niemeier-lattices/workbench/continuation/endpoint-costs/sources/NOTICE.md b/research/niemeier-lattices/workbench/continuation/endpoint-costs/sources/NOTICE.md new file mode 100644 index 0000000000..eccbbad196 --- /dev/null +++ b/research/niemeier-lattices/workbench/continuation/endpoint-costs/sources/NOTICE.md @@ -0,0 +1,16 @@ +# Attribution and scope + +Four verified provider types are offered to every call: PR571's prefix word with twelve multi-cut cold kernels (PR568's unchanged constructor), the same word with forty width-five entrance promotions (PR569's unchanged frame constructor) before those kernels, PR575's plain joint-prefix provider, and PR568's eight-cold provider. A window MILP with an exact per-pattern model chooses the dispatch and a literal owner map, improved by a 2-swap ejection chain on idle remainders. PR583's six source-bound kernel frame coalescences (its unchanged constructor and negative controls) are applied to every PR571-prefix provider word (ck and atm3), followed by two further coalescences that each first move one addition past two commuting additions (package code/build_gate_moves.py, extending PR583's constructor); widths are unchanged, so the free4f literal owner map is re-admitted with the coalesced locals. + +Pinned external sources (unchanged; `verify.py` checks their manifests and every file it uses): +- PR568 (`research/complex-multicold`, commit `71c99ac935a33bb7b59a1b4a7af8725a939b0b90`). +- PR571 (`research/structural-gain`, commit `b7bdc5a8de18fcf418cfb9c7d83a2b1097a5e4a3`). +- PR569 (`research/larger-gain`, commit `897213a855b7fb8a1b8b0e552a2f5c42e6c1b326`). +- PR575 (`research/complex-joint-prefix`, commit `eac6f7e378ab622dac50a2fd4b7ef6eab014de44`). +- PR583 (`research/kernel-frame-coalescing`, commit `89be768f0157f5a3d8e4db69cf1748768605eac9`). + +Provider construction is described as typed steps in `data/SPEC.json` (`build`); `verify.py` runs only allowlisted step types with their paired omission audits and checks every input and output content hash. `code/` holds the spec-driven global admission scripts adapted from PR571's `frontier-composition/global-code/` (OpenAI Codex assisted, Apache-2.0): every expected value is read from `data/SPEC.json` and compared only after independent recomputation. `code/generated_identity.py` and `code/generated_identity_controls.py`, and the decompressed-content artifact identities in the bind, audit, invoice, assembly and phase-timing scripts, are taken unchanged from PR586 by chafreaky (`research/promoted-kernel-coalescing`, commit `4846fee5b06af770e9b3a10c36200f673d656725`, OpenAI Codex assisted, Apache-2.0), including its proof-content and reference/metadata mutation controls; the normalization of descriptive width metadata after frame changes follows the same PR. + +The owner map was nominated by a HiGHS window MILP (adapted from PR569's `global/nominate_width7.py`) with lazy cross-time Hall cuts; it is not an admission dependency. + +Prepared by Claude Code (Anthropic) research agents. Apache-2.0 applies; LICENSE is included. All notices, licenses, source pins and AI disclosures of the pinned sources and their ancestry are retained in those packages and are not replaced by this notice. All inherited conditional assumptions remain conditional. diff --git a/research/niemeier-lattices/workbench/continuation/endpoint-costs/sources/README.md b/research/niemeier-lattices/workbench/continuation/endpoint-costs/sources/README.md new file mode 100644 index 0000000000..41b292240d --- /dev/null +++ b/research/niemeier-lattices/workbench/continuation/endpoint-costs/sources/README.md @@ -0,0 +1,66 @@ +# complex-571-hpeq6: conditional κ 0.001612722852614221 + +## Definition + +The witness `hpeq6` is the PR558 geometry (112 data columns, h = 9, 504 helpers, 8 centres, 9 stages × 8 replicas = 72 calls) served by 2 provider words, each a pinned base word followed by typed build steps: **ck** (53 calls) = PR571's `frontier-composition/combined-prefix-word/FULL-WORD.json.gz` → `greedy_kernels` (12 multi-cut cold kernels, PR568's constructor) → `frame_coalesce` (6 frame coalescences, PR583's constructor) → `gate_move_coalesce` (2 coalescences, each after moving an addition past commuting additions; 2 moves) → `hyperplane_end` (38 helpers end at a hyperplane U instead of FULL); **atm3** (19 calls) = PR571's `frontier-composition/combined-prefix-word/FULL-WORD.json.gz` → `promote` (40 entrance promotions, PR569's constructor) → `greedy_kernels` (12 multi-cut cold kernels, PR568's constructor) → `frame_coalesce` (6 frame coalescences, PR583's constructor) → `gate_move_coalesce` (2 coalescences, each after moving an addition past commuting additions; 2 moves) → `hyperplane_end` (38 helpers end at a hyperplane U instead of FULL). Each step runs a pinned, unchanged constructor plus its paired omission or negative-control audit; a hyperplane-ended helper starts at its entrance frame E, ends at the hyperplane U, and pays residual width |U| − |E| (every other helper ends at FULL, so its width stays h − |E|). The dispatch (which provider serves each (stage, replica) call), the asynchronous call order and the literal owner map are `data/PLAN.json` (`provider_calls`, `order`) and `data/OWNERS.json.gz`, both nominated by a window MILP with lazy cross-time Hall cuts; the resulting paid profile is {1:182818, 2:34192, 3:22559, 4:14584, 5:4033, 6:4032, 8:576, 12:1, 48:1969} with live 7168, physical 7682 and deficit 2560, giving κ = 1612722852614221/1000000000000000000 = 0.001612722852614221 (conservative model 0.001612722849763179). + +## Files + +| file | role | +|---|---| +| `data/WITNESS.json` | **inputs only**: geometry, pinned sources, the provider build chains above, dispatch and call order, and PLAN/OWNERS/recipes by SHA-256 | +| `data/EXPECTED.json` | **expected values only**, all derived by recomputation (`spec.py generate`): per-step word hashes and control counts, per-provider artifacts, programs and profiles, global profile and counts, κ. Bound to WITNESS.json by hash | +| `data/PLAN.json`, `data/OWNERS.json.gz` | dispatch, call order and literal owner map (inputs, pinned by WITNESS.json) | +| `data/recipes/` | the package's own step recipes (inputs, pinned by WITNESS.json); `data/recipes/jp8-step4-hyperplane_end.json` is used only by `data/cand.json` (a menu provider the owner solve did not dispatch), not by the witness | +| `data/cand.json` | the search candidate (schema candidate/1), an input of the search side: providers on offer, dispatch policy and owner-search settings, with package-relative and `NAME:` source references. Regenerating the package from it with the proposal pipeline reproduces SPEC, PLAN, OWNERS, recipes, WITNESS and EXPECTED: byte for byte with the localized providers it was made with, and otherwise up to PLAN's raw hash of each phase receipt (it records seconds) and the hashes that follow it. The verifiers do not read it | +| `data/search/WARM-OWNERS.json.gz` | search input named by `data/cand.json` (`owners.warm_owners`): the owner map the owner solve was warm-started from. The shipped `data/OWNERS.json.gz` is the solve's result; the verifiers do not read this file | +| `data/SPEC.json`, `verify.py`, `code/` | the original combined spec and package verifier, unchanged; they still verify the same record | + +`data/PLAN.json` carries `"status": "LITERAL_OWNER_NOMINATION_PENDING_ADMISSION"`. That label was written by the owner nominator before admission and is kept unchanged because WITNESS.json pins PLAN by hash; admission is what the verifiers below establish, and the label carries no meaning for them. + +## Verification + +The pinned sources are not merged; fetch their pinned commits first: + +```sh +git fetch origin pull/568/head:pr568 pull/571/head:pr571 pull/569/head:pr569 pull/575/head:pr575 pull/583/head:pr583 +git worktree add --detach ../pr568 71c99ac935a33bb7b59a1b4a7af8725a939b0b90 +git worktree add --detach ../pr571 b7bdc5a8de18fcf418cfb9c7d83a2b1097a5e4a3 +git worktree add --detach ../pr569 897213a855b7fb8a1b8b0e552a2f5c42e6c1b326 +git worktree add --detach ../pr575 eac6f7e378ab622dac50a2fd4b7ef6eab014de44 +git worktree add --detach ../pr583 89be768f0157f5a3d8e4db69cf1748768605eac9 +``` + +Generic verifier (`research/witness-spec-tools`, PR587 with the `frame_coalesce`, `gate_move_coalesce` and `hyperplane_end` step types). It builds everything from `data/WITNESS.json` and only compares against `data/EXPECTED.json`: + +```sh +T=research/witness-spec-tools +python3 $T/spec.py check --package research/complex-571-hpeq6 \ + --source multicold=../pr568/research/complex-multicold \ + --source structural_gain=../pr571/research/structural-gain \ + --source larger_gain=../pr569/research/larger-gain \ + --source joint_prefix=../pr575/research/complex-joint-prefix \ + --source frame_coalescing=../pr583/research/kernel-frame-coalescing +python3 $T/verify.py --package research/complex-571-hpeq6 --work /tmp/complex-571-hpeq6 \ + --source multicold=../pr568/research/complex-multicold \ + --source structural_gain=../pr571/research/structural-gain \ + --source larger_gain=../pr569/research/larger-gain \ + --source joint_prefix=../pr575/research/complex-joint-prefix \ + --source frame_coalescing=../pr583/research/kernel-frame-coalescing +# kappa 0.001612722852614221 +``` + +`verify.py` checks the package MANIFEST and every source, PLAN, OWNERS and recipe pin; replays PR558; rebuilds each provider word through its build chain (each step's input and output content hash pinned, 9 steps); localizes each word (hyperplane-ended words through the generalised native chain); then runs binding, the physical owner audit with 23 controller mutants, a fresh finite invoice, both exact moment engines, 114 assembly controls, the phase-timing controls and the generated-identity controls. + +The original package verifier gives the same result from `data/SPEC.json`: + +```sh +python3 -B research/complex-571-hpeq6/verify.py --work /tmp/complex-571-hpeq6-spec \ + --source multicold=../pr568/research/complex-multicold \ + --source structural_gain=../pr571/research/structural-gain \ + --source larger_gain=../pr569/research/larger-gain \ + --source joint_prefix=../pr575/research/complex-joint-prefix \ + --source frame_coalescing=../pr583/research/kernel-frame-coalescing +``` + +Both use only the Python standard library and require assertions (no `-O`). The result is conditional on the inherited assumptions listed in NOTICE.md. diff --git a/research/niemeier-lattices/workbench/continuation/endpoint-costs/sources/SOURCES.json b/research/niemeier-lattices/workbench/continuation/endpoint-costs/sources/SOURCES.json new file mode 100644 index 0000000000..ca9c5e389a --- /dev/null +++ b/research/niemeier-lattices/workbench/continuation/endpoint-costs/sources/SOURCES.json @@ -0,0 +1,38 @@ +{ + "README.md": { + "url": "https://raw.githubusercontent.com/jeremykothe2/integer-mult-bounds/dfb7b827990d7b586b3cd778e3d750c050dd4778/research/complex-571-hpeq6/README.md", + "sha256": "9e0fcb6d72a4c204b73086b2fd86e882f82a8f824260e75d04776c6ee1f81c33" + }, + "build_hpe.py": { + "url": "https://raw.githubusercontent.com/jeremykothe2/integer-mult-bounds/dfb7b827990d7b586b3cd778e3d750c050dd4778/research/complex-571-hpeq6/code/build_hpe.py", + "sha256": "7a1f37c40f067f9b3ef4711d2ce3843fc10dfcc14b8cc290c3d6f6893c8224ae" + }, + "hpe_adapters_local.py": { + "url": "https://raw.githubusercontent.com/jeremykothe2/integer-mult-bounds/dfb7b827990d7b586b3cd778e3d750c050dd4778/research/complex-571-hpeq6/code/hpe_adapters_local.py", + "sha256": "b5d976f7496eac0d9c7ed97b3a028efcf6950ec989f1789b8c6f5d2b325c40b3" + }, + "hpe_audit_and_transport.py": { + "url": "https://raw.githubusercontent.com/jeremykothe2/integer-mult-bounds/dfb7b827990d7b586b3cd778e3d750c050dd4778/research/complex-571-hpeq6/code/hpe_audit_and_transport.py", + "sha256": "9020d9f295c37836619b59f58c8e67d0dc14bafe8b53aa2c9e98cb5053cbbe9b" + }, + "hpe_emit_local.py": { + "url": "https://raw.githubusercontent.com/jeremykothe2/integer-mult-bounds/dfb7b827990d7b586b3cd778e3d750c050dd4778/research/complex-571-hpeq6/code/hpe_emit_local.py", + "sha256": "7222e564cabb7ff15b8b4833385342f19a49d3037130f8c21afd52b74638fd40" + }, + "ck-step3-hyperplane_end.json": { + "url": "https://raw.githubusercontent.com/jeremykothe2/integer-mult-bounds/dfb7b827990d7b586b3cd778e3d750c050dd4778/research/complex-571-hpeq6/data/recipes/ck-step3-hyperplane_end.json", + "sha256": "d136602543927ff4470f1761bafececa11377ad2662244981e269277c212c68a" + }, + "main-general-clifford-frames.tex": { + "url": "https://raw.githubusercontent.com/CrocSwap/integer-mult-bounds/5996f9043c27bafa90b23e1c937de92a738f1a76/notes/general-clifford-frames.tex", + "sha256": "c9bb9655b594583c6b4bab8cb485233f1d1e588f83858cf1e4c3530cee8fed54" + }, + "EXPECTED.json": { + "url": "https://raw.githubusercontent.com/jeremykothe2/integer-mult-bounds/dfb7b827990d7b586b3cd778e3d750c050dd4778/research/complex-571-hpeq6/data/EXPECTED.json", + "sha256": "55e972cdc22fe3b8503af590abde3860fcc8a35d7d0f1a8564238727a40c8b5c" + }, + "NOTICE.md": { + "url": "https://raw.githubusercontent.com/jeremykothe2/integer-mult-bounds/dfb7b827990d7b586b3cd778e3d750c050dd4778/research/complex-571-hpeq6/NOTICE.md", + "sha256": "e83191ed8db9cb4d5aebd80435ef3e5557ca79df0584e4ffbca9935739cb8980" + } +} diff --git a/research/niemeier-lattices/workbench/continuation/endpoint-costs/sources/build_hpe.py b/research/niemeier-lattices/workbench/continuation/endpoint-costs/sources/build_hpe.py new file mode 100644 index 0000000000..f306369ed4 --- /dev/null +++ b/research/niemeier-lattices/workbench/continuation/endpoint-costs/sources/build_hpe.py @@ -0,0 +1,113 @@ +"""Hyperplane-ended helpers (HPE) constructor. Apache-2.0. Prepared with Claude Code assistance. + +usage: build_hpe.py --input FULL-WORD.json.gz --recipe RECIPE.json --out DIR + +RECIPE: {"changes": [{"helper": h, "cleanup_index": i, "frame": [raw vectors]}, ...], "output_content_sha256": hex}. +Each change must be an admissible early-cleanup move: + * h is an entering helper (start_extra) touched by exactly one cleanup gate, cleanup[i]; + * cleanup[i] is the first cleanup gate on each of its registers (so it commutes with the whole skipped cleanup prefix); + * frame == join of the registers' last A/B frames and has dimension < 9. +The output word gets early_cleanup [[i, U]], helper_final {h: U} and the exact paid profile (only the two registers' +final chains change: h ends at U instead of FULL; the partner climbs through U). Exactness of the reordered program, +the frame chains and the profile are re-checked independently by hpe_audit_and_transport.py; the paired control audit +is controls_hpe_atc.py.""" +import argparse, gzip, hashlib, json +from collections import Counter +from pathlib import Path + + +def span(vs): + p = {} + for x in vs: + for k in sorted(p, reverse=True): + if x >> k & 1: x ^= p[k] + if x: + k = x.bit_length() - 1 + for j in p: + if p[j] >> k & 1: p[j] ^= x + p[k] = x + return [p[k] for k in sorted(p, reverse=True)] + + +def regs(e): return [e[1]] + ([e[2]] if e[0] == 'add' else []) + + +def last_frames(w): + v, R = w['report']['v'], w['report']['R']; n = 2*v + R + cur = [[] for _ in range(n)] + for i, p in enumerate(w['ports']): cur[i] = [p] + for r, f in w.get('start_extra', {}).items(): cur[int(r)] = list(f) + for part in ('A', 'B'): + for e in w[part]: + for r in regs(e): cur[r] = list(e[-1]) + return cur + + +def apply(w, changes): + assert 'early_cleanup' not in w and 'helper_final' not in w, 'input already has hyperplane-ended helpers' + v, R = w['report']['v'], w['report']['R']; n = 2*v + R + cur = last_frames(w); count = Counter(); first = {} + for i, e in enumerate(w['cleanup']): + for r in regs(e): count[r] += 1; first.setdefault(r, i) + lp = Counter({int(r): c for r, c in w['report']['local_profile'].items()}) + seen = set() + for z in changes: + h, i, U = z['helper'], z['cleanup_index'], z['frame'] + assert 2*v <= h < n and str(h) in w['start_extra'], ('not an entering helper', h) + assert count[h] == 1 and first[h] == i, ('helper must have exactly this one cleanup gate', h) + e = w['cleanup'][i]; rs = regs(e) + assert all(first[r] == i for r in rs), ('cleanup gate is not first on its registers', i) + assert not seen & set(rs), 'registers shared between changes'; seen |= set(rs) + assert span([x for r in rs for x in cur[r]]) == U and len(U) < 9, ('frame is not the join of the last frames', h) + for r in rs: # old: last -> FULL; new: last -> U (-> FULL for the partner) + d0 = 9 - len(span(cur[r])) + if d0: lp[d0] -= 1 + d1 = len(U) - len(span(cur[r])) + if d1: lp[d1] += 1 + if r != h and len(U) < 9: lp[9 - len(U)] += 1 + w['early_cleanup'] = [[z['cleanup_index'], z['frame']] for z in changes] + w['helper_final'] = {str(z['helper']): z['frame'] for z in changes} + w['report']['local_profile'] = {str(r): c for r, c in sorted(lp.items()) if c} + w['report']['local_rank_mass'] = sum(r*c for r, c in lp.items()) + if 'width_inventory' in w['report']: # descriptive metadata: one width-w helper becomes width w-(9-dim U) + inv = Counter({int(k): c for k, c in w['report']['width_inventory'].items()}) + for z in changes: + e = len(w['start_extra'][str(z['helper'])]); inv[9 - e] -= 1; inv[len(z['frame']) - e] += 1 + assert min(inv.values()) >= 0 + w['report']['width_inventory'] = {str(k): c for k, c in sorted(inv.items()) if c} + assert min(lp.values()) >= 0 + return w + + +def candidates(w): + v, R = w['report']['v'], w['report']['R']; n = 2*v + R + cur = last_frames(w); count = Counter(); first = {} + for i, e in enumerate(w['cleanup']): + for r in regs(e): count[r] += 1; first.setdefault(r, i) + out = [] + for h in range(2*v, n): + if count[h] != 1 or str(h) not in w.get('start_extra', {}): continue + i = first[h]; rs = regs(w['cleanup'][i]) + if any(first[r] != i for r in rs): continue + U = span([x for r in rs for x in cur[r]]) + if len(U) < 9: out.append(dict(helper=h, cleanup_index=i, frame=U)) + return out + + +if __name__ == '__main__': + ap = argparse.ArgumentParser(); ap.add_argument('--input', type=Path, required=True); ap.add_argument('--recipe', type=Path) + ap.add_argument('--out', type=Path); ap.add_argument('--nominate', type=Path, help='write a recipe with every candidate') + a = ap.parse_args() + w = json.loads(gzip.decompress(a.input.read_bytes())) + if a.nominate: + a.nominate.write_text(json.dumps(dict(changes=candidates(w), output_content_sha256=''), indent=1) + '\n', newline='\n') + raise SystemExit(0) + r = json.loads(a.recipe.read_bytes()) + w = apply(w, r['changes']) + raw = json.dumps(w, sort_keys=True, separators=(',', ':')).encode(); h = hashlib.sha256(raw).hexdigest() + assert h == r['output_content_sha256'], h + a.out.mkdir(parents=True, exist_ok=False) + (a.out/'FULL-WORD.json.gz').write_bytes(gzip.compress(raw, mtime=0)) + (a.out/'HPE-CONSTRUCTION.json').write_text(json.dumps(dict(status='PASS_HYPERPLANE_ENDED_HELPERS', helpers=[z['helper'] for z in r['changes']], + local_profile=w['report']['local_profile']), indent=1) + '\n', newline='\n') + print('PASS', len(r['changes']), 'hyperplane-ended helpers') diff --git a/research/niemeier-lattices/workbench/continuation/endpoint-costs/sources/ck-step3-hyperplane_end.json b/research/niemeier-lattices/workbench/continuation/endpoint-costs/sources/ck-step3-hyperplane_end.json new file mode 100644 index 0000000000..91495222a2 --- /dev/null +++ b/research/niemeier-lattices/workbench/continuation/endpoint-costs/sources/ck-step3-hyperplane_end.json @@ -0,0 +1,537 @@ +{ + "changes": [ + { + "helper": 448, + "cleanup_index": 686, + "frame": [ + 256, + 128, + 64, + 32, + 16, + 8, + 6, + 1 + ] + }, + { + "helper": 449, + "cleanup_index": 685, + "frame": [ + 256, + 128, + 64, + 32, + 16, + 8, + 6, + 1 + ] + }, + { + "helper": 450, + "cleanup_index": 684, + "frame": [ + 256, + 128, + 64, + 32, + 24, + 4, + 2, + 1 + ] + }, + { + "helper": 451, + "cleanup_index": 683, + "frame": [ + 256, + 128, + 64, + 32, + 24, + 4, + 2, + 1 + ] + }, + { + "helper": 452, + "cleanup_index": 682, + "frame": [ + 256, + 128, + 64, + 32, + 18, + 10, + 6, + 1 + ] + }, + { + "helper": 453, + "cleanup_index": 681, + "frame": [ + 256, + 128, + 64, + 32, + 18, + 10, + 6, + 1 + ] + }, + { + "helper": 454, + "cleanup_index": 680, + "frame": [ + 256, + 128, + 64, + 32, + 18, + 10, + 6, + 1 + ] + }, + { + "helper": 455, + "cleanup_index": 679, + "frame": [ + 256, + 128, + 64, + 32, + 18, + 10, + 6, + 1 + ] + }, + { + "helper": 456, + "cleanup_index": 678, + "frame": [ + 256, + 128, + 64, + 32, + 24, + 4, + 2, + 1 + ] + }, + { + "helper": 457, + "cleanup_index": 677, + "frame": [ + 256, + 128, + 64, + 32, + 24, + 4, + 2, + 1 + ] + }, + { + "helper": 458, + "cleanup_index": 676, + "frame": [ + 256, + 128, + 64, + 32, + 16, + 8, + 6, + 1 + ] + }, + { + "helper": 459, + "cleanup_index": 675, + "frame": [ + 256, + 128, + 64, + 32, + 16, + 8, + 6, + 1 + ] + }, + { + "helper": 460, + "cleanup_index": 670, + "frame": [ + 256, + 128, + 64, + 32, + 16, + 8, + 6, + 1 + ] + }, + { + "helper": 461, + "cleanup_index": 669, + "frame": [ + 256, + 128, + 64, + 32, + 16, + 8, + 6, + 1 + ] + }, + { + "helper": 470, + "cleanup_index": 660, + "frame": [ + 256, + 128, + 64, + 32, + 16, + 8, + 6, + 1 + ] + }, + { + "helper": 471, + "cleanup_index": 659, + "frame": [ + 256, + 128, + 64, + 32, + 16, + 8, + 6, + 1 + ] + }, + { + "helper": 472, + "cleanup_index": 654, + "frame": [ + 256, + 128, + 64, + 32, + 16, + 8, + 6, + 1 + ] + }, + { + "helper": 473, + "cleanup_index": 653, + "frame": [ + 256, + 128, + 64, + 32, + 16, + 8, + 6, + 1 + ] + }, + { + "helper": 482, + "cleanup_index": 644, + "frame": [ + 256, + 128, + 64, + 32, + 16, + 8, + 6, + 1 + ] + }, + { + "helper": 483, + "cleanup_index": 643, + "frame": [ + 256, + 128, + 64, + 32, + 16, + 8, + 6, + 1 + ] + }, + { + "helper": 580, + "cleanup_index": 497, + "frame": [ + 256, + 128, + 64, + 32, + 16, + 8, + 6, + 1 + ] + }, + { + "helper": 581, + "cleanup_index": 496, + "frame": [ + 256, + 128, + 64, + 32, + 16, + 8, + 6, + 1 + ] + }, + { + "helper": 590, + "cleanup_index": 481, + "frame": [ + 256, + 128, + 64, + 32, + 16, + 8, + 6, + 1 + ] + }, + { + "helper": 591, + "cleanup_index": 480, + "frame": [ + 256, + 128, + 64, + 32, + 16, + 8, + 6, + 1 + ] + }, + { + "helper": 592, + "cleanup_index": 475, + "frame": [ + 256, + 128, + 64, + 32, + 16, + 8, + 6, + 1 + ] + }, + { + "helper": 593, + "cleanup_index": 474, + "frame": [ + 256, + 128, + 64, + 32, + 16, + 8, + 6, + 1 + ] + }, + { + "helper": 602, + "cleanup_index": 465, + "frame": [ + 256, + 128, + 64, + 32, + 16, + 8, + 6, + 1 + ] + }, + { + "helper": 603, + "cleanup_index": 464, + "frame": [ + 256, + 128, + 64, + 32, + 16, + 8, + 6, + 1 + ] + }, + { + "helper": 604, + "cleanup_index": 459, + "frame": [ + 256, + 128, + 64, + 32, + 16, + 8, + 6, + 1 + ] + }, + { + "helper": 605, + "cleanup_index": 458, + "frame": [ + 256, + 128, + 64, + 32, + 16, + 8, + 6, + 1 + ] + }, + { + "helper": 606, + "cleanup_index": 457, + "frame": [ + 256, + 128, + 64, + 32, + 24, + 4, + 2, + 1 + ] + }, + { + "helper": 607, + "cleanup_index": 456, + "frame": [ + 256, + 128, + 64, + 32, + 24, + 4, + 2, + 1 + ] + }, + { + "helper": 609, + "cleanup_index": 454, + "frame": [ + 256, + 128, + 64, + 32, + 18, + 10, + 6, + 1 + ] + }, + { + "helper": 611, + "cleanup_index": 452, + "frame": [ + 256, + 128, + 64, + 32, + 18, + 10, + 6, + 1 + ] + }, + { + "helper": 612, + "cleanup_index": 451, + "frame": [ + 256, + 128, + 64, + 32, + 24, + 4, + 2, + 1 + ] + }, + { + "helper": 613, + "cleanup_index": 450, + "frame": [ + 256, + 128, + 64, + 32, + 24, + 4, + 2, + 1 + ] + }, + { + "helper": 614, + "cleanup_index": 449, + "frame": [ + 256, + 128, + 64, + 32, + 16, + 8, + 6, + 1 + ] + }, + { + "helper": 615, + "cleanup_index": 448, + "frame": [ + 256, + 128, + 64, + 32, + 16, + 8, + 6, + 1 + ] + } + ], + "output_content_sha256": "b7a1c01f3444f50296feb276faccc2652073687aa4d86c8d267125f3b544b5b6" +} diff --git a/research/niemeier-lattices/workbench/continuation/endpoint-costs/sources/hpe_adapters_local.py b/research/niemeier-lattices/workbench/continuation/endpoint-costs/sources/hpe_adapters_local.py new file mode 100644 index 0000000000..f77c09a0ca --- /dev/null +++ b/research/niemeier-lattices/workbench/continuation/endpoint-costs/sources/hpe_adapters_local.py @@ -0,0 +1,95 @@ +"""Paid residual wrappers for entering helpers (Lemma A): generation and independent decoding. Apache-2.0. +Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code. The wrapper construction follows PR #458 +(chafreaky): a helper that starts at frame E and ends at FULL has the physical residual S_E = T_FULL T_E^-1 (apply T_E^-1 first); +S_E = L_E (C_[w] x I) R_E with R_E applied first; the wrappers R_E^-1 before the whole local invocation and L_E^-1 after its +full-frame cleanup turn the residual into the coordinate factor C_[w] x I and restore the other 9 - w coordinates identically. +usage: adapters_gk.py CANONICAL-WORD.json.gz RESIDUAL-ADAPTERS.json OUT-LEMMA-A.json +Writes the recipes (generation uses clifford_exact.lower_child) and then DECODES the serialized monomials with the primitive +constructors of clifford_exact (not lower_child) and checks, for every entrance frame, that L_E^-1 S_E R_E^-1 has exactly the +tableau and zero-state vector (global phase included) of fourier(9, range(w)); mutated recipes (wrong phase, swapped sides, wrong +translation, wrong width) must be rejected.""" +import copy, gzip, json, sys +from pathlib import Path +# Package-path adaptation only; the exact wrapper mathematics is unchanged. +import os +R = Path(os.environ['MULTICOLD_SOURCE']) / 'vendor/pr558/vendor/pr552/complex-global/vendor531' # HPE: the pinned vendor531 clifford_exact +sys.path.insert(0, str(R / 'vendor/clifford')) +from clifford_exact import canonical_frame, lower_child, fourier, diag, perm, xgate, Op, identity +n = 9; FULL = tuple(1 << i for i in range(n)) +def monomial_op(rec): + """decode a serialized monomial (matrix rows, quadratic rows, Z4 linear part, translation, phase)""" + out = diag(n, tuple(rec['quadratic_rows']), tuple(rec['linear_Z4'])).then(perm(n, tuple(rec['matrix_rows']))).then(xgate(n, rec['offset'])) + return out.then(Op(n, identity(n), [('phase', rec['phase'])])) +def S_of(U, END=FULL): + # HPE: a helper that starts at E=U and ends at END has residual T_END T_E^-1 (END = FULL is the inherited case) + return canonical_frame(n, tuple(U)).adjoint().then(canonical_frame(n, tuple(END))) +def wrapped_ok(U, rec, END=FULL): + """exactly: L^-1 S R^-1 = C_[w] x I (apply order: R^-1, S, L^-1), w = dim END - dim E""" + w = len(END) - len(U); S = S_of(U, END); Rm = monomial_op(rec['right']); Lm = monomial_op(rec['left']) + net = Rm.adjoint().then(S).then(Lm.adjoint()); target = fourier(n, range(w)) + return rec['rank'] == w and net.tab == target.tab and net.zero_state() == target.zero_state() +def _contains(a, b): + a = list(a) + for x in b: + for y in sorted(a, reverse=True): + if x >> (y.bit_length() - 1) & 1: x ^= y + if x: return False + return True +def main(): + c = json.load(gzip.open(sys.argv[1])); frames = c['frames']; v = c['v'] + starts = sorted({c['start'][r] for r in range(2 * v, len(c['start'])) if c['start'][r] != 0 and c['final'][r] == 1}) + # HPE: (entrance, terminal) pairs of helpers ending at a proper subspace; key 'E:U'. Inherited words have none. + ends = sorted({(c['start'][r], c['final'][r]) for r in range(2 * v, len(c['start'])) if c['final'][r] != 1}) + assert all(e != 0 and len(frames[u]) < n and _contains(frames[u], frames[e]) for e, u in ends) + recipes = {} + for f in starts: + U = frames[f]; rec, compiled = lower_child(S_of(U)) + recipes[str(f)] = dict(frame=f, vectors=list(U), entrance_rank=len(U), rank=rec['rank'], left=rec['left'], right=rec['right']) + for e, u in ends: + E, U = frames[e], frames[u]; rec, compiled = lower_child(S_of(E, U)) + recipes['%d:%d' % (e, u)] = dict(frame=e, terminal=u, vectors=list(E), terminal_vectors=list(U), entrance_rank=len(E), + terminal_rank=len(U), rank=rec['rank'], left=rec['left'], right=rec['right']) + Path(sys.argv[2]).write_text(json.dumps(dict(status='RESIDUAL_ADAPTER_RECIPES', recipes=recipes), sort_keys=True, separators=(',', ':')) + '\n') + # ---- independent decode of the serialized file + data = json.loads(Path(sys.argv[2]).read_text())['recipes'] + dot = lambda a, b: bin(a & b).count('1') & 1 + detail = [] + for f in starts: + rec = data[str(f)]; U = frames[f] + assert wrapped_ok(U, rec), ('wrapper product is not C_[w] x I', f) + detail.append(dict(frame=f, entrance_rank=len(U), residual_width=n - len(U), vectors=list(U), all_even_weight=all(bin(x).count('1') % 2 == 0 for x in U), + totally_isotropic=all(dot(a, b) == 0 for a in U for b in U))) + hdetail = [] + for e, u in ends: # HPE: independent decode, residual width = dim U - dim E + rec = data['%d:%d' % (e, u)]; E, U = frames[e], frames[u] + assert wrapped_ok(E, rec, U), ('HPE wrapper product is not C_[w] x I', e, u) + hdetail.append(dict(frame=e, terminal=u, entrance_rank=len(E), terminal_rank=len(U), residual_width=len(U) - len(E))) + controls = [] + def reject(name, fn): + f = starts[0]; rec = copy.deepcopy(data[str(f)]); fn(rec) + if wrapped_ok(frames[f], rec): raise AssertionError('mutated wrapper accepted: ' + name) + controls.append(name) + reject('wrong global phase', lambda r: r['right'].__setitem__('phase', (r['right']['phase'] + 1) % 4)) + reject('wrong translation', lambda r: r['left'].__setitem__('offset', r['left']['offset'] ^ 1)) + reject('wrong quadratic form', lambda r: r['right'].__setitem__('quadratic_rows', [x ^ 2 if i == 0 else x for i, x in enumerate(r['right']['quadratic_rows'])]) if False else r['right']['linear_Z4'].__setitem__(0, (r['right']['linear_Z4'][0] + 2) % 4)) + reject('sides swapped', lambda r: (r.__setitem__('left', r['right']), r.__setitem__('right', r['left']))) + reject('wrong width', lambda r: r.__setitem__('rank', r['rank'] + 1)) + if ends: # HPE controls on the first (E, U) pair + e, u = ends[0]; rec0 = data['%d:%d' % (e, u)] + def hreject(name, ok): + if ok: raise AssertionError('HPE mutation accepted: ' + name) + controls.append(name) + hreject('hpe: U-wrapper decoded against a FULL ending', wrapped_ok(frames[e], rec0)) + hreject('hpe: width misreported as 9 - dim E', wrapped_ok(frames[e], dict(rec0, rank=n - len(frames[e])), frames[u])) + other = next(f for f in range(len(frames)) if f != u and len(frames[f]) == len(frames[u]) and _contains(frames[f], frames[e])) + hreject('hpe: wrong terminal hyperplane', wrapped_ok(frames[e], rec0, frames[other])) + fe = next(((e2, u2) for e2, u2 in ends if str(e2) in data), None) + if fe: # an entrance shared with FULL-ended helpers: R_{E,FULL} must not serve as R_{E,U} + r2 = data['%d:%d' % fe] + hreject('hpe: pre wrapper resolved to the FULL recipe R_{E,FULL}', wrapped_ok(frames[fe[0]], dict(r2, right=data[str(fe[0])]['right']), frames[fe[1]])) + hreject('hpe: wrong global phase', wrapped_ok(frames[e], dict(rec0, left=dict(rec0['left'], phase=(rec0['left']['phase'] + 1) % 4)), frames[u])) + res = dict(status='PASS_RESIDUAL_WRAPPERS_ARE_EXACT', frames=len(detail), **({'hyperplane_end_pairs': hdetail} if ends else {}), wrappers_reject=controls, all_even_weight=all(d['all_even_weight'] for d in detail), + all_totally_isotropic=all(d['totally_isotropic'] for d in detail), detail=detail) + Path(sys.argv[3]).write_text(json.dumps(res, indent=1) + '\n') + print('Lemma A (exact): L^-1 S_E R^-1 = C_[w] x I for', len(detail), 'entrance frames, global phase included;', len(controls), 'wrapper mutations rejected; all-even-weight:', res['all_even_weight']) +if __name__ == '__main__': main() diff --git a/research/niemeier-lattices/workbench/continuation/endpoint-costs/sources/hpe_audit_and_transport.py b/research/niemeier-lattices/workbench/continuation/endpoint-costs/sources/hpe_audit_and_transport.py new file mode 100644 index 0000000000..845680279b --- /dev/null +++ b/research/niemeier-lattices/workbench/continuation/endpoint-costs/sources/hpe_audit_and_transport.py @@ -0,0 +1,249 @@ +"""Rebuild arbitrary-dirt compensation and transport a changed local word. + +Prepared for icekylinx with GPT-6 Astra assistance. Apache-2.0. +Adapted from PR425/433 (icekylinx), PR484's audit_gk.py +(DreamingOfClouds), and Round19's 112-port transport. This auditor does not +import a constructor, a nominated compensation table, or a global price. +Only the changed scalar word and its actual frame chains are checked here. +""" +import argparse +import copy +import gzip +import hashlib +import json +import time +from collections import Counter +from fractions import Fraction as Q +from pathlib import Path + + +def read(path): + data = path.read_bytes() + return json.loads(gzip.decompress(data) if path.suffix == '.gz' else data) + + +def content_hash(path): + data = path.read_bytes() + return hashlib.sha256(gzip.decompress(data) if path.suffix == '.gz' else data).hexdigest() + + +def basis(vs): + pivots = {} + for x in vs: + for k in sorted(pivots, reverse=True): + if x >> k & 1: + x ^= pivots[k] + if x: + k = x.bit_length() - 1 + for j in pivots: + if pivots[j] >> k & 1: + pivots[j] ^= x + pivots[k] = x + return tuple(pivots[k] for k in sorted(pivots, reverse=True)) + + +def apply(e, rows): + if e[0] == 'scale': + _, target, a, b, *_ = e + q = Q(a, b) + assert q + rows[target] = {j: q*x for j, x in rows[target].items()} + else: + _, target, source, a, b, *_ = e + assert target != source + q = Q(a, b) + for j, x in list(rows[source].items()): + value = rows[target].get(j, Q(0)) + q*x + if value: + rows[target][j] = value + else: + rows[target].pop(j, None) + + +def inverse(e): + e = copy.deepcopy(e) + if e[0] == 'add': + e[3] *= -1 + else: + e[2], e[3] = e[3], e[2] + return e + + +def dual(e): + e = inverse(e) + if e[0] == 'add': + e[1], e[2] = e[2], e[1] + return e + + +def run(raw_path, routing_path, out): + start_time = time.monotonic() + word, routing = read(raw_path), read(routing_path) + v, h, helpers = word['report']['v'], word['report']['h'], word['report']['R'] + assert h == 9 and v == 112 and routing['v'] == v + n = 2*v + helpers + ident = lambda: [{r: Q(1)} for r in range(n)] + main = word['A'] + word['cut'] + word['B'] + assert all(not v <= e[2] < 2*v for e in word['A'] + word['B'] + if e[0] == 'add' and not v <= e[1] < 2*v) + matrix = ident() + for e in main: + apply(e, matrix) + dirty = [{} for _ in range(n)] + for r in range(v, 2*v): + dirty[r] = {j: x for j, x in matrix[r].items() if j >= 2*v} + for e in word['precondition']: + apply(e, dirty) + compensation = [['add', r, j, -q.numerator, q.denominator, []] + for r in range(v, 2*v) for j, q in sorted(dirty[r].items())] + entering = set(map(int, word.get('start_extra', {}))) + assert all(e[2] not in entering for e in compensation) + cleanup = [inverse(e) for e in reversed(word['A'] + word['B']) + if not v <= e[1] < 2*v] + assert [e[:-1] for e in cleanup] == [e[:-1] for e in word['cleanup']] + # HPE: hyperplane-ended helpers. 'early_cleanup' = [[cleanup index, frame vectors], ...] are cleanup gates executed + # right after B (before the final climbs) at the given frame; 'helper_final' = {helper: frame vectors} replaces FULL as + # that helper's terminal frame. The scalar program is the reordered one and is replayed exactly below, so a move that + # does not commute with the skipped cleanup prefix is rejected by the arbitrary-dirt endpoint assertion. + early = [(int(i), f) for i, f in word.get('early_cleanup', [])] + helper_final = {int(r): f for r, f in word.get('helper_final', {}).items()} + early_idx = [i for i, _ in early] + assert len(set(early_idx)) == len(early_idx) and all(0 <= i < len(cleanup) for i in early_idx) + assert all(2*v <= r < n for r in helper_final) + assert all(r in entering for r in helper_final), 'hyperplane end only for entering helpers (wrapper exists)' + tail = [word['cleanup'][i][:-1] + [f] for i, f in early] + rest = [e for i, e in enumerate(word['cleanup']) if i not in set(early_idx)] + program = word['precondition'] + compensation + main + tail + rest + for backward in (False, True): + actual = ident() + for e in program: + apply(dual(e) if backward else e, actual) + expected = ident() + for t in range(v): + expected[t if backward else v+t][v+t if backward else t] = Q(-1 if backward else 1) + assert actual == expected, ('arbitrary-dirt endpoint', backward) + # The native scalar compiler allows dyadic additions and signed binary scales. + for e in program: + q = Q(e[3], e[4]) if e[0] == 'add' else Q(e[2], e[3]) + assert q.denominator & (q.denominator-1) == 0 + if e[0] == 'scale': + a = abs(q.numerator) + assert a and a & (a-1) == 0 + + tau = lambda x: x ^ (43 if (x & 43).bit_count() % 2 else 0) + ports = routing['ports'] + assert len(set(ports)) == v and set(map(tau, word['ports'])) == set(ports) + mapping = {i: ports.index(tau(p)) for i, p in enumerate(word['ports'])} + mapping.update({v+i: v+mapping[i] for i in range(v)}) + mapping.update({r: r for r in range(2*v, n)}) + full = tuple(1 << i for i in range(h-1, -1, -1)) + frames, frame_ids = [(), full], {(): 0, full: 1} + + def fid(vectors): + frame = basis(vectors) + if frame not in frame_ids: + frame_ids[frame] = len(frames) + frames.append(frame) + return frame_ids[frame] + + def transform(e): + e = copy.deepcopy(e) + e[1] = mapping[e[1]] + if e[0] == 'add': + e[2] = mapping[e[2]] + if e[-1] is not None: + e[-1] = fid([tau(x) for x in e[-1]]) + return e + + canonical = {key: [transform(e) for e in word[key]] + for key in ('A', 'B', 'cut', 'cleanup', 'precondition')} + canonical['compensation'] = [transform(e) for e in compensation] + if tail: + canonical['tail'] = [transform(e) for e in tail] # HPE (absent for inherited words: identical output) + canonical['cleanup'] = [transform(e) for e in rest] # HPE (full cleanup minus the early gates) + canonical['start'] = [fid([p]) for p in ports] + [0]*(v+helpers) + for r, frame in word.get('start_extra', {}).items(): + assert 2*v <= int(r) < n + canonical['start'][int(r)] = fid([tau(x) for x in frame]) + canonical['final'] = ([1]*v + + [fid(x for x in range(1, 1 << h) if (x & p).bit_count() % 2 == 0) for p in ports] + + [1]*helpers) + for r, frame in helper_final.items(): # HPE + canonical['final'][r] = fid([tau(x) for x in frame]) + canonical.update(h=h, v=v, R=helpers, ports=ports, frames=frames, + centers=[mapping[r] for r in word['centers']], + center_frames=[fid(tau(x) for x in frame) for frame in word['center_frames']]) + levels = {mapping[v+int(t)]-v: fid(tau(x) for x in frame) + for t, frame in word.get('target_levels', {}).items()} + canonical['target_levels'] = {str(t): f for t, f in levels.items()} + current, paid, edges = canonical['start'].copy(), Counter(), set() + + def climb(r, f): + old = current[r] + assert basis(frames[old] + frames[f]) == frames[f], ('nonmonotone', r, old, f) + if old != f: + degree = len(frames[f]) - len(frames[old]) + assert degree > 0 + paid[degree] += 1 + edges.add((old, f)) + current[r] = f + + for part in ('A', 'B'): + if part == 'B': + assert all(current[r] == f for r, f in zip(canonical['centers'], canonical['center_frames'])) + assert all(current[v+t] == levels.get(t, 0) for t in range(v)) + for r, frame in zip(canonical['centers'], canonical['center_frames']): + served = {levels.get(e[1]-v, 0) for e in canonical['cut'] if e[2] == r} + old = frame + for f in sorted(served, key=lambda f: -len(frames[f])): + assert basis(frames[old] + frames[f]) == frames[old] + if old != f: + paid[len(frames[old])-len(frames[f])] += 1 + edges.add((old, f)) + edges.add((f, old)) + old = f + for e in canonical[part]: + for r in [e[1]] + ([e[2]] if e[0] == 'add' else []): + climb(r, e[-1]) + for e in canonical.get('tail', []): # HPE: early cleanup gates at their common frame + for r in [e[1]] + ([e[2]] if e[0] == 'add' else []): + climb(r, e[-1]) + for r, f in enumerate(canonical['final']): + climb(r, f) + # HPE: the remaining cleanup runs at FULL, so it may touch only registers whose terminal frame is FULL; a + # hyperplane-ended helper is touched by nothing after its tail gates. + for e in canonical['cleanup']: + assert all(canonical['final'][r] == 1 for r in [e[1]] + ([e[2]] if e[0] == 'add' else [])), ('cleanup on non-FULL end', e) + assert all(len(frames[canonical['final'][r]]) < h for r in helper_final) + assert dict(paid) == {int(r): count for r, count in word['report']['local_profile'].items()} + canonical['local_profile'] = dict(sorted(paid.items())) + canonical['edges'] = sorted(edges) + out.mkdir(parents=True, exist_ok=False) + payload = json.dumps(canonical, sort_keys=True, separators=(',', ':')).encode() + (out/'CANONICAL-WORD.json.gz').write_bytes(gzip.compress(payload, mtime=0)) + (out/'FULL-WORD.json.gz').write_bytes(gzip.compress(json.dumps(word, sort_keys=True, separators=(',', ':')).encode(), mtime=0)) + receipt = dict(status='PASS_CHANGED_WORD_ALL_COLUMNS_AND_NATIVE_FRAMES', + input_content_sha256=content_hash(raw_path), routing_content_sha256=content_hash(routing_path), + canonical_content_sha256=hashlib.sha256(payload).hexdigest(), + live_columns_each_orientation=n, helpers=helpers, + compensation_additions=len(compensation), scalar_operations=len(program), + center_rank_sum=sum(map(len, word['center_frames'])), + local_profile=dict(sorted(paid.items())), local_rank_mass=sum(r*z for r, z in paid.items()), + widths=[len(frames[canonical['final'][r]])-len(frames[canonical['start'][r]]) for r in range(2*v, n)], + frames=len(frames), directed_edges=len(edges), + tau=43, target_port_order_matches_baseline=True, + seconds=time.monotonic()-start_time, + scope='Fresh exact forward and inverse-transpose scalar replay on every arbitrary input; all native frame chains and dyadic primitives. Phase lowering, wrappers, global owners and global assembly are separate.') + if tail or helper_final: + receipt.update(hyperplane_ended_helpers=sorted(helper_final), early_cleanup_gates=len(tail)) + (out/'SCALAR-FRAME-AUDIT.json').write_text(json.dumps(receipt, indent=2)+'\n') + print(json.dumps({k: value for k, value in receipt.items() if k != 'widths'}, indent=2), flush=True) + + +if __name__ == '__main__': + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument('--raw', type=Path, required=True) + parser.add_argument('--routing', type=Path, required=True) + parser.add_argument('--out', type=Path, required=True) + args = parser.parse_args() + run(args.raw, args.routing, args.out) diff --git a/research/niemeier-lattices/workbench/continuation/endpoint-costs/sources/hpe_emit_local.py b/research/niemeier-lattices/workbench/continuation/endpoint-costs/sources/hpe_emit_local.py new file mode 100644 index 0000000000..aad715c157 --- /dev/null +++ b/research/niemeier-lattices/workbench/continuation/endpoint-costs/sources/hpe_emit_local.py @@ -0,0 +1,161 @@ +# GK: modified copy of PR #433's complex_emit.py (icekylinx with Codex assistance; PR #425) by DreamingOfClouds with Claude Opus 5.5 +# (Anthropic), via Claude Code. Every change is marked "GK:". On a word without start_extra / target_levels the behaviour +# and every written file equal PR #433's (regression anchor in verify.py). +import sys as _sys, os as _os; _sys.path.insert(0, _os.path.dirname(_os.path.abspath(__file__))) +# Adapted from PR425 by icekylinx with Codex assistance. +"""Emit both local programs, integrate the fixed nine-stage geometry, and bill. + +The expanded scalar interface is justified in PROOF.txt. This script does not +claim a Lean proof or discharge the inherited all-size/analytic interfaces. +""" +import copy,gzip,hashlib,importlib.util,json,math +from collections import Counter +from fractions import Fraction as Q +from pathlib import Path + +OUT=Path(_sys.argv[1]);cp=OUT/'CANONICAL-WORD.json.gz' +c=json.load(gzip.open(cp)) +frames=c['frames'];v,R,h=c['v'],c['R'],c['h'];n=2*v+R +def digest(x):return hashlib.sha256(json.dumps(x,sort_keys=True,separators=(',',':')).encode()).hexdigest() +def basis(vs): + b={} + for x in vs: + for p in sorted(b,reverse=True): + if x>>p&1:x^=b[p] + if x: + p=x.bit_length()-1 + for q in b: + if b[q]>>p&1:b[q]^=x + b[p]=x + return tuple(b[p] for p in sorted(b,reverse=True)) +def dual(e): + e=list(e) + if e[0]=='add':e[1],e[2]=e[2],e[1];e[3]*=-1 + else:e[2],e[3]=e[3],e[2] + return e +def scalar(e,back): + e=dual(e) if back else list(e) + if e[0]=='add':return ['add',['L',e[1]],['L',e[2]],e[3],e[4]] + return ['scale',['L',e[1]],e[2],e[3]] +def primitive_bound(e): + if e[0] in ('clear','copy'):return 1 + q=Q(e[3],e[4]) if e[0]=='add' else Q(e[2],e[3]) + a,b=abs(q.numerator),q.denominator + assert b&(b-1)==0 + if e[0]=='scale': + assert a&(a-1)==0 + return abs(a.bit_length()-b.bit_length())+int(q<0) + # Exact binary Horner multiplication on a cleared private stream, repeated + # halving, signed accumulation into the target, then clear the private stream. + return 3+(a.bit_length()-1)+a.bit_count()+(b.bit_length()-1) + +def emit(back): + out=[];cur=c['start'].copy() + def event(e):out.append(e) + def scal(e,f):event(['scalar',e,f]) + event(['phase','zero_frame_precondition']) + for e in c['precondition']+c['compensation']: + regs=[e[1]]+([e[2]] if e[0]=='add' else []) + assert all(c['start'][r]==0 for r in regs) + scal(scalar(e,back),0) + event(['phase','entry_adapters']) + # GK: a helper that starts at a nonzero entrance frame has NO entry adapter; its paid wrapper R_E^-1 (Lemma A) precedes every gate + # that touches it and L_E^-1 follows the full-frame cleanup. Words without entrances emit PR #433's events. + entering=[r for r,f in enumerate(c['start']) if r>=2*v and f!=0] + hpe={r for r in range(2*v,n) if c['final'][r]!=1} # HPE: hyperplane-ended helpers (terminal frame != FULL) + assert hpe<=set(entering) + for r,f in enumerate(c['start']): + if r in entering:event(['residual_adapter','pre',r,f]+([c['final'][r]] if c['final'][r]!=1 else [])) # HPE: R_{E,U}^-1 names U + else:event(['adapter','old_to_new',r,f]) + def climb(r,f,part): + old=cur[r] + if old==f:return + assert basis(frames[old]+frames[f])==tuple(frames[f]) + rank=len(frames[f])-len(frames[old]);assert rank>0 + event(['climb',part,r,old,f,rank]);cur[r]=f + # HPE: 'cleanup_tail' = early cleanup gates at their own frames; emitted only when the word has them, so a word + # without hyperplane-ended helpers yields exactly the inherited event stream. + for part in ('A','centers','B')+(('cleanup_tail',) if c.get('tail') else ())+('final',): + event(['phase',part]) + if part in ('A','B','cleanup_tail'): + for e in (c['tail'] if part=='cleanup_tail' else c[part]): + regs=[e[1]]+([e[2]] if e[0]=='add' else []) + for r in regs:climb(r,e[-1],part) + scal(scalar(e,back),e[-1]) + elif part=='centers': + # GK: targets sit at their nonzero-frame-compensation level (0 at the defaults); a centre copy descends + # cf -> served levels (descending chain) -> lowest served level and scatters into each target at its level + lv=lambda r:c.get('target_levels',{}).get(str(r-v),0) + assert all(cur[r]==lv(r) for r in range(v,2*v)) + for k,r in enumerate(c['centers']): + cf=c['center_frames'][k];assert cur[r]==cf + terms=[e for e in c['cut'] if e[2]==r] + path=[cf]+sorted({lv(e[1]) for e in terms}-{cf},key=lambda x:-len(frames[x])) + C=['C',k] + if not back: + scal(['clear',C],cf);scal(['copy',C,['L',r]],cf) + for a in range(len(path)): + if a>0:event(['center',k,path[a-1],path[a],len(frames[path[a-1]])-len(frames[path[a]])]) + for e in terms: + if lv(e[1])==path[a]:scal(['add',['L',e[1]],C,e[3],e[4]],path[a]) + scal(['clear',C],path[-1]) + else: + scal(['clear',C],path[-1]) + for a in range(len(path)-1,-1,-1): + for e in terms: + if lv(e[1])==path[a]:scal(['add',C,['L',e[1]],e[3],e[4]],path[a]) + if a>0:event(['center',k,path[a],path[a-1],len(frames[path[a-1]])-len(frames[path[a]])]) + scal(['add',['L',r],C,-1,1],cf);scal(['clear',C],cf) + else: + for r,f in enumerate(c['final']):climb(r,f,part) + event(['phase','exit_adapters']) + for r,f in enumerate(c['final']): + if r not in hpe:event(['adapter','new_to_old',r,f]) # HPE: no old-convention exit for a helper that ends at U; its post wrapper is L_{E,U}^-1 + event(['phase','full_frame_cleanup']) + for e in c['cleanup']: + regs=[e[1]]+([e[2]] if e[0]=='add' else []) + assert all(c['final'][r]==1 for r in regs) + scal(scalar(e,back),1) + for r in entering:event(['residual_adapter','post',r,c['start'][r]]+([c['final'][r]] if r in hpe else [])) # HPE: 5th field = terminal frame U # GK: L_E^-1 after the full-frame cleanup + event(['phase','done']) + return out + +def check_local(events,back): + # Literal scalar replay of the emitted stream, including center lifetimes. + state={('L',r):{r:Q(1)} for r in range(n)};live=[];paid=Counter();prims=0;macros=0 + for e in events: + if e[0]=='climb':assert not live;paid[e[5]]+=1 + elif e[0]=='center':assert live==[e[1]];paid[e[4]]+=1 + elif e[0]=='scalar': + op=e[1];tag=op[0];t=tuple(op[1]);macros+=1;prims+=2*primitive_bound(op) + if tag=='clear': + state[t]={} + if t[0]=='C' and t[1] in live:live.remove(t[1]) + elif tag=='copy':state[t]=dict(state.get(tuple(op[2]),{}));live.append(t[1]) + elif tag=='scale':state[t]={j:Q(op[2],op[3])*x for j,x in state[t].items()} + else: + s=tuple(op[2]);q=Q(op[3],op[4]);row=state.setdefault(t,{}) + if t[0]=='C' and t[1] not in live:live.append(t[1]) + for j,x in list(state.get(s,{}).items()): + z=row.get(j,Q(0))+q*x + if z:row[j]=z + else:row.pop(j,None) + assert not live and all(not row for r,row in state.items() if r[0]=='C') + for r in range(n): + expected={r:Q(1)} + if back and r> bit & 1: + anf[:, index] ^= anf[:, index ^ (1 << bit)] +h.require(not anf[:, 0].any() and not anf[:, 15].any(), 'Even sign parity and canonical representatives') +cubic = anf[:, [7, 11, 13, 14]] @ np.array([1, 2, 4, 8], dtype=np.uint8) +bases = [h.top_basis.copy() for k in range(4)] +for offset in range(0, len(h.spinors), 128): + end = min(offset+128, len(h.spinors)) + decoder = (h.spinors[offset:end] @ h.rows.T) % 16 == 0 + spinor_decoder = decoder[:, h.sparse_count:].copy() + for k, basis in enumerate(bases): + decoder[:, h.sparse_count:] = spinor_decoder & ( + (cubic[offset:end, None] >> k) == (cubic[None, :] >> k)) + for packed in np.packbits(decoder, axis=1, bitorder='little'): + h.insert(basis, int.from_bytes(packed.tobytes(), 'little')) + +results = [dict(cubic_dimensions=k, spinor_block_count=1 << (4-k), + ports=len(h.rows), rank=len(basis), full_center_bill=16*len(basis), + asymptotic_surplus=len(h.rows)-16*len(basis)) + for k, basis in enumerate(bases)] +Path(__file__).with_name('HYBRID-SECTIONS.json').write_text(json.dumps(results, indent=2)+'\n') +print(json.dumps(results, indent=2), flush=True) diff --git a/research/niemeier-lattices/workbench/continuation/section-search/sections.py b/research/niemeier-lattices/workbench/continuation/section-search/sections.py new file mode 100644 index 0000000000..9a35aaf273 --- /dev/null +++ b/research/niemeier-lattices/workbench/continuation/section-search/sections.py @@ -0,0 +1,71 @@ +"""Exact binary decoder ranks for Reed--Muller sections of D16+ spinors. + +Local research by DominikScholz with OpenAI GPT-6 Astra in T3 Code. +No complete supplier or kappa claim. +""" +from itertools import combinations +from pathlib import Path +import json +import time + + +def require(ok, message): + if not ok: + raise ValueError(message) + + +def evaluation(monomial): + return sum(int(all(x >> j & 1 for j in monomial)) << x for x in range(16)) + + +def section(k): + monomials = ([(i,) for i in range(4)] + list(combinations(range(4), 2)) + + list(combinations(range(4), 3))[:k]) + masks = [evaluation(m) for m in monomials] + words = [0] + for mask in masks: + words += [w ^ mask for w in words] + n = len(words) + require(len(set(words)) == n, 'Independent function evaluations') + require(all(w & 1 == 0 and w.bit_count() % 2 == 0 for w in words), + 'Canonical even spinor signs') + kernel = [int(w == 0 or w.bit_count() == 8) for w in words] + first = sum(c << i for i, c in enumerate(kernel)) + swaps = [] + for bit in range(len(masks)): + shift = 1 << bit + mask = sum(((1 << shift) - 1) << start for start in range(0, n, 2 * shift)) + swaps.append((shift, mask)) + basis = {} + previous_gray = 0 + row = first + for index in range(n): + gray = index ^ (index >> 1) + if index: + bit = (gray ^ previous_gray).bit_length() - 1 + shift, mask = swaps[bit] + row = ((row & mask) << shift) | ((row >> shift) & mask) + previous_gray = gray + value = row + while value: + pivot = value.bit_length() - 1 + if pivot in basis: + value ^= basis[pivot] + else: + basis[pivot] = value + break + return dict(cubic_dimensions=k, ports=n, dimension=16, + orthogonality_degree=sum(kernel)-1, rank=len(basis), + full_center_bill=16*len(basis), + asymptotic_rank_surplus=n-16*len(basis), + monomials=[list(m) for m in monomials]) + + +if __name__ == '__main__': + result = [] + for k in range(5): + start = time.monotonic() + row = section(k) + result.append(row) + print(row, 'seconds', time.monotonic()-start, flush=True) + Path(__file__).with_name('SECTIONS.json').write_text(json.dumps(result, indent=2)+'\n') diff --git a/research/niemeier-lattices/workbench/continuation/spinor-routes/CLIQUE-HALF.json b/research/niemeier-lattices/workbench/continuation/spinor-routes/CLIQUE-HALF.json new file mode 100644 index 0000000000..63608447e7 --- /dev/null +++ b/research/niemeier-lattices/workbench/continuation/spinor-routes/CLIQUE-HALF.json @@ -0,0 +1,8 @@ +[ + { + "coordinate_count_permuted": 8, + "cliques_tested": 5040, + "leading_rank": 75, + "odd_relation_found": false + } +] diff --git a/research/niemeier-lattices/workbench/continuation/spinor-routes/CLIQUE-LEADING.json b/research/niemeier-lattices/workbench/continuation/spinor-routes/CLIQUE-LEADING.json new file mode 100644 index 0000000000..79fae82e3c --- /dev/null +++ b/research/niemeier-lattices/workbench/continuation/spinor-routes/CLIQUE-LEADING.json @@ -0,0 +1 @@ +{"sampled_cliques": 910, "leading_span_rank": 909, "odd_relation_size": 433, "selected_cliques": [[0, 7874, 4980, 3510, 10152, 14698, 13532, 10782, 13863, 10469, 9555, 15249, 4495, 3917, 763, 7225], [0, 10571, 14564, 4527, 13660, 7191, 3512, 9459, 13962, 8129, 3694, 10021, 982, 10909, 15154, 4729], [0, 6896, 1150, 7822, 7443, 2019, 6509, 925, 11833, 13513, 10823, 12471, 13098, 10714, 14164, 11684], [0, 3193, 9518, 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16193, 16196, 16257, 16288, 16320], "binary_rank": 2942, "degree_census": {"0": 0, "1": 0, "2": 0, "3": 0, "4": 0, "5": 976, "6": 1496, "7": 1712, "8": 1255, "9": 0, "10": 0, "11": 0, "12": 0, "13": 0, "14": 0}} diff --git a/research/niemeier-lattices/workbench/continuation/spinor-routes/CLIQUE-SMALL.json b/research/niemeier-lattices/workbench/continuation/spinor-routes/CLIQUE-SMALL.json new file mode 100644 index 0000000000..c4485eac0a --- /dev/null +++ b/research/niemeier-lattices/workbench/continuation/spinor-routes/CLIQUE-SMALL.json @@ -0,0 +1,38 @@ +[ + { + "coordinate_count_permuted": 3, + "cliques_tested": 6, + "leading_rank": 5, + "odd_relation_found": false + }, + { + "coordinate_count_permuted": 4, + "cliques_tested": 24, + "leading_rank": 14, + "odd_relation_found": false + }, + { + "coordinate_count_permuted": 5, + "cliques_tested": 120, + "leading_rank": 28, + "odd_relation_found": false + }, + { + "coordinate_count_permuted": 6, + "cliques_tested": 720, + "leading_rank": 48, + "odd_relation_found": false + }, + { + "coordinate_count_permuted": 7, + "cliques_tested": 5040, + "leading_rank": 75, + "odd_relation_found": false + }, + { + "coordinate_count_permuted": 8, + "cliques_tested": 40320, + "leading_rank": 275, + "odd_relation_found": false + } +] diff --git a/research/niemeier-lattices/workbench/continuation/spinor-routes/CROSS-FACTOR.json b/research/niemeier-lattices/workbench/continuation/spinor-routes/CROSS-FACTOR.json new file mode 100644 index 0000000000..04e30735bd --- /dev/null +++ b/research/niemeier-lattices/workbench/continuation/spinor-routes/CROSS-FACTOR.json @@ -0,0 +1,8 @@ +{ + "binary_rank": 470, + "all_rows_replayed_exactly": true, + "decoder_source_support_rational_rank_lower_bounds_mod101": { + "16": 470 + }, + "scope": "The chosen natural row basis with lexicographic Gaussian pivots; not all factorizations." +} diff --git a/research/niemeier-lattices/workbench/continuation/spinor-routes/CROSS-FACTOR.npz 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"rank_minimal_full_aggregate_private_copy_descent_lower_bound": 11412, + "spinor_full_center_bill": 9216, + "spinor_nominal_surplus": 7168, + "fused_packet_quotient_rank": 16, + "fused_rank_minimal_column_width": 16, + "fused_rank": 1024, + "fused_rank_minimal_center_bill": 16384, + "scope": "The residual bound applies to rank-minimal factorizations implemented by full-frame aggregates followed by private copies into common target caps. The fused bill applies to rank-minimal scalar factorizations of the specified fused packet kernel. Neither is a universal circuit lower bound." +} diff --git a/research/niemeier-lattices/workbench/continuation/spinor-routes/PROOF.txt b/research/niemeier-lattices/workbench/continuation/spinor-routes/PROOF.txt new file mode 100644 index 0000000000..3e495e8422 --- /dev/null +++ b/research/niemeier-lattices/workbench/continuation/spinor-routes/PROOF.txt @@ -0,0 +1,104 @@ +Scoped spinor frame bounds +========================= +Derived by Codex, with kernel/schedule interfaces supplied by the parent +and budget tracks. These are bounds for specified implementations, not +universal lower bounds for spinor circuits. + +Conventions. G=F2^14 indexes lines +v(x)=(1,(-1)^x1,...,(-1)^x14,(-1)^sum(x))/2 in Q^16. +C is the convolution kernel 1 at0 and at Hamming weights7,8. H is the +4-dimensional linear-evaluation subgroup; each H-coset consists of16 +mutually orthogonal lines. Let J_H be its indicator and D=C+J_H. Previously +certified ranks are rank(C)=576, rank(J_H)=1024, rank(D)=1268. + +A. Hadamard baseline plus matching layers +---------------------------------------- +Identify F2[G] with F2[z1,...,z14]/(zi^2), using zi=1+gi for a group basis +whose first4 generators span H. Then J_H=z1z2z3z4. A matching layer on a +coset of a7-dimensional translation subgroup W has kernel g_a J_W and +augmentation-filtration degree>=7. Any sum of such layers leaves J_H's +degree4 initial term unchanged. + +Multiplication by z1z2z3z4 has rank2^10=1024: the1024 monomials disjoint +from {1,2,3,4} map to distinct monomials. Order their columns and images +by degree. The same rows/columns in the perturbed multiplication matrix +have an identity diagonal block at each degree; all additional entries +have strictly higher output degree. This is a triangular1024-square +minor with determinant1. Hence the new kernel's rank is at least1024. + +Scope: Hadamard-baseline kernels plus these affine7-subgroup matching +kernels. Different baselines are not covered. + +B. Every nonzero vector in im(D) has address-support span>=9 +---------------------------------------------------------- +As a Boolean truth function, + + C=e0+e1+e2+e3+e4+e5+e6, + +where er is the elementary square-free polynomial on14 bits. The identity +is verified by the15 possible Hamming weights. Thus im(C) consists of +Boolean functions of degree<=6. For w=Dg and h in H, + + Delta_h w = Delta_h(Cg), + +since J_H g is H-invariant. Translation differentiation lowers Boolean +degree by at least1, so a nonzero Delta_h w has degree<=5 and Hamming +weight>=2^(14-5)=512. This Boolean-polynomial weight bound follows by +induction on the number of variables, splitting into two slices in the +last variable: either the slices agree, or their nonzero difference has +degree one lower and supplies that many nonzero entries across the slices. + +A k-dimensional real subspace contains at most2^k sign-cube vertices: +choose k coordinate projections injective on the subspace. Pairing +antipodal vertices therefore leaves at most2^(k-1) canonical spinor lines. +If supp(w) spanned dimension<=8, its size would be<=128. Each Delta_h w +would then have weight<=256<512 and would vanish. Hence w would be +H-invariant. A nonempty H-invariant support contains an entire H-coset, +which spans16 dimensions. This is a contradiction. + +For any rank-minimal factorization D=AB with1268 decoder columns in A, +im(A)=im(D), so each nonzero decoder column has target-support span>=9. +Its common target cap has codimension>=9. A private copy from FULL16 into +that cap must descend by at least9 ranks. Total descent is therefore +at least1268*9=11412, exceeding spinor full-center surplus +16384-576*16=7168. + +Scope: rank-minimal1268-column factorizations implemented as full-frame +aggregates and private copies into common target caps. Nonminimal +factorizations, other bases with additional columns, fused or partial +schedules, mutable sources, and additional target motion are not ruled +out by this proof. + +C. Fused Hadamard-clique / one-matching kernel +-------------------------------------------- +For the compatible half-incidence used by the budget track's fused word, +each complementary8D packet pair U,U-perp contains16 clique halves on +each side. Each half comprises8 orthogonal lines spanning its side. +After collapsing the constant values on each half, the scalar block is + + T=[[I16,A],[A,I16]], A=I16+J16. + +A^2=I16 because J16^2=0 overF2. Thus im(T)={(z,Az)}, and a nonzero z gives +a nonzero Az. Every nonzero block image column contains an entire half +on each side. Its address support contains U and U-perp and spans16. +If a global column has multiple packet components, any nonzero component +already supplies that full span. Every nonzero global image column thus +has support span16. + +The global kernel has rank1024. In any rank-minimal factorization, every +nonzero decoder column lies in its image, so all1024 center factors need +width16. The rank-minimal center bill is1024*16=16384, exactly N. This +kernel and the matching fusion therefore have no positive full-width +center surplus. + +Scope: the specified fused kernel and rank-minimal scalar factorizations. +The compatible physical half-incidence and arbitrary-dirt replay are +established by the separate budget-track word. No lower bound for all +redundant factorizations or all possible circuits is asserted here. + +Finite premises and checks +-------------------------- +frame_bounds.py / FRAME-BOUNDS.json check the degree-six identity, exact +Hadamard Gram matrix, A^2=I, rank(T)=16, and im(T)={(z,Az)}. Normal Python +and -I -O both pass. The1268 convolution rank comes from the earlier +workbench/bit-routes/clique_residual.py receipt. diff --git a/research/niemeier-lattices/workbench/continuation/spinor-routes/RESULTS.txt b/research/niemeier-lattices/workbench/continuation/spinor-routes/RESULTS.txt new file mode 100644 index 0000000000..c004aae168 --- /dev/null +++ b/research/niemeier-lattices/workbench/continuation/spinor-routes/RESULTS.txt @@ -0,0 +1,170 @@ +D16 spinor continuation: exact local results and implementation scopes +==================================================================== + +No kappa improvement or complete improved supplier is claimed. These files preserve the local research, now packaged for reproducible replay; +the computations do not establish a new supplier. + +Conventions +----------- +For x in F2^14, the rational address line is represented by + + v(x) = (1, (-1)^x1, ..., (-1)^x14, (-1)^sum(x))/2. + +There are N=16384 lines in dimension16. Their squared norms are4, and +v(x).v(y)=0 exactly when wt(x+y) is7 or8. The scalar fitting kernel over +F2 is C(x)=1 at x=0 and at weights7,8, and zero elsewhere. Its previously +certified binary rank is576, giving full-width center bill9216 and nominal +surplus N-9216=7168 before further paid costs. + +H consists of the16 evaluations of linear functions on F2^4, with the +zero coordinate and last parity coordinate handled by the above convention: + + H(a)_(j-1) = parity(a & j), j=1,...,14. + +The H-cosets are1024 orthogonal16-port Hadamard bases. J_H denotes the +convolution kernel that is1 on H. Its rank is1024. The residual +D=C+J_H has rank1268, certified in +workbench/bit-routes/{clique_residual.py,CLIQUE-RESIDUAL.json}. + +1. A basis-independent bound for the rank-minimal residual-copy route +------------------------------------------------------------------- +The truth function C has Boolean algebraic normal form + + C=e0+e1+e2+e3+e4+e5+e6, + +where er is the elementary square-free polynomial in the14 index bits. +The equality follows by checking the15 possible Hamming weights. Thus +all translations of C and all vectors in im(C) have Boolean degree<=6. + +Let w be any nonzero vector in im(D). For h in H, + + Delta_h(w)=Delta_h(Cg) + +for some g, because J_H*g is H-invariant. A translation derivative of a +degree<=6 Boolean polynomial has degree<=5. Every nonzero degree<=5 +Boolean polynomial on14 variables has at least2^(14-5)=512 nonzero +values. This elementary Reed--Muller distance bound follows by induction: +split in the last variable; a nonzero derivative of degree d-1 accounts +for at least2^(n-d) nonzero positions across the two slices, and if that +derivative vanishes the identical slices double the n-1 variable bound. + +Suppose the spinor lines in supp(w) span a real subspace of dimension k<=8. +A k-dimensional real subspace intersects the sign cube in at most2^k +vertices: choose k coordinate projections injective on that subspace. +Antipodal pairing leaves at most2^(k-1)<=128 of our canonical lines. +Consequently every Delta_h(w) has support size<=256, so it must vanish. +Then w is H-invariant. A nonempty H-invariant support contains a whole +H-coset, hence16 mutually orthogonal lines spanning the entire16-space. +This contradicts k<=8. + +Therefore EVERY nonzero column in im(D) has rational support span>=9. +For a rank-minimal factorization D=AB with1268 columns in A, im(A)=im(D). +A FULL16 aggregate that is privately copied into the common target cap +for one such decoder column must descend by at least9 ranks. The total +is therefore at least + + 1268*9 =11412 >7168. + +This rejects every rank-minimal1268-column full-aggregate/private-copy +implementation of this residual, not just its natural columns. It does +NOT reject redundant factorizations, mutable-source schedules, partial +or fused transforms, intermediate target descents, or all circuits. + +frame_bounds.py checks the complete truth table by weight and the exact +16x16 Hadamard Gram matrix. FRAME-BOUNDS.json records these finite premises. +The rank1268 is an input from the separate full convolution-rank receipt, +not recomputed by frame_bounds.py. + +2. Matching additions cannot lower the Hadamard baseline rank +----------------------------------------------------------- +In the F2 group algebra, set zi=1+gi for a group basis extending H. Then +zi^2=0 and J_H=z1*z2*z3*z4. A complementary8D packet-matching layer has +kernel g_a*J_W for a7-dimensional translation subgroup W. Its augmentation +filtration degree is at least7. Any XOR sum of those layers therefore +leaves the degree4 initial term of J_H unchanged. + +Multiplication by the initial term has rank2^(14-4)=1024. Ordering domain +and codomain by degree makes its independent pivots a triangular minor +in the perturbed multiplication matrix. Hence the perturbed kernel has +rank>=1024. This is a bound for this Hadamard-baseline plus affine7-subgroup +matching family; it says nothing universal about different baselines. + +3. Exact width of the useful one-matching fused construction +---------------------------------------------------------- +The budget agent constructed a complete clique16 adjacency schedule with +34 dirty helpers per clique, and a compatible matching can be fused at +no additional paid profile. This paragraph audits its scalar kernel, +not the separate word replay. + +In each complementary8D packet pair U,U-perp, the Hadamard cliques cut +into16 halves on each side, each half consisting of8 orthogonal lines +spanning that side. Collapsing constant values on those halves gives + + T = [[I16, A], [A, I16]], A=I16+J16. + +Since A^2=I16, im(T)={(z,Az)}. Every nonzero z has Az nonzero. Therefore +every nonzero image column uses at least one entire8-port half in U and +at least one entire8-port half in U-perp. Its rational support spans16. +An active packet-pair component already forces this span even if the +column also uses other components. + +Thus every column of EVERY rank-minimal1024-column factorization of the +fused kernel has full16-dimensional address support. Its rank-minimal +center bill is exactly1024*16=N. The matching fusion is a legitimate +schedule improvement but produces no positive full-width center surplus. +The statement does not lower-bound every redundant scalar factorization. + +frame_bounds.py independently checks A^2=I, rank(T)=16, and the image +identity. The physical half-incidence and complete arbitrary-dirt word +belong to the budget agent's separate receipt. + +4. Full D16 sparse--spinor cross block +------------------------------------ +The14,576 sparse norm-four lines against the16,384 spinor lines have +orthogonality incidence matrix X of exact binary rank470. + +cross_rank.py regenerates all its rows from the regenerated doubled integer +coordinates workbench/bit-routes/d16-ports.npz, computes X by exact +integer dot products, and records470 independent natural sparse rows in +CROSS-RANK.json. cross_factor.py expresses ALL14,576 rows in those470 +original rows and independently replays every resulting equality. + +For this explicitly chosen lexicographic natural-row factorization, +each of its470 decoder supports spans the full16-dimensional rational +space. This is established by finite-field101 rank16, which exhibits +nonzero integer minors and therefore proves rational rank16. Such decoder +columns have common target cap0. Direct FULL16 private copies would cost +16*470 per orientation, or15040 across both orientations. This figure is +not a complete word or a viable total price. + +CROSS-FACTOR.json records the verified statement; CROSS-FACTOR.npz stores +the witness rows and coefficient matrix. This width result is about this +one factorization, not all470-rank factorizations. + +5. Bounded alternate-baseline search +----------------------------------- +An odd sum of Hadamard J_H kernels is a valid fitting kernel. There is no +universal obstruction to cancelling their degree4 initial forms: +clique_leading.py finds an exact433-clique odd relation among910 sampled +cliques, whose initial4-form span has rank909. The resulting kernel has +2720 nonzero entries in its convolution row, lowest augmentation degree5, +and exact binary rank2942. Its full-width center cost is worse than N. +CLIQUE-LEADING.json contains all selected cliques and the resulting kernel; +clique_rank.py performs the complete packed convolution elimination. + +The bounded structured searches clique_small.py and clique_half.py find +no odd leading-term relation in the tested small coordinate-permutation +orbits. This is a finite search outcome, not a theorem for all cliques. +These searches do not produce an improved supplier or a cheap word. + +Source context +-------------- +The Boolean/Hadamard-block exploration was motivated in part by the +primary paper A. R. Calderbank, P. J. Cameron, W. M. Kantor and J. J. Seidel, +"Z4-Kerdock Codes, Orthogonal Spreads, and Extremal Euclidean Line-Sets", +Proc. London Math. Soc.75 (1997), 436-480: +https://www.math.ucdavis.edu/~deloera/MISC/LA-BIBLIO/trunk/Calderbank/kantorcalder.pdf +https://doi.org/10.1112/S0024611597000403 +The exact kernel identities, width arguments, schedules' cost scopes, +and finite computations above are derived here, not attributed to that +paper. diff --git a/research/niemeier-lattices/workbench/continuation/spinor-routes/clique_half.py b/research/niemeier-lattices/workbench/continuation/spinor-routes/clique_half.py new file mode 100644 index 0000000000..ae95a2163b --- /dev/null +++ b/research/niemeier-lattices/workbench/continuation/spinor-routes/clique_half.py @@ -0,0 +1,48 @@ +from pathlib import Path +import itertools,json +import numpy as np +OUT=Path(__file__).resolve().parent + +def require(x,m): + if not x:raise ValueError(m) +def rank(vs): + b={} + for v in vs: + v=int(v) + while v: + k=v.bit_length()-1 + if k in b:v^=b[k] + else:b[k]=v;break + return len(b) +ix=np.array(list(itertools.combinations(range(14),4)),dtype=np.int16) +lut=np.array([rank(t)==4 for t in itertools.product(range(16),repeat=4)],dtype=np.uint8).reshape((16,)*4) +results=[] +for m in [8]: + basis={};chosen=[];known=set();out=None + for order in itertools.permutations(range(8,16)): + cols=list(range(1,8))+list(order);cs=np.array(cols[:14],dtype=np.int16) + gens=[sum(((int(c)>>j)&1)<>i&1];kernel=set() + for i in ids:kernel.symmetric_difference_update(chosen[i]) + require(0 in kernel,'diagonal') + out={'coordinate_count_permuted':m,'cliques_tested':trial+1,'leading_rank':len(basis),'relation_size':len(ids),'selected_cliques':[chosen[i]for i in ids],'kernel_support':sorted(kernel)} + print({k:v for k,v in out.items()if k not in('selected_cliques','kernel_support')},'support',len(kernel),flush=True) + (OUT/f'CLIQUE-HALF-{m}.json').write_text(json.dumps(out)+'\n');break + if out is None: + out={'coordinate_count_permuted':m,'cliques_tested':len(chosen),'leading_rank':len(basis),'odd_relation_found':False};print(out,flush=True) + results.append({k:v for k,v in out.items()if k not in('selected_cliques','kernel_support')}) + if out.get('relation_size'):break +(OUT/'CLIQUE-HALF.json').write_text(json.dumps(results,indent=2)+'\n') diff --git a/research/niemeier-lattices/workbench/continuation/spinor-routes/clique_leading.py b/research/niemeier-lattices/workbench/continuation/spinor-routes/clique_leading.py new file mode 100644 index 0000000000..968a3541b6 --- /dev/null +++ b/research/niemeier-lattices/workbench/continuation/spinor-routes/clique_leading.py @@ -0,0 +1,52 @@ +from pathlib import Path +import itertools, json, random +import numpy as np +OUT=Path(__file__).resolve().parent + +def require(x,m): + if not x: raise ValueError(m) + +def gf2rank(vs): + b={} + for v in vs: + v=int(v) + while v: + k=v.bit_length()-1 + if k in b:v^=b[k] + else:b[k]=v;break + return len(b) + +ix=np.array(list(itertools.combinations(range(14),4)),dtype=np.int16) +lut=np.array([gf2rank((a,b,c,d))==4 for a,b,c,d in itertools.product(range(16),repeat=4)],dtype=np.uint8).reshape((16,)*4) +rng=random.Random(148) +basis={}; chosen=[]; lead=[] +for trial in range(3000): + cols=list(range(1,16));rng.shuffle(cols) + cs=np.array(cols[:14],dtype=np.int16) + bits=lut[cs[ix[:,0]],cs[ix[:,1]],cs[ix[:,2]],cs[ix[:,3]]] + vec=int.from_bytes(np.packbits(bits,bitorder='little').tobytes(),'little') + gens=[sum(((int(c)>>j)&1)<>i&1] + kernel=set() + for i in ids:kernel.symmetric_difference_update(chosen[i]) + require(0 in kernel,'diagonal') + require(all(x==0 or x.bit_count()in(7,8)for x in kernel),'support') + result={'sampled_cliques':trial+1,'leading_span_rank':len(basis),'odd_relation_size':len(ids),'selected_cliques':[chosen[i]for i in ids],'kernel_support':sorted(kernel)} + (OUT/'CLIQUE-LEADING.json').write_text(json.dumps(result)+'\n') + print({k:v for k,v in result.items()if k not in ('selected_cliques','kernel_support')},'kernel_support',len(kernel),flush=True) + break + if trial%100==0:print('trial',trial,'rank',len(basis),flush=True) +else: + (OUT/'CLIQUE-LEADING.json').write_text(json.dumps({'sampled_cliques':3000,'leading_span_rank':len(basis),'odd_relation':False})+'\n') diff --git a/research/niemeier-lattices/workbench/continuation/spinor-routes/clique_rank.py b/research/niemeier-lattices/workbench/continuation/spinor-routes/clique_rank.py new file mode 100644 index 0000000000..418626fea9 --- /dev/null +++ b/research/niemeier-lattices/workbench/continuation/spinor-routes/clique_rank.py @@ -0,0 +1,26 @@ +from pathlib import Path +import json +OUT=Path(__file__).resolve().parent +D=14;N=1<>bit&1:v[i]^=v[i|(1<>bit&1:row=((row&mask)<>shift)&mask) + while row: + q=row.bit_length()-1 + if q in basis:row^=basis[q] + else:basis[q]=row;break + if x%1024==0:print(x,len(basis),flush=True) +data['binary_rank']=len(basis);data['degree_census']={k:sum(c and i.bit_count()==k for i,c in enumerate(v))for k in range(15)} +p.write_text(json.dumps(data)+'\n');print('FINAL',len(basis)) diff --git a/research/niemeier-lattices/workbench/continuation/spinor-routes/clique_small.py b/research/niemeier-lattices/workbench/continuation/spinor-routes/clique_small.py new file mode 100644 index 0000000000..bddb18fdd3 --- /dev/null +++ b/research/niemeier-lattices/workbench/continuation/spinor-routes/clique_small.py @@ -0,0 +1,48 @@ +from pathlib import Path +import itertools,json +import numpy as np +OUT=Path(__file__).resolve().parent + +def require(x,m): + if not x:raise ValueError(m) +def rank(vs): + b={} + for v in vs: + v=int(v) + while v: + k=v.bit_length()-1 + if k in b:v^=b[k] + else:b[k]=v;break + return len(b) +ix=np.array(list(itertools.combinations(range(14),4)),dtype=np.int16) +lut=np.array([rank(t)==4 for t in itertools.product(range(16),repeat=4)],dtype=np.uint8).reshape((16,)*4) +results=[] +for m in range(3,9): + basis={};chosen=[];known=set();out=None + for order in itertools.permutations(range(1,m+1)): + cols=list(order)+list(range(m+1,16));cs=np.array(cols[:14],dtype=np.int16) + gens=[sum(((int(c)>>j)&1)<>i&1];kernel=set() + for i in ids:kernel.symmetric_difference_update(chosen[i]) + require(0 in kernel,'diagonal') + out={'coordinate_count_permuted':m,'cliques_tested':trial+1,'leading_rank':len(basis),'relation_size':len(ids),'selected_cliques':[chosen[i]for i in ids],'kernel_support':sorted(kernel)} + print({k:v for k,v in out.items()if k not in('selected_cliques','kernel_support')},'support',len(kernel),flush=True) + (OUT/f'CLIQUE-SMALL-{m}.json').write_text(json.dumps(out)+'\n');break + if out is None: + out={'coordinate_count_permuted':m,'cliques_tested':len(chosen),'leading_rank':len(basis),'odd_relation_found':False};print(out,flush=True) + results.append({k:v for k,v in out.items()if k not in('selected_cliques','kernel_support')}) + if out.get('relation_size'):break +(OUT/'CLIQUE-SMALL.json').write_text(json.dumps(results,indent=2)+'\n') diff --git a/research/niemeier-lattices/workbench/continuation/spinor-routes/cross_factor.py b/research/niemeier-lattices/workbench/continuation/spinor-routes/cross_factor.py new file mode 100644 index 0000000000..57971556de --- /dev/null +++ b/research/niemeier-lattices/workbench/continuation/spinor-routes/cross_factor.py @@ -0,0 +1,42 @@ +from pathlib import Path +WORKBENCH = Path(__file__).resolve().parents[2] +PACKAGE = WORKBENCH.parent +import json +import numpy as np +OUT=Path(__file__).resolve().parent +P=np.load((WORKBENCH / 'bit-routes/d16-ports.npz'))['ports'];S=P[:14576];T=P[14576:] +basis={};witness=[]; coeff=[]; original=[]; independent=[] +for st in range(0,len(S),128): + rows=np.packbits(S[st:st+128]@T.T==0,axis=1,bitorder='little') + for i,r in enumerate(rows,st): + v=int.from_bytes(r.tobytes(),'little');original.append(v);c=0 + while v: + k=v.bit_length()-1 + if k in basis: + vv,cc=basis[k];v^=vv;c^=cc + else: + new=1<>j&1 for j in range(R)]for c in coeff],dtype=np.uint8) +widths=[rank_mod(S[D[:,j].astype(bool)])for j in range(R)] +if any(w!=16 for w in widths):raise ValueError('decoder support width') +res={'binary_rank':R,'all_rows_replayed_exactly':True,'decoder_source_support_rational_rank_lower_bounds_mod101':{str(k):widths.count(k)for k in set(widths)},'scope':'The chosen natural row basis with lexicographic Gaussian pivots; not all factorizations.'} +np.savez_compressed(OUT/'CROSS-FACTOR.npz',witness=witness,coeff=D) +(OUT/'CROSS-FACTOR.json').write_text(json.dumps(res,indent=2)+'\n');print(res) diff --git a/research/niemeier-lattices/workbench/continuation/spinor-routes/cross_rank.py b/research/niemeier-lattices/workbench/continuation/spinor-routes/cross_rank.py new file mode 100644 index 0000000000..a1f1e351ce --- /dev/null +++ b/research/niemeier-lattices/workbench/continuation/spinor-routes/cross_rank.py @@ -0,0 +1,20 @@ +from pathlib import Path +WORKBENCH = Path(__file__).resolve().parents[2] +PACKAGE = WORKBENCH.parent +import json +import numpy as np +OUT=Path(__file__).resolve().parent +P=np.load((WORKBENCH / 'bit-routes/d16-ports.npz'))['ports'];S=P[:14576];T=P[14576:] +basis={};witness=[] +for st in range(0,len(S),128): + rows=np.packbits(S[st:st+128]@T.T==0,axis=1,bitorder='little') + for i,r in enumerate(rows,st): + v=int.from_bytes(r.tobytes(),'little') + while v: + k=v.bit_length()-1 + if k in basis:v^=basis[k] + else:basis[k]=v;witness.append(i);break +if len(basis)!=470:raise ValueError('unexpected cross rank') +result={'rows':14576,'columns':16384,'rank':len(basis),'row_basis_sparse_ids':witness} +(OUT/'CROSS-RANK.json').write_text(json.dumps(result,indent=2)+'\n') +print('rank',len(basis)) diff --git a/research/niemeier-lattices/workbench/continuation/spinor-routes/frame_bounds.py b/research/niemeier-lattices/workbench/continuation/spinor-routes/frame_bounds.py new file mode 100644 index 0000000000..7fcbca6f4b --- /dev/null +++ b/research/niemeier-lattices/workbench/continuation/spinor-routes/frame_bounds.py @@ -0,0 +1,54 @@ +"""Finite premises for the accompanying exact, scoped frame bounds.""" +from pathlib import Path +from math import comb +import json +import numpy as np +OUT=Path(__file__).resolve().parent +D=14;N=1<>i)&1) for i in range(14)]+[1-2*(x.bit_count()&1)] for x in H],dtype=np.int64) +require(np.array_equal(V@V.T,16*np.eye(16,dtype=np.int64)),'H clique full rational rank') +# This is the exact 32-dimensional quotient of the fused packet-pair +# block after collapsing the constant eight-entry halves. +I=np.eye(16,dtype=np.uint8);J=np.ones((16,16),dtype=np.uint8);A=I^J +require(np.array_equal((A@A)%2,I),'I+J16 must be an involution') +T=np.block([[I,A],[A,I]]) +rank_T=binary_rank(int.from_bytes(np.packbits(r,bitorder='little').tobytes(),'little') for r in T) +require(rank_T==16,'fused quotient rank') +require(np.array_equal(T[16:],(A@T[:16])%2),'fused image must be (z,Az)') +receipt={ + 'ports':N,'address_dimension':16, + 'C_truth_by_weight':truth,'C_ANF':'e0+e1+e2+e3+e4+e5+e6', + 'C_ANF_degree':6,'nonzero_H_derivative_minimum_weight':512, + 'rational_dimension_at_most_8_spinor_line_bound':128, + 'residual_D_rank_previously_certified':1268, + 'every_nonzero_residual_image_column_rational_span_at_least':9, + 'rank_minimal_full_aggregate_private_copy_descent_lower_bound':11412, + 'spinor_full_center_bill':9216,'spinor_nominal_surplus':7168, + 'fused_packet_quotient_rank':rank_T, + 'fused_rank_minimal_column_width':16, + 'fused_rank':1024,'fused_rank_minimal_center_bill':16384, + 'scope':'The residual bound applies to rank-minimal factorizations implemented by full-frame aggregates followed by private copies into common target caps. The fused bill applies to rank-minimal scalar factorizations of the specified fused packet kernel. Neither is a universal circuit lower bound.'} +(OUT/'FRAME-BOUNDS.json').write_text(json.dumps(receipt,indent=2)+'\n') +print(json.dumps(receipt,indent=2)) diff --git a/research/niemeier-lattices/workbench/continuation/tensor-routes/NATIVE-TENSOR-LIFT.json b/research/niemeier-lattices/workbench/continuation/tensor-routes/NATIVE-TENSOR-LIFT.json new file mode 100644 index 0000000000..011c0c70a2 --- /dev/null +++ b/research/niemeier-lattices/workbench/continuation/tensor-routes/NATIVE-TENSOR-LIFT.json @@ -0,0 +1,7449 @@ +{ + "center_chronology": "Each native rank8 center lives in a16D lifted frame E+image(U). Its private copy descends to E, costing8; targets have already grown to E. Different fixed-root blocks have different E, but never share data roles or require regrouping.", + "clique_cover": { + "identity": "H=(G_entrywise_square-J)/4 =3I-(1/4)sum_C 1_C1_C^T", + "number": 135, + "orthogonal_edges": 3780, + "root_multiplicity": 9, + "scope": "Exact dyadic matrix decomposition only. Independent clique lifts need repeated source incidences and do not constitute a free common-frame circuit.", + "selected": [ + [ + 0, + 1, + 6, + 11, + 20, + 29, + 106, + 119 + ], + [ + 0, + 12, + 30, + 44, + 57, + 93, + 104, + 110 + ], + [ + 0, + 13, + 41, + 58, + 69, + 75, + 84, + 116 + ], + [ + 0, + 18, + 39, + 65, + 70, + 92, + 99, + 107 + ], + [ + 0, + 19, + 33, + 47, + 50, + 80, + 85, + 118 + ], + [ + 0, + 21, + 38, + 45, + 64, + 74, + 95, + 117 + ], + [ + 0, + 22, + 32, + 51, + 68, + 86, + 105, + 108 + ], + [ + 0, + 27, + 40, + 56, + 71, + 81, + 94, + 109 + ], + [ + 0, + 28, + 31, + 46, + 59, + 87, + 98, + 115 + ], + [ + 1, + 12, + 38, + 60, + 73, + 91, + 102, + 107 + ], + [ + 1, + 13, + 32, + 43, + 52, + 78, + 89, + 118 + ], + [ + 1, + 18, + 30, + 49, + 54, + 96, + 103, + 109 + ], + [ + 1, + 19, + 41, + 62, + 67, + 79, + 82, + 115 + ], + [ + 1, + 21, + 40, + 53, + 72, + 76, + 97, + 110 + ], + [ + 1, + 22, + 31, + 42, + 61, + 83, + 100, + 116 + ], + [ + 1, + 27, + 39, + 48, + 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A hyperplane-ended helper remains hyperplane-ended after lifting. Arbitrary dirt cancellation and cleanup follow from the identical scalar word and the isometric common-frame embedding; endpoint wrappers and spectator ownership must remain paid.", + "inherited_prototype": { + "isometry_to_native": [ + 502, + 42, + 296, + 104, + 334, + 458, + 346, + 1, + 188 + ], + "path": "research/frame-sa504/inputs/hamming/PROTOTYPE.json", + "root_map": [ + 57, + 62, + 61, + 58, + 59, + 60, + 63, + 56, + 89, + 94, + 25, + 30, + 29, + 26, + 93, + 90, + 91, + 92, + 27, + 28, + 31, + 24, + 95, + 88, + 49, + 54, + 81, + 86, + 17, + 22, + 21, + 18, + 85, + 82, + 53, + 50, + 105, + 110, + 41, + 46, + 45, + 42, + 109, + 106, + 73, + 78, + 9, + 14, + 13, + 10, + 77, + 74, + 75, + 76, + 11, + 12, + 15, + 8, + 79, + 72, + 107, + 108, + 43, + 44, + 47, + 40, + 111, + 104, + 97, + 102, + 33, + 38, + 37, + 34, + 101, + 98, + 65, + 70, + 1, + 6, + 5, + 2, + 69, + 66, + 67, + 68, + 3, + 4, + 7, + 0, + 71, + 64, + 99, + 100, + 35, + 36, + 39, + 32, + 103, + 96, + 51, + 52, + 83, + 84, + 19, + 20, + 23, + 16, + 87, + 80, + 55, + 48 + ], + "root_signs": [ + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1, + -1, + 1 + ], + "scope": "The112 punctured roots are an exact signed reindexing of the inherited Hamming-coordinate prototype. Signed data conjugation transports its Gram factor and entire native scalar identity.", + "sha256": "327d2a868043e846381aa58a33053c24425e8cad522ff401b10ec8af1d649a3f" + }, + "lifted": { + "center_bill": 14336, + "center_copies": { + "8": 1792 + }, + "dimension": 17, + "extra_data_climbs": { + "8": 50176 + }, + "native_blocks": 224, + "native_histogram_multiplier": 224, + "ports": 25088, + "prefix_coalescence": "Optional exact improvement: defer each extra8D data climb until its first scalar incidence, combining it with an immediately preceding native initial edge if there is one. This receipt conservatively keeps the8D climbs separate." + }, + "native": { + "center_bill": 64, + "center_support_spans": [ + [ + 1, + 7, + 9, + 28, + 36, + 68, + 236, + 487 + ], + [ + 1, + 3, + 9, + 28, + 36, + 68, + 236, + 487 + ], + [ + 1, + 3, + 4, + 9, + 36, + 120, + 236, + 487 + ], + [ + 1, + 4, + 9, + 18, + 36, + 120, + 236, + 487 + ], + [ + 1, + 3, + 4, + 9, + 24, + 60, + 176, + 415 + ], + [ + 1, + 4, + 9, + 24, + 60, + 78, + 176, + 415 + ], + [ + 1, + 6, + 8, + 16, + 60, + 96, + 128, + 319 + ], + [ + 1, + 3, + 8, + 16, + 60, + 96, + 128, + 319 + ] + ], + "center_widths": [ + 8, + 8, + 8, + 8, + 8, + 8, + 8, + 8 + ], + "dimension": 9, + "ports": 112, + "real_centers": 8 + }, + "omitted_root_indices": [ + 0, + 1, + 6, + 11, + 20, + 29, + 106, + 119 + ], + "retained_root_indices": [ + 2, + 3, + 4, + 5, + 7, + 8, + 9, + 10, + 12, + 13, + 14, + 15, + 16, + 17, + 18, + 19, + 21, + 22, + 23, + 24, + 25, + 26, + 27, + 28, + 30, + 31, + 32, + 33, + 34, + 35, + 36, + 37, + 38, + 39, + 40, + 41, + 42, + 43, + 44, + 45, + 46, + 47, + 48, + 49, + 50, + 51, + 52, + 53, + 54, + 55, + 56, + 57, + 58, + 59, + 60, + 61, + 62, + 63, + 64, + 65, + 66, + 67, + 68, + 69, + 70, + 71, + 72, + 73, + 74, + 75, + 76, + 77, + 78, + 79, + 80, + 81, + 82, + 83, + 84, + 85, + 86, + 87, + 88, + 89, + 90, + 91, + 92, + 93, + 94, + 95, + 96, + 97, + 98, + 99, + 100, + 101, + 102, + 103, + 104, + 105, + 107, + 108, + 109, + 110, + 111, + 112, + 113, + 114, + 115, + 116, + 117, + 118 + ], + "scalar_kernel": "After an invertible H2 relative-sign change on each pair of identical binary labels: E_A tensor I_B on the odd-A sector and I_A tensor E_B on the odd-B sector, where E=root_Gram/2.", + "scope": "A complete common-frame lift of any inherited punctured E8 native scalar word, conditional on that word and its paid endpoint-normalization interface. This file does not replay the inherited native word or certify a moment gain.", + "status": "PASS_EXACT_NATIVE_TENSOR_LIFT_INTERFACE" +} diff --git a/research/niemeier-lattices/workbench/continuation/tensor-routes/ORTHOGONAL-CLIQUE-COVER.json b/research/niemeier-lattices/workbench/continuation/tensor-routes/ORTHOGONAL-CLIQUE-COVER.json new file mode 100644 index 0000000000..04a0f20831 --- /dev/null +++ b/research/niemeier-lattices/workbench/continuation/tensor-routes/ORTHOGONAL-CLIQUE-COVER.json @@ -0,0 +1,147 @@ +{ + "all_cliques": 2025, + "orthogonal_edges": 3780, + "edge_multiplicity": 15, + "search_timeout_ms": 45000, + "exact_cover_status": "sat", + "selected_indices": [ + 0, + 43, + 61, + 72, + 82, + 100, + 106, + 117, + 119, + 167, + 178, + 188, + 205, + 222, + 223, + 235, + 242, + 257, + 305, + 307, + 323, + 330, + 342, + 370, + 377, + 387, + 391, + 432, + 449, + 456, + 473, + 480, + 485, + 500, + 517, + 555, + 562, + 580, + 585, + 604, + 621, + 635, + 642, + 671, + 691, + 695, + 710, + 719, + 730, + 736, + 763, + 784, + 798, + 811, + 817, + 837, + 843, + 850, + 864, + 887, + 903, + 911, + 927, + 938, + 947, + 950, + 979, + 998, + 1005, + 1028, + 1030, + 1047, + 1067, + 1078, + 1084, + 1092, + 1105, + 1116, + 1136, + 1141, + 1148, + 1158, + 1177, + 1193, + 1197, + 1218, + 1222, + 1234, + 1263, + 1265, + 1276, + 1291, + 1298, + 1312, + 1325, + 1337, + 1352, + 1358, + 1363, + 1378, + 1392, + 1403, + 1431, + 1447, + 1460, + 1490, + 1524, + 1535, + 1543, + 1579, + 1592, + 1593, + 1606, + 1644, + 1655, + 1713, + 1726, + 1764, + 1775, + 1786, + 1793, + 1815, + 1840, + 1850, + 1861, + 1879, + 1904, + 1908, + 1935, + 1973, + 1982, + 2002, + 2007, + 2018, + 2023 + ], + "root_multiplicities": [ + 9 + ] +} diff --git a/research/niemeier-lattices/workbench/continuation/tensor-routes/README.txt b/research/niemeier-lattices/workbench/continuation/tensor-routes/README.txt new file mode 100644 index 0000000000..bb6197f8e7 --- /dev/null +++ b/research/niemeier-lattices/workbench/continuation/tensor-routes/README.txt @@ -0,0 +1,36 @@ +verify_tensor.py checks all 224 binary isometric embeddings of the +punctured E8 native-word interface into dimension 17, the eight-dimensional +spectator complements, rank-eight center descents, and the exact signed +port transport to the inherited Hamming prototype. Its source hash and +checkout-relative path are recorded in NATIVE-TENSOR-LIFT.json. + +This is a complete common-frame lift conditional on the inherited native +scalar word and paid endpoint-normalization interface. The native word +itself is not replayed here. The construction does not establish an +improved exponent; the separate endpoint-cost package prices it. + +The frozen clique fixture verifies a selected family of 135 orthogonal +eight-root cliques partitioning every orthogonal edge exactly once. It +gives a dyadic matrix identity, not a free circuit for that identity. + +Run replay.py with a fresh output directory outside the checkout. NumPy +is required. The replay writes its complete regenerated receipt only to +that directory and compares parsed JSON with the frozen receipt. No Z3 +installation is needed for verification. cliques.py is the optional +discovery program: it requires z3-solver and writes a possibly different +valid cover to a fresh external output directory. Its solve is bounded +at 45 seconds; discovery output is not the frozen verification source. + +Prepared by DominikScholz with OpenAI GPT-6 Astra assistance in T3 Code. The inherited +Hamming-coordinate prototype comes from research/frame-sa504 and retains +the community provenance cited by the enclosing package. + +Verification from the repository root (Python 3.11+ with NumPy): + + python3 -B research/niemeier-lattices/workbench/continuation/tensor-routes/replay.py /tmp/niemeier-tensor-normal + python3 -B -I -O research/niemeier-lattices/workbench/continuation/tensor-routes/replay.py /tmp/niemeier-tensor-optimized + +Both output directories must be new and outside the checkout. The -I -O +invocation propagates isolation and disabled assertions to every child. +The certificate replay takes approximately 0.1 seconds on the packaging +machine. No native supplier replay is implied by this runtime. diff --git a/research/niemeier-lattices/workbench/continuation/tensor-routes/cliques.py b/research/niemeier-lattices/workbench/continuation/tensor-routes/cliques.py new file mode 100644 index 0000000000..04811fc893 --- /dev/null +++ b/research/niemeier-lattices/workbench/continuation/tensor-routes/cliques.py @@ -0,0 +1,47 @@ +import sys,json,time,argparse +from pathlib import Path +import numpy as np +import z3 +PACKAGE = Path(__file__).resolve().parents[3] +sys.path.insert(0, str(PACKAGE / 'code')) +import e8_cubed as e +parser=argparse.ArgumentParser(description='Optional Z3 discovery; not needed for certificate replay.') +parser.add_argument('output_directory',type=Path) +args=parser.parse_args() +p=args.output_directory.resolve() +e.require(not p.is_relative_to(PACKAGE.parents[1]),'Discovery output must be outside the checkout') +p.mkdir(parents=True,exist_ok=False) +r,f=e.e8_shells();adj=(r@r.T)==0 +neighbors=[sum(1<>j)&1 for j in range(n)]for row in rows],dtype=np.uint8) + +roots,fourth=e.e8_shells() +G=(roots@roots.T)//4 +cover=json.loads((HERE/'ORTHOGONAL-CLIQUE-COVER.json').read_text()) +cliques=json.loads((HERE/'orthogonal-cliques.json').read_text()) +selected=[cliques[i]for i in cover['selected_indices']] +inc=np.zeros((120,120),dtype=np.int16) +for c in selected: + e.require(len(c)==8 and len(set(c))==8,'Eight distinct roots per selected frame') + e.require(np.array_equal(G[np.ix_(c,c)],2*np.eye(8,dtype=np.int16)),'Real orthogonal root frame') + inc[np.ix_(c,c)]+=1 +expected=(G==0).astype(np.int16)+9*np.eye(120,dtype=np.int16) +e.require(np.array_equal(inc,expected),'Exact edge partition by135 cliques') +e.require(np.array_equal(G*G-1,12*np.eye(120,dtype=np.int16)-inc),'Dyadic H decomposition') +# Puncture one root frame. Each surviving root is odd-adjacent to exactly4 frame roots. +omitted=selected[0] +keep=[i for i in range(120)if i not in omitted] +r=roots[keep] +native_labels,native_packed,_=e.binary_labels(roots,r) +native_centers=(r@roots[omitted].T)//4 +center_spans=[] +for j in range(8): + support=native_packed[native_centers[:,j]!=0] + span=binary_basis(support) + e.require(len(span)==8 and len(support)==56,'Punctured E8 center rank8') + center_spans.append(list(span)) +e.require(np.array_equal(2*(r@r.T)//4,native_centers@native_centers.T),'E8 eight-center scalar factor') +e.require(np.all((native_centers!=0).sum(axis=1)==4),'Four nonzero coordinates per punctured root') +# Match the inherited Hamming-coordinate prototype, including signed line orientation. +prototype_relative=Path('research/frame-sa504/inputs/hamming/PROTOTYPE.json') +prototype_path=PACKAGE.parents[1]/prototype_relative +prototype=json.loads(prototype_path.read_text()) +prototype_rows=[tuple(row)for row in prototype['root_lines'][8:]] +port_map=[];port_sign=[] +for row in native_centers: + sign=1 if next(int(x)for x in row if x)>0 else -1 + port_map.append(prototype_rows.index(tuple(map(int,sign*row)))) + port_sign.append(sign) +e.require(len(set(port_map))==112,'Signed bijection to inherited Hamming root lines') +prototype_packed=[prototype['binary_ports'][8+i]for i in port_map] +prototype_bits=bits(prototype_packed,9) +proto_chosen=[];proto_basis={} +for i,p in enumerate(prototype_packed): + if e.insert_binary(proto_basis,p):proto_chosen.append(i) +prototype_isometry=(e.inverse_binary(prototype_bits[proto_chosen])@native_labels[proto_chosen])&1 +e.require(np.array_equal((prototype_isometry@prototype_isometry.T)&1,np.eye(9,dtype=np.uint8)),'Inherited prototype label isometry') +e.require(np.array_equal((prototype_bits@prototype_isometry)&1,native_labels),'All inherited prototype port labels transported') +# Same-label +/- pairs, represented explicitly in canonical root ordering. +ports=[] +for a in r: + for b in r: + ports.extend([np.concatenate((a,b)),np.concatenate((a,-b))]) +ports=np.array(ports,dtype=np.int16) +labels,packed,_=e.binary_labels(roots,ports) +e.require(np.array_equal(labels[::2],labels[1::2]),'Free relative-sign Walsh pairs have identical labels') +# Choose nine independent native labels once. Every fixed-root group is their isometric image. +chosen=[];b={} +for i,p in enumerate(native_packed): + if e.insert_binary(b,p):chosen.append(i) +e.require(len(chosen)==9,'Native labels span9') +inv=e.inverse_binary(native_labels[chosen]) +embeddings=[] +for orientation in range(2): + for fixed in range(112): + ids=np.array([2*(varying*112+fixed)+orientation if orientation==0 else 2*(fixed*112+varying)+orientation for varying in range(112)]) + image=(inv@labels[ids[chosen]])&1 + e.require(np.array_equal((image@image.T)&1,np.eye(9,dtype=np.uint8)),'Fixed-root embedding is an isometry') + e.require(np.array_equal((native_labels@image)&1,labels[ids]),'Every fixed-root port is embedded correctly') + complement=(np.eye(17,dtype=np.uint8)^(image.T@image&1)) + epacked=binary_basis(pack(complement)) + E=bits(epacked,17) + e.require(len(epacked)==8 and not((E@image.T)&1).any(),'Eight-dimensional spectator complement') + e.require(len(binary_basis(list(pack(image))+list(epacked)))==17,'Native full plus spectator equalsFULL17') + e.require(not((labels[ids]@E.T)&1).any(),'Spectator lies in every target cap in its fixed-root block') + for center_span in center_spans: + lifted=(bits(center_span,9)@image)&1 + e.require(len(binary_basis(list(pack(lifted))+list(epacked)))==16,'Native center lift dimension16') + embeddings.append(dict(orientation=orientation,fixed_root_index=keep[fixed],native_image=list(map(int,pack(image))),spectator=list(map(int,epacked)))) +# The two Walsh sectors use different fixed-coordinate kernels. Neither requires cross-B regrouping. +e.require(len(embeddings)==224 and len(ports)==25088,'Symmetric punctured mixed-port counts') +receipt=dict(status='PASS_EXACT_NATIVE_TENSOR_LIFT_INTERFACE', + scope='A complete common-frame lift of any inherited punctured E8 native scalar word, conditional on that word and its paid endpoint-normalization interface. This file does not replay the inherited native word or certify a moment gain.', + omitted_root_indices=omitted,retained_root_indices=keep, + inherited_prototype=dict(path=prototype_relative.as_posix(),sha256=hashlib.sha256(prototype_path.read_bytes()).hexdigest(),root_map=port_map,root_signs=port_sign,isometry_to_native=list(map(int,pack(prototype_isometry))),scope='The112 punctured roots are an exact signed reindexing of the inherited Hamming-coordinate prototype. Signed data conjugation transports its Gram factor and entire native scalar identity.'), + native=dict(ports=112,dimension=9,real_centers=8,center_widths=[8]*8,center_bill=64,center_support_spans=center_spans), + lifted=dict(ports=25088,dimension=17,native_blocks=224,center_copies={8:1792},center_bill=14336,extra_data_climbs={8:50176},native_histogram_multiplier=224,prefix_coalescence='Optional exact improvement: defer each extra8D data climb until its first scalar incidence, combining it with an immediately preceding native initial edge if there is one. This receipt conservatively keeps the8D climbs separate.'), + scalar_kernel='After an invertible H2 relative-sign change on each pair of identical binary labels: E_A tensor I_B on the odd-A sector and I_A tensor E_B on the odd-B sector, where E=root_Gram/2.', + data_chronology='Within each fixed-root block, grow source line p to E+

by rank8 and target0 to E by rank8. Run the inherited9D word using every frame U replaced by E+image(U). Finish atFULL17 for sources and p-perp for targets. Reverse the same-label Walsh change at the endpoints. Blocks are disjoint within each sector.', + helper_chronology='Native helper entrance U and final V lift to E+image(U) and E+image(V); ranks both increase by8, so residual width dim(V)-dim(U) and every native edge rank are unchanged. A hyperplane-ended helper remains hyperplane-ended after lifting. Arbitrary dirt cancellation and cleanup follow from the identical scalar word and the isometric common-frame embedding; endpoint wrappers and spectator ownership must remain paid.', + center_chronology='Each native rank8 center lives in a16D lifted frame E+image(U). Its private copy descends to E, costing8; targets have already grown to E. Different fixed-root blocks have different E, but never share data roles or require regrouping.', + embeddings=embeddings, + clique_cover=dict(selected=selected,number=135,root_multiplicity=9,orthogonal_edges=3780,identity='H=(G_entrywise_square-J)/4 =3I-(1/4)sum_C 1_C1_C^T',scope='Exact dyadic matrix decomposition only. Independent clique lifts need repeated source incidences and do not constitute a free common-frame circuit.')) +print(json.dumps(receipt,indent=2,sort_keys=True)) diff --git a/research/niemeier-lattices/workbench/helper-routes/COMPENSATION-RANK.json b/research/niemeier-lattices/workbench/helper-routes/COMPENSATION-RANK.json new file mode 100644 index 0000000000..2a562ac04b --- /dev/null +++ b/research/niemeier-lattices/workbench/helper-routes/COMPENSATION-RANK.json @@ -0,0 +1,143 @@ +{ + "compensation_rank": 30960, + "kernel_spectrum": [ + { + "eigenvalue": "39", + "multiplicity": 224 + }, + { + "eigenvalue": "45", + "multiplicity": 560 + }, + { + "eigenvalue": "each root of lambda^2+186*lambda-6075", + "multiplicity_each": 16 + }, + { + "eigenvalue": "0", + "multiplicity": 30144 + } + ], + "linear_radial_matrix": [ + [ + "9", + "-240" + ], + [ + "-18", + "-195" + ] + ], + "moment_checks": [ + { + "moment": "30", + "powers": [ + 2 + ], + "shell": "roots" + }, + { + "moment": "18", + "powers": [ + 4 + ], + "shell": "roots" + }, + { + "moment": "6", + "powers": [ + 2, + 2 + ], + "shell": "roots" + }, + { + "moment": "15", + "powers": [ + 6 + ], + "shell": "roots" + }, + { + "moment": "3", + "powers": [ + 4, + 2 + ], + "shell": "roots" + }, + { + "moment": "1", + "powers": [ + 2, + 2, + 2 + ], + "shell": "roots" + }, + { + "moment": "540", + "powers": [ + 2 + ], + "shell": "norm4" + }, + { + "moment": "648", + "powers": [ + 4 + ], + "shell": "norm4" + }, + { + "moment": "216", + "powers": [ + 2, + 2 + ], + "shell": "norm4" + }, + { + "moment": "1080", + "powers": [ + 6 + ], + "shell": "norm4" + }, + { + "moment": "216", + "powers": [ + 4, + 2 + ], + "shell": "norm4" + }, + { + "moment": "72", + "powers": [ + 2, + 2, + 2 + ], + "shell": "norm4" + } + ], + "moment_symmetry_proof": "Both E8 shells are closed under coordinate permutations and even coordinate sign changes. Any odd-exponent moment of degree at most6 changes sign under an even sign flip, since fewer than8 coordinates occur. The six checked even patterns therefore establish all moments needed through degree6.", + "ports": 30960, + "proof": "Under the two E8 orthogonal groups, the independent degree-three feature space splits into harmonic cubics in each block (2*112), linear times traceless quadratic across blocks (2*8*35), and two radial linear copies for each block (2*2*8). The exact moments give eigenvalues39,45 and the displayed two-by-two radial matrix. None equals1 (the radial polynomial is -5888 at1). Thus I-K is invertible on every nonzero kernel eigenspace and is the identity on its30144-dimensional nullspace.", + "real_kernel_rank": 816, + "row_square_checks": [ + { + "multiplicity": 2160, + "row_square_sum": "69" + }, + { + "multiplicity": 28800, + "row_square_sum": "72" + } + ], + "scoped_consequence": "Any sum of scalar rank-one weighted rectangles representing I-K requires at least30960 terms. This does not lower-bound a general reusable arithmetic DAG or its number of physical helpers.", + "status": "PASS_FULL_COMPENSATION_RANK", + "trace": "30960", + "trace_square": "2222640" +} diff --git a/research/niemeier-lattices/workbench/helper-routes/ENCODER-PARTIAL.json b/research/niemeier-lattices/workbench/helper-routes/ENCODER-PARTIAL.json new file mode 100644 index 0000000000..53184e7e61 --- /dev/null +++ b/research/niemeier-lattices/workbench/helper-routes/ENCODER-PARTIAL.json @@ -0,0 +1,2070 @@ +{ + "center_copy_rank_bill": 12752, + "compressed_affine_identity": "C=M(C0+X); D[b,P,i]=D0[b,P,i]+2*qP(b)*C[selected_i,b]; Z[i,P]=Z0[i,P]+sum_b D[b,P,i]. Subtract the same expression with X=0 at target frame0 before gathering. The remaining fresh coefficient is2*R^T X Q. Reverse the additions and the verified dyadic mixer atFULL17 to restore all old C,D,Z.", + "frame_records": [ + { + "frame": [ + 3, + 5, + 9, + 17, + 33, + 65, + 129, + 513, + 1024, + 2048, + 4352, + 8192, + 16640, + 33024, + 65537 + ], + "orientation": 0, + "pauli": [ + 0, + 1 + ] + }, + { + "frame": [ + 3, + 5, + 9, + 17, + 33, + 65, + 129, + 256, + 513, + 2048, + 5120, + 9216, + 16384, + 33792, + 65537 + ], + "orientation": 0, + "pauli": [ + 0, + 2 + ] + }, + { + "frame": [ + 3, + 5, + 9, + 17, + 33, + 65, + 129, + 513, + 1280, + 2048, + 4096, + 8448, + 16640, + 32768, + 65537 + ], + "orientation": 0, + "pauli": [ + 0, + 3 + ] + }, + { + "frame": [ + 3, + 5, + 9, + 17, + 33, + 65, + 129, + 256, + 513, + 2048, + 4096, + 9216, + 16384, + 32768, + 65537 + ], + "orientation": 0, + "pauli": [ + 0, + 4 + ] + }, + { + "frame": [ + 3, + 5, + 9, + 17, + 33, + 65, + 129, + 513, + 1280, + 2048, + 4352, + 8448, + 16640, + 33024, + 65537 + ], + "orientation": 0, + "pauli": [ + 0, + 5 + ] + }, + { + "frame": [ + 3, + 5, + 9, + 17, + 33, + 65, + 129, + 256, + 513, + 1024, + 2048, + 8192, + 16384, + 36864, + 65537 + ], + "orientation": 0, + "pauli": [ + 0, + 6 + ] + }, + { + "frame": [ + 3, + 5, + 9, + 17, + 33, + 65, + 129, + 513, + 1024, + 2048, + 4096, + 8192, + 16640, + 32768, + 65537 + ], + "orientation": 0, + "pauli": [ + 0, + 7 + ] + }, + { + "frame": [ + 3, + 5, + 9, + 17, + 33, + 65, + 129, + 256, + 513, + 1025, + 2048, + 8193, + 16384, + 36864, + 65537 + ], + "orientation": 0, + "pauli": [ + 1, + 0 + ] + }, + { + "frame": [ + 3, + 5, + 9, + 17, + 33, + 65, + 129, + 256, + 513, + 2048, + 4097, + 9216, + 16384, + 32769, + 65537 + ], + "orientation": 0, + "pauli": [ + 1, + 2 + ] + }, + { + "frame": [ + 3, + 5, + 9, + 17, + 33, + 65, + 129, + 256, + 513, + 2048, + 5121, + 9216, + 16384, + 33793, + 65537 + ], + "orientation": 0, + "pauli": [ + 1, + 4 + ] + }, + { + "frame": [ + 1, + 2, + 4, + 8, + 16, + 32, + 64, + 128, + 256, + 512, + 2048, + 9216, + 16384, + 36864, + 65536 + ], + "orientation": 0, + "pauli": [ + 1, + 6 + ] + }, + { + "frame": [ + 3, + 5, + 9, + 17, + 33, + 65, + 129, + 513, + 1025, + 2048, + 4352, + 8193, + 16640, + 33024, + 65537 + ], + "orientation": 0, + "pauli": [ + 2, + 0 + ] + }, + { + "frame": [ + 1, + 2, + 4, + 8, + 16, + 32, + 64, + 128, + 512, + 2048, + 4352, + 9216, + 16640, + 33024, + 65536 + ], + "orientation": 0, + "pauli": [ + 2, + 1 + ] + }, + { + "frame": [ + 3, + 5, + 9, + 17, + 33, + 65, + 129, + 513, + 1281, + 2048, + 4352, + 8449, + 16640, + 33024, + 65537 + ], + "orientation": 0, + "pauli": [ + 2, + 4 + ] + }, + { + "frame": [ + 3, + 5, + 9, + 17, + 33, + 65, + 129, + 257, + 513, + 2048, + 4097, + 9216, + 16385, + 32769, + 65537 + ], + "orientation": 0, + "pauli": [ + 2, + 5 + ] + }, + { + "frame": [ + 3, + 5, + 9, + 17, + 33, + 65, + 129, + 513, + 1025, + 2048, + 4096, + 8193, + 16640, + 32768, + 65537 + ], + "orientation": 0, + "pauli": [ + 3, + 0 + ] + }, + { + "frame": [ + 3, + 5, + 9, + 17, + 33, + 65, + 129, + 257, + 513, + 2048, + 4096, + 9216, + 16385, + 32768, + 65537 + ], + "orientation": 0, + "pauli": [ + 3, + 3 + ] + }, + { + "frame": [ + 3, + 5, + 9, + 17, + 33, + 65, + 129, + 513, + 1281, + 2048, + 4096, + 8449, + 16640, + 32768, + 65537 + ], + "orientation": 0, + "pauli": [ + 3, + 4 + ] + }, + { + "frame": [ + 1, + 2, + 4, + 8, + 16, + 32, + 64, + 128, + 512, + 2048, + 4096, + 9216, + 16640, + 32768, + 65536 + ], + "orientation": 0, + "pauli": [ + 3, + 7 + ] + }, + { + "frame": [ + 3, + 5, + 9, + 17, + 33, + 65, + 129, + 513, + 1025, + 2048, + 4353, + 8193, + 16640, + 33025, + 65537 + ], + "orientation": 0, + "pauli": [ + 4, + 0 + ] + }, + { + "frame": [ + 1, + 2, + 4, + 8, + 16, + 32, + 64, + 128, + 512, + 2048, + 5376, + 9216, + 16640, + 34048, + 65536 + ], + "orientation": 0, + "pauli": [ + 4, + 1 + ] + }, + { + "frame": [ + 3, + 5, + 9, + 17, + 33, + 65, + 129, + 513, + 1281, + 2048, + 4097, + 8449, + 16640, + 32769, + 65537 + ], + "orientation": 0, + "pauli": [ + 4, + 2 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The simpler816 full-width-center encoder requires only816 helpers and is financially stronger unless this narrow encoder work can be shared with compensation.", + "mixer": { + "determinant": "256", + "inverse": [ + [ + "7/4", + "1/4", + "1/4", + "1/4", + "1/4", + "1/4", + "1/4", + "-1/4" + ], + [ + "5/2", + "1/2", + "1/2", + "1/2", + "1/2", + "1/2", + "0", + "0" + ], + [ + "5/4", + "1/4", + "1/4", + "1/4", + "1/4", + "1/4", + "1/4", + "1/4" + ], + [ + "1", + "0", + "0", + "0", + "0", + "0", + "0", + "0" + ], + [ + "2", + "1/2", + "1/2", + "1/2", + "1/2", + "0", + "0", + "0" + ], + [ + "3/2", + "1/2", + "1/2", + "1/2", + "0", + "0", + "0", + "0" + ], + [ + "1", + "1/2", + "1/2", + "0", + "0", + "0", + "0", + "0" + ], + [ + "1/2", + "1/2", + "0", + "0", + "0", + "0", + "0", + "0" + ] + ], + "reduce_B_to_identity": [ + [ + "swap", + 0, + 6 + ], + [ + "scale", + 0, + "1/2" + ], + [ + "add", + 7, + 0, + "2" + ], + [ + "swap", + 1, + 5 + ], + [ + "scale", + 1, + "1/2" + ], + [ + "add", + 0, + 1, + "1" + ], + [ + "add", + 7, + 1, + "2" + ], + [ + "swap", + 2, + 7 + ], + [ + "scale", + 2, + "1/4" + ], + [ + "add", + 0, + 2, + "-1" + ], + [ + "scale", + 3, + "-1" + ], + [ + "add", + 0, + 3, + "3/4" + ], + [ + "add", + 1, + 3, + "1/2" + ], + [ + "add", + 2, + 3, + "1/4" + ], + [ + "add", + 4, + 3, + "1" + ], + [ + "add", + 5, + 3, + "1" + ], + [ + "add", + 6, + 3, + "-1" + ], + [ + "add", + 7, + 3, + "1" + ], + [ + "scale", + 4, + "1/2" + ], + [ + "add", + 0, + 4, + "1/2" + ], + [ + "add", + 1, + 4, + "1" + ], + [ + "add", + 2, + 4, + "1/2" + ], + [ + "scale", + 5, + "-1/2" + ], + [ + "add", + 0, + 5, + "5/2" + ], + [ + "add", + 1, + 5, + "3" + ], + [ + "add", + 2, + 5, + "3/2" + ], + [ + "add", + 3, + 5, + "2" + ], + [ + "add", + 4, + 5, + "2" + ], + [ + "add", + 6, + 5, + "-2" + ], + [ + "add", + 7, + 5, + "2" + ], + [ + "swap", + 6, + 7 + ], + [ + "scale", + 6, + "1/2" + ], + [ + "add", + 0, + 6, + "1/2" + ], + [ + "add", + 1, + 6, + "1" + ], + [ + "add", + 2, + 6, + "1/2" + ], + [ + "add", + 4, + 6, + "1" + ], + [ + "add", + 5, + 6, + "1" + ], + [ + "scale", + 7, + "1/2" + ], + [ + "add", + 0, + 7, + "7/2" + ], + [ + "add", + 1, + 7, + "5" + ], + [ + "add", + 2, + 7, + "5/2" + ], + [ + "add", + 3, + 7, + "2" + ], + [ + "add", + 4, + 7, + "4" + ], + [ + "add", + 5, + 7, + "3" + ], + [ + "add", + 6, + 7, + "2" + ] + ], + "reduction_pivots": [ + "2", + "2", + "4", + "-1", + "2", + "-2", + "2", + "2" + ], + "selected_root_rows": [ + 0, + 2, + 1, + 42, + 6, + 12, + 20, + 30 + ] + }, + "paid_center_copies": { + "15": 560, + "17": 256 + }, + "source_count": 30960, + "source_profile": [ + 16 + ], + "status": "PASS_ENCODER_PARTIAL_WORD_ONLY", + "target_during_this_partial_word": 0 +} diff --git a/research/niemeier-lattices/workbench/helper-routes/MOBIUS-FRAMES.json b/research/niemeier-lattices/workbench/helper-routes/MOBIUS-FRAMES.json new file mode 100644 index 0000000000..8e2c12ffcc --- /dev/null +++ b/research/niemeier-lattices/workbench/helper-routes/MOBIUS-FRAMES.json @@ -0,0 +1,127 @@ +{ + "adjacency_binary_rank": 16384, + "adjacency_degree": 6435, + "adjacency_invertibility_proof": "The orthogonality adjacency is convolution by the6435 translations of weights7 and8 in the elementary abelian2-group. In characteristic2, every distinct pair of translations occurs twice in its square, while each translation squares to the identity. Thus A^2=6435 I=I.", + "ambient_dimension": 16, + "butterfly_layers": [ + { + "footprint_size": 2, + "layer": 1, + "rational_frame_dimension": 2, + "wires": 16384 + }, + { + "footprint_size": 4, + "layer": 2, + "rational_frame_dimension": 4, + "wires": 16384 + }, + { + "footprint_size": 8, + "layer": 3, + "rational_frame_dimension": 5, + "wires": 16384 + }, + { + "footprint_size": 16, + "layer": 4, + "rational_frame_dimension": 6, + "wires": 16384 + }, + { + "footprint_size": 32, + "layer": 5, + "rational_frame_dimension": 7, + "wires": 16384 + }, + { + "footprint_size": 64, + "layer": 6, + "rational_frame_dimension": 8, + "wires": 16384 + }, + { + "footprint_size": 128, + "layer": 7, + "rational_frame_dimension": 9, + "wires": 16384 + }, + { + "footprint_size": 256, + "layer": 8, + "rational_frame_dimension": 10, + "wires": 16384 + }, + { + "footprint_size": 512, + "layer": 9, + "rational_frame_dimension": 11, + "wires": 16384 + }, + { + "footprint_size": 1024, + "layer": 10, + "rational_frame_dimension": 12, + "wires": 16384 + }, + { + "footprint_size": 2048, + "layer": 11, + "rational_frame_dimension": 13, + "wires": 16384 + }, + { + "footprint_size": 4096, + "layer": 12, + "rational_frame_dimension": 14, + "wires": 16384 + }, + { + "footprint_size": 8192, + "layer": 13, + "rational_frame_dimension": 15, + "wires": 16384 + }, + { + "footprint_size": 16384, + "layer": 14, + "rational_frame_dimension": 16, + "wires": 16384 + } + ], + "final_frame_dimension_every_role": 16, + "full_shell": { + "center_bill": 13040, + "effective_bill": 29424, + "extra_copy_rank1": 16384, + "ports": 30960, + "remaining_asymptotic_surplus": 1536 + }, + "mandatory_full_to_cap_rank_mass": 16384, + "ports": 16384, + "proof": "After each complete coordinate butterfly layer, both operands must share the union of their incoming frames. Induction gives every role the complete Boolean face in all coordinates already processed. The port coordinates on a k-face are the constant character, k single-coordinate characters, and their product; their ranks are1,2,k+2 as checked modulo3. Sign translations are diagonal real isometries, so the canonical-face calculation applies to every face. At the last layer every wire reaches dimension16. Scalar cancellations in the Darboux shear or inverse Mobius stage cannot reduce a visited physical frame for free.", + "rational_face_rank_sequence": [ + 1, + 2, + 4, + 5, + 6, + 7, + 8, + 9, + 10, + 11, + 12, + 13, + 14, + 15, + 16 + ], + "scope": "This excludes the straightforward in-place14-layer Mobius / middle-shear / inverse-Mobius realization under the stated center models. It does not exclude fused, copied, partially transformed, or differently framed circuits.", + "scoped_failure": "The16384 necessary full-to-cap output descents exceed the spinor per-call asymptotic surplus7168 before any other helpers. Output reassignment cannot help because every role is full and every target cap has dimension15; the adjacency outputs are independent.", + "spinor_center_bill": 9216, + "spinor_center_rank": 576, + "spinor_per_call_asymptotic_surplus": 7168, + "status": "PASS_SCOPED_MOBIUS_FRAME_OBSTRUCTION", + "target_cap_dimension": 15 +} diff --git a/research/niemeier-lattices/workbench/helper-routes/README.txt b/research/niemeier-lattices/workbench/helper-routes/README.txt new file mode 100644 index 0000000000..53f50f0888 --- /dev/null +++ b/research/niemeier-lattices/workbench/helper-routes/README.txt @@ -0,0 +1,30 @@ +The default replay verifies three finite certificates: + +* encoder_receipt.py: exact dyadic encoder, old-helper compensation and + restoration for the narrow Pauli factor. This is a partial word. It does + not implement I-K or certify a complete supplier. +* compensation_rank.py: exact moments and the full rank of I-K on the E8 + squared norm-four shell. The rank-one rectangle consequence does not + apply to arbitrary reusable arithmetic circuits. +* mobius_frames.py: physical frame growth in the direct, unfused D16 + spinor Mobius schedule. Its obstruction does not exclude other schedules. + +Run replay.py with a fresh output directory outside the checkout. It +requires NumPy, propagates -I and -O to the child checks, and compares +parsed JSON with all three frozen receipts. The source checkout is not +modified. pure_blocks.py is retained as optional exploratory history; +its printed diagnostic is not a claimed certificate or a default check. + +Prepared by DominikScholz with OpenAI Codex assistance. The inherited E8 +geometry, frame interfaces and community provenance remain those cited +by the enclosing package. No new multiplication exponent is claimed. + +Verification from the repository root (Python 3.11+ with NumPy): + + python3 -B research/niemeier-lattices/workbench/helper-routes/replay.py /tmp/niemeier-helper-normal + python3 -B -I -O research/niemeier-lattices/workbench/helper-routes/replay.py /tmp/niemeier-helper-optimized + +Both output directories must be new and outside the checkout. The -I -O +invocation propagates isolation and disabled assertions to every child. +The three-certificate replay takes approximately one second on the +packaging machine. diff --git a/research/niemeier-lattices/workbench/helper-routes/compensation_rank.py b/research/niemeier-lattices/workbench/helper-routes/compensation_rank.py new file mode 100644 index 0000000000..b10e428cfa --- /dev/null +++ b/research/niemeier-lattices/workbench/helper-routes/compensation_rank.py @@ -0,0 +1,44 @@ +"""Exact moment certificate for invertibility of I-K on the E8 squared norm4 shell.""" +from pathlib import Path +import sys,json +from fractions import Fraction as F +import numpy as np +PACKAGE = Path(__file__).resolve().parents[2] +sys.path.insert(0, str(PACKAGE / 'code')) +import e8_cubed as e +roots,fourth,ports,profiles=e.construct_shell();ports=ports[(profiles&4)==0,:16].astype(np.int64) +checks=[] +for name,rows,norm in [('roots',roots,2),('norm4',fourth,4)]: + rows=rows.astype(np.int64);N=len(rows) + for powers,wick in [([2],1),([4],3),([2,2],1),([6],15),([4,2],3),([2,2,2],1)]: + degree=sum(powers);numerator=int(np.prod(np.stack([rows[:,i]**power for i,power in enumerate(powers)]),axis=0).sum());actual=F(numerator,2**degree) + denominator=1 + for j in range(degree//2):denominator*=8+2*j + expected=F(N*norm**(degree//2)*wick,denominator) + e.require(actual==expected,'Exact spherical moment representative') + checks.append(dict(shell=name,powers=powers,moment=str(actual))) +# Every odd-exponent moment below degree8 vanishes under even coordinate sign flips; +# coordinate permutation symmetry covers all permutations of the even patterns checked above. +lambda3=F(1080*4**3+28800*2**3,48*160) +lambdamix=F(2*30*12,16) +e.require(lambda3==39 and lambdamix==45,'Harmonic-sector eigenvalues') +C=[[F(1,60),-F(1,30)],[-F(1,30),-F(13,480)]];mass=[540,7200] +M=[[C[i][j]*mass[j]for j in range(2)]for i in range(2)] +trace=M[0][0]+M[1][1];det=M[0][0]*M[1][1]-M[0][1]*M[1][0] +e.require(trace==-186 and det==-6075,'Linear-radial sector characteristic polynomial') +N=len(ports);rank=224+560+2*16 +e.require(rank==816 and 39*224+45*560+trace*16==N,'Kernel rank and exact trace') +trace2=39**2*224+45**2*560+(trace**2-2*det)*16 +row_squares=[] +for index,count in [(0,2160),(2160,28800)]: + raw=ports@ports[index];e.require(not(raw%4).any(),'Integral shell inner products');t=raw//4;num=t**3-4*t + row_squares.append((count,F(int(num@num),48**2))) +e.require(sum(count*value for count,value in row_squares)==trace2,'Independent exact trace-of-square check using the two shell orbits') +print(json.dumps(dict(status='PASS_FULL_COMPENSATION_RANK',ports=N,real_kernel_rank=816,compensation_rank=N, + moment_checks=checks, + moment_symmetry_proof='Both E8 shells are closed under coordinate permutations and even coordinate sign changes. Any odd-exponent moment of degree at most6 changes sign under an even sign flip, since fewer than8 coordinates occur. The six checked even patterns therefore establish all moments needed through degree6.', + kernel_spectrum=[dict(eigenvalue='39',multiplicity=224),dict(eigenvalue='45',multiplicity=560),dict(eigenvalue='each root of lambda^2+186*lambda-6075',multiplicity_each=16),dict(eigenvalue='0',multiplicity=N-816)], + linear_radial_matrix=[[str(x)for x in row]for row in M], + trace=str(N),trace_square=str(trace2),row_square_checks=[dict(multiplicity=count,row_square_sum=str(value))for count,value in row_squares], + proof='Under the two E8 orthogonal groups, the independent degree-three feature space splits into harmonic cubics in each block (2*112), linear times traceless quadratic across blocks (2*8*35), and two radial linear copies for each block (2*2*8). The exact moments give eigenvalues39,45 and the displayed two-by-two radial matrix. None equals1 (the radial polynomial is -5888 at1). Thus I-K is invertible on every nonzero kernel eigenspace and is the identity on its30144-dimensional nullspace.', + scoped_consequence='Any sum of scalar rank-one weighted rectangles representing I-K requires at least30960 terms. This does not lower-bound a general reusable arithmetic DAG or its number of physical helpers.'),indent=2,sort_keys=True)) diff --git a/research/niemeier-lattices/workbench/helper-routes/encoder_receipt.py b/research/niemeier-lattices/workbench/helper-routes/encoder_receipt.py new file mode 100644 index 0000000000..b34cf42f36 --- /dev/null +++ b/research/niemeier-lattices/workbench/helper-routes/encoder_receipt.py @@ -0,0 +1,95 @@ +"""Finite checks for a legal, but currently uneconomic, Pauli encoder partial word.""" +from pathlib import Path +from fractions import Fraction +from collections import Counter +import sys,json,hashlib +import numpy as np +PACKAGE = Path(__file__).resolve().parents[2] +sys.path.insert(0, str(PACKAGE / 'code')) +import e8_cubed as e + +def basis(values): + b={} + for value in values:e.insert_binary(b,value) + for low in sorted(b): + for high in b: + if high>low and (b[high]>>low)&1:b[high]^=b[low] + return tuple(b[p]for p in sorted(b)) + +def contains(big,small): + return len(basis(big+small))==len(big) + +roots,fourth,ports,profiles=e.construct_shell();ports=ports[(profiles&4)==0,:16] +labels,packed,_=e.binary_labels(roots,ports) +selected=[0,2,1,42,6,12,20,30] +B=roots[selected].T.astype(np.int64);work=[[Fraction(int(x))for x in row]+[Fraction(int(i==j))for j in range(8)]for i,row in enumerate(B)];pivots=[];ops=[];det=Fraction(1) +for col in range(8): + pivot=next(row for row in range(col,8)if work[row][col]) + if pivot!=col:work[col],work[pivot]=work[pivot],work[col];ops.append(['swap',col,pivot]);det=-det + value=work[col][col];det*=value;pivots.append(str(value));e.require(abs(value.numerator)&(abs(value.numerator)-1)==0 and value.denominator&(value.denominator-1)==0,'Dyadic unit pivot') + work[col]=[x/value for x in work[col]];ops.append(['scale',col,str(1/value)]) + for row in range(8): + if row==col or not work[row][col]:continue + value=work[row][col];work[row]=[x-value*y for x,y in zip(work[row],work[col])];ops.append(['add',row,col,str(-value)]) +Binv=[[row[8+j]for j in range(8)]for row in work] +e.require(abs(det)==256,'Selected root minor determinant') +e.require(all(sum(Fraction(int(B[i,k]))*Binv[k][j]for k in range(8))==int(i==j)for i in range(8)for j in range(8)),'Exact mixer inverse') +# Every gate in reverse reduction realizes B; then the other112 carrier inputs are added. +M=np.eye(120,dtype=np.int64) +for coordinate,carrier in enumerate(selected):M[carrier]=roots[:,coordinate] +for coordinate,carrier in enumerate(selected):e.require(np.array_equal(M[carrier],roots[:,coordinate]),'Selected mixer row is the root-coordinate total') +paulis=[];Q=[] +for a in range(8): + for b in range(8): + if (a,b)==(0,0)or(a&b).bit_count()%2:continue + values=(roots.astype(np.int64)*roots[:,[j^a for j in range(8)]]*[(-1)**((j&b).bit_count()%2)for j in range(8)]).sum(axis=1) + e.require(set(map(int,values))=={-8,0,8} and int((values!=0).sum())==24,'Pauli root support') + paulis.append([a,b]);Q.append(values) +Q=np.stack(Q,axis=1) +e.require(np.all((Q!=0).sum(axis=1)==7),'Exactly seven Pauli destinations per fixed quadratic root') +frame_records=[] +for orientation in range(2): + groupframes=[] + for fixed in range(120): + port_ids=np.array([2160+2*(linear*120+fixed)+orientation if orientation==0 else 2160+2*(fixed*120+linear)+1 for linear in range(120)]) + F=basis(packed[port_ids]);e.require(len(F)==9,'Fixed quadratic root input frame dimension9') + for coordinate in range(8):e.require(basis(packed[port_ids[roots[:,coordinate]!=0]])==F,'Each selected total spans the same input frame') + groupframes.append(F) + for p in range(35): + U=basis([x for fixed in np.flatnonzero(Q[:,p])for x in groupframes[int(fixed)]]) + e.require(len(U)==15,'Pauli output frame dimension15') + e.require(all(contains(U,groupframes[int(fixed)])for fixed in np.flatnonzero(Q[:,p])),'Every fanout climb is nested9 to15') + frame_records.append(dict(orientation=orientation,pauli=paulis[p],frame=list(U))) +# Dense integer diagnostic of the compressed universal affine identities. +rng=np.random.default_rng(170816);worst=0 +for orientation in range(2): + X=rng.integers(-8,9,size=(120,120),dtype=np.int64);C0=rng.integers(-8,9,size=(120,120),dtype=np.int64) + C=M@(C0+X);D0=rng.integers(-8,9,size=(120,35,8),dtype=np.int64);Z0=rng.integers(-8,9,size=(8,35),dtype=np.int64) + D=D0.copy();Z=Z0.copy() + for fixed,p in np.argwhere(Q!=0): + D[fixed,p]+=2*Q[fixed,p]*C[selected,fixed] + Z[:,p]+=D[fixed,p] + dirt=Z0.copy();MC0=M@C0 + for fixed,p in np.argwhere(Q!=0):dirt[:,p]+=D0[fixed,p]+2*Q[fixed,p]*MC0[selected,fixed] + e.require(np.array_equal(Z-dirt,2*roots.T@X@Q),'Encoder output after old-dirt cancellation') + # Reverse every addition atFULL. This identity is coefficient-wise, not an approximate check. + for fixed,p in reversed(np.argwhere(Q!=0).tolist()):Z[:,p]-=D[fixed,p] + for fixed,p in reversed(np.argwhere(Q!=0).tolist()):D[fixed,p]-=2*Q[fixed,p]*C[selected,fixed] + e.require(np.array_equal(Z,Z0)and np.array_equal(D,D0),'Fanout and center dirt restored') + # Explicit inverse: subtract untouched-column contributions, then apply the verified8x8 inverse. + Cback=C.copy();other=[j for j in range(120)if j not in selected] + top=C[selected]-roots[other].T@C[other] + for i,carrier in enumerate(selected): + for column in range(120): + value=sum(Binv[i][j]*int(top[j,column])for j in range(8));e.require(value.denominator==1,'Inverse returns integral original carrier values');Cback[carrier,column]=value.numerator + Cback-=X;e.require(np.array_equal(Cback,C0),'All input-carrier dirt restored') +helper_profiles={"input_[1,8,8]":28800,"fanout_[9,6,2]":13440,"Pauli_center_[15,2]":560,"full_center_[17]":256} +receipt=dict(status='PASS_ENCODER_PARTIAL_WORD_ONLY',helpers=43056,helper_profiles=helper_profiles, + source_profile=[16],source_count=30960,target_during_this_partial_word=0, + paid_center_copies={15:560,17:256},center_copy_rank_bill=12752, + mixer=dict(selected_root_rows=selected,determinant=str(det),reduction_pivots=pivots,reduce_B_to_identity=ops,inverse=[[str(x)for x in row]for row in Binv]), + frame_records=frame_records, + compressed_affine_identity='C=M(C0+X); D[b,P,i]=D0[b,P,i]+2*qP(b)*C[selected_i,b]; Z[i,P]=Z0[i,P]+sum_b D[b,P,i]. Subtract the same expression with X=0 at target frame0 before gathering. The remaining fresh coefficient is2*R^T X Q. Reverse the additions and the verified dyadic mixer atFULL17 to restore all old C,D,Z.', + frame_schedule='Input copies at their one-dimensional port lines; carriers climb to fixed-root9 frames and execute the invertible120-carrier mixer there. Fanouts enter at the9 frame, climb to the containing Pauli15 frame, feed the center, then finish atFULL17. Centers enter at15, send a private15-to0 scatter copy, and finish at17. Full-width centers gather once sources reach17, and send private17-to0 copies. All arbitrary initial helper responses are subtracted from targets at0 before any helper moves; all non-target scalar gates are undone at17.', + limitations='This constructs only the exact K encoder/scatter/restoration partial word. It does not implement I-K, does not complete target endpoints, and does not establish a new kappa. The simpler816 full-width-center encoder requires only816 helpers and is financially stronger unless this narrow encoder work can be shared with compensation.') +print(json.dumps(receipt,indent=2,sort_keys=True)) diff --git a/research/niemeier-lattices/workbench/helper-routes/mobius_frames.py b/research/niemeier-lattices/workbench/helper-routes/mobius_frames.py new file mode 100644 index 0000000000..ed85fd48c2 --- /dev/null +++ b/research/niemeier-lattices/workbench/helper-routes/mobius_frames.py @@ -0,0 +1,41 @@ +"""Exact paid-frame obstruction for the unfused D16 spinor Mobius circuit.""" +from pathlib import Path +import sys,json +from math import comb +import numpy as np +PACKAGE = Path(__file__).resolve().parents[2] +sys.path.insert(0, str(PACKAGE / 'code')) +import e8_cubed as e +n=14;N=1<>i)&1)for i in range(n)]+[(-1)**x.bit_count()]for x in range(N)],dtype=np.int16) +sequence=[] +for k in range(n+1): + face=ports[:1<>bit)&1:continue + high=low|(1<=3 an odd stage count. Put m=1+L(h-1). The data stock before helpers +is W0=(L-1)N. Each original data path has nested increments totaling h-1. +Each ordinary helper runs from frame zero to the full h-frame; its nested +increments total h. The private center-copy descent contributes C per call. +The inherited paired data idle slots are retained and paid at their actual +ranks, with no idle hosting credit. Long idle ranks are split into balanced +children satisfying m^2>2r^2. + +Set q=floor(m/h), p=m-qh. Replicate the entire data template by a multiple +T of q. At each stage tile all TR helper residuals into width-h slots, q +slots per full physical bank. There are TR/q banks per stage, each with a +paid rank-p pad when p>0. The ordinary zero-entrance helper has no wrapper +spectators. Different slots are disjoint; replicas are invoked separately. +The effective stock per replica is + + W = (L-1)N + LR/q. + +All helper paths plus pads have rank mass LR(h+p/q)=LmR/q. Hence + + D=mW-total_rank_mass=(L-1)N-LC + +is unchanged by the additional restored helpers. A helper must not be +charged as added child work without also adding its banked stock. Conversely, +adding stock without its residual and pad children would be equally wrong. + +For alpha=kappa/(1-kappa), the exact per-helper pressure addition is + + L [sum_j r_j((m/r_j)^alpha-1) + +(p/q)((m/p)^alpha-1)], + +where sum_j r_j=h, and the second term is omitted if p=0. All quoted +thresholds use outward rational logarithm/exponential bounds, not floating +sign tests. The complete rank-mass identity is checked. Fractional stock in +the report is shorthand for the exactly integral T-replicated construction. +The geometry of the actual address cover, finite arithmetic, dirty circuit, +and compiler still have to be supplied for this new h and port set. + +Splitting a path increases pressure: x^(1-alpha) is subadditive for +0=1. At the +displayed limits its numerator is always 16384, and its denominator is +minimized at k=8. Thus every rectangle satisfies + + ab/(a+b-1) <= 16384/255. + +The numerical maximum does not assert that a 128-by-128 rectangle meeting +the parity restrictions exists. It is an upper bound, which is enough. + +Compiler assumption and role count +---------------------------------- +The tested independent rectangle factory allocates a binary addition tree +with a-1 internal additions and b distinct terminal output-use carriers. +Under its retained role accounting c+q, it costs a+b-1 helper roles. No +carrier is reused between rectangles, and no output-use carrier is shared +between distinct target uses. This is a compiler cost assumption, not an +information-theoretic lower bound for arbitrary rectangle implementations. + +Even when rectangles overlap and their contributions cancel over F2, +every nonzero matrix entry must occur in at least one rectangle. Therefore +sum ab >= 105431040, and the independent factory needs + + R = sum(a+b-1) >= 105431040 * 255/16384 + = 6435*255 = 1640925 helpers. + +Ordinary-role moment obstruction +-------------------------------- +Now retain the ordinary h=16 helper interface: each helper runs from frame +zero to the full local frame, with separately paid local children inside +that role. Its nested increments r_j sum to 16 and each child has rank at +most 16. For 0= 16*((m/16)^alpha-1). + +Cross-role child fusion into a rank 32 or larger transform is not admitted +by this scoped estimate. Neither are partial or virtual helper endpoints. + +For the paired controller m=15L+1 with odd L>=3, the complete banked mass +identity gives D=L(N-C)-N. All other pressure is nonnegative, so + + P_total/L >= 16 R *((46/16)^alpha-1), + D/L < N-C. + +At the pinned frontier kappa=0.001612719427075359, with +alpha=kappa/(1-kappa), exact rational interval arithmetic gives + + 16 R *((46/16)^alpha-1) > 44825.4388635721. + +This exceeds the full-shell allowance N-C=30960-13040=17920 and the +spinor-only allowance 16384-9216=7168. It even exceeds the full-shell +port count 30,960 with every center cost omitted. Thus this ordinary +independent rectangle factory fails for every odd stage count before +charging any active data, center or idle pressure. + +Convenient global necessary kappa upper bounds are: + + full-shell bit with C=13,040: 0.000645676583105551; + spinor-only bit with C=9,216: 0.000258423602940770; + full shell with C=0: 0.001114721384339448. + +These are weaker than optimizing the actual invoice but are sufficient +for the all-stage exclusion. Idle hosting does not remove the separately +paid local helper children. Shared DAG roles, reused output carriers, +fused transforms, changed endpoint widths and different controllers can +escape these assumptions and are not excluded. + +Reproduction +------------ +verify.py checks every possible source-span dimension, the exact edge +and helper counts, and the rational moment signs. A one-helper negative +control must fail the same sufficient rejection test. Both normal Python +and python -I -O pass. The inherited rational enclosure source and its +SHA256 are recorded in RESULT.json. No repository files are modified. diff --git a/research/niemeier-lattices/workbench/paid-controller/RECTANGLE-LOWER/RESULT.json b/research/niemeier-lattices/workbench/paid-controller/RECTANGLE-LOWER/RESULT.json new file mode 100644 index 0000000000..62049860ce --- /dev/null +++ b/research/niemeier-lattices/workbench/paid-controller/RECTANGLE-LOWER/RESULT.json @@ -0,0 +1,182 @@ +{ + "frontier": "1612719427075359/1000000000000000000", + "maximum_edges_per_helper": "16384/255", + "moment_cases": { + "full_shell_bit": { + "C": 13040, + "N": 30960, + "all_stage_kappa_upper_bound_interval_scaled_1e18": [ + 645676583105550, + 645676583105551 + ], + "minimum_ambient_dimension": 46, + "minimum_odd_stage": 3, + "normalized_deficit_strict_upper_bound": 17920, + "normalized_helper_pressure_interval_scaled_1e12": [ + 44825438863572107, + 44825438863572108 + ], + "ordinary_helpers_lower_bound": 1640925, + "verdict": "FAILS_FRONTIER_FOR_ALL_ODD_STAGES" + }, + "full_shell_even_without_centers": { + "C": 0, + "N": 30960, + "all_stage_kappa_upper_bound_interval_scaled_1e18": [ + 1114721384339447, + 1114721384339448 + ], + "minimum_ambient_dimension": 46, + "minimum_odd_stage": 3, + "normalized_deficit_strict_upper_bound": 30960, + "normalized_helper_pressure_interval_scaled_1e12": [ + 44825438863572107, + 44825438863572108 + ], + "ordinary_helpers_lower_bound": 1640925, + "verdict": "FAILS_FRONTIER_FOR_ALL_ODD_STAGES" + }, + "spinor_bit": { + "C": 9216, + "N": 16384, + "all_stage_kappa_upper_bound_interval_scaled_1e18": [ + 258423602940769, + 258423602940770 + ], + "minimum_ambient_dimension": 46, + "minimum_odd_stage": 3, + "normalized_deficit_strict_upper_bound": 7168, + "normalized_helper_pressure_interval_scaled_1e12": [ + 44825438863572107, + 44825438863572108 + ], + "ordinary_helpers_lower_bound": 1640925, + "verdict": "FAILS_FRONTIER_FOR_ALL_ODD_STAGES" + } + }, + "negative_control": "A single helper does not satisfy the same sufficient obstruction test.", + "ordered_degree": 6435, + "ordered_edges": 105431040, + "ordinary_helpers_lower_bound": 1640925, + "rectangle_rows": [ + { + "edges_per_helper_upper_bound": "1", + "source_size_upper_bound": 1, + "source_span_dimension": 1, + "target_size_upper_bound": 16384, + "target_span_dimension_upper_bound": 15 + }, + { + "edges_per_helper_upper_bound": "16384/8193", + "source_size_upper_bound": 2, + "source_span_dimension": 2, + "target_size_upper_bound": 8192, + "target_span_dimension_upper_bound": 14 + }, + { + "edges_per_helper_upper_bound": "16384/4099", + "source_size_upper_bound": 4, + "source_span_dimension": 3, + "target_size_upper_bound": 4096, + "target_span_dimension_upper_bound": 13 + }, + { + "edges_per_helper_upper_bound": "16384/2055", + "source_size_upper_bound": 8, + "source_span_dimension": 4, + "target_size_upper_bound": 2048, + "target_span_dimension_upper_bound": 12 + }, + { + "edges_per_helper_upper_bound": "16384/1039", + "source_size_upper_bound": 16, + "source_span_dimension": 5, + "target_size_upper_bound": 1024, + "target_span_dimension_upper_bound": 11 + }, + { + "edges_per_helper_upper_bound": "16384/543", + "source_size_upper_bound": 32, + "source_span_dimension": 6, + "target_size_upper_bound": 512, + "target_span_dimension_upper_bound": 10 + }, + { + "edges_per_helper_upper_bound": "16384/319", + "source_size_upper_bound": 64, + "source_span_dimension": 7, + "target_size_upper_bound": 256, + "target_span_dimension_upper_bound": 9 + }, + { + "edges_per_helper_upper_bound": "16384/255", + "source_size_upper_bound": 128, + "source_span_dimension": 8, + "target_size_upper_bound": 128, + "target_span_dimension_upper_bound": 8 + }, + { + "edges_per_helper_upper_bound": "16384/319", + "source_size_upper_bound": 256, + "source_span_dimension": 9, + "target_size_upper_bound": 64, + "target_span_dimension_upper_bound": 7 + }, + { + "edges_per_helper_upper_bound": "16384/543", + "source_size_upper_bound": 512, + "source_span_dimension": 10, + "target_size_upper_bound": 32, + "target_span_dimension_upper_bound": 6 + }, + { + "edges_per_helper_upper_bound": "16384/1039", + "source_size_upper_bound": 1024, + "source_span_dimension": 11, + "target_size_upper_bound": 16, + "target_span_dimension_upper_bound": 5 + }, + { + "edges_per_helper_upper_bound": "16384/2055", + "source_size_upper_bound": 2048, + "source_span_dimension": 12, + "target_size_upper_bound": 8, + "target_span_dimension_upper_bound": 4 + }, + { + "edges_per_helper_upper_bound": "16384/4099", + "source_size_upper_bound": 4096, + "source_span_dimension": 13, + "target_size_upper_bound": 4, + "target_span_dimension_upper_bound": 3 + }, + { + "edges_per_helper_upper_bound": "16384/8193", + "source_size_upper_bound": 8192, + "source_span_dimension": 14, + "target_size_upper_bound": 2, + "target_span_dimension_upper_bound": 2 + }, + { + "edges_per_helper_upper_bound": "1", + "source_size_upper_bound": 16384, + "source_span_dimension": 15, + "target_size_upper_bound": 1, + "target_span_dimension_upper_bound": 1 + } + ], + "scope": [ + "Each rectangle has mutually orthogonal source and target spans.", + "Each independent addition-tree/output-carrier rectangle is charged a+b-1 roles.", + "No carrier or DAG role is shared between rectangles; no free multi-target output-carrier reuse is assumed.", + "Each ordinary helper starts at zero and ends at the full h=16 frame.", + "Paid local children stay within one helper role, rank at most16; no cross-role child fusion.", + "Partial or virtual endpoints, fused transforms and alternate scalar factorizations are outside this bound." + ], + "source": { + "path": "code/costs.py", + "sha256": "885303fde89db6153b9cf74e229e5ababe511f971c2f928f93355ac4740631e8" + }, + "status": "PASS_SCOPED_RECTANGLE_CARRIER_OBSTRUCTION", + "vertices": 16384 +} diff --git a/research/niemeier-lattices/workbench/paid-controller/RECTANGLE-LOWER/verify.py b/research/niemeier-lattices/workbench/paid-controller/RECTANGLE-LOWER/verify.py new file mode 100644 index 0000000000..de610c2861 --- /dev/null +++ b/research/niemeier-lattices/workbench/paid-controller/RECTANGLE-LOWER/verify.py @@ -0,0 +1,68 @@ +from pathlib import Path +import sys +sys.dont_write_bytecode = True +sys.path.insert(0, str(Path(__file__).resolve().parents[1])) +from portable import PACKAGE, output_path, source_name +"""Scoped independent-rectangle carrier lower bound, with exact moment signs.""" +from fractions import Fraction as F +from math import comb +from pathlib import Path +import importlib.util,hashlib,json +SOURCE=PACKAGE / 'code/costs.py' +s=importlib.util.spec_from_file_location('cost_bounds',SOURCE);c=importlib.util.module_from_spec(s);s.loader.exec_module(c) + + +def moment_obstruction(N,C,R): + h=16;minimum_m=3*(h-1)+1;a=c.FRONTIER/(1-c.FRONTIER) + lo,hi=c.power_bounds(F(minimum_m,h),a) + lower=h*R*(lo-1);upper=h*R*(hi-1) + c.require(lower>N-C,'Independent ordinary helpers alone exclude every odd stage') + # All-stage ceiling obtained by discarding all other paid pressure and + # the negative -N/L term in D/L. This is deliberately a weak bound. + nl,nh=c.logarithm(1+F(N-C,h*R));dl,dh=c.logarithm(F(minimum_m,h)) + al,ah=nl/dh,nh/dl + return dict(N=N,C=C,ordinary_helpers_lower_bound=R, + minimum_odd_stage=3,minimum_ambient_dimension=minimum_m, + normalized_helper_pressure_interval_scaled_1e12=c.scaled(lower,upper,10**12), + normalized_deficit_strict_upper_bound=N-C, + all_stage_kappa_upper_bound_interval_scaled_1e18=c.scaled(al/(1+al),ah/(1+ah)), + verdict='FAILS_FRONTIER_FOR_ALL_ODD_STAGES') + + +def run(): + rows=[] + for k in range(1,16): + A,B=2**(k-1),2**(15-k) + rows.append(dict(source_span_dimension=k,target_span_dimension_upper_bound=16-k, + source_size_upper_bound=A,target_size_upper_bound=B, + edges_per_helper_upper_bound=str(F(A*B,A+B-1)))) + maximum=max(F(row['edges_per_helper_upper_bound']) for row in rows) + c.require(maximum==F(16384,255),'Exact maximum rectangle efficiency') + c.require([r['source_span_dimension'] for r in rows if F(r['edges_per_helper_upper_bound'])==maximum]==[8], + 'Unique dimensional maximum') + vertices=2**14;degree=comb(16,8)//2;edges=vertices*degree + required=F(edges)/maximum + c.require(required.denominator==1 and required==1640925,'Integral helper lower bound') + R=int(required) + cases={name:moment_obstruction(N,C,R) for name,N,C in + [('full_shell_bit',30960,13040),('spinor_bit',16384,9216),('full_shell_even_without_centers',30960,0)]} + try:moment_obstruction(30960,13040,1) + except ValueError as error:c.require('helpers alone' in str(error),'Control fails at the intended check') + else:raise ValueError('One-helper negative control unexpectedly rejected') + return dict(status='PASS_SCOPED_RECTANGLE_CARRIER_OBSTRUCTION',vertices=vertices, + ordered_degree=degree,ordered_edges=edges,rectangle_rows=rows, + maximum_edges_per_helper=str(maximum),ordinary_helpers_lower_bound=R, + moment_cases=cases,frontier=str(c.FRONTIER), + source=dict(path=source_name(SOURCE),sha256=hashlib.sha256(SOURCE.read_bytes()).hexdigest()), + scope=['Each rectangle has mutually orthogonal source and target spans.', + 'Each independent addition-tree/output-carrier rectangle is charged a+b-1 roles.', + 'No carrier or DAG role is shared between rectangles; no free multi-target output-carrier reuse is assumed.', + 'Each ordinary helper starts at zero and ends at the full h=16 frame.', + 'Paid local children stay within one helper role, rank at most16; no cross-role child fusion.', + 'Partial or virtual endpoints, fused transforms and alternate scalar factorizations are outside this bound.'], + negative_control='A single helper does not satisfy the same sufficient obstruction test.') + +if __name__=='__main__': + out=run();output_path('paid-controller/RECTANGLE-LOWER/RESULT.json').write_text(json.dumps(out,indent=2,sort_keys=True)+'\n') + print(out['status'],out['ordinary_helpers_lower_bound']) + for name,row in out['moment_cases'].items():print(name,row['all_stage_kappa_upper_bound_interval_scaled_1e18']) diff --git a/research/niemeier-lattices/workbench/paid-controller/RESULT.json b/research/niemeier-lattices/workbench/paid-controller/RESULT.json new file mode 100644 index 0000000000..ec10f6ceb0 --- /dev/null +++ b/research/niemeier-lattices/workbench/paid-controller/RESULT.json @@ -0,0 +1,1706 @@ +{ + "1": { + "single_h": { + "best_L3_to101": { + "stages": 5, + "m": 81, + "q": 4, + "pad": 13, + "max_helpers": 146036, + "helpers_per_port": 4.716925064599483, + "stock_at_max": "306385", + "deficit": 60080, + "profile": [ + 17 + ], + "virtual": false + }, + "L3": { + "stages": 3, + "m": 49, + "q": 2, + "pad": 15, + "max_helpers": 139740, + "helpers_per_port": 4.513565891472868, + "stock_at_max": "271530", + "deficit": 23664, + "profile": [ + 17 + ], + "virtual": false + }, + "L5": { + "stages": 5, + "m": 81, + "q": 4, + "pad": 13, + "max_helpers": 146036, + "helpers_per_port": 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scalar words or supplier admission.""" +from fractions import Fraction as F +from pathlib import Path +import importlib.util,json,math +P=PACKAGE / 'code/costs.py' +s=importlib.util.spec_from_file_location('niemeier_costs',P);c=importlib.util.module_from_spec(s);s.loader.exec_module(c) + +def parts(total,cap): + if total==0:return [] + k=(total+cap-1)//cap + return [total//k+(i2r^2') + c.require(sum(x for x,_ in terms)==m*W-D,'base rank-mass identity') + return dict(ports=n,h=h,m=m,stages=L,deficit=D,stock=F(W),terms=terms,cap=cap) + +def helper(r,profile,virtual=False): + w=sum(profile);h=r['h'];m=r['m'];e=h-w + c.require(0=3. For any number of restored banked helpers, +the inherited paired model has deficit D=L(N-C)-N. Write delta=(N-C)/N. +Positive D first occurs at odd Lmin=21 in the bit case and 45 in the complex +case; every smaller odd stage count has nonpositive deficit. + +For alpha=kappa/(1-kappa)>0 the unavoidable active data climbs alone have +pressure + + P_active=2NLd*((m/d)^alpha-1). + +Splitting those climbs can only increase this pressure. At every L>=Lmin, + + P_total/(NL) >= 2d*((Lmin+1/d)^alpha-1), + D/(NL) < delta. + +Consequently a necessary condition at every positive-deficit stage is + + alpha < log(1+delta/(2d))/log(Lmin+1/d). + +Exact rational log bounds give the following upper limits on kappa: + + bit full shell: 0.000541880687595200; + complex full shell: 0.000186502121809012. + +Both are below the pinned frontier 0.001612719427075359. These bounds +already discard every center's pressure, every helper's pressure and all +idle costs. They therefore prove failure at every odd stage count without +an optimizer or a finite stage cutoff. They are convenient upper bounds, +not claims to be the optimal roots of the complete necessary model. + +For the spinor-only bit shell, C=576*16+16384=25600>N16384, so D<0 at every +stage immediately. + +verify.py replays the rational inequalities with explicit checks under +both ordinary and optimized Python. Its negative controls remove the +16384 extra descents; the same rejection test must then fail, preventing +this schedule-specific obstruction from being asserted for the original +unaugmented factors. No PR, commit or released-package edit was made. diff --git a/research/niemeier-lattices/workbench/paid-controller/unfused/RESULT.json b/research/niemeier-lattices/workbench/paid-controller/unfused/RESULT.json new file mode 100644 index 0000000000..47bd4da103 --- /dev/null +++ b/research/niemeier-lattices/workbench/paid-controller/unfused/RESULT.json @@ -0,0 +1,65 @@ +{ + "cases": { + "bit_full_shell": { + "N": 30960, + "center_bill": 29424, + "center_bill_per_port": "613/645", + "center_histogram": { + "1": 16384, + "16": 815 + }, + "center_roles": 815, + "deficit_at_previous_odd_stage": -1776, + "first_odd_stage_with_positive_deficit": 21, + "fraction_of_frontier_upper_bound": "2753968060945828268907925010290013177000000000000000000/8196228238254226650007634922948801479602716271872338721", + "frontier": "1612719427075359/1000000000000000000", + "gap": 1536, + "global_necessary_kappa_upper_bound_interval_scaled_1e18": [ + 541880687595199, + 541880687595200 + ], + "h": 16, + "normalized_active_pressure_lower_bound_at_frontier": "7402724226358739120629972245329922423/50000000000000000000000000000000000000", + "normalized_deficit_strict_upper_bound": "32/645", + "scope": "Necessary bound using active data climbs alone, with no helper costs, center pressure, idle pressure or physical-hosting penalties. Not the optimal root of the full necessary model.", + "status": "FAILS_FRONTIER_FOR_EVERY_ODD_STAGE" + }, + "bit_spinor_only": { + "N": 16384, + "center_bill": 25600, + "h": 16, + "proof": "D=L(N-C)-N<0", + "status": "NONPOSITIVE_DEFICIT_FOR_ALL_STAGES" + }, + "complex_full_shell": { + "N": 30960, + "center_bill": 30256, + "center_bill_per_port": "1891/1935", + "center_histogram": { + "1": 16384, + "17": 816 + }, + "center_roles": 816, + "deficit_at_previous_odd_stage": -688, + "first_odd_stage_with_positive_deficit": 45, + "fraction_of_frontier_upper_bound": "7103419626448543054130157924247485941000000000000000000/61424624659087503218761135415251730967671075616151632801", + "frontier": "1612719427075359/1000000000000000000", + "gap": 704, + "global_necessary_kappa_upper_bound_interval_scaled_1e18": [ + 186502121809011, + 186502121809012 + ], + "h": 17, + "normalized_active_pressure_lower_bound_at_frontier": "7712743516677781535821243138435501049/39062500000000000000000000000000000000", + "normalized_deficit_strict_upper_bound": "44/1935", + "scope": "Necessary bound using active data climbs alone, with no helper costs, center pressure, idle pressure or physical-hosting penalties. Not the optimal root of the full necessary model.", + "status": "FAILS_FRONTIER_FOR_EVERY_ODD_STAGE" + } + }, + "negative_controls": "Removing the 16384 output descents does not satisfy this rejection test.", + "source": { + "path": "code/costs.py", + "sha256": "885303fde89db6153b9cf74e229e5ababe511f971c2f928f93355ac4740631e8" + }, + "status": "PASS_EXACT_UNFUSED_MOBIUS_OBSTRUCTION" +} diff --git a/research/niemeier-lattices/workbench/paid-controller/unfused/verify.py b/research/niemeier-lattices/workbench/paid-controller/unfused/verify.py new file mode 100644 index 0000000000..293d08af2a --- /dev/null +++ b/research/niemeier-lattices/workbench/paid-controller/unfused/verify.py @@ -0,0 +1,55 @@ +from pathlib import Path +import sys +sys.dont_write_bytecode = True +sys.path.insert(0, str(Path(__file__).resolve().parents[1])) +from portable import PACKAGE, output_path, source_name +"""Exact all-odd-stage obstruction for the tested unfused Mobius route.""" +from fractions import Fraction as F +import hashlib,importlib.util,json +from pathlib import Path +SOURCE=PACKAGE / 'code/costs.py' +s=importlib.util.spec_from_file_location('bounds',SOURCE);c=importlib.util.module_from_spec(s);s.loader.exec_module(c) + +def certify(N,h,centers,copies): + C=h*centers+copies;gap=N-C + if gap<=0: + return dict(N=N,h=h,center_bill=C,status='NONPOSITIVE_DEFICIT_FOR_ALL_STAGES',proof='D=L(N-C)-N<0') + Lmin=3 + while Lmin*gap<=N:Lmin+=2 + delta=F(gap,N);d=h-1;a=c.FRONTIER/(1-c.FRONTIER) + base=F(d*Lmin+1,d) + p_lower,p_upper=c.power_bounds(base,a) + forced_lower=2*d*(p_lower-1) + c.require(forced_lower>delta,'Active climbs alone rule out every positive-deficit stage') + # For every positive-deficit stage, active pressure/(LN) is at least + # 2d*((Lmin+1/d)^alpha-1), while D/(LN)