diff --git a/SOURCES.json b/SOURCES.json index f7117f0f98..7ab049d96a 100644 --- a/SOURCES.json +++ b/SOURCES.json @@ -500,5 +500,58 @@ ], "scope": "Conditional on retained analytic, uniform-recursion and fixed-tape contracts; complete local paired-cube schedule, exact core sharing and scalar charge supplied in the proof.", "curation": "Selected paired-coordinate cube only, compact frozen carrier arcs and bit subset, exact suppliers and assembly. Excludes fallback shared_core results, paired-five motifs, other candidate graphs, search screens and PDFs." + }, + "paired_cube_physical_bit_extension": { + "contributor": "eumemic", + "assistance": "Prepared with Anthropic Claude assistance.", + "predecessor_pr": 144, + "predecessor_commit": "c8b22bc5c10dba497ac25804e27d9647d818e2ff", + "kappa": "5878747/10000000000", + "proof": "notes/paired-cube-note.tex", + "certificate": "certificates/paired-cube-network.json", + "change": "PR144 paired cubes at p=11 with a triple module restricted from a pinned searched complex DAG and a pinned searched pair-disjoint module with merged face-2/edge-02 outputs, physical frame descent and compensated late-read reuse on the complex word (scripts/paired_cube_physical.py), and a paired-cube bit word with partner-pair source mixing and an annealed pair module (research/paired-cube-bit); phase stop 1e-9 as re-instantiated by geckods (PR153, PR158); PR144 checks and assembly otherwise unchanged.", + "adopted": [ + { + "pr": 144, + "contributor": "icekylinx", + "commit": "c8b22bc5c10dba497ac25804e27d9647d818e2ff", + "role": "Paired-cube decoder, modules, shared cores, bit ledger, finite bridge and assembly." + }, + { + "pr": 131, + "contributor": "eumemic", + "role": "Physical frame descent and compensated reuse on Cayley covers (method)." + }, + { + "pr": 124, + "contributor": "jamesyc", + "role": "Compensated birth-read reuse." + }, + { + "pr": 143, + "contributor": "eumemic", + "role": "Deadline (late) compensation reads." + }, + { + "pr": 128, + "contributor": "an664", + "role": "Completed-core sharing." + }, + { + "pr": 97, + "contributor": "Zhihao Chen (jacklightChen)", + "role": "Bit-side compile conventions (deferred readouts, plain or lifted frames)." + }, + { + "contributor": "Swapnil Jain", + "role": "Round-seven bit lineage (PR62 interval strips) from which the bit pair modules are cut." + }, + { + "pr": 158, + "contributor": "geckods", + "role": "Re-instantiated free assembly constants (phase stop, PR153; atom exponent, PR158)." + } + ], + "scope": "Conditional on the retained PR144 analytic, uniform-recursion and fixed-tape contracts." } } diff --git a/certificates/paired-cube-complex-input.json b/certificates/paired-cube-complex-input.json index 2f6fe0ae7b..e6c1ce77da 100644 --- a/certificates/paired-cube-complex-input.json +++ b/certificates/paired-cube-complex-input.json @@ -1,86 +1,82 @@ { - "R": 26417, - "W_per_vertex": 29937, - "c": 24767, + "R": 16011, + "W_per_vertex": 18651, + "c": 17735, "child_histogram": { - "1": 79374, - "10": 5829, - "11": 3261, - "12": 3291, - "13": 2028, - "14": 6972, - "15": 2478, - "16": 6789, - "17": 4794, - "18": 1194, - "19": 11760, - "2": 81787, - "20": 11880, - "21": 2640, - "22": 2712, - "23": 0, - "24": 0, - "3": 34704, - "4": 29763, - "5": 22455, - "6": 8535, - "60": 4840, - "7": 7929, - "8": 7881, - "9": 3603 + "1": 60462, + "10": 2250, + "11": 1146, + "12": 1383, + "13": 1095, + "14": 3231, + "15": 1902, + "16": 588, + "17": 8448, + "18": 8910, + "19": 1980, + "2": 58161, + "20": 2046, + "21": 0, + "22": 0, + "3": 24618, + "4": 20595, + "5": 14559, + "54": 2970, + "6": 6093, + "7": 4881, + "8": 4743, + "9": 2571 }, "copied_centers_already": true, - "deficit_per_vertex": 1936, - "h": 24, - "loss": 528, - "m": 72, - "matched": 6074, + "deficit_per_vertex": 1320, + "h": 22, + "loss": 440, + "m": 66, + "matched": 6201, "matching_frames": "coordinate", - "phase1_operations": 13616, - "q": 7724, - "rank_per_vertex": 2153528, + "phase1_operations": 10150, + "q": 4477, + "rank_per_vertex": 1229646, "remaining_internal_histogram": [ - 11459, - 19418, - 24329, - 11568, - 9921, - 7485, - 2845, - 2643, - 2627, - 1201, - 1943, - 1087, - 1097, - 676, - 2324, - 826, - 2263, - 1598, - 398, - 3920, - 440, - 880, - 904, + 6855, + 14874, + 17187, + 8206, + 6865, + 4853, + 2031, + 1627, + 1581, + 857, + 750, + 382, + 461, + 365, + 1077, + 634, + 196, + 2816, + 330, + 660, + 682, 0, 0 ], "selected_rank_histogram": { - "20": 4840 + "18": 2970 }, - "selected_roles": 4840, + "selected_roles": 2970, "source_data_histogram": { - "1": 1760, - "2": 1760, - "20": 1760 + "1": 1320, + "18": 1320, + "2": 1320 }, "target_data_histogram": { - "0": 12760, - "1": 5280, - "20": 1760 + "0": 8250, + "1": 3960, + "18": 1320 }, - "total_M_operations": 49424, - "untouched_non_source_roles": 15417, - "v": 1760 + "total_M_operations": 32426, + "untouched_non_source_roles": 7819, + "v": 1320 } diff --git a/certificates/paired-cube-input.json b/certificates/paired-cube-input.json index 368d040cd0..0e9dc69013 100644 --- a/certificates/paired-cube-input.json +++ b/certificates/paired-cube-input.json @@ -1,66 +1,57 @@ { - "absorption_gap": "49772927594887408131062764367/500461342287379865539365770570000000000", - "assembly_minimum": "23067114022986563163422943/50046134228737986553936577057", + "absorption_gap": "37351770714269060570360194191023257/500588566790582931049971823558856690000000000", + "assembly_minimum": "29428339087721461768518344114331/50058856679058293104997182355885669", "bit_child_multiplicities": { - "1": 193950, - "10": 474, - "11": 1365, - "12": 585, - "13": 1059, - "14": 330, - "15": 2580, - "16": 3180, - "17": 4935, - "18": 3246, - "19": 9987, - "2": 200588, - "20": 7161, - "21": 1242, - "22": 75, - "3": 40116, - "4": 33642, - "48": 72, - "5": 18126, - "51": 131, - "54": 806, - "57": 3871, - "6": 4875, - "60": 1307, - "63": 1596, - "66": 1760, - "7": 2250, - "8": 3489, - "9": 1482 + "1": 120972, + "10": 2160, + "11": 2016, + "12": 4464, + "13": 1080, + "14": 2232, + "15": 216, + "16": 2808, + "17": 2736, + "18": 3888, + "19": 3672, + "2": 73336, + "20": 7260, + "21": 8580, + "22": 15912, + "3": 29832, + "4": 8208, + "5": 12096, + "6": 4392, + "60": 5720, + "7": 4824, + "8": 2736, + "9": 648 }, - "bit_moment_gap": "9683053551624097783276572314019807322107841/67308782636558677514166530014945948467200000000000000", + "bit_moment_gap": "34879094578234633698904410970360032171054091/523539292246066669784729944875262279680000000000000000", "complex_child_multiplicities": { - "1": 79374, - "10": 5829, - "11": 3261, - "12": 3291, - "13": 2028, - "14": 6972, - "15": 2478, - "16": 6789, - "17": 4794, - "18": 1194, - "19": 11760, - "2": 81787, - "20": 11880, - "21": 2640, - "22": 2712, - "3": 34704, - "4": 29763, - "5": 22455, - "6": 8535, - "60": 4840, - "7": 7929, - "8": 7881, - "9": 3603 + "1": 81498, + "10": 2211, + "11": 948, + "12": 1482, + "13": 2913, + "14": 1194, + "15": 4449, + "16": 2736, + "17": 1518, + "18": 7920, + "19": 2130, + "2": 52044, + "20": 6006, + "3": 23037, + "4": 11535, + "5": 13140, + "6": 5832, + "7": 3990, + "8": 4575, + "9": 2577 }, - "complex_group_order": 18646466226026462592952011788951531754110931481867826045386715225513339911071134170601952718216417488289150428632052202980863033609242977381747250264990084715296201847651908198681766675753160791649285107510691238985963706255089989456699920961342595095228775420538279717562788772643958616644258216248819271553295126895131993564379512367852197664617342277934678865080413621612751262266453859055432749295232782952753053108052418519375708005607816485667984223061786810719121526682978029691520838623314161409960947186213220855798991391576957539912643800234313922722815599751095827998877881904497464356201696239917156093128550867728502903876632148399699575860894114797305318253105534652030783324407335764468260764573859627847585742189622249940283054201241600000000000000000000, - "complex_moment_gap": "35983219312333093839004994137969/179068943294158619192045723836257040072704", - "kappa": "4609169/10000000000", - "row_coefficient": 20445, + "complex_group_order": 3525886442745164056544063644145716736004044180364723200424782744512582104634391857878947474919000911664177132185103526533556870749183073596734571342973795793383290849303191539662953799849547227737400544457729391514661713176468240351330298583217125220869576729661326266259024901649005584114421035252391760900092184489369973370724377894349644301151172572225522957784648909234781611948223712944954177723427440753012339451839407462242698998122862136268155486957700758485307238973641035062973415368109606762999662774702118665848434384838590914531021276158042617166069588021894697108433312726842277534281982740181708602162296504975360000000000000000000, + "complex_moment_gap": "46965161945076382039329824844403/229279866220935923833050583634816916258816", + "kappa": "5878747/10000000000", + "row_coefficient": 12320, "row_degree": 70000 } diff --git a/certificates/paired-cube-network.json b/certificates/paired-cube-network.json index 11afbc33aa..a858f00d43 100644 --- a/certificates/paired-cube-network.json +++ b/certificates/paired-cube-network.json @@ -1,334 +1,351 @@ { "assembly": { - "absorption_gap": "49772927594887408131062764367/500461342287379865539365770570000000000", + "absorption_gap": "37351770714269060570360194191023257/500588566790582931049971823558856690000000000", "margins": { - "Gaussian_arithmetic": "27680687058255018102013446063422943/40036907382990389243149261645600000000", - "bulk_exposure": "4613422943/10000000000000", - "compact_phase_layer": "23067114022986563163422943/50046134228737986553936577057", - "coordinate_movement": "4613422943/10000000000000", - "dimension": "49999999500000000000000000000/50046134228737986553936577057", - "original_prefix": "2306761448432885054328848236577057/5004613422873798655393657705700000000", - "scalar_work": "36907732944255014411275091663422943/40036907382990389243149261645600000000" + "Gaussian_arithmetic": 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} }, - "kappa": "4609169/10000000000", + "kappa": "5878747/10000000000", "predecessor_commit": "6a9970a530119174507904e23592fd59ede19a5d", "scope": "Exact inventories, supplier moments, full fallback/router charge and semantic assembly. 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The complex word adds physical frame descent and compensated late-read +reuse (`scripts/paired_cube_physical.py`); the bit supplier is the paired-cube bit word in +`research/paired-cube-bit`. From the repository root with Python 3.11+: @@ -13,16 +15,21 @@ make paired-cube-verify The selected incremental target uses only the Python standard library. CI runs it in its own `paired-cube` matrix group. The checks are: -- Rebuild the selected signed H-channel graph from exact zero restrictions of - the pinned PR #117 positive DAG. Replay the frozen 6,074 carrier arcs, +- Rebuild the selected signed H-channel graph (p = 11) from exact zero + restriction of a pinned complex DAG and a pinned pair-disjoint module in + `references/paired-cube/sources`, with merged face-2/edge-02 outputs. Replay the frozen 6,201 carrier arcs, coordinate frames, full backward intersections, signed physical mixer, - center closure and 4,840 rank-20 partial gauges. Independent checks cover + center closure and 2,970 rank-18 partial gauges. Independent checks cover every scalar coefficient of `H+K+B=I`, the original-source K involution and inverse, actual frame containment and reverse target chains. -- Reconstruct the bit subset from the existing PR #97 witness and readout - order. Check the retained 9,543 and omitted 2,022 slots, moved zero-frame - dirty reads, changed first transitions and target subsequences. Unchanged - complete-basis and rational-frame suites remain inherited checks. +- Apply the frozen physical frames and late-read reuse pairs and check chains, + chronology, target order and an aliased dirty-scratch replay + (`scripts/paired_cube_physical.py`). +- Check the paired-cube bit word with its standalone checker + (`research/paired-cube-bit/check_paired_cube_bit.py`): exact frames, decoder + identity, chains, partner-pair chronology, nondegeneracy, F2 replay and + ledger. `python3 research/paired-cube-bit/paired_cube_bit_word.py --p 12 --check` + regenerates its outputs byte for byte. - Rebuild both shared-core child lists and certify strict moments, full rare-class fallback, stopped bit saving, actual finite group/router charge, semantic guard, row stock, 47 strict inequalities and seven assembly margins. diff --git a/notes/paired-cube-assembly.tex b/notes/paired-cube-assembly.tex index d44227eb55..dd1be755a3 100644 --- a/notes/paired-cube-assembly.tex +++ b/notes/paired-cube-assembly.tex @@ -1,54 +1,56 @@ \subsection{Strict suppliers and full finite group} The selected local complex profile has \[ - h=24,\quad v=1760,\quad c=24767,\quad q_{\rm roots}=7724, - \quad |\mathcal M|=6074,\quad R=26417,\quad\ell=528. + h=22,\quad v=1320,\quad c=17735,\quad q_{\rm roots}=4477, + \quad |\mathcal M|=6201,\quad R=16011,\quad\ell=440. \] The exact signed graph, frozen carrier arcs, coordinate frames and chronological gauge list are reconstructed by \path{scripts/paired_cube_producer.py}. Its complete shared inventory has -$m_c=72$, $w_c=29937$, $s_c=2153528$, deficit 1936 and largest child 60. -For $a_c=4856569/10^{10}$ the rational moment satisfies +After physical frame descent and compensated late-read reuse, it has +$m_c=66$, $w_c=15681$, $s_c=1033626$, deficit 1320 and largest child 20. +For $a_c=5885669/10^{10}$ the rational moment satisfies \[ - \frac1{w_c}\sum_r n_r(r/72)^{1-a_c} - <1-2.0094617553\times10^{-10}. + \frac1{w_c}\sum_r n_r(r/66)^{1-a_c} + <1-2.0483770650\times10^{-10}. \] -The bit subset has $m_b=69$, $w_b=32408$, $s_b=2234128$ and deficit 2024. -At coarse saving $a_0=4617656/10^{10}$ its moment includes the entire -$32\cdot69^2$-child local-ring fallback on a fraction $10^{-16}$ of every -ideal edge. Its gap exceeds $1.4386017949\times10^{-10}$; the rank moment +The paired-cube bit word (\path{research/paired-cube-bit}) has $m_b=72$, +$w_b=26888$, $s_b=1934000$ and deficit 1936. +At coarse saving $a_0=5936323/10^{10}$ its moment includes the entire +$32\cdot72^2$-child local-ring fallback on a fraction $10^{-16}$ of every +ideal edge. Its gap exceeds $6.6621732302\times10^{-11}$; the rank moment also contracts. With the retained atom exponent $10^{-3}$, \[ a_b=(1-10^{-3})a_0+10^{-3}\frac{384599}{10^{10}} - =\frac{4613422943}{10^{13}}. + =\frac{5930771276}{10^{13}}. \] Both adapter and internally borrowed-row tolls are lower order since $a_b<10^{-3}<1-a_b$. The weighted compiler, residue bound, full fallback and ordinary outer wrapper are retained from \path{notes/three-stage-cover-bit.tex} and -\path{notes/three-stage-cover-rows.tex}, now with ambient dimension 69. +\path{notes/three-stage-cover-rows.tex}, now with ambient dimension 72. Sharing routers are fixed right multiplications, batched on the same complete high matrix field; the fresh-selector contract is unchanged. -For the complex cover $V=|O(72,2)|$ the retained even-dimensional group +For the complex cover $V=|O(66,2)|$ the retained even-dimensional group formula gives \[ - V=2^{1296}\prod_{i=1}^{35}(2^{2i}-1). + V=2^{1089}\prod_{i=1}^{32}(2^{2i}-1). \] -It has 2,556 bits, and full role stock $W=Vw_c$ has 2,571 bits. +It has 2,145 bits, and full role stock $W=Vw_c$ has 2,159 bits. Its full rank is $s=Vs_c$ and $N=Vv$. -The least halving degree for $72\to60$ is four, contributing 10,284 to +The least halving degree for $66\to20$ is one, contributing 2,159 to the row coefficient. Including the conservative preceding bit reserves, \[ - 10284+9909+252=20445,\qquad - 70000-\frac{51}{25}\,20445=\frac{141461}{5}>0. + 2159+9909+252=12320,\qquad + 70000-\frac{51}{25}\,12320=\frac{224336}{5}>0. \] Thus the fixed external reserve $p^{70000}$ suffices. The q-adic bit selector stock is borrowed and restored internally, not multiplied into this external polynomial reserve. \subsection{Literal charge, sharing routers and exact precision} -Let $M=49424$ be the actual local mixer-operation count. Put +Let $M=32426$ be the actual local mixer-operation count. Put \[ L=4(c+v)+10v+4hv+4h^2+8h+8+2h +8Rv(M+16)+32v. @@ -56,7 +58,7 @@ \subsection{Literal charge, sharing routers and exact precision} The last term pays both original-source involutions and their inverses. Every mixer update has signed unit coefficients. Starting from the identity, each entry of its exact matrix is therefore bounded by $2^M$. -There are $q_{\rm roots}=7724$ decoder roots; after clearing denominator +There are $q_{\rm roots}=4477$ decoder roots; after clearing denominator six, their scatter coefficients have absolute numerator at most three. Thus each numerator of $6JM$ is bounded by $3q_{\rm roots}2^M<2^{M+15}$. Including the sign, $M+16$ bits suffice. @@ -92,7 +94,7 @@ \subsection{Literal charge, sharing routers and exact precision} retains the inherited $O(V_{\rm bits}\log d)$ charge. \subsection{Exact retained semantic assembly} -Put $a=a_b$, $b=a_c$, $\eta=10^{-8}$, $\beta=10^{-6}$ and +Put $a=a_b$, $b=a_c$, $\eta=10^{-8}$, $\beta=10^{-9}$ and \[ \tau=1-a,\quad\sigma=1-b,\quad q_0=a(1-2\eta),\quad c_0=q_0(1+\eta),\quad @@ -108,14 +110,14 @@ \subsection{Exact retained semantic assembly} 1-\epsilon(1+c_0),\ a,\ g,\ a,\ \min(1-\epsilon-\delta,r-\delta),\ 1-\epsilon-\delta,\ \epsilon. \] -Their minimum $g$ exceeds $4609169/10^{10}$ by more than -$9.9454090434\times10^{-11}$. +Their minimum $g$ exceeds $5878747/10^{10}$ by more than +$7.4615708771\times10^{-11}$. \begin{theorem} Under the retained analytic, semantic-precision, exact-recovery, uniform-recursion and fixed-tape interfaces, the selected construction gives \[ T(n)=O\!\left(n(\log n)^{1-\kappa}\right),\qquad - \boxed{\kappa=\frac{4609169}{10000000000}=4.609169\times10^{-4}}. + \boxed{\kappa=\frac{5878747}{10000000000}=5.878747\times10^{-4}}. \] \end{theorem} \begin{proof} diff --git a/notes/paired-cube-bit.tex b/notes/paired-cube-bit.tex index dd84535b21..a321105e8e 100644 --- a/notes/paired-cube-bit.tex +++ b/notes/paired-cube-bit.tex @@ -2,60 +2,34 @@ % Construction with GPT-6 Astra; integration with OpenAI Codex. % Retained PR97 ledger: Zhihao Chen; frozen scalar/frame witness: Swapnil Jain. % Their original licenses, credits and assistance disclosures remain pinned. -\subsection{Selected inherited bit gauges} -The bit supplier uses the pinned PR97 word at $h=23$, with $v=1771$ data -ports and $R=28866$ dirty roles. Its compressed witness, frame reader, -physical-ledger receipt and original notices remain under -\texttt{references/partial-gauge/pr97/}. The finite selection in -\texttt{references/paired-cube/pr97-subset/selection.json} retains $9543$ -of the $11565$ certified entrance gauges, in their original readout order, -and omits $2022$. This selects actual inherited frames, rather than -optimizing an unattached multiset of ranks. - -For an omitted role with entrance frame $\sigma_s$ of dimension $f_s$, -perform its old-value correction at the common initial zero frame, -before any positive-rank operation. Its exact scalar coefficients are -the coefficients of the original word's adjoint, including copied-center -readouts. Both ends of each moved dirty read have frame zero, so arbitrary -dirty inputs are allowed. Its subsequent first-use frame is unchanged. -All ordinary scalar operations, positive-rank $V$ injections, source-data -paths, copied centers and later role frames retain their pinned values. -Retained dirty readouts form the original subsequence. Deleting members -of a certified nested target chain preserves actual subspace nesting; -dimension monotonicity alone would not establish that assertion. - -If the original first-use dimension of an omitted role is $r_s$, its -first auxiliary transition changes from $r_s-f_s$ to $r_s$, and its -shared-core exterior child of width $3f_s$ disappears. Target chains -retain precisely the positive integer adjoint supports of the chosen -readouts. This is the inherited conservative support convention; an even -coefficient would merely give a zero bit read with a retained frame. -The source histogram and all $23$ copied-center transitions of width $22$ -are unchanged. The selected gauge dimensions have multiplicities +\subsection{Paired-cube bit word} +The bit supplier is a bit-side analog of the paired-cube decoder at $p=12$ +($h=24$, $v=1760$, $R=23368$), in \path{research/paired-cube-bit}. Its +pair-disjoint module (438 additions) is annealed from a witness cut. Ports are +transversal triples of coordinate pairs; the mod-2 decoder is \[ - (22,1760),\ (21,1596),\ (20,1307),\ (19,3871),\ - (18,806),\ (17,131),\ (16,72). + x_T=\sum_{c\in T}\mathrm{star}(c)+\sum_{|S\cap T|=1}x_S , \] +with point stars of span $h-2$ (centre loss $\ell=h(h-2)=528$), the face and +edge channels of PR144's pair and all-but-one modules, and partner-pair sums +mixed on the source registers. \path{research/paired-cube-bit/LEMMA.md} proves +the mixing step: the carrier and passive chains $[1,h-4,2]$ and $[1,h-2]$ +are nested, every XOR happens at one common frame, the delivery happens at the +receivers' level-2 cap, and the mix completes inside its core, so it is +sequential rather than lockstep. The word is compiled with PR144's ledger on +rational frames, with 5720 chronological gauges of dimension 20. -In the cross-stage orthogonal sharing architecture, $m=3h=69$ and -$W=2v+R=32408$ per cover vertex. There are three copies of each local -auxiliary, source, target and copied-center transition, one exterior child -of width $3f_s$ per selected gauge, and $2v$ final children of width $2$. -Consequently the selected profile has +In the cross-stage orthogonal sharing architecture, $m=3h=72$ and +$W=2v+R=26888$ per cover vertex, and \[ - \ell=h(h-1)=506,\qquad - \sum_r r C_r=2234128=Wm-2024,\qquad - 2024=2v-3\ell. + \sum_r r C_r=1934000=Wm-1936,\qquad 1936=2v-3\ell. \] -Its largest child is $66$. The strict supplier moment below also charges -the retained rare-class fallback and stopped atom wrapper. +Its largest child is $60$. The strict supplier moment below also charges the +retained rare-class fallback and stopped atom wrapper. -\texttt{scripts/paired\_cube\_bit.py} reconstructs this profile from the -existing pinned inputs. It checks their hashes, the exact selected slot -partition and inherited order, the integer target-incidence sweep, the -initial zero-frame corrections, and the changed transition histogram -against a direct reconstruction of the modified role chains. The -incremental review finds $72$ changed transition types and $9594$ moved -old-value reads. It repeats neither the complete-basis scalar replay nor -the inherited exact rational frame and lifted-witness audits. The -original frozen witnesses and scripts remain unchanged. +The standalone checker \path{research/paired-cube-bit/check_paired_cube_bit.py}, +run by \path{scripts/paired_cube_network.py}, recomputes the exact frames, the +decoder identity, every role and target chain, the partner-pair chronology, +$G$-nondegeneracy, a literal F2 replay with arbitrary dirty scratch and the +ledger. The generator regenerates every output byte for byte, and five +mutation controls are rejected. diff --git a/notes/paired-cube-construction.tex b/notes/paired-cube-construction.tex index 4428af9c6d..61af913c59 100644 --- a/notes/paired-cube-construction.tex +++ b/notes/paired-cube-construction.tex @@ -130,13 +130,20 @@ \subsection{Arbitrary dirty contents and the frozen witness} have been restored. Exact frame representatives and rank-zero phase adapters are those of the arbitrary-subspace Clifford interface. -For the selected $p=12$ graph, 24,767 active additions and 7,724 root -uses, with 6,074 actual monotone carrier links, give +The selected graph has $p=11$. Its triple module is an exact restriction +of a pinned searched complex DAG, and its pair module is a pinned searched +pair-disjoint module (\path{references/paired-cube/sources}), both in place +of PR117's. For every port $T=(I,b)$ the face-2 and edge-02 outputs are +merged into one signed node $P\pm Q$ per $(b_0,b_2)$, read with coefficient +$1/2$ by both ports $b_1=0,1$ (\texttt{merge\_outputs}); the decoder +contribution is unchanged and every addition keeps disjoint supports. +Its 17,735 active additions and 4,477 root uses, with 6,201 actual +monotone carrier links, give \[ - h=24,\quad v=1760,\quad R=26417,\quad\ell=528. + h=22,\quad v=1320,\quad R=16011,\quad\ell=440. \] -The frozen chronological gauge witness selects 4,840 roles, each of -dimension 20. Its graph, matching/frame witness and gauge-selection file, +The frozen chronological gauge witness selects 2,970 roles, each of +dimension 18. Its graph, matching/frame witness and gauge-selection file, not an abstract circuit-rank bound, determine the paid internal histogram. -The selected source histogram is $[2,20,1]$ per port; the target histogram -is $[20,1,1,1]$ per port. Zero-width adapters remain paid scalar work. +The selected source histogram is $[2,18,1]$ per port; the target histogram +is $[18,1,1,1]$ per port. Zero-width adapters remain paid scalar work. diff --git a/notes/paired-cube-note.tex b/notes/paired-cube-note.tex index b72c67f72f..7a522059bc 100644 --- a/notes/paired-cube-note.tex +++ b/notes/paired-cube-note.tex @@ -9,7 +9,7 @@ \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \title{Paired cubes and shared completed cores for integer multiplication\\ -\large A conditional saving $\kappa=4609169/10^{10}$} +\large A conditional saving $\kappa=5878747/10^{10}$} \author{icekylinx\\\small Construction with substantial OpenAI GPT-6 Astra assistance;\\ \small exposition and repository integration with Codex assistance} \date{9 October 2026} @@ -20,7 +20,7 @@ on the original source registers. Complete arbitrary-dirty cores share one auxiliary bank across three orthogonal address blocks. The exact selected ledgers, bit gauge subset and retained semantic assembly yield -$T(n)=O(n(\log n)^{1-0.0004609169})$ under the inherited interfaces. +$T(n)=O(n(\log n)^{1-0.0005878747})$ under the inherited interfaces. \end{abstract} \section{Scope and contributions} diff --git a/notes/paired-cube-sharing.tex b/notes/paired-cube-sharing.tex index 56e729572d..b4b58f1158 100644 --- a/notes/paired-cube-sharing.tex +++ b/notes/paired-cube-sharing.tex @@ -79,12 +79,18 @@ \subsection{The complete paid ledger} \end{align*} For the selected complex word, \[ - m=72,\quad W_0=29937,\quad - \sum_r rn_r=2153528,\quad mW_0-\sum_r rn_r=1936. + m=66,\quad W_0=15681,\quad + \sum_r rn_r=1033626,\quad mW_0-\sum_r rn_r=1320. \] -The 4,840 exterior children have width 60; all local internal children -have width at most 22, and final data children have width two. -Hence the largest child is 60. Full stock is $W=|\mathcal G|W_0$; +Physical frame descent (\path{scripts/paired_cube_physical.py}) moves +7,880 common operation frames between the span of their previous frames +and the intersection of their next frames. Each of the 2,970 gauged +roles then occupies the physical slot of a dead donor whose last frame +lies in its gauge. Its old-value read moves to the deadline immediately +before its first operation, and the slot keeps the donor's zero start. +So no exterior correction remains: every local internal child has width +at most 20, and final data children have width two. Hence the largest +child is 20, with 13,041 physical roles. Full stock is $W=|\mathcal G|W_0$; the group factor cancels from the normalized moment, not from routing, scalar work, row stock or the finite semantic guard. diff --git a/references/paired-cube/physical/README.md b/references/paired-cube/physical/README.md new file mode 100644 index 0000000000..5d5f0d7dbc --- /dev/null +++ b/references/paired-cube/physical/README.md @@ -0,0 +1,19 @@ +# Physical frames and compensated reuse on the p = 11 complex word + +Frozen inputs for `scripts/paired_cube_physical.py`, which applies them to the word regenerated by #144's producer. + +- `frames.json` lists the 7,880 operations whose common frame moves. For each one it gives the canonical basis of the + new frame. Every other operation keeps #144's full backward-intersection frame. + - Each moved frame lies between the span of the two roles' previous frames and the intersection of their next + frames. This is physical frame descent under #130's general Clifford frames, as in #131. +- `pairs.json` lists 2,970 `[donor, recipient, read_before_op]` triples. + - Each recipient is a gauged role. It lives in its donor's physical slot, as in #124's compensated birth-cut reuse. + - Its old-value read moves to a deadline immediately before its first operation, as in #143's late compensation + reads. + - The donor's last operation precedes that read, and the donor's last frame lies in the recipient's gauge. + +Prepared by eumemic with Anthropic Claude assistance. `scripts/paired_cube_physical.py` checks: +- value spans, chain nesting with the spliced handoffs, and pair chronology; +- target-chain order in actual read time; +- the telescoping deficit; +- an exact aliased dirty-scratch replay mod 2^61 − 1 with three rejected controls. diff --git a/references/paired-cube/physical/frames.json b/references/paired-cube/physical/frames.json new file mode 100644 index 0000000000..4052c82a13 --- /dev/null +++ b/references/paired-cube/physical/frames.json @@ -0,0 +1 @@ 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diff --git a/references/paired-cube/physical/pairs.json b/references/paired-cube/physical/pairs.json new file mode 100644 index 0000000000..3365316fc9 --- /dev/null +++ b/references/paired-cube/physical/pairs.json @@ -0,0 +1 @@ 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diff --git a/references/paired-cube/selected-module/SOURCE.json b/references/paired-cube/selected-module/SOURCE.json index 0ba9215073..f663d87de8 100644 --- a/references/paired-cube/selected-module/SOURCE.json +++ b/references/paired-cube/selected-module/SOURCE.json @@ -1,6 +1,6 @@ { "source_commit": "cbb05ce504d571546d9b7794c186a613c659c3bf", "inherited_dag_sha256": "3c034d0aae388ef567a454826f4f48b26fd8a94c71e8ffed4835271b349a783b", - "matching_arcs_sha256": "22c90ed078d765e58ac824d6f9558cc4620961b48644a32c0115bdb9852de771", - "graph_sha256": "7a7a58df595dc245ebe8d2673abdd8c4eaae98e767885240715a61f43a2f4dff" + "matching_arcs_sha256": "cbc3e1e0f332916511d775427c7f8dfa4441a8450e3f4a2ce9a30d3284bb74b3", + "graph_sha256": "59a77f9b593d7b0f627c85db302dc3a6d47ff091d86c1ea0b4e32eef1964ca8c" } diff --git a/references/paired-cube/selected-module/matching-arcs.json b/references/paired-cube/selected-module/matching-arcs.json index e9238210ab..2fb88665c3 100644 --- a/references/paired-cube/selected-module/matching-arcs.json +++ b/references/paired-cube/selected-module/matching-arcs.json @@ -1 +1 @@ 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diff --git a/references/paired-cube/sources/SOURCE.json b/references/paired-cube/sources/SOURCE.json new file mode 100644 index 0000000000..63042f331c --- /dev/null +++ b/references/paired-cube/sources/SOURCE.json @@ -0,0 +1,7 @@ +{ + "files": { + "h20_g1.json.gz": "2da21138a3887b0b190b234033c08740e81e145dd0db059e8f24c45d52fa41e9", + "pmod_C35_5.483127e-4.json": "55e1979722255b3b70acf0b93b3a62a1e4e603a56276b254858a5b99974f47a9" + }, + "note": "h20_g1: #117-format complex witness by eumemic (Claude assistance), restricted by restricted_triples_from; pmod_C35_5.483127e-4.json: direct pair-disjoint module (PMOD lane, eumemic, Claude assistance), loaded by pair_module_from. Apache-2.0." +} diff --git a/references/paired-cube/sources/h20_g1.json.gz b/references/paired-cube/sources/h20_g1.json.gz new file mode 100644 index 0000000000..a6c992708e Binary files /dev/null and b/references/paired-cube/sources/h20_g1.json.gz differ diff --git a/references/paired-cube/sources/pmod_C35_5.483127e-4.json b/references/paired-cube/sources/pmod_C35_5.483127e-4.json new file mode 100644 index 0000000000..907cd6ccd2 --- /dev/null +++ b/references/paired-cube/sources/pmod_C35_5.483127e-4.json @@ -0,0 +1 @@ +{"input_count": 45, "args": [null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, null, [3, 4], [17, 2], [10, 2], [0, 12], [4, 12], [41, 8], [32, 31], [16, 8], [8, 7], [7, 52], [54, 15], [29, 28], [50, 16], [11, 12], [9, 1], [28, 22], [1, 22], [10, 9], [34, 55], 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355, 333, 362, 401, 357, 385, 325, 373, 303, 328, 350]} \ No newline at end of file diff --git a/research/paired-cube-bit/FORMAT.md b/research/paired-cube-bit/FORMAT.md new file mode 100644 index 0000000000..91bb631a3e --- /dev/null +++ b/research/paired-cube-bit/FORMAT.md @@ -0,0 +1,132 @@ +# File formats (paired-cube bit word, `P` in {11, 12, 13}, `h = 2P`) + +All files are plain JSON with sorted keys, written by `paired_cube_bit_word.py --p P`. +`--check` regenerates them byte-identically. The exceptions are `word_pP.json.gz`, a compact internal copy +written with gzip `mtime = 0`, and `profile_pP.json`. Nodes, ports and roles are zero-based integers. + +## Conventions + +- **Ports.** `labels[i]` is the triple of coordinates of port `i` (for example coordinate `2k + b` is + member `b` of pair `k`). Port `i` is also graph node `i`, a source leaf. +- **Label.** `chi_S` in `Q^h` is the 0/1 indicator of `labels[S]`. +- **Target covector.** The target covector of port `T` is `3*chi_T - 1`. The cap of `T` is + `{u : (3 chi_T - 1).u = 0}`, the H0-orthogonal complement of `chi_T` for `H0 = (I - J/9)/2`. +- **Nondegeneracy.** Nondegeneracy is tested for `G = I - J/9` (equivalently `H0`). + - A frame with integer basis `B` is nondegenerate iff the integer matrix `9 B B^T - (B1)(B1)^T` is + nonsingular. + - A frame given by integer annihilator rows `A` is nondegenerate iff the dual matrix + `(9-h) A A^T + (A1)(A1)^T` is nonsingular. +- **Scalars.** The payload is F2: every coefficient is 1 (mod 2), and every op is an XOR. + +## `graph_pP.json` — scalar circuit and decoder + +- `p`, `h`, `v`, `labels`. +- `args[x]`: `null` for the source leaves (`x < v`); otherwise `[a, b]`, where node `x` is the sum of + nodes `a` and `b`. Supports are disjoint, so node `x` equals the XOR of the sources in its support. +- `roots[j]`: an object with fields `node`, `targets` (a list of ports), `kind` and `coefficient`: + - `kind` is `side` or `center`; `coefficient` is always 1. + - Side roots carry `channel`, one of `face0`, `edge12`, `face1`, `edge01`, `face2`, `edge02` or + `partner`. A `partner` root's node is a source leaf (single-source root). + - Centre roots carry `coordinate c`. Their node is `star(c)` (the sum of all ports containing `c`), and + their targets are all ports containing `c`. + - Root `j` adds its node's value into `Y_t` for every `t` in `targets`. Broadcast roots have several + targets: `face0` reaches 4 targets and `edge12` reaches 2. +- `partner_mix[k]`: an object `{carrier, passive, receivers:[a, b]}`. It adds `x_carrier + x_passive` to + `Y_a` and `Y_b` through source-register mixing (see `kchron`). +- `decoder`: the identity being implemented, + + Y_T += x_T = sum_{c in T} star(c) + sum_{|S cap T| = 1} x_S (mod 2). + + Stars are read with coefficient 1 for each `c in T`. The side part is the XOR of all side roots + (including the partner singles) received by `T`, plus the partner-pair sum, and it equals + `sum_{|S cap T|=1} x_S`. + +## `word_pP.json` — compiled physical word (shaped like #144's `selection.json`) + +- `ops[i] = [dest, control, node]`. This means `z[dest] ^= z[control]`, at the frame of `node`. + - For an addition node, `dest` holds one operand and becomes the sum, while `control` keeps the other + operand. + - For a copy (an extra use of a value), `dest` is the fresh role and `control` is the value's role. + - ops are listed in execution order. +- `sources[str(leaf)] = role`: the auxiliary role receiving `z[role] ^= x_leaf` (the V injection at the + rank-one frame ``). +- `rootroles[j]`: the role that is read into the targets of `roots[j]`. +- `phase1`: indices of the ops in the centre closure, recomputed by the checker. +- `gauges[k] = {role, first, targets, dim, frame}`. Each entry is a deferred old-value read of `role`'s + dirty content, at frame `frame` (an exact annihilator id in `frames`, of dimension `dim`): + - `first` is the node of the role's first op; + - `targets` is the role's response support; + - selection order is listed, and the time order per target is the reverse of it, which is ascending. +- `order`: compile order of the active nodes; `arcs`: the frozen carrier links `[donor, use]`, with `use` + either `["op", node, operand_slot]` or `["root", j]`. +- `plain`: nodes whose frame is the plain span of their support. +- `node_frame[str(node)]`: frame id of every active node. The frame of op `[dest, control, node]` is + `node_frame[node]`. +- `root_frame[j]`: frame id of root `j`. For side roots this is the common cap of the targets; for centre + roots it is the star span, of dimension `h-2`. +- `source_frame[S]`: frame id of ``. `full_frame`: id of `Q^h` (id 0, no annihilator rows). +- `conventions` and `schedule`: text restating the above. + +## `frames_pP.json` — exact frames + +- `frames[str(id)]` is an object `{dim, a}` or `{dim, b}`. Its rows are primitive integer vectors of + length `h`, the rows of the rational RREF scaled to integers: + - `a`: annihilator rows; the frame is their common kernel, of dimension `dim = h - len(a)`. + - `b`: basis rows; the frame is their span, of dimension `dim = len(b)`. +- The representation with fewer rows is stored. The checker computes the other one exactly by integer + elimination and verifies `A.B^T = 0` and `rank A + rank B = h`. +- `P_used_for_ranks`: the generator computes ranks modulo the Mersenne prime `2^127 - 1`. Every minor of + the integer generator matrices involved is below this prime by Hadamard's bound for `h <= 26`, so + modular ranks are rational ranks. The exported integer rows are recovered by rational reconstruction and + are re-verified exactly by the checker, which uses no modular arithmetic. + +## `kchron_pP.json` — partial-K chronology + +`entries[k]` lists, for each partner pair `{carrier c, passive d}` and its receivers `[a, b]`, the +following fields: + +- `mix_frame`: id of `M = span{chi_c, chi_d}` (dimension 2, H0-Gram `I_2`). +- `deliver_frame`: id of the receivers' shared cap `C`, the frame of `roots[deliver_after_root]`, whose + targets are exactly `[a, b]` (the `edge12` root). +- `deliver_after_root`: the delivery happens right after this root's read, while `Y_a` and `Y_b` are at `C`. +- `undo_frame`: the full frame. +- `carrier_chain`: `[, M, C, full]`, with ranks `[1, h-4, 2]`. +- `passive_chain`: `[, M, full]`, with ranks `[1, h-2]`. + +Chronology: + +1. Run all V injections at the rank-one frames. +2. Raise `X_c` and `X_d` to `M`, then set `X_c ^= X_d`. +3. Raise `X_c` to `C`. After root `deliver_after_root` is read, set `Y_a ^= X_c` and `Y_b ^= X_c`. +4. After all target reads, raise `X_c` and `X_d` to full and set `X_c ^= X_d`. +5. Run the inverse producer and the V removal. + +## `profile_pP.json` — PR #144 bit ledger + +- `h, v, R, c, q, matched, loss` (copied-centre loss `= h(h-2)`). +- `m = 3h`, `W_per_vertex = 2v + R`, `rank_per_vertex`, `deficit_per_vertex = 2v - 3*loss`. +- `selected_rank_histogram` (gauge dimensions); `remaining_internal_histogram` (local auxiliary children + after gauging, centre copies included); `source_data_histogram`; `target_data_histogram`. +- `child_histogram`, which is assembled as: + - 3 times the local, source and target histograms; + - plus one child of width `3*dim` per gauge; + - plus `2v` children of width 2. + +## `certificate_pP.json` — exact moment + +The method is PR #144's `bit_certificate`: + +- an exact upper moment with `log_upper` and `exp_upper`, rounded to the grid `2^-120`; +- the full `32 m^2` local-ring fallback on a fraction `1e-16` of every edge. + +`coarse_saving` is the largest point `k/10^10` that passes, and `next_grid_point` is rejected. +`effective_bit_saving = (1 - 1e-3) * coarse + 1e-3 * 384599/1e10`. + +## `data/` + +- `pair_module_p{11,12,13}.json`: pinned pair-disjoint modules with provenance, re-derivable by + `derive_pair_modules.py`. + - p=11 and p=12 are cut from an h=21 bit witness (sha256 in the file). + - p=13 is PR117's `restricted_pairs(12)`. +- `arcs_p{11,12,13}.json`: frozen carrier matchings. `--solve-matching` recomputes them deterministically + with Hopcroft-Karp. diff --git a/research/paired-cube-bit/LEMMA.md b/research/paired-cube-bit/LEMMA.md new file mode 100644 index 0000000000..4879c26839 --- /dev/null +++ b/research/paired-cube-bit/LEMMA.md @@ -0,0 +1,159 @@ +# Partner-pair sums on bit source registers + +This lemma covers the only new step that the paired-cube bit word adds to PR #144's +completed-core interface: the partial analog of #144's original-source involution K. + +## Setting and conventions + +- **Ports.** Coordinates `[h]`, `h = 2p`, are split into `p` pairs. A port is a triple `S` that takes one + coordinate from each of three distinct pairs. The label is `chi_S` in `Q^h`. +- **Form.** `H0 = (I - J/9)/2` and `G = 2 H0`. Then `H0(chi_S, chi_T) = (|S cap T| - 1)/2`. In particular + `H0(chi_S, chi_S) = 1`, and `S` is H0-orthogonal to `T` iff `|S cap T| = 1`. +- **Caps.** The cap of a target `T` is `cap(T) = ker(3 chi_T - 1) = chi_T^{perp_H0}`. A group's cap is the + intersection of its members' caps. +- **Cubes and parity classes.** A cube is the set of eight ports on one triple of coordinate pairs, indexed + by selectors `e in {0,1}^3`. For two ports of one cube, `|S cap T| = 3 - d(e_S, e_T)`, where `d` is the + Hamming distance. A parity class is the set of four ports in a cube whose selector sums have the same + parity. +- **Partner pair.** The partner of `T` is the port `T'` that has the same selector as `T` on cube pair 0 + and the opposite selectors on cube pairs 1 and 2. Each parity class `{a, b, c, d}` splits into two + partner pairs, `{a, b}` and `{c, d}`. +- **Physical rules.** These are #144's rules on the bit side: + - Each register (data `X_S`, `Y_T`, or auxiliary) follows an ascending chain of G-nondegenerate frames. + - An XOR between two registers is performed at an identical frame representative. + - A register may be added into `Y_T` only at a frame that occurs in `Y_T`'s nested chain and lies in + `cap(T)`. + - Source injections `z += V x` happen while `X_S` is at its rank-one frame ``. + +## Facts + +**F1.** The four ports of a parity class are pairwise H0-orthogonal, since `d = 2` gives `|S cap T| = 1`. +For a partner pair `{c, d}`, the space `M = span{chi_c, chi_d}` therefore has H0-Gram matrix `I_2`. So `M` +is nondegenerate and 2-dimensional, and `M` lies in `cap(a) cap cap(b)` for the opposite pair `{a, b}`. + +**F2.** The level-2 group of `a` is `{a, b}`. Its cap `C = cap(a) cap cap(b)` has dimension `h - 2` and is +the frame of a side root (the edge12 channel) read into exactly `Y_a` and `Y_b`. It satisfies + + capF(a) <= C <= cap(a), + +where `capF(a)` is the cap of `a`'s level-1 group (the four cube ports sharing `a`'s selector on cube +pair 0). So every target chain + + 0 -> (gauge frames) -> capF -> C -> cap(T) + +is ascending. The checker verifies this exactly for every target. + +## Construction + +The construction runs once per parity class and partner pair. The carrier `c` is the pair member with the +smaller selector vector and `d` is the passive member; the receivers are `{a, b}`. + +1. All V injections from `X_c` and `X_d` happen first, at `` and ``, as in #144. Then + raise both `X_c` and `X_d` to `M`. Each pays one child of width 1. +2. At `M`, set `X_c <- X_c + X_d`. +3. Raise `X_c` to `C`, one child of width `h - 4`. When `Y_a` and `Y_b` are at `C` (right after their + level-2 root read), set `Y_a <- Y_a + X_c` and `Y_b <- Y_b + X_c`. +4. After all target reads, raise `X_c` to `F = Q^h` (width 2) and `X_d` to `F` (width `h - 2`). Then set + `X_c <- X_c + X_d`. + +Data ledger: the carrier's chain is `[1, h-4, 2]` and the passive's is `[1, h-2]`. Both total `h - 1`, +exactly like an unmixed source chain. No auxiliary role is used. + +## Lemma + +Steps 1–4 have the following effects: + +- They add `x_c + x_d` to `y_a` and to `y_b`. +- They restore `X_c = x_c` and `X_d = x_d`. +- They obey every frame rule. +- They read or write no register other than `X_c`, `X_d`, `Y_a` and `Y_b`. + +Together with the auxiliary word and its arbitrary-dirty restoration, the complete local word adds + + (J M V + K) x = x (mod 2), + +where `K` is the all-ones block on `{a,b} x {c,d}` of every class. This is the mod-2 decoder identity + + x_T = sum_{c in T} star(c) + sum_{|S cap T| = 1} x_S. + +## Proof + +**Scalars.** After step 2, `X_c = x_c + x_d` over F2. Step 3 adds this value to `y_a` and `y_b`. Step 4 +subtracts `x_d` again, which equals adding it in F2, so `X_c = x_c`. `X_d` is never written. + +**Frames.** + +- `X_c` follows ` <= M <= C <= F` and `X_d` follows ` <= M <= F`. These are ascending by F1 + and F2. +- Each XOR takes place at one common frame: + - step 2 at `M`, where both source registers sit; + - step 3 at `C`, where `X_c` and the targets sit; + - step 4 at `F`. +- The delivered register's support labels are `{chi_c, chi_d}`. They span `M`, which lies in `C`, which + lies in `cap(a)` and `cap(b)`. This is #144's rule that a physical frame contains the span of its support + and lies in the common target cap. +- `M` has Gram matrix `I_2`. `C` is a root frame and `F` is the whole space. The checker certifies + G-nondegeneracy of every frame with integer determinants. + +**No cross-talk.** This step is sequential, not lockstep. + +- The steps are a sequence of completed XORs on four named registers. No auxiliary role or dirty scratch + register is involved. +- Each source belongs to exactly one partner pair, and each target receives from exactly one pair. Different + pairs act on disjoint register sets, so they commute. +- The auxiliary word reads source registers only at its V injections, which precede step 1, and at its + source-subtraction cleanup. Following #144's original-source ordering, the cleanup runs at the full frame + after step 4, when `X_c` and `X_d` again hold `x_c` and `x_d`. +- The targets receive their other contributions at other times: centre copies at frame 0, gauged old-value + reads, and side roots. Additions into `Y` commute. +- None of the shared-register interleaving of #132/#139 occurs. No register ever holds a mixture of two + cores' data, because the mix is undone inside the same completed core. + +**Dirty restoration.** The literal auxiliary sequence + + y -= J M z; z += V x; z <- M z; y += J z; z <- M^-1 z; z -= V x + +restores arbitrary `z` and adds `J M V x`. Steps 1–4 touch no auxiliary register and add `K x`. + +**Completed cores and reversal.** The mixing is internal to a completed local core and leaves the data +registers' logical values and endpoints unchanged. The core's exact operator `U = F_A T_sigma^{-1}` (#144, +paired-cube-sharing) is therefore unchanged. Its complemented time reverse undoes step 4, then steps 3 and +2, at the reflected frames, so the cross-stage sharing argument applies verbatim. + +## Why this is only a partial K (obstruction) + +The source registers that may feed `Y_a` are constrained as follows: + +- A register's frame always contains its own label and every label mixed into it. +- A register can feed several targets only at one common level of their nested chains. +- `H0(chi_a, chi_a) = 1`, so `chi_a` never lies in `cap(a)`. + +Consequences: + +- No register that ever contained `chi_a` can feed `a`. +- The three class-mates of a port never form one level (levels are halves or partner pairs), and two + different single caps are never nested. +- The cap of a level-2 pair and the cap of a third single are not nested either, because `3 chi_S' - 1` is + not in the span of the pair's covectors. + +So the partner term `x_b -> y_a` cannot come from a source register. It is delivered by a single-source +root from `b`'s injected leaf register. + +Moreover, on any fully orthogonal pair of port groups, the mod-2 block that the decoder needs is all-ones +(rank 1). Source mixing can therefore deliver only sums, never a Hadamard-type block like #144's `(P - A)/2`. +Within a cube, bit-orthogonality joins ports of the same parity class (two copies of K4), not ports of +opposite classes (K_{4,4}) as on the complex side. For this reason #144's involution itself has no bit analog. + +## Machine checks + +`check_paired_cube_bit.py` verifies the following: + +- `|c cap d| = 1` for every carrier and passive pair. +- `dim M = 2`, `chi_c, chi_d in M`, and `M` is nondegenerate. +- `M` lies in `C`, and `C` lies in the caps of the receivers. +- `C` is the frame of the receivers' level-2 root, and the delivery is scheduled right after that root. +- The carrier and passive chains are nested. +- Every source lies in exactly one pair. +- The literal F2 replay restores `X`. + +The mutation controls `broken_mix`, `outside_cap` and `below_level` are rejected. diff --git a/research/paired-cube-bit/README.md b/research/paired-cube-bit/README.md new file mode 100644 index 0000000000..7b7da62fab --- /dev/null +++ b/research/paired-cube-bit/README.md @@ -0,0 +1,98 @@ +# Paired-cube bit word with partner-pair source mixing + +This package is a bit-side analog of PR #144's paired-cube decoder, for integration on upstream main +`d1d6c07`. #144 is merged there. Standard library only; run every script without `-O`. + +| p | h | v | R | R/v | loss h(h-2) | deficit 2v-3l | W=2v+R | gauges | coarse a0 (cert) | effective a_b | +|---|---|---|---|---|---|---|---|---|---|---| +| 11 | 22 | 1320 | 18964 | 14.37 | 440 | 1320 | 21604 | 4290 | 5502556/10^10 | 5.4974380e-4 | +| **12** | **24** | **1760** | **25624** | **14.56** | **528** | **1936** | **29144** | **5720 (dim 20)** | **5566382/10^10** | **5.5612002e-4** | +| 13 | 26 | 2288 | 33514 | 14.65 | 624 | 2704 | 38090 | 7436 | 5541517/10^10 | 5.5363601e-4 | + +The accounting is PR #144's bit ledger: + +- `m = 3h` and `W = 2v + R`; +- three copies of the local, source-data and target-data children; +- an exterior child of width `3 dim sigma` per gauge; +- `2v` finishing children of width 2; +- full `32 m^2` fallback. + +The effective saving is `(1 - 1e-3) a0 + 1e-3 * 384599/1e10`. For comparison: + +- PR #144's bit supplier has coarse `4617656/10^10`. +- The h=21 Fibonacci-strip bit word (PR I) has coarse `5201637/10^10`. + +So p=12 is +7.0% over the h=21 word and +20.5% over #144's bit supplier. + +## What the construction is + +The word is a mod-2 decoder on transversal triples: + + x_T = sum_{c in T} star(c) + sum_{|S cap T|=1} x_S. + +- **Ports and centres.** Coordinates are grouped into pairs, and ports are triples with one coordinate from + each of three pairs; `p = 12` gives `v = 8*C(12,3)`. The point stars over Q have span `h - 2`, so the + centre loss is `h(h - 2)`. +- **Side channels.** Disjoint cubes contribute nothing mod 2, so there is no disjoint-triple module. The + face channels are #144's pair modules, with the matching selector. The edge channels are #144's + all-but-one modules on `E00 + E11` and `E01 + E10`. +- **Within a cube.** Two of the three within-cube terms come from partner-pair sums mixed on the SOURCE + registers (`LEMMA.md`). The partner term is a single-source root. +- **Compile.** The PR #144 ledger on rational frames, with these choices: + - target cap `ker(3 chi_T - 1)`; + - coordinate-padded maximum matching, with frozen arcs; + - plain frames `span(n)` for additions of span dimension at most 8, the bit compiler's lifted-set rule; + - PR #144 chronological gauges with a G-nondegenerate sigma. + +## Run + + python3 paired_cube_bit_word.py --p 12 # regenerate out/*_p12.json (about 50 s) + python3 paired_cube_bit_word.py --p 12 --check # byte-identical regeneration of all p=12 outputs + python3 check_paired_cube_bit.py --dir out --p 12 # independent checker + 5 mutation controls (about 20 s) + python3 moment_certificate.py out/profile_p12.json --tree REPO --output out/certificate_p12.json + python3 derive_pair_modules.py --witness data/source_witness_21.json.gz --tree REPO --check # module pins + +Replace 12 with 11 or 13 for the variants (their outputs are not packaged; p = 12 is certified). + +## Status (PROVEN / COMPUTED / CONJECTURED) + +**PROVEN, written:** `LEMMA.md`, the partner-pair mixing step. It covers the frames, the sequential order +with no cross-talk, and why a full #144-type involution is impossible on the bit side. + +**COMPUTED, checked by `check_paired_cube_bit.py` with exact integer arithmetic:** + +- The 19,973 exported frames. +- The decoder identity, including the stars and `sum_{|S cap T|=1}` terms. +- Root, centre and source geometry. +- Every role chain nested, and every target chain nested. +- Phase one equals the centre closure, and the gauge target sets equal the response supports. +- The partner-pair chronology. +- G-nondegeneracy. +- A literal F2 replay with arbitrary dirty scratch. +- An independent recount that equals the saved child histogram. +- All controls rejected. + +**COMPUTED, exact rationals:** the moment certificate. The grid point passes and its successor is rejected. + +**Inherited, not re-checked here:** + +- the three-stage cover and completed-core sharing on the bit side (#130/#144 notes); +- the bit stage geometry per triple port (any triple has H0-norm 1); +- the weighted local-ring compile and fallback; +- the semantic assembly; +- the analytic and tape interfaces. + +PR97/Swapnil's checkers assume all `C(h,3)` triples and single-target outputs, so they do not apply to +this subset-port, broadcast-root word. `check_paired_cube_bit.py` replaces them. + +## Credits + +- Paired-cube decoder, modules and ledger: icekylinx PR #144, with OpenAI GPT-6 Astra and Codex assistance. +- PR117 DAG: eumemic, with Claude assistance. +- Bit-side compile conventions (deferred readouts, plain or lifted frames): Swapnil Jain / Zhihao Chen + PR97, and the h=21 bit compiler (eumemic, with Anthropic Claude assistance). +- p=11 and p=12 pair modules: cut from an h=21 Fibonacci-strip witness (eumemic, with Anthropic Claude + assistance), which descends from + ikeboy PR62's interval strips. +- Completed-core sharing: an664 PR128. +- The bit analog, partner-pair mixing, rational compiler and checker: eumemic, with Anthropic Claude assistance. diff --git a/research/paired-cube-bit/check_paired_cube_bit.py b/research/paired-cube-bit/check_paired_cube_bit.py new file mode 100644 index 0000000000..69d535a6c2 --- /dev/null +++ b/research/paired-cube-bit/check_paired_cube_bit.py @@ -0,0 +1,465 @@ +#!/usr/bin/env python3 +"""Standalone checker for subset-port bit words with broadcast roots (paired-cube bit word with partner-pair +source mixing). Standard library only; independent of the generator (own exact integer linear algebra, own +register simulation). Refuses to run under python -O. + +Inputs (plain JSON written by paired_cube_bit_word.py): graph_pP.json, word_pP.json, frames_pP.json, +kchron_pP.json, profile_pP.json. Checks: + 1. exact frames: every frame record (integer annihilator rows 'a' or integer basis rows 'b') is completed to + the other representation by exact integer elimination; A.B^T = 0 exactly and rank A + rank B = h; + 2. the mod-2 decoder: supports of all additions are disjoint; for every target the deliveries (star roots, + side roots, partner singles, partner-pair sums) XOR to {T}; stars read are exactly the c in T and the + side part is exactly sum_{|S cap T|=1} x_S; + 3. geometry: source frames are the lines ; side root frames lie in every receiver's cap + ker(3 chi_T - 1) and equal the common cap of the group; centre frames are the star spans; the frame of + every addition contains its operands' frames; every root node's frame lies in its root frame; + 4. every role chain (start -> each op frame -> root frame -> full) is nested, including gauged starts + (sigma inside the first op frame, gauged roles untouched in phase one); every target chain (gauge reads + ascending, side root caps in root order, partner-pair delivery at the level-2 cap, final cap) is nested; + the partner-pair chronology: mix frame span{chi_c,chi_d} with H0-Gram I_2, contained in the receivers' + level-2 cap, carrier chain [, mix, cap, full] and passive chain [, mix, full] nested; + 5. G = I - J/9 nondegenerate on every frame used (exact integer determinant test); + 6. literal F2 replay with arbitrary dirty scratch: y = y0 + x, scratch and sources restored; + 7. independent ledger recount from the role chains equals the saved child histogram (PR #144 bit ledger); + 8. mutation controls (must all be REJECTED): broken partner mix, delivery outside the shared cap (full + frame), delivery below the level (face cap without span{chi_c,chi_d}), dropped star, non-nested role chain. +Usage: python3 check_paired_cube_bit.py --dir out --p 12 +""" +import argparse, json, random, sys +from collections import Counter, defaultdict +from math import gcd +from operator import mul +from pathlib import Path + +if sys.flags.optimize: + raise SystemExit('refusing to run under python -O: assertions are part of the check') + + +class Fail(Exception): + pass + + +def need(cond, msg): + if not cond: + raise Fail(msg) + + +# ------------------------------------------------------------------ exact integer linear algebra +def prim(v): + g = 0 + for x in v: + g = gcd(g, abs(x)) + if g > 1: + v = [x // g for x in v] + f = next((x for x in v if x), 0) + return tuple(-x for x in v) if f < 0 else tuple(v) + + +def reduce_rows(rows, h): + """Integer reduced echelon form: returns (rows, pivot columns); each pivot column is zero outside its row.""" + rows = [list(r) for r in rows] + out, pivs = [], [] + for r in rows: + for pr, pc in zip(out, pivs): + if r[pc]: + a, b = pr[pc], r[pc] + r = [x * a - y * b for x, y in zip(r, pr)] + r = list(prim(r)) + c = next((j for j in range(h) if r[j]), None) + if c is None: + continue + for i, (pr, pc) in enumerate(zip(out, pivs)): + if pr[c]: + a, b = r[c], pr[c] + out[i] = list(prim([x * a - y * b for x, y in zip(pr, r)])) + out.append(r); pivs.append(c) + return out, pivs + + +def kernel(rows, h): + red, pivs = reduce_rows(rows, h) + ps = set(pivs) + basis = [] + for f in range(h): + if f in ps: + continue + D = 1 + for r, c in zip(red, pivs): + if r[f]: + D = D * abs(r[c]) // gcd(D, abs(r[c])) + v = [0] * h + v[f] = D + for r, c in zip(red, pivs): + if r[f]: + v[c] = -r[f] * D // r[c] + basis.append(prim(v)) + return basis, len(red) + + +def rank(rows, h): + return len(reduce_rows(rows, h)[0]) + + +def dot(a, b): + return sum(map(mul, a, b)) + + +# ------------------------------------------------------------------ checker +class Checker: + def __init__(self, d, p, mutate=None): + self.mut = mutate + rd = lambda name: json.loads((d / ('%s_p%d.json' % (name, p))).read_text()) + self.g, self.w, self.fr, self.k, self.prof = rd('graph'), rd('word'), rd('frames'), rd('kchron'), rd('profile') + self.h, self.v = self.g['h'], self.g['v'] + if mutate == 'dropped_star': + j = next(i for i, r in enumerate(self.g['roots']) if r['kind'] == 'center') + self.g['roots'][j]['targets'] = [] + if mutate == 'broken_mix': + self.k['entries'][0]['skip_mix'] = True + if mutate == 'outside_cap': + # raise the mixed carrier past the receivers' shared cap (to the full frame) and deliver there + e = self.k['entries'][0] + e['deliver_frame'] = self.w['full_frame'] + e['carrier_chain'][2] = e['deliver_frame'] + if mutate == 'below_level': + # deliver at the receivers' level-1 (face) cap, which does not contain span{chi_c, chi_d} + e = self.k['entries'][0] + face = next(j for j, r in enumerate(self.g['roots']) if r['kind'] == 'side' and set(e['receivers']) < set(r['targets'])) + e['deliver_frame'] = self.w['root_frame'][face] + e['carrier_chain'][2] = e['deliver_frame'] + if mutate == 'non_nested': + # move one addition's frame down to the frame of its smallest operand's source line + x = next(int(k) for k, f in sorted(self.w['node_frame'].items(), key=lambda kv: int(kv[0])) if int(k) >= self.v and str(int(k)) not in set(map(str, self.w['plain']))) + self.w['node_frame'][str(x)] = self.w['source_frame'][0] + + # ---------------- frames + def frames(self): + h = self.h + self.A, self.B, self.dimf = {}, {}, {} + for key, rec in self.fr['frames'].items(): + f = int(key) + if 'a' in rec: + A = [tuple(r) for r in rec['a']] + need(rank(A, h) == len(A), 'independent annihilator rows %d' % f) + B, _ = kernel(A, h) + else: + B = [tuple(r) for r in rec['b']] + need(rank(B, h) == len(B), 'independent basis rows %d' % f) + A, _ = kernel(B, h) + need(len(A) + len(B) == h and len(B) == rec['dim'], 'dimensions of frame %d' % f) + need(all(dot(a, b) == 0 for a in A for b in B), 'A.B = 0 for frame %d' % f) + self.A[f], self.B[f], self.dimf[f] = A, B, len(B) + self.cc = {} + return len(self.A) + + def sub(self, f1, f2): + """frame f1 contained in frame f2 (exact)""" + if f1 == f2: + return True + key = (f1, f2) + r = self.cc.get(key) + if r is None: + B1, A2 = self.B[f1], self.A[f2] + r = len(B1) <= len(self.B[f2]) and all(dot(a, b) == 0 for a in A2 for b in B1) + self.cc[key] = r + return r + + def in_frame(self, vec, f): + return all(dot(a, vec) == 0 for a in self.A[f]) + + def nondeg(self, f): + h = self.h + B, A = self.B[f], self.A[f] + if not B or len(B) == h: + return True + if len(B) <= len(A): + s = [sum(b) for b in B] + M = [[9 * dot(B[i], B[j]) - s[i] * s[j] for j in range(len(B))] for i in range(len(B))] + return rank(M, len(B)) == len(B) + s = [sum(a) for a in A] + M = [[(9 - h) * dot(A[i], A[j]) + s[i] * s[j] for j in range(len(A))] for i in range(len(A))] + return rank(M, len(A)) == len(A) + + # ---------------- decoder + def decoder(self): + g, v = self.g, self.v + args = g['args'] + lab = [frozenset(l) for l in g['labels']] + sup = [1 << i for i in range(v)] + [0] * (len(args) - v) + for x in range(v, len(args)): + a, b = args[x] + need(not sup[a] & sup[b], 'support-disjoint addition %d' % x) + sup[x] = sup[a] | sup[b] + self.sup = sup + acc, stars, side = [0] * v, [set() for _ in range(v)], [0] * v + for r in g['roots']: + need(r.get('coefficient', 1) == 1, 'mod-2 unit coefficients') + for t in r['targets']: + acc[t] ^= sup[r['node']] + if r['kind'] == 'center': + stars[t].add(r['coordinate']) + else: + side[t] ^= sup[r['node']] + for e in self.k['entries']: + for t in e['receivers']: + acc[t] ^= (1 << e['carrier']) | (1 << e['passive']) + side[t] ^= (1 << e['carrier']) | (1 << e['passive']) + for t in range(v): + need(acc[t] == 1 << t, 'decoder identity at target %d' % t) + need(stars[t] == set(lab[t]), 'stars read by target %d' % t) + orth = sum(1 << s for s in range(v) if len(lab[s] & lab[t]) == 1) + need(side[t] == orth, 'side part = sum over |S cap T| = 1 at target %d' % t) + for r in g['roots']: + if r['kind'] == 'center': + c = r['coordinate'] + need(sup[r['node']] == sum(1 << s for s in range(v) if c in lab[s]), 'star support %d' % c) + + # ---------------- geometry and chains + def geometry(self): + g, w, h, v = self.g, self.w, self.h, self.v + lab = g['labels'] + chi = [tuple(1 if j in l else 0 for j in range(h)) for l in lab] + cov = [tuple((3 if j in l else 0) - 1 for j in range(h)) for l in lab] + nf = {int(k): f for k, f in w['node_frame'].items()} + rf = w['root_frame'] + full = w['full_frame'] + need(self.dimf[full] == h, 'full frame') + for s in range(v): + f = w['source_frame'][s] + need(self.dimf[f] == 1 and self.in_frame(chi[s], f), 'source line ') + args = g['args'] + for x, f in nf.items(): + if x < v: + need(self.in_frame(chi[x], f), 'leaf label inside its frame') + else: + for y in args[x]: + need(self.sub(nf[y], f), 'operand frame inside node frame (%d <- %d)' % (x, y)) + loss = 0 + for j, r in enumerate(g['roots']): + f = rf[j] + need(self.sub(nf[r['node']], f), 'root node frame inside root frame %d' % j) + if r['kind'] == 'center': + c = r['coordinate'] + S = [chi[s] for s in range(v) if c in lab[s]] + need(all(self.in_frame(x, f) for x in S) and rank(S, h) == self.dimf[f], 'centre frame = star span') + loss += self.dimf[f] + else: + need(all(dot(cov[t], b) == 0 for t in r['targets'] for b in self.B[f]), 'root frame inside caps') + need(self.dimf[f] == h - rank([cov[t] for t in r['targets']], h), 'root frame = common cap') + need(loss == self.prof['loss'], 'copied-centre loss') + self.chi, self.cov, self.nf, self.rf = chi, cov, nf, rf + + def chains(self): + """Role chains of the physical word and target chains; returns the recounted child histogram.""" + g, w, h, v = self.g, self.w, self.h, self.v + nf, rf, full = self.nf, self.rf, w['full_frame'] + ops, rootroles = w['ops'], w['rootroles'] + R = 1 + max(max(a, b) for a, b, _ in ops) + R = max(R, 1 + max(rootroles), 1 + max(w['sources'].values())) + need(R == self.prof['R'], 'role count') + start = {} + for leaf, s in w['sources'].items(): + start[s] = w['source_frame'][int(leaf)] + phase1 = set(w['phase1']) + touched_p1 = set() + for i in phase1: + touched_p1.update(ops[i][:2]) + # phase one = closure of the centre roots' last ops under per-role precedence (recomputed) + prev_op, pred = {}, [] + for i, (a, b, _) in enumerate(ops): + pred.append((prev_op.get(a, -1), prev_op.get(b, -1))) + prev_op[a] = prev_op[b] = i + stack = [prev_op[s] for r, s in zip(g['roots'], rootroles) if r['kind'] == 'center' and s in prev_op] + closure = set() + while stack: + i = stack.pop() + if i not in closure: + closure.add(i); stack.extend(j for j in pred[i] if j >= 0) + need(closure == phase1, 'phase one is the centre closure') + # conservative response support of every role (union incidence through the XOR word) + co = [0] * R + for j, (r, s) in enumerate(zip(g['roots'], rootroles)): + for t in r['targets']: + co[s] |= 1 << t + for a, b, _ in reversed(ops): + co[b] |= co[a] + gauge_of = {} + for z in w['gauges']: + need(sum(1 << t for t in z['targets']) == co[z['role']], 'gauge targets = response support of its role') + s = z['role'] + need(s not in start, 'gauged role is not a source role') + need(s not in touched_p1, 'gauged role untouched in phase one') + need(self.dimf[z['frame']] == z['dim'] and z['dim'] > 0, 'gauge dimension') + need(self.nondeg(z['frame']), 'gauge frame nondegenerate') + start[s] = z['frame']; gauge_of[s] = z + seq = defaultdict(list) + for i, (a, b, x) in enumerate(ops): + f = nf[x] + seq[a].append(f); seq[b].append(f) + for j, s in enumerate(rootroles): + seq[s].append(rf[j]) + H = Counter() + for s in range(R): + chain = [start.get(s)] + seq[s] + [full] + prev = chain[0] + dprev = 0 if prev is None else self.dimf[prev] + if prev is not None and s not in gauge_of: + H[1] += 1 # source role: injection frame (entrance child of width 1) + for f in chain[1:]: + if prev is not None: + need(self.sub(prev, f), 'nested role chain (role %d)' % s) + d = self.dimf[f] + need(d >= dprev, 'ascending role chain (role %d)' % s) + if d > dprev: + H[d - dprev] += 1 + prev, dprev = f, d + for j, r in enumerate(g['roots']): + if r['kind'] == 'center': + H[self.dimf[rf[j]]] += 1 # copied-centre transform + # partner-pair chronology + src = Counter() + lab = [set(l) for l in g['labels']] + for e in self.k['entries']: + c, d = e['carrier'], e['passive'] + need(len(lab[c] & lab[d]) == 1, 'carrier and passive orthogonal (H0-Gram I_2)') + mf, cf = e['mix_frame'], e['deliver_frame'] + need(self.dimf[mf] == 2 and self.in_frame(self.chi[c], mf) and self.in_frame(self.chi[d], mf), 'mix frame = span{chi_c, chi_d}') + need(self.nondeg(mf), 'mix frame nondegenerate') + need(self.sub(mf, cf), 'mix frame inside the receivers shared cap') + need(all(dot(self.cov[t], b) == 0 for t in e['receivers'] for b in self.B[cf]), 'delivery frame inside receiver caps') + need(e['carrier_chain'][0] == w['source_frame'][c] and e['passive_chain'][0] == w['source_frame'][d], 'chains start at source lines') + need(e['carrier_chain'][1:] == [mf, cf, w['full_frame']] and e['passive_chain'][1:] == [mf, w['full_frame']], 'chain shape') + for ch in (e['carrier_chain'], e['passive_chain']): + for f1, f2 in zip(ch, ch[1:]): + need(self.sub(f1, f2), 'nested source-register chain') + for f1, f2 in zip(ch, ch[1:]): + src[self.dimf[f2] - self.dimf[f1]] += 1 + # target chains: gauge reads (time order = reverse selection order), side caps in root order, + # partner-pair delivery at the level-2 cap right after its root, final cap + events = defaultdict(list) + for z in reversed(w['gauges']): + for t in z['targets']: + events[t].append(z['frame']) + deliveries = defaultdict(list) + for e in self.k['entries']: + deliveries[e['deliver_after_root']].append(e) + for j, r in enumerate(g['roots']): + if r['kind'] == 'center': + continue + for t in r['targets']: + events[t].append(rf[j]) + for e in deliveries.get(j, []): + need(sorted(e['receivers']) == sorted(r['targets']), 'delivery after the receivers own level-2 root') + for t in e['receivers']: + events[t].append(e['deliver_frame']) + Y = Counter() + cap = {} + for t in range(v): + capf = None + prev, dprev = None, 0 + for f in events[t]: + if prev is not None: + need(self.sub(prev, f), 'nested target chain (target %d)' % t) + need(all(dot(self.cov[t], b) == 0 for b in self.B[f]), 'target frame inside cap (target %d)' % t) + if self.dimf[f] > dprev: + Y[self.dimf[f] - dprev] += 1 + prev, dprev = f, self.dimf[f] + need(dprev <= h - 1, 'target chain below the cap') + Y[h - 1 - dprev] += 1 if h - 1 - dprev else 0 + # every source register is in exactly one mixing pair + members = Counter() + for e in self.k['entries']: + members[e['carrier']] += 1; members[e['passive']] += 1 + need(all(members[s] == 1 for s in range(v)), 'each source in exactly one partner pair') + for f in set(self.nf.values()) | set(self.rf): + need(self.nondeg(f), 'G-nondegenerate frame %d' % f) + return H, Y, src, R + + # ---------------- replay + def replay(self, seeds=3): + g, w, v = self.g, self.w, self.v + ops, rootroles = w['ops'], w['rootroles'] + R = self.prof['R'] + srcmap = {int(k): s for k, s in w['sources'].items()} + roots = g['roots'] + for seed in range(seeds): + rng = random.Random(1000 + seed) + x = [rng.getrandbits(1) for _ in range(v)] + y0 = [rng.getrandbits(1) for _ in range(v)] + z0 = [rng.getrandbits(1) for _ in range(R)] + X, Yv, z = x[:], y0[:], z0[:] + # y <- y - J M z (old-value response of the dirty scratch) + zd = z0[:] + for a, b, _ in ops: + zd[a] ^= zd[b] + for j, r in enumerate(roots): + for t in r['targets']: + Yv[t] ^= zd[rootroles[j]] + for leaf, s in srcmap.items(): # z <- z + V x (at the rank-one source lines) + z[s] ^= X[leaf] + for a, b, _ in ops: # z <- M z + z[a] ^= z[b] + for j, r in enumerate(roots): # y <- y + J z + for t in r['targets']: + Yv[t] ^= z[rootroles[j]] + for e in self.k['entries']: # partner-pair sums on the source registers + if not e.get('skip_mix'): + X[e['carrier']] ^= X[e['passive']] + for t in e['receivers']: + Yv[t] ^= X[e['carrier']] + for e in self.k['entries']: # undo at the full frame + if not e.get('skip_mix'): + X[e['carrier']] ^= X[e['passive']] + for a, b, _ in reversed(ops): # z <- M^-1 z + z[a] ^= z[b] + for leaf, s in srcmap.items(): # z <- z - V x + z[s] ^= X[leaf] + need(all(Yv[t] == y0[t] ^ x[t] for t in range(v)), 'replay: y = y0 + x (seed %d)' % seed) + need(z == z0 and X == x, 'replay: scratch and sources restored (seed %d)' % seed) + + def run(self): + nfr = self.frames() + self.decoder() + self.geometry() + H, Y, src, R = self.chains() + self.replay() + h, v = self.h, self.v + m, W = 3 * h, 2 * v + R + C = Counter() + for part in (H, Y, src): + for r, c in part.items(): + if r and c: + C[r] += 3 * c + for z in self.w['gauges']: + C[3 * z['dim']] += 1 + C[2] += 2 * v + saved = {int(k): c for k, c in self.prof['child_histogram'].items()} + need(dict(C) == saved, 'independent recount equals the saved child histogram') + mass = sum(r * c for r, c in C.items()) + need(W == self.prof['W_per_vertex'] and W * m - mass == 2 * v - 3 * self.prof['loss'] == self.prof['deficit_per_vertex'], + 'W = 2v + R and deficit 2v - 3 ell') + return dict(frames=nfr, roles=R, W=W, m=m, deficit=W * m - mass, children=sum(C.values())) + + +def main(): + ap = argparse.ArgumentParser(description=__doc__) + ap.add_argument('--dir', type=Path, default=Path(__file__).resolve().parent / 'out') + ap.add_argument('--p', type=int, default=12) + ap.add_argument('--no-controls', action='store_true') + a = ap.parse_args() + res = Checker(a.dir, a.p).run() + print('PASS p=%d: %s' % (a.p, res)) + if a.no_controls: + return + for m in ('broken_mix', 'outside_cap', 'below_level', 'dropped_star', 'non_nested'): + try: + Checker(a.dir, a.p, mutate=m).run() + except Fail as e: + print('control %-13s REJECTED: %s' % (m, e)) + continue + raise SystemExit('FAIL: control %s was accepted' % m) + print('PASS: all 5 mutation controls rejected') + + +if __name__ == '__main__': + main() diff --git a/research/paired-cube-bit/data/arcs_p11.json b/research/paired-cube-bit/data/arcs_p11.json new file mode 100644 index 0000000000..4ce0080071 --- /dev/null +++ b/research/paired-cube-bit/data/arcs_p11.json @@ -0,0 +1 @@ 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0000000000..e9aae9b923 Binary files /dev/null and b/research/paired-cube-bit/data/source_witness_21.json.gz differ diff --git a/research/paired-cube-bit/derive_pair_modules.py b/research/paired-cube-bit/derive_pair_modules.py new file mode 100644 index 0000000000..8a37c88d0e --- /dev/null +++ b/research/paired-cube-bit/derive_pair_modules.py @@ -0,0 +1,122 @@ +#!/usr/bin/env python3 +"""Derive the pinned pair-disjoint modules (data/pair_module_p{11,12,13}.json). + +A pair-disjoint module on n points has inputs x_{ab} (pairs of [n]) and, for every pair {i,j}, the output +sum of x_{ab} over pairs disjoint from {i,j}. For fixed c, the bit outputs out(c,T) = sum_{S cap T = {c}} x_S +of a bit producer ARE such a query on the points other than c, so an exact zero restriction of a bit witness +to the triples {c,a,b} with a,b in a chosen n-set Z gives a module (support hash-consing, unary additions +contracted, inactive nodes pruned), exactly as PR #144's modules.restricted_pairs does for the PR117 DAG. + +p=11, p=12: cut from the h=21 Fibonacci-strip bit witness (p=12: pair_module_p12_cut.json, the seed of the + annealed pair_module_p12.json) (sha256 pinned below), c=14, + Z = 10 resp. 11 consecutive points (the minimum-addition cut, first in (c, cyclic start) order). +p=13: PR #144 modules.restricted_pairs(12) (PR117 DAG, pinned in the repository). +Usage: python3 derive_pair_modules.py --witness PATH/witness_21.json.gz --tree REPO_ROOT [--check] +""" +import argparse, gzip, hashlib, json, sys +from itertools import combinations +from pathlib import Path +HERE = Path(__file__).resolve().parent +WITNESS_SHA256 = None # filled by first run, then pinned in data/pair_module_provenance.json + + +def cut(w, c, Z): + Z = list(Z); n = len(Z); pos = {z: i for i, z in enumerate(Z)} + pairs = list(combinations(range(n), 2)); pid = {pp: i for i, pp in enumerate(pairs)} + args_w = {int(k): v for k, v in w['args'].items()} + leaf = {int(k): v for k, v in w['leaf'].items()} + args = [None] * len(pairs); support = [1 << i for i in range(len(pairs))]; by = {s: i for i, s in enumerate(support)} + image = {} + for x in sorted(args_w): + a = args_w[x] + if a is None: + T = leaf[x] + o = [t for t in T if t != c] + image[x] = pid[tuple(sorted(pos[t] for t in o))] if c in T and all(t in pos for t in o) else None + continue + ia, ib = image.get(a[0]), image.get(a[1]) + if ia is None or ib is None: + image[x] = ia if ib is None else ib + continue + assert not support[ia] & support[ib] + s = support[ia] | support[ib] + if s not in by: + by[s] = len(args); args.append([ia, ib]); support.append(s) + image[x] = by[s] + outs = {} + for cc, T, node in w['outputs']: + o = [t for t in T if t != c] + if cc == c and all(t in pos for t in o): + outs[tuple(sorted(pos[t] for t in o))] = image[node] + roots = [] + for (i, j) in pairs: + r = outs[(i, j)] + expected = sum(1 << k for k, (a, b) in enumerate(pairs) if i not in (a, b) and j not in (a, b)) + assert r is not None and support[r] == expected + roots.append(r) + active, stack = set(range(len(pairs))), list(roots) + while stack: + x = stack.pop() + if x not in active: + active.add(x) + if args[x] is not None: + stack.extend(args[x]) + ids = sorted(active); ren = {x: i for i, x in enumerate(ids)} + return dict(kind='pair_disjoint', n=n, input_count=len(pairs), input_labels=[list(p_) for p_ in pairs], + args=[None if args[x] is None else [ren[y] for y in args[x]] for x in ids], roots=[ren[x] for x in roots]) + + +def best_cut(w, n): + best = None + for c in range(w['h']): + others = [t for t in range(w['h']) if t != c] + for start in range(len(others)): + Z = sorted(others[(start + k) % len(others)] for k in range(n)) + m = cut(w, c, Z) + adds = len(m['args']) - m['input_count'] + if best is None or adds < best[0]: + best = (adds, c, Z, m) + return best + + +def dumps(o): + return json.dumps(o, sort_keys=True, separators=(',', ':')) + '\n' + + +def main(): + ap = argparse.ArgumentParser(description=__doc__) + ap.add_argument('--witness', type=Path, required=True) + ap.add_argument('--tree', type=Path, required=True) + ap.add_argument('--check', action='store_true') + a = ap.parse_args() + raw = a.witness.read_bytes() + wsha = hashlib.sha256(raw).hexdigest() + w = json.loads(gzip.decompress(raw)) + sys.path.insert(0, str(a.tree / 'scripts')) + from paired_cube.modules import restricted_pairs + dag = a.tree / 'references/three-stage-cover/pr117/dag.json.gz' + files = {} + prov = dict(witness_sha256=wsha, witness='h=21 Fibonacci-strip bit witness (data/source_witness_21.json.gz)', + pr117_dag_sha256=hashlib.sha256(dag.read_bytes()).hexdigest()) + for p in (11, 12): + adds, c, Z, m = best_cut(w, p - 1) + m['provenance'] = dict(source='bit witness cut', witness_sha256=wsha, c=c, Z=Z, additions=adds) + files['pair_module_p%d.json' % p if p == 11 else 'pair_module_p12_cut.json'] = dumps(m) + m = restricted_pairs(12) + m = dict(kind='pair_disjoint', n=12, input_count=m['input_count'], input_labels=[list(x) for x in m['input_labels']], + args=m['args'], roots=m['roots'], + provenance=dict(source='PR144 modules.restricted_pairs(12) of the PR117 DAG', pr117_dag_sha256=prov['pr117_dag_sha256'], + additions=len(m['args']) - m['input_count'])) + files['pair_module_p13.json'] = dumps(m) + files['pair_module_provenance.json'] = dumps(prov) + for name, text in files.items(): + path = HERE / 'data' / name + if a.check: + assert path.read_text() == text, name + else: + path.write_text(text) + print(('PASS' if a.check else 'wrote'), {k: len(v) for k, v in files.items()}) + + +if __name__ == '__main__': + main() diff --git a/research/paired-cube-bit/moment_certificate.py b/research/paired-cube-bit/moment_certificate.py new file mode 100644 index 0000000000..7a80fe24f1 --- /dev/null +++ b/research/paired-cube-bit/moment_certificate.py @@ -0,0 +1,130 @@ +#!/usr/bin/env python3 +"""Exact rational moment certificate for the paired-cube bit word under PR #144's bit accounting. + +Accounting (PR #144 notes/paired-cube-bit.tex and scripts/paired_cube_network.py, bit_certificate): + m = 3h, W = 2v + R per cover vertex; children = 3 x (local internal + source data + target data) + one + exterior child of width 3*dim(sigma) per selected gauge + 2v final children of width 2 (read from the profile). + Coarse saving a0 = k/10^10 passes iff, with rational upper bounds log_upper / exp_upper (32-term atanh series + with geometric tail; exp(u) <= 1+u+u^2/(2(1-u/3)); both rounded up to the grid 2^-120), + 1 - sum_r n_r r/(W m) * exp_upper(a0 * log_upper(m/r)) - added > 0, + added = 10^-16 * (32 m^2 * #children)/(W m) * exp_upper(a0 * log_upper(m)) (full local-ring fallback), + and the rank moment also contracts. Effective bit saving a_b = (1 - 10^-3) a0 + 10^-3 * 384599/10^10. +The certificate reports the largest k on the 10^-10 grid that passes and checks that k+1 is rejected. +log_upper / exp_upper / up are copied verbatim from PR #144 scripts/three_stage_cover_network.py (icekylinx, +Apache-2.0); with --tree REPO the copies are checked against the repository functions. +Usage: python3 moment_certificate.py out/profile_p12.json [--tree REPO] [--output cert.json] +""" +import argparse, json, sys +from fractions import Fraction as Q +from pathlib import Path + +GRID = 1 << 120 +BAD = Q(1, 10 ** 16) +ATOM = Q(1, 1000) +OLD = Q(384599, 10 ** 10) +STEP = 10 ** 10 + + +def require(c, msg): + if not c: + raise SystemExit('FAIL: ' + msg) + + +def up(x): + return Q((x.numerator * GRID + x.denominator - 1) // x.denominator, GRID) + + +def log_upper(x): + power = 0 + while x >= 2: + x /= 2 + power += 1 + + def small(y): + z = (y - 1) / (y + 1) + return 2 * sum((z ** (2 * j + 1) / (2 * j + 1) for j in range(32)), Q(0)) + 2 * z ** 65 / (65 * (1 - z * z)) + return up(power * small(Q(2)) + small(x)) + + +def exp_upper(u): + require(0 <= u < 3, 'Exponential enclosure') + return up(1 + u + u * u / (2 * (1 - u / 3))) + + +def test(profile, k, logs, logm): + m, W = profile['m'], profile['W_per_vertex'] + H = {int(r): n for r, n in profile['child_histogram'].items()} + a0 = Q(k, STEP) + upper = sum(Q(n * r, W * m) * exp_upper(a0 * logs[r]) for r, n in H.items()) + edges = sum(H.values()) + fallback = 32 * m * m + added = BAD * Q(fallback * edges, W * m) * exp_upper(a0 * logm) + rank_upper = Q(sum(r * n for r, n in H.items())) + BAD * fallback * edges + ok = 1 - upper - added > 0 and rank_upper < W * m + return ok, dict(coarse=a0, moment_upper=upper, added_bad_moment_upper=added, strict_gap=1 - upper - added, + rank_mass_upper=rank_upper, fallback_children_per_edge=fallback, edge_count=edges) + + +def certify(profile): + h, v, R, ell = (profile[k] for k in ('h', 'v', 'R', 'loss')) + m, W = profile['m'], profile['W_per_vertex'] + require(m == 3 * h and W == 2 * v + R, 'PR144 bit accounting m=3h, W=2v+R') + H = {int(r): n for r, n in profile['child_histogram'].items()} + require(all(0 < r < m and n > 0 for r, n in H.items()), 'proper children') + mass = sum(r * n for r, n in H.items()) + require(mass == profile['rank_per_vertex'] and W * m - mass == 2 * v - 3 * ell == profile['deficit_per_vertex'], + 'telescoping deficit 2v - 3 ell') + logs = {r: log_upper(Q(m, r)) for r in H} + logm = log_upper(Q(m)) + lo, hi = 0, 10 ** 7 # a0 in [0, 1e-3) + require(test(profile, lo, logs, logm)[0], 'zero saving passes') + require(not test(profile, hi, logs, logm)[0], 'upper bracket rejected') + while hi - lo > 1: + mid = (lo + hi) // 2 + if test(profile, mid, logs, logm)[0]: + lo = mid + else: + hi = mid + ok, best = test(profile, lo, logs, logm) + nok, nxt = test(profile, lo + 1, logs, logm) + require(ok and not nok, 'largest grid point and rejected successor') + a0 = Q(lo, STEP) + ab = (1 - ATOM) * a0 + ATOM * OLD + require(ATOM > ab and ATOM < 1 - ab, 'subordinate adapter and row tolls') + require(Q(2 * m ** 3, 2 ** 80) < BAD, 'fixed prime bad-class allowance') + s = lambda q: '%d/%d' % (q.numerator, q.denominator) + return dict(h=h, v=v, R=R, loss=ell, m=m, W_per_vertex=W, rank_per_vertex=mass, deficit_per_vertex=W * m - mass, + coarse_saving=s(a0), coarse_saving_float=float(a0), strict_gap=s(best['strict_gap']), + strict_gap_float=float(best['strict_gap']), moment_upper=s(best['moment_upper']), + added_bad_moment_upper=s(best['added_bad_moment_upper']), rank_mass_upper=s(best['rank_mass_upper']), + next_grid_point=s(Q(lo + 1, STEP)), next_grid_gap_float=float(nxt['strict_gap']), next_grid_rejected=True, + atom_exponent=s(ATOM), ordinary_leaf_saving=s(OLD), effective_bit_saving=s(ab), effective_bit_saving_float=float(ab), + bad_fraction=s(BAD), fallback_children_per_edge=best['fallback_children_per_edge'], edge_count=best['edge_count'], + logarithm_upper_bounds={str(r): s(logs[r]) for r in sorted(logs)}, rounding_denominator=GRID, + method='PR144 bit_certificate: exact rational upper moment with full 32m^2 fallback on 1e-16 of every edge') + + +def main(): + require(not sys.flags.optimize, 'run without -O') + ap = argparse.ArgumentParser(description=__doc__) + ap.add_argument('profile', type=Path) + ap.add_argument('--tree', type=Path) + ap.add_argument('--output', type=Path) + a = ap.parse_args() + if a.tree: + sys.path.insert(0, str(a.tree / 'scripts')) + import three_stage_cover_network as t + for x in (Q(72, 5), Q(72), Q(66, 20), Q(78, 7)): + require(t.log_upper(x) == log_upper(x), 'log_upper equals repository function') + for u in (Q(1, 1000), Q(123, 10 ** 5)): + require(t.exp_upper(u) == exp_upper(u), 'exp_upper equals repository function') + cert = certify(json.loads(a.profile.read_text())) + text = json.dumps(cert, indent=1, sort_keys=True) + '\n' + if a.output: + a.output.write_text(text) + print('PASS coarse a0 = %s (%.10e), next grid point rejected; effective a_b = %s (%.10e)' % ( + cert['coarse_saving'], cert['coarse_saving_float'], cert['effective_bit_saving'], cert['effective_bit_saving_float'])) + + +if __name__ == '__main__': + main() diff --git a/research/paired-cube-bit/out/certificate_p12.json b/research/paired-cube-bit/out/certificate_p12.json new file mode 100644 index 0000000000..fe9eac40b5 --- /dev/null +++ b/research/paired-cube-bit/out/certificate_p12.json @@ -0,0 +1,54 @@ +{ + "R": 23368, + "W_per_vertex": 26888, + "added_bad_moment_upper": "479420651296216381267426376991805682565303/174513097415355556594909981625087426560000000000000000", + "atom_exponent": "1/1000", + "bad_fraction": "1/10000000000000000", + "coarse_saving": "5936323/10000000000", + "coarse_saving_float": 0.0005936323, + "deficit_per_vertex": 1936, + "edge_count": 319788, + "effective_bit_saving": "1482692819/2500000000000", + "effective_bit_saving_float": 0.0005930771276, + "fallback_children_per_edge": 165888, + "h": 24, + "logarithm_upper_bounds": { + "1": "5684664334020965693831331240677363777/1329227995784915872903807060280344576", + "10": "1312001882868133107061488896308511211/664613997892457936451903530140172288", + "11": "2497314806457063515100369249596123659/1329227995784915872903807060280344576", + "12": "2381656848210652194302493466502442645/1329227995784915872903807060280344576", + "13": "1137630920156279732341064381896544223/664613997892457936451903530140172288", + "14": "272094431126872922090605886446602277/166153499473114484112975882535043072", + "15": "130315512045308539392073308841221045/83076749736557242056487941267521536", + "16": "499815445905580118231488502496862473/332306998946228968225951765070086144", + "17": "479669459767510575280471038897699499/332306998946228968225951765070086144", + "18": "1842701275199322610452688615344956943/1329227995784915872903807060280344576", + "19": "885416805516287189171116420374863439/664613997892457936451903530140172288", + "2": "2381656848210652194302493466502442653/664613997892457936451903530140172288", + "20": "1702653128136604908896633484944543951/1329227995784915872903807060280344576", + "21": "818899938001826896437521120207666249/664613997892457936451903530140172288", + "22": "1575964168857402209874024941923645187/1329227995784915872903807060280344576", + "3": "4224358123409974804755182081847399587/1329227995784915872903807060280344576", + "4": "3841963058821643083378642625332406835/1329227995784915872903807060280344576", + "5": "1772677201667963759674661050144750447/664613997892457936451903530140172288", + "6": "825751871452578374882209443543730279/332306998946228968225951765070086144", + "60": "121173458762807009910242163057289889/664613997892457936451903530140172288", + "7": "3098106086614644681951191399245296687/1329227995784915872903807060280344576", + "8": "730153105305495444538074579414982091/332306998946228968225951765070086144", + "9": "1382025956399491957839516461508717707/664613997892457936451903530140172288" + }, + "loss": 528, + "m": 72, + "method": "PR144 bit_certificate: exact rational upper moment with full 32m^2 fallback on 1e-16 of every edge", + "moment_upper": "214441694089113318708267688768534769491421/214441694103988907943825385420907429756928", + "next_grid_gap_float": -1.019629330445217e-10, + "next_grid_point": "1484081/2500000000", + "next_grid_rejected": true, + "ordinary_leaf_saving": "384599/10000000000", + "rank_mass_upper": "2360839843756475707/1220703125000", + "rank_per_vertex": 1934000, + "rounding_denominator": 1329227995784915872903807060280344576, + "strict_gap": "34879094578234633698904410970360032171054091/523539292246066669784729944875262279680000000000000000", + "strict_gap_float": 6.662173230321985e-11, + "v": 1760 +} diff --git a/research/paired-cube-bit/out/frames_p12.json b/research/paired-cube-bit/out/frames_p12.json new file mode 100644 index 0000000000..514ece3e03 --- /dev/null +++ b/research/paired-cube-bit/out/frames_p12.json @@ -0,0 +1 @@ +{"P_used_for_ranks":"2^127-1","conventions":{"frame_ids":"frame id = id of the canonical annihilator; record \"a\" lists integer annihilator rows (frame = common kernel), record \"b\" lists integer basis rows; id 0 (empty \"a\") is the full space Q^h","gram":"G = I - J/9 on Q^h (H0 = G/2, H0(chi_S,chi_T) = (|S cap T|-1)/2); nondegeneracy tested for G","nodes":"zero-based graph nodes; nodes < v are the source leaves (port index)","ops":"[dest, control, node]: dest ^= control at the frame of node (copies: dest is the fresh role)","target_covector":"3*chi_T - 1 (target cap = {u : (3 chi_T - 1).u = 0} = H0-orthogonal complement of 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diff --git a/research/paired-cube-bit/out/graph_p12.json b/research/paired-cube-bit/out/graph_p12.json new file mode 100644 index 0000000000..9711c4ffe4 --- /dev/null +++ b/research/paired-cube-bit/out/graph_p12.json @@ -0,0 +1 @@ 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2: Y_T += sum of root nodes whose targets contain T (star roots: T reads star(c) for c in T; side roots; partner singles) + x_carrier + x_passive for each partner_mix entry receiving T; equals x_T = sum_{c in T} star(c) + sum_{|S cap T|=1} 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diff --git a/research/paired-cube-bit/out/kchron_p12.json b/research/paired-cube-bit/out/kchron_p12.json new file mode 100644 index 0000000000..aa31a1974b --- /dev/null +++ b/research/paired-cube-bit/out/kchron_p12.json @@ -0,0 +1 @@ +{"conventions":{"frame_ids":"frame id = id of the canonical annihilator; record \"a\" lists integer annihilator rows (frame = common kernel), record \"b\" lists integer basis rows; id 0 (empty \"a\") is the full space Q^h","gram":"G = I - J/9 on Q^h (H0 = G/2, H0(chi_S,chi_T) = (|S cap T|-1)/2); nondegeneracy tested for G","nodes":"zero-based graph nodes; nodes < v are the source leaves (port index)","ops":"[dest, control, node]: dest ^= control at the frame of node (copies: dest is the fresh role)","target_covector":"3*chi_T - 1 (target cap = {u : (3 chi_T - 1).u = 0} = H0-orthogonal complement of 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reads of non-gauged roles (frame 0); V injections at ; ops in order (phase1 ops are those listed); centre copies read at 0; gauged old-value reads at their gauge frames (ascending per target); side root reads in root order; partial-K deliveries right after root deliver_after_root; undo at full frame; inverse ops; V 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diff --git a/research/paired-cube-bit/out/word_p12.json.gz b/research/paired-cube-bit/out/word_p12.json.gz new file mode 100644 index 0000000000..5b3ccb8d7c Binary files /dev/null and b/research/paired-cube-bit/out/word_p12.json.gz differ diff --git a/research/paired-cube-bit/paired_cube_bit_word.py b/research/paired-cube-bit/paired_cube_bit_word.py new file mode 100644 index 0000000000..bf2848f0f3 --- /dev/null +++ b/research/paired-cube-bit/paired_cube_bit_word.py @@ -0,0 +1,861 @@ +#!/usr/bin/env python3 +"""Paired-cube BIT local word with partner-pair source mixing: deterministic generator and rank compiler. + +Ports are the triples of [h] (h = 2p) taking one coordinate from each of three distinct coordinate pairs +(v = 8*C(p,3)); bit labels chi_S in Q^h with H0 = (I - J/9)/2, so H0(chi_S, chi_T) = (|S cap T| - 1)/2. +Payload F2. Mod-2 decoder, for every target T: + x_T = sum_{c in T} star(c) (2p copied point-star centres, frames of dim h-2) + + face channels P'(J,i,eta_i) (cubes sharing exactly pair i, matching selector) + + edge channels Q'(J,i,j,mode) (cubes sharing pairs i,j, exactly one selector match) + + partner x_{T'}, T' = T with the selectors of cube pairs 1,2 flipped (single-source root) + + partner-pair sum of the opposite pair of T's parity class, mixed on two SOURCE registers and + delivered at the shared cap of {T, T'} (no auxiliary role). +Compiler: the PR #144 frames.py/gauges.py ledger on rational frames (target cap ker(3 chi_T - 1), centre frame = +span of the star), coordinate-padded maximum carrier matching (frozen arcs), plain frames span(n) for additions of +span dimension <= PLAIN (closed under operands; a linked donor's span must lie in the plain target), PR #144 +chronological partial gauges with a G-nondegeneracy filter, PR #144 bit ledger (m = 3h, W = 2v + R). +Linear algebra is exact: ranks are computed modulo the Mersenne prime 2^127-1, which exceeds the Hadamard bound +of every minor that occurs (see README.md); G-nondegeneracy is certified by integer Gram determinants. + +Usage: python3 paired_cube_bit_word.py --p 12 [--out DIR] [--check] [--solve-matching] +Standard library only. Run without -O. +""" +import argparse, gzip, hashlib, json, math, os, sys, time +from collections import Counter, defaultdict, deque +from itertools import combinations, product +from pathlib import Path + +HERE = Path(__file__).resolve().parent +P = (1 << 127) - 1 +PLAIN = 8 +MODULES = {11: 'pair_module_p11.json', 12: 'pair_module_p12.json', 13: 'pair_module_p13.json'} + + +def require(cond, msg): + if not cond: + raise SystemExit('FAIL: ' + msg) + + +# ----------------------------------------------------------------------------- exact subspaces of F_P^h +def rref(rows, h): + piv = {} + for r in rows: + x = [a % P for a in r] + for c, y in piv.items(): + if x[c]: + f = x[c] + x = [(a - f * b) % P for a, b in zip(x, y)] + nz = next((j for j in range(h) if x[j]), None) + if nz is None: + continue + inv = pow(x[nz], P - 2, P) + x = [(a * inv) % P for a in x] + for c in list(piv): + y = piv[c] + if y[nz]: + f = y[nz] + piv[c] = [(a - f * b) % P for a, b in zip(y, x)] + piv[nz] = x + return tuple(tuple(piv[c]) for c in sorted(piv)) + + +def kernel(S, h): + pivs = [next(j for j in range(h) if row[j]) for row in S] + ps = set(pivs) + out = [] + for f in range(h): + if f in ps: + continue + u = [0] * h + u[f] = 1 + for row, c in zip(S, pivs): + u[c] = (-row[f]) % P + out.append(u) + return rref(out, h) + + +class Table: + """Interned subspaces (canonical RREF) with memoized joins; id 0 is the zero space.""" + def __init__(self, h): + self.h = h + self.rows = [()] + self.index = {(): 0} + self.jc = {} + self.pc = {} + + def intern(self, rows): + S = rref(rows, self.h) + i = self.index.get(S) + if i is None: + i = len(self.rows) + self.rows.append(S) + self.index[S] = i + return i + + def dim(self, i): + return len(self.rows[i]) + + def join(self, i, j): + if i == j or j == 0: + return i + if i == 0: + return j + if i > j: + i, j = j, i + k = self.jc.get((i, j)) + if k is None: + A, B = self.rows[i], self.rows[j] + k = i if len(A) == self.h else j if len(B) == self.h else self.intern(A + B) + self.jc[(i, j)] = k + return k + + def join_many(self, ids): + k = 0 + for i in ids: + k = self.join(k, i) + return k + + def contained(self, i, j): + return self.join(i, j) == j + + def perp(self, i): + k = self.pc.get(i) + if k is None: + k = self.intern(kernel(self.rows[i], self.h)) + self.pc[i] = k + return k + + +class Geometry: + """Bit geometry: source line ; target cap annihilated by 3 chi_T - 1 (the H0 = (I-J/9)/2 cap).""" + def __init__(self, h, labels): + self.h = h + self.T = Table(h) + self.src = [self.T.intern([[1 if j in lab else 0 for j in range(h)]]) for lab in labels] + self.tcov = [self.T.intern([[(3 if j in lab else 0) - 1 for j in range(h)]]) for lab in labels] + self.coord = [self.T.intern([[1 if k == j else 0 for k in range(h)]]) for j in range(h)] + + def join(self, a, b): return self.T.join(a, b) + def join_many(self, ids): return self.T.join_many(ids) + def dim(self, a): return self.T.dim(a) + def perp(self, a): return self.T.perp(a) + def contained(self, a, b): return self.T.contained(a, b) + + +def gram_nondegenerate_frame(rows, h): + """G = I - J/9 nondegenerate on the F_P-subspace with basis rows (exact sound test: an integer determinant + nonzero mod P is nonzero over Q; frames here are reductions of rational frames of the same dimension).""" + k = len(rows) + if k == 0: + return True + inv9 = pow(9, P - 2, P) + sums = [sum(s) % P for s in rows] + M = [[(sum(a * b for a, b in zip(rows[i], rows[j])) - inv9 * sums[i] * sums[j]) % P for j in range(k)] for i in range(k)] + return len(rref(M, k)) == k + + +# ----------------------------------------------------------------------------- graph +class BitGraph: + def __init__(self, p): + self.p, self.h = p, 2 * p + self.cubes = list(combinations(range(p), 3)) + self.labels = [tuple(2 * i + b for i, b in zip(I, bits)) for I in self.cubes for bits in product(range(2), repeat=3)] + self.a = [None] * len(self.labels) + self.s = [1 << i for i in range(len(self.labels))] + self.intern = {} + self.source = {(I, bits): 8 * j + k for j, I in enumerate(self.cubes) for k, bits in enumerate(product(range(2), repeat=3))} + self.A, self.G = {}, {} + self.root, self.centers, self.mix = [], [], [] + + def add(self, a, b): + if a is None: + return b + if b is None: + return a + if a > b: + a, b = b, a + if (a, b) not in self.intern: + require(not self.s[a] & self.s[b], 'support-disjoint addition') + self.intern[a, b] = len(self.a) + self.a.append([a, b]) + self.s.append(self.s[a] | self.s[b]) + return self.intern[a, b] + + def sum(self, xs): + xs = [x for x in xs if x is not None] + while len(xs) > 1: + xs = [self.add(xs[i], xs[i + 1]) if i + 1 < len(xs) else xs[i] for i in range(0, len(xs), 2)] + return xs[0] if xs else None + + def module(self, data, inputs): + image = list(inputs) + n = data['input_count'] + require(len(image) == n, 'module arity') + for pair in data['args'][n:]: + image.append(self.add(image[pair[0]], image[pair[1]])) + return [image[x] for x in data['roots']] + + def local_channels(self): + for I in self.cubes: + src = lambda bits: self.source[I, tuple(bits)] + edges = {} + for ii, jj in combinations(range(3), 2): + kk = 3 - ii - jj + for a, b in product(range(2), repeat=2): + b0 = [0] * 3 + b0[ii], b0[jj] = a, b + b1 = b0[:] + b1[kk] = 1 + edges[ii, jj, a, b] = self.add(src(b0), src(b1)) + self.G[I, I[ii], I[jj], 0] = self.add(edges[ii, jj, 0, 0], edges[ii, jj, 1, 1]) + self.G[I, I[ii], I[jj], 1] = self.add(edges[ii, jj, 0, 1], edges[ii, jj, 1, 0]) + for ii in range(3): + jj = next(j for j in range(3) if j != ii) + for a in range(2): + ids = [edges[(ii, jj, a, b) if ii < jj else (jj, ii, b, a)] for b in range(2)] + self.A[I, I[ii], a] = self.add(*ids) + + def finish(self, pair, allbut): + p = self.p + self.local_channels() + Pm, Qm = {}, {} + for i in range(p): + pairs = list(combinations([a for a in range(p) if a != i], 2)) + for bit in range(2): + for K, node in zip(pairs, self.module(pair, [self.A[tuple(sorted((i, *K))), i, bit] for K in pairs])): + Pm[tuple(sorted((i, *K))), i, bit] = node + for i, j in combinations(range(p), 2): + others = [a for a in range(p) if a not in (i, j)] + for mode in range(2): + for k, node in zip(others, self.module(allbut, [self.G[tuple(sorted((i, j, k))), i, j, mode] for k in others])): + Qm[tuple(sorted((i, j, k))), i, j, mode] = node + allb = list(product(range(2), repeat=3)) + + def emit(I, bl, node, channel): + self.root.append(dict(node=node, targets=[self.source[I, b] for b in bl], kind='side', channel=channel)) + for I in self.cubes: + for a in range(2): + emit(I, [b for b in allb if b[0] == a], Pm[I, I[0], a], 'face0') + for a in range(2): + for m12 in range(2): + emit(I, [b for b in allb if b[0] == a and (b[1] ^ b[2]) == m12], Qm[I, I[1], I[2], 1 - m12], 'edge12') + for b in allb: + emit(I, [b], Pm[I, I[1], b[1]], 'face1') + emit(I, [b], Qm[I, I[0], I[1], 1 - (b[0] ^ b[1])], 'edge01') + emit(I, [b], Pm[I, I[2], b[2]], 'face2') + emit(I, [b], Qm[I, I[0], I[2], 1 - (b[0] ^ b[2])], 'edge02') + emit(I, [b], self.source[I, (b[0], 1 - b[1], 1 - b[2])], 'partner') + # partner-pair source mixing: in each parity class the pair with selector a on cube pair 0 + # (carrier = smaller bits, passive = its partner) is delivered to the opposite pair of the class. + for par in range(2): + for a in range(2): + pair_ = sorted(b for b in allb if sum(b) % 2 == par and b[0] == a) + recv = sorted(b for b in allb if sum(b) % 2 == par and b[0] == 1 - a) + self.mix.append(dict(carrier=self.source[I, pair_[0]], passive=self.source[I, pair_[1]], + receivers=[self.source[I, r] for r in recv])) + for i in range(p): + for e in range(2): + self.centers.append(self.sum(self.A[I, i, e] for I in self.cubes if i in I)) + for coordinate, node in enumerate(self.centers): + self.root.append(dict(node=node, targets=[t for t, lab in enumerate(self.labels) if coordinate in lab], + kind='center', coordinate=coordinate)) + return dict(p=p, h=self.h, v=len(self.labels), labels=[list(l) for l in self.labels], args=self.a, + roots=self.root, partner_mix=self.mix) + + +def check_decoder(g): + v, args = g['v'], g['args'] + sup = [1 << i for i in range(v)] + [0] * (len(args) - v) + for x in range(v, len(args)): + a, b = args[x] + require(not sup[a] & sup[b], 'disjoint supports') + sup[x] = sup[a] | sup[b] + acc = [0] * v + for r in g['roots']: + for t in r['targets']: + acc[t] ^= sup[r['node']] + for mx in g['partner_mix']: + for t in mx['receivers']: + acc[t] ^= (1 << mx['carrier']) | (1 << mx['passive']) + return sum(acc[t] != 1 << t for t in range(v)) + + +# ----------------------------------------------------------------------------- matching +def hopcroft_karp(adj, nleft, nright): + INF = 1 << 30 + mL, mR = [-1] * nleft, [-1] * nright + while True: + dist = [INF] * nleft + dq = deque(u for u in range(nleft) if mL[u] < 0) + for u in dq: + dist[u] = 0 + found = False + while dq: + u = dq.popleft() + for w in adj[u]: + m = mR[w] + if m < 0: + found = True + elif dist[m] == INF: + dist[m] = dist[u] + 1 + dq.append(m) + if not found: + return mL + it = [0] * nleft + for u0 in range(nleft): + if mL[u0] >= 0: + continue + stack = [u0] + while stack: + u = stack[-1] + if it[u] < len(adj[u]): + w = adj[u][it[u]] + it[u] += 1 + m = mR[w] + if m < 0: + ws = w + for uu in reversed(stack): + prev = mL[uu] + mL[uu], mR[ws] = ws, uu + ws = prev + break + if dist[m] == dist[u] + 1: + stack.append(m) + else: + dist[u] = INF + stack.pop() + + +# ----------------------------------------------------------------------------- compiler +def compile_word(g, frozen=None, plain_k=PLAIN, log=lambda *a: None): + h, v = g['h'], g['v'] + geo = Geometry(h, [tuple(l) for l in g['labels']]) + args = [None] + [None if a is None else [x + 1 for x in a] for a in g['args']] # node 0 reserved + roots = [dict(r, node=r['node'] + 1) for r in g['roots']] + n, q = len(args), len(roots) + sup = [0] * n + for i in range(v): + sup[i + 1] = 1 << i + spans = [0] * n + for i in range(v): + spans[i + 1] = geo.src[i] + for x in range(v + 1, n): + a, b = args[x] + spans[x] = geo.join(spans[a], spans[b]) + sup[x] = sup[a] | sup[b] + rframe, rann, ell = [], [], 0 + Y, targetH = [0] * v, Counter() + for r in roots: + x = r['node'] + if r['kind'] == 'center': + U = spans[x]; A = geo.perp(U); ell += geo.dim(U) + else: + A = geo.join_many(geo.tcov[t] for t in r['targets']); U = geo.perp(A) + for t in r['targets']: + require(geo.contained(Y[t], U), 'target retreat') + targetH[geo.dim(U) - geo.dim(Y[t])] += 1 + Y[t] = U + require(geo.contained(spans[x], U), 'root physical compatibility') + rframe.append(U); rann.append(A) + for t in range(v): + require(geo.contained(Y[t], geo.perp(geo.tcov[t])), 'target cap') + targetH[h - 1 - geo.dim(Y[t])] += 1 + active, todo = set(range(1, v + 1)), [r['node'] for r in roots] + while todo: + x = todo.pop() + if x not in active: + active.add(x) + if args[x]: + todo.extend(args[x]) + succ0, cons = [[] for _ in args], [[] for _ in args] + for x in sorted(active): + if args[x]: + for y in args[x]: + succ0[y].append(x) + for j, r in enumerate(roots): + cons[r['node']].append(rann[j]) + preann, initial = [0] * n, [0] * n + for x in sorted(active, reverse=True): + preann[x] = geo.join_many(cons[x] + [preann[y] for y in succ0[x]]) + cover = 0 + s = sup[x] + while s: + b = s & -s; s ^= b + for j in g['labels'][b.bit_length() - 1]: + cover |= 1 << j + initial[x] = geo.join_many([preann[x]] + [geo.coord[j] for j in range(h) if not cover >> j & 1]) + order = sorted(active, key=lambda x: (h - geo.dim(initial[x]), x)) + position = {x: i for i, x in enumerate(order)} + uses, usevalue, usetarget, useann, usecode = [[] for _ in args], [], [], [], [] + for x in order: + if args[x]: + for j, y in enumerate(args[x]): + uses[y].append(len(usevalue)); usevalue.append(y); usetarget.append(x); useann.append(initial[x]); usecode.append(['op', x - 1, j]) + for j, r in enumerate(roots): + x = r['node'] + uses[x].append(len(usevalue)); usevalue.append(x); usetarget.append(n + j); useann.append(rann[j]); usecode.append(['root', j]) + donors = [x for x in order if args[x]] + code_index = {tuple(c): u for u, c in enumerate(usecode)} + if frozen is not None: + arcs = {d + 1: code_index[tuple(c)] for d, c in frozen} + else: + dindex = {x: i for i, x in enumerate(donors)} + adj = [[] for _ in donors] + for x in donors: + for y in args[x]: + for u in uses[y]: + t = usetarget[u] + if t == x or (t < n and position[t] <= position[x]): + continue + if geo.contained(useann[u], initial[x]): + adj[dindex[x]].append(u) + mL = hopcroft_karp(adj, len(donors), len(usevalue)) + arcs = {donors[i]: u for i, u in enumerate(mL) if u >= 0} + for x, u in arcs.items(): + t = usetarget[u] + require(usevalue[u] in args[x] and (t >= n or position[t] > position[x]), 'arc order') + require(geo.contained(useann[u], initial[x]), 'arc frame') + succ, direct = [[] for _ in args], [[] for _ in args] + for x in order: + if args[x]: + for y in args[x]: + succ[y].append(x) + for j, r in enumerate(roots): + direct[r['node']].append(rann[j]) + for x, u in arcs.items(): + t = usetarget[u] + (succ[x].append(t) if t < n else direct[x].append(rann[t - n])) + ann, rank = [0] * n, [0] * n + for x in reversed(order): + ann[x] = geo.join_many(direct[x] + [ann[y] for y in succ[x]]) + rank[x] = h - geo.dim(ann[x]) + require(geo.contained(ann[x], geo.perp(spans[x])), 'span inside frame') + # plain frames span(n): leaves and small additions, closed under operands; a linked donor must be plain and + # its span must lie in the plain target's span + cand = set(x for x in active if not args[x]) + for x in sorted(active): + if args[x] and geo.dim(spans[x]) <= plain_k and all(y in cand for y in args[x]): + cand.add(x) + arcin, users = defaultdict(list), defaultdict(list) + for x, u in arcs.items(): + if usetarget[u] < n: + arcin[usetarget[u]].append(x) + for x in active: + if args[x]: + for y in args[x]: + users[y].append(x) + changed = True + while changed: + changed = False + for t in sorted(cand): + if t in cand and any(x not in cand or not geo.contained(spans[x], spans[t]) for x in arcin[t]): + cand.discard(t); changed = True + st = [t] + while st: + z = st.pop() + for w in users[z]: + if w in cand: + cand.discard(w); st.append(w) + plain = sorted(cand) + for x in plain: + ann[x] = geo.perp(spans[x]); rank[x] = h - geo.dim(ann[x]) + H = Counter() + for x in order: + r, deg = rank[x], len(uses[x]) + require(deg > 0, 'unused node') + H[r] += deg - 1 + if args[x]: + H[h - r] += 1 + for y in args[x]: + require(geo.contained(ann[x], ann[y]), 'operand nesting') + H[r - rank[y]] += 1 + else: + H[1] += 1; H[r - 1] += 1 + for j, root in enumerate(roots): + x = root['node']; r = rank[x]; rt = geo.dim(rframe[j]) + require(r <= rt, 'root climb') + if root['kind'] == 'center': + require(r == rt, 'centre frame') + H[r] += 1; H[h - r] += 1 + else: + H[rt - r] += 1; H[h - rt] += 1 + for x, u in arcs.items(): + val, t = usevalue[u], usetarget[u] + rv, rd = rank[val], rank[x] + rt = rank[t] if t < n else geo.dim(rframe[t - n]) + require(rt >= rd >= rv, 'arc ranks') + H[h - rd] -= 1; H[rv] -= 1; H[rt - rv] -= 1; H[rt - rd] += 1 + require(min(H.values()) >= 0, 'nonnegative ledger') + R = len(donors) + q - len(arcs) + require(sum(r * c for r, c in H.items()) == h * R + ell, 'mass identity') + prof = dict(h=h, v=v, R=R, q=q, c=len(donors), matched=len(arcs), loss=ell, plain=len(plain)) + wit = dict(geo=geo, args=args, roots=roots, n=n, order=order, arcs=arcs, ann=ann, rank=rank, uses=uses, + usevalue=usevalue, usetarget=usetarget, usecode=usecode, rframe=rframe, rann=rann, spans=spans, + plain=plain, H=H, targetH=targetH) + return prof, wit + + +def nondeg_id(geo, f, cache): + if f not in cache: + cache[f] = gram_nondegenerate_frame(list(geo.T.rows[f]), geo.h) + return cache[f] + + +def select_gauges(g, prof, wit, trial_a=0.00065): + """PR #144 gauges.select, exactly, plus the bit G-nondegeneracy filter on sigma.""" + geo = wit['geo']; h, v = prof['h'], prof['v'] + args, roots, n, order, ann, arcs = wit['args'], wit['roots'], wit['n'], wit['order'], wit['ann'], wit['arcs'] + usevalue, usetarget = wit['usevalue'], wit['usetarget'] + incoming = set(arcs.values()) + uses, opuse, rootuse = [[] for _ in args], {}, {} + for u, (val, t) in enumerate(zip(usevalue, usetarget)): + uses[val].append(u) + if t < n: + opuse[t, 0 if args[t][0] == val else 1] = u + else: + rootuse[t - n] = u + assign, sources, ops, R = {}, {}, [], 0 + for x in order: + if args[x]: + aa, bb = args[x] + dest, control = assign[opuse[x, 0]], assign[opuse[x, 1]] + if x in arcs: + u = arcs[x] + if usevalue[u] == aa: + dest, control = control, dest + require(u not in assign, 'arc use assigned once') + assign[u] = control + ops.append((dest, control, x)) + else: + dest = R; R += 1; sources[x] = dest + free = [u for u in uses[x] if u not in incoming] + require(free, 'free use') + for j, u in enumerate(free): + if j == 0: + assign[u] = dest + else: + t = R; R += 1; assign[u] = t; ops.append((t, dest, x)) + require(R == prof['R'], 'role count') + rootroles = [assign[rootuse[j]] for j in range(len(roots))] + prev, pred, first = [-1] * R, [], [None] * R + for i, (a, b, x) in enumerate(ops): + pred.append((prev[a], prev[b])); prev[a] = prev[b] = i + if first[a] is None: + first[a] = x + if first[b] is None: + first[b] = x + stack = [prev[s] for r, s in zip(roots, rootroles) if r['kind'] == 'center' and prev[s] >= 0] + phase = set() + while stack: + i = stack.pop() + if i not in phase: + phase.add(i); stack.extend(j for j in pred[i] if j >= 0) + touched = set(sources.values()) + for i in phase: + touched.update(ops[i][:2]) + co = [0] * R + for r, s in zip(roots, rootroles): + co[s] |= sum(1 << t for t in r['targets']) + for a, b, x in reversed(ops): + co[b] |= co[a] + limit = [None] * v + for r in roots: + if r['kind'] != 'center': + A = geo.join_many(geo.tcov[t] for t in r['targets']) + for t in r['targets']: + if limit[t] is None: + limit[t] = A + for t in range(v): + if limit[t] is None: + limit[t] = geo.tcov[t] + H = Counter(wit['H']) + gauges, selected, cache = Counter(), [], {} + cands = sorted((s for s in range(R) if s not in touched and first[s] is not None), + key=lambda s: (geo.dim(ann[first[s]]), bin(co[s]).count('1'), s)) + + def excess(t): + return t * math.expm1(trial_a * math.log(3 * h / t)) if t else 0. + for s in cands: + A, targets, bits = ann[first[s]], [], co[s] + while bits: + low = bits & -bits; t = low.bit_length() - 1; bits ^= low; targets.append(t) + A = geo.join(A, limit[t]) + if geo.dim(A) == h: + break + if geo.dim(A) == h: + continue + r, d = h - geo.dim(ann[first[s]]), h - geo.dim(A) + delta = 3 * (excess(r - d) - excess(r)) + excess(3 * d) + delta += 3 * math.fsum(excess(d) + excess(h - geo.dim(limit[t]) - d) - excess(h - geo.dim(limit[t])) for t in targets) + if delta >= -1e-12 or not nondeg_id(geo, geo.perp(A), cache): + continue + H[r] -= 1; H[r - d] += 1; gauges[d] += 1 + require(H[r] >= 0, 'gauge ledger') + for t in targets: + limit[t] = A + selected.append(dict(role=s, first=first[s] - 1, targets=targets, dim=d, ann=A)) + Yh, cur = Counter(), [0] * v + for z in reversed(selected): + U = geo.perp(z['ann']) + for t in z['targets']: + require(geo.contained(cur[t], U), 'gauge target chain') + Yh[geo.dim(U) - geo.dim(cur[t])] += 1; cur[t] = U + for r in roots: + if r['kind'] != 'center': + U = geo.perp(geo.join_many(geo.tcov[t] for t in r['targets'])) + for t in r['targets']: + require(geo.contained(cur[t], U), 'side target chain') + Yh[geo.dim(U) - geo.dim(cur[t])] += 1; cur[t] = U + for t in range(v): + Yh[h - 1 - geo.dim(cur[t])] += 1 + word = dict(ops=[[a, b, x - 1] for a, b, x in ops], sources={str(x - 1): s for x, s in sources.items()}, + rootroles=rootroles, phase1=sorted(phase), gauges=[{k: z[k] for k in ('role', 'first', 'targets', 'dim')} for z in selected], + _gauge_ann=[z['ann'] for z in selected]) + return gauges, H, Yh, word + + +def profile(prof, gauges, H, Y): + h, v, R, ell = prof['h'], prof['v'], prof['R'], prof['loss'] + m, W = 3 * h, 2 * v + R + source = Counter({1: v, h - 4: v // 2, 2: v // 2, h - 2: v // 2}) + C = Counter() + for part in (H, source, Y): + for r, c in part.items(): + if r and c: + C[r] += 3 * c + for d, c in gauges.items(): + C[3 * d] += c + C[2] += 2 * v + mass = sum(r * c for r, c in C.items()) + require(W * m - mass == 2 * v - 3 * ell, 'shared-core telescoping deficit') + srt = lambda c: {str(k): c[k] for k in sorted(c) if c[k]} + return dict(h=h, v=v, R=R, c=prof['c'], q=prof['q'], matched=prof['matched'], loss=ell, plain_frames=prof['plain'], + m=m, W_per_vertex=W, rank_per_vertex=mass, deficit_per_vertex=W * m - mass, + selected_roles=sum(gauges.values()), selected_rank_histogram=srt(gauges), + remaining_internal_histogram=srt(H), source_data_histogram=srt(source), target_data_histogram=srt(Y), + child_histogram=srt(C), maxchild=max(C)) + + +def float_root(child, W, m): + items = [(int(t), c) for t, c in child.items()] + lo, hi = 0., .1 + for _ in range(80): + a = (lo + hi) / 2 + if math.fsum(c * (t / m) ** (1 - a) for t, c in items) < W: + lo = a + else: + hi = a + return lo + + +def ratrec(r): + """Rational reconstruction a/b of a residue mod P with |a|, b < sqrt(P/2).""" + r %= P + if r == 0: + return 0, 1 + bound = math.isqrt(P // 2) + r0, r1, s0, s1 = P, r, 0, 1 + while r1 > bound: + qq = r0 // r1 + r0, r1 = r1, r0 - qq * r1 + s0, s1 = s1, s0 - qq * s1 + require(abs(s1) <= bound and s1 != 0, 'rational reconstruction') + a, b = (r1, s1) if s1 > 0 else (-r1, -s1) + require((a - r * b) % P == 0, 'reconstruction residue') + return a, b + + +def primitive(row): + from fractions import Fraction + fr = [Fraction(*ratrec(x)) for x in row] + den = 1 + for f in fr: + den = den * f.denominator // math.gcd(den, f.denominator) + ints = [int(f * den) for f in fr] + gg = 0 + for x in ints: + gg = math.gcd(gg, abs(x)) + ints = [x // gg for x in ints] if gg else ints + first = next((x for x in ints if x), 1) + return [-x for x in ints] if first < 0 else ints + + +def frame_record(geo, ann_id): + """Exact integer description of the frame perp(ann): 'a' = integer annihilator rows (frame = common kernel) + when codim <= dim, else 'b' = integer basis rows. Both are primitive rows of the rational RREF.""" + h = geo.h + A = geo.T.rows[ann_id] + codim = len(A) + if codim <= h - codim: + rows = [primitive(r) for r in A] + return dict(dim=h - codim, a=rows) + B = geo.T.rows[geo.perp(ann_id)] + return dict(dim=h - codim, b=[primitive(r) for r in B]) + + +def dumps(obj): + return json.dumps(obj, sort_keys=True, separators=(',', ':')) + '\n' + + +def build(p, frozen_arcs=None, log=print): + mod = json.loads((HERE / 'data' / MODULES[p]).read_text()) + abo = all_but_one(p - 2) + g = BitGraph(p).finish(mod, abo) + require(check_decoder(g) == 0, 'mod-2 decoder identity') + t0 = time.time() + prof, wit = compile_word(g, frozen=frozen_arcs) + log('compiled p=%d in %.1fs: R=%d arcs=%d plain=%d' % (p, time.time() - t0, prof['R'], prof['matched'], prof['plain'])) + gauges, H, Y, word = select_gauges(g, prof, wit) + prf = profile(prof, gauges, H, Y) + arcs = sorted([x - 1, wit['usecode'][u]] for x, u in wit['arcs'].items()) + cache = {} + geo = wit['geo'] + deg = sum(1 for x in wit['order'] if not nondeg_id(geo, geo.perp(wit['ann'][x]), cache)) + deg += sum(1 for U in wit['rframe'] if not nondeg_id(geo, U, cache)) + require(deg == 0, 'G-nondegenerate node and root frames') + out = dict(graph=dict(p=p, h=g['h'], v=g['v'], labels=g['labels'], args=g['args'], roots=g['roots'], + partner_mix=g['partner_mix'], pair_module_sha256=hashlib.sha256((HERE / 'data' / MODULES[p]).read_bytes()).hexdigest()), + compile=dict(plain_threshold=PLAIN, order=[x - 1 for x in wit['order']], plain=[x - 1 for x in wit['plain']], arcs=arcs), + word={k: v for k, v in word.items() if not k.startswith('_')}) + exp = export(p, g, prof, wit, word, prf) + return out, prf, arcs, exp + + +def export(p, g, prof, wit, word, prf): + """Plain-JSON gate files: graph, word (selection.json shape), exact frames, partial-K chronology.""" + geo, h, v = wit['geo'], g['h'], g['v'] + n, roots = wit['n'], wit['roots'] + ann = wit['ann'] + frames = {} + + def fid(a): + if a not in frames: + frames[a] = None + return a + node_frame = {str(x - 1): fid(ann[x]) for x in wit['order']} + root_frame = [fid(a) for a in wit['rann']] + gauges = [] + for z, A in zip(word['gauges'], word['_gauge_ann']): + gauges.append(dict(z, frame=fid(A))) + lab = g['labels'] + src_ann = [fid(geo.perp(geo.src[i])) for i in range(v)] + full = fid(0) + level2 = {} + for j, r in enumerate(g['roots']): + if r['kind'] == 'side' and len(r['targets']) == 2: + level2[tuple(sorted(r['targets']))] = j + kchron = [] + for mx in g['partner_mix']: + c, d = mx['carrier'], mx['passive'] + mixf = fid(geo.perp(geo.join(geo.src[c], geo.src[d]))) + j = level2[tuple(sorted(mx['receivers']))] + capf = root_frame[j] + require(geo.contained(capf, mixf), 'mix frame inside receivers cap') + kchron.append(dict(carrier=c, passive=d, receivers=mx['receivers'], mix_frame=mixf, deliver_frame=capf, + deliver_after_root=j, undo_frame=full, + carrier_chain=[src_ann[c], mixf, capf, full], passive_chain=[src_ann[d], mixf, full], + carrier_ranks=[1, h - 4, 2], passive_ranks=[1, h - 2])) + table = {str(a): frame_record(geo, a) for a in sorted(frames)} + conv = dict(target_covector='3*chi_T - 1 (target cap = {u : (3 chi_T - 1).u = 0} = H0-orthogonal complement of chi_T)', + gram='G = I - J/9 on Q^h (H0 = G/2, H0(chi_S,chi_T) = (|S cap T|-1)/2); nondegeneracy tested for G', + frame_ids='frame id = id of the canonical annihilator; record "a" lists integer annihilator rows (frame =' + ' common kernel), record "b" lists integer basis rows; id 0 (empty "a") is the full space Q^h', + ops='[dest, control, node]: dest ^= control at the frame of node (copies: dest is the fresh role)', + nodes='zero-based graph nodes; nodes < v are the source leaves (port index)') + graph = dict(p=p, h=h, v=v, labels=g['labels'], args=g['args'], + roots=[dict(r, coefficient=1) for r in g['roots']], partner_mix=g['partner_mix'], + decoder='mod 2: Y_T += sum of root nodes whose targets contain T (star roots: T reads star(c) for c in T;' + ' side roots; partner singles) + x_carrier + x_passive for each partner_mix entry receiving T;' + ' equals x_T = sum_{c in T} star(c) + sum_{|S cap T|=1} x_S') + wordj = dict(ops=word['ops'], sources=word['sources'], rootroles=word['rootroles'], phase1=word['phase1'], + gauges=gauges, order=[x - 1 for x in wit['order']], arcs=sorted([x - 1, wit['usecode'][u]] for x, u in wit['arcs'].items()), + plain=[x - 1 for x in wit['plain']], node_frame=node_frame, root_frame=root_frame, + source_frame=src_ann, full_frame=full, conventions=conv, + schedule='old-value reads of non-gauged roles (frame 0); V injections at ; ops in order (phase1 ops' + ' are those listed); centre copies read at 0; gauged old-value reads at their gauge frames' + ' (ascending per target); side root reads in root order; partial-K deliveries right after' + ' root deliver_after_root; undo at full frame; inverse ops; V removal') + return dict(graph=graph, word=wordj, frames=dict(h=h, P_used_for_ranks='2^127-1', conventions=conv, frames=table), + kchron=dict(h=h, entries=kchron, conventions=conv)) + + +def all_but_one(n): + """PR #144 modules.all_but_one (balanced binary tree upward sums and complementary downward sums).""" + args, support = [None] * n, [1 << i for i in range(n)] + by = {s: i for i, s in enumerate(support)} + + def add(a, b): + if a is None: + return b + if b is None: + return a + s = support[a] | support[b] + if s not in by: + by[s] = len(args); args.append([a, b]); support.append(s) + return by[s] + + def tree(lo, hi): + if hi - lo == 1: + return (lo, None, None) + mid = (lo + hi) // 2 + L, Rr = tree(lo, mid), tree(mid, hi) + return (add(L[0], Rr[0]), L, Rr) + roots = [None] * n + + def walk(t, outside): + x, L, Rr = t + if L is None: + roots[x] = outside + return + walk(L, add(outside, Rr[0])); walk(Rr, add(outside, L[0])) + walk(tree(0, n), None) + active, stack = set(range(n)), list(roots) + while stack: + x = stack.pop() + if x not in active: + active.add(x) + if args[x] is not None: + stack.extend(args[x]) + ids = sorted(active); ren = {x: i for i, x in enumerate(ids)} + return dict(input_count=n, args=[None if args[x] is None else [ren[y] for y in args[x]] for x in ids], roots=[ren[x] for x in roots]) + + +def main(): + require(not sys.flags.optimize, 'run without -O') + ap = argparse.ArgumentParser(description=__doc__) + ap.add_argument('--p', type=int, default=12) + ap.add_argument('--out', type=Path, default=HERE / 'out') + ap.add_argument('--check', action='store_true', help='regenerate from frozen arcs and compare byte-identically') + ap.add_argument('--solve-matching', action='store_true', help='recompute the carrier matching (Hopcroft-Karp)') + a = ap.parse_args() + a.out.mkdir(parents=True, exist_ok=True) + arcs_path = HERE / 'data' / ('arcs_p%d.json' % a.p) + frozen = None if a.solve_matching or not arcs_path.exists() else json.loads(arcs_path.read_text()) + out, prf, arcs, exp = build(a.p, frozen) + root = float_root(prf['child_histogram'], prf['W_per_vertex'], prf['m']) + word_bytes = gzip.compress(dumps(out).encode(), mtime=0) + prof_bytes = dumps(prf).encode() + names = {'word_p%d.json.gz' % a.p: word_bytes, 'profile_p%d.json' % a.p: prof_bytes} + for k, obj in exp.items(): + names['%s_p%d.json' % (k, a.p)] = dumps(obj).encode() + if a.check: + for name, data in names.items(): + old = (a.out / name).read_bytes() + if name.endswith('.gz'): # gzip bytes depend on the zlib build; compare the compressed content + old, data = gzip.decompress(old), gzip.decompress(data) + require(old == data, 'byte-identical regeneration of ' + name) + require(frozen is not None, 'frozen arcs present') + print('PASS p=%d: word and profile regenerate byte-identically (sha256 %s, %s)' % ( + a.p, hashlib.sha256(word_bytes).hexdigest()[:16], hashlib.sha256(prof_bytes).hexdigest()[:16])) + else: + for name, data in names.items(): + (a.out / name).write_bytes(data) + if frozen is None: + arcs_path.write_text(dumps(arcs)) + print('wrote p=%d: R=%d R/v=%.3f loss=%d deficit=%d W=%d gauges=%d float a_b=%.7e' % ( + a.p, prf['R'], prf['R'] / prf['v'], prf['loss'], prf['deficit_per_vertex'], prf['W_per_vertex'], + prf['selected_roles'], root)) + + +if __name__ == '__main__': + main() diff --git a/scripts/paired_cube/modules.py b/scripts/paired_cube/modules.py index f590aa7edd..f510169ebc 100644 --- a/scripts/paired_cube/modules.py +++ b/scripts/paired_cube/modules.py @@ -123,3 +123,136 @@ def restricted_triples(p): source_author='eumemic', contract='Root indexed by J sums input I exactly when I and J are disjoint.') return result + + +# ---- Generalized restrictions for other p / other #117-format sources (eumemic, Claude assistance; Apache-2.0). +# Same contracts and checks as restricted_triples/restricted_pairs above; the defaults of those functions are +# keep = 0..p-1 (triples) and keep = 0..n-1, pin = 23, out = 22 (pairs) on the PR117 h24 witness. +def _source_witness(path): + return json.loads(gzip.decompress(Path(path).read_bytes())) + +def restricted_triples_from(path, keep): + w = _source_witness(path) + keep = sorted(keep); p = len(keep); rename = {x: j for j, x in enumerate(keep)} + labels = list(combinations(range(p), 3)); index = {t: i+1 for i, t in enumerate(labels)} + args = [(0, 0)] + [(0, 0)] * len(labels); supports = [0] + [1 << i for i in range(len(labels))] + intern = {s: i for i, s in enumerate(supports) if s} + old_labels = list(combinations(range(w['h']), 3)) + def new(t): return tuple(rename[x] for x in t) if all(x in rename for x in t) else None + image = [0] + [index.get(new(t), 0) for t in old_labels] + for aa, bb in zip(w['args'][::2], w['args'][1::2]): + a, b = image[aa], image[bb] + if not a or not b: + image.append(a or b); continue + assert not supports[a] & supports[b] + s = supports[a] | supports[b] + if s not in intern: + intern[s] = len(args); args.append((a, b)); supports.append(s) + image.append(intern[s]) + roots = [image[x] for t, x in zip(old_labels, w['D']) if new(t) is not None] + for J, r in zip(labels, roots): + assert r and supports[r] == sum(1 << i for i, I in enumerate(labels) if not set(I) & set(J)) + active = set(range(len(labels)+1)); stack = roots[:] + while stack: + x = stack.pop() + if x in active: continue + active.add(x); stack.extend(args[x]) + order = sorted(active); rename2 = {x: i for i, x in enumerate(order)} + return dict(kind='disjoint_triples', p=p, input_labels=labels, target_labels=labels, input_count=len(labels), + args=[[rename2[a], rename2[b]] for x in order for a, b in [args[x]]], roots=[rename2[x] for x in roots], + source='exact zero restriction of a pinned #117-format witness to kept points', kept=keep, + contract='Root indexed by J sums input I exactly when I and J are disjoint.') + +def restricted_pairs_from(path, keep, pin, out): + w = _source_witness(path) + keep = sorted(keep); n = len(keep) + assert pin not in keep and out not in keep and pin != out + rename = {x: j for j, x in enumerate(keep)} + oldtriples = list(combinations(range(w['h']), 3)); pairs = list(combinations(range(n), 2)) + pid = {q: i for i, q in enumerate(pairs)} + args = [None]*len(pairs); support = [1 << i for i in range(len(pairs))]; by = {s: i for i, s in enumerate(support)} + image = [None] + for t in oldtriples: + rest = tuple(x for x in t if x != pin) + ok = pin in t and len(rest) == 2 and all(x in rename for x in rest) + image.append(pid[tuple(sorted(rename[x] for x in rest))] if ok else None) + for aa, bb in zip(w['args'][::2], w['args'][1::2]): + a, b = image[aa], image[bb] + if a is None or b is None: + image.append(a if b is None else b); continue + assert not support[a] & support[b] + s = support[a] | support[b] + if s not in by: + by[s] = len(args); args.append([a, b]); support.append(s) + image.append(by[s]) + oldroots = {t: x for t, x in zip(oldtriples, w['D'])} + roots = [image[oldroots[tuple(sorted((keep[i], keep[j], out)))]] for i, j in pairs] + for (i, j), r in zip(pairs, roots): + expected = sum(1 << k for k, (a, b) in enumerate(pairs) if i not in (a, b) and j not in (a, b)) + assert r is not None and support[r] == expected + active = set(range(len(pairs))); stack = list(roots) + while stack: + x = stack.pop() + if x in active: continue + active.add(x) + if args[x] is not None: stack.extend(args[x]) + ids = sorted(active); ren = {x: i for i, x in enumerate(ids)} + args = [None if args[x] is None else [ren[y] for y in args[x]] for x in ids] + return dict(kind='pair_disjoint', n=n, input_count=len(pairs), input_labels=pairs, output_labels=pairs, + args=args, roots=[ren[x] for x in roots], kept=keep, pin=pin, out=out, + source='zero restriction of a pinned #117-format witness: inputs {a,b,pin}, outputs D_{i,j,out}') + + +# ---- Direct pair-module source (PMOD lane; eumemic, Claude assistance; Apache-2.0). +def pair_module_from(path, n): + """Zero-based pair-disjoint module given as JSON {input_count, args, roots}; same contract as restricted_pairs(n): + additions combine disjoint supports and root (i,j) sums exactly the inputs {a,b} disjoint from {i,j}.""" + d = json.loads(Path(path).read_text()) + pairs = list(combinations(range(n), 2)) + assert d['input_count'] == len(pairs) == len(d['roots']) + support = [] + for x, a in enumerate(d['args']): + if x < len(pairs): + assert a is None + support.append(1 << x) + else: + assert a is not None and 0 <= a[0] < x and 0 <= a[1] < x and not support[a[0]] & support[a[1]] + support.append(support[a[0]] | support[a[1]]) + for (i, j), r in zip(pairs, d['roots']): + assert support[r] == sum(1 << k for k, (a, b) in enumerate(pairs) if i not in (a, b) and j not in (a, b)) + return dict(kind='pair_disjoint', n=n, input_count=len(pairs), input_labels=pairs, output_labels=pairs, + args=[None if a is None else list(a) for a in d['args']], roots=list(d['roots']), + source='Direct pair module JSON; contract checked above') + + +# ---- Output merging (eumemic, Claude assistance; Apache-2.0). +def merge_outputs(g, G, variant): + """Replace #144's single-target channels of each port T=(I,b) -- face2 P[I,I2,1-b2] (1/2), edge02 + Q[I,I0,I2,b0^b2] ((2b0-1)/2), edge12 Q[I,I1,I2,b1^b2] ((2b1-1)/2) -- by signed sums read with coefficient 1/2. + w02: face2 + (2b0-1)*edge02, one node per (b0,b2) shared by the ports b1=0,1; edge12 stays single. The decoder + contribution is unchanged and Graph.add asserts disjoint supports.""" + from fractions import Fraction + specs = {'w02': [('face2', 'edge02')], 'w12': [('face2', 'edge12')], + 's': [('face2', 'edge02', 'edge12')]}[variant] + side = [r for r in g['roots'] if r['kind'] == 'side'] + centre = [r for r in g['roots'] if r['kind'] != 'side'] + chan = {c: k for k, spec in enumerate(specs) for c in spec} + found = {} + keep = [] + for r in side: + if r['channel'] in chan: + assert all(Fraction(c) == Fraction(r['coefficients'][0]) for c in r['coefficients']) + found[chan[r['channel']], tuple(r['targets']), r['channel']] = (r['node'], Fraction(r['coefficients'][0])) + else: + keep.append(r) + new = [] + for k, tg in sorted({(k, tg) for k, tg, _ in found}): + parts = [found[k, tg, ch] for ch in specs[k]] + node, c0 = parts[0] + for nd, c in parts[1:]: + sg = c / c0 + assert sg in (1, -1) + node = G.add(node, nd, int(sg)) + new.append(dict(node=node, targets=list(tg), coefficients=[str(c0)] * len(tg), kind='side', channel=variant)) + new.sort(key=lambda r: -len(r['targets'])) # multi-target roots before singles (target chains ascend) + return dict(g, roots=keep + new + centre, args=G.a, signs=G.signs) diff --git a/scripts/paired_cube/verify.py b/scripts/paired_cube/verify.py index 9b0d6be233..c6c2ee47ad 100644 --- a/scripts/paired_cube/verify.py +++ b/scripts/paired_cube/verify.py @@ -147,7 +147,7 @@ def each(bits): assert contained(A,current[t]);Y[len(current[t])-len(A)]+=1;current[t]=A for t,q in enumerate(inputs):assert contained((q,),current[t]);Y[len(current[t])-1]+=1 assert dict(Y)=={int(k):c for k,c in record['target_data_histogram'].items()} - assert record['selected_roles']==len(seen)==4840 and birth==Counter({20:4840}) + assert record['selected_roles']==len(seen)==2970 and birth==Counter({18:2970}) return dict(signed_scalar_pairs_checked=scalar_pairs,scalar_H_exact=True,K_involution_and_inverse_exact=True, decoder_identity_exact=True,conservative_source_supports=True,K_source_itinerary_verified=True, full_backward_intersections=True,physical_signed_M_replayed=True, diff --git a/scripts/paired_cube_network.py b/scripts/paired_cube_network.py index 63dab6d4e3..c6e59ed543 100644 --- a/scripts/paired_cube_network.py +++ b/scripts/paired_cube_network.py @@ -16,7 +16,22 @@ import sys from certify import require -from paired_cube_bit import reconstruct +from paired_cube_physical import checked_record + + +def bitcube_row(): + """Paired-cube bit word with partner-pair source mixing (research/paired-cube-bit, p = 12). Its standalone + checker recomputes the exact frames, the mod-2 decoder identity, every role and target chain, the + partner-pair chronology, G-nondegeneracy, a literal F2 replay and the ledger recount; the generator and + mutation controls run in research/paired-cube-bit.""" + folder = ROOT/'research/paired-cube-bit' + sys.path.insert(0,str(folder)) + from check_paired_cube_bit import Checker + checked = Checker(folder/'out',12).run() + row = json.loads((folder/'out/profile_p12.json').read_text()) + require((checked['roles'],checked['W'],checked['m'],checked['deficit']) == + (row['R'],row['W_per_vertex'],row['m'],row['deficit_per_vertex']),'Checked bit word profile') + return row from three_stage_cover_network import log_upper, exp_upper from structured_bulk_assembly import assembly, halving, js @@ -24,15 +39,15 @@ sys.set_int_max_str_digits(0) ROOT = Path(__file__).resolve().parents[1] -AC = Q(4856569,10**10) -COARSE = Q(4617656,10**10) +AC = Q(5885669,10**10) +COARSE = Q(5936323,10**10) ATOM = Q(1,1000) OLD = Q(384599,10**10) AB = (1-ATOM)*COARSE+ATOM*OLD BAD = Q(1,10**16) -PHASE_STOP = Q(1,10**6) +PHASE_STOP = Q(1,10**9) ASSEMBLY_BIT = min(AB,(1-PHASE_STOP)*AC-Q(1,10**10)) -KAPPA = Q(4609169,10**10) +KAPPA = Q(5878747,10**10) def clean(hist): @@ -48,9 +63,9 @@ def shared_profile(row,complex_word): require(0 < r < h and n > 0,'Proper local gauges') H[3*r] += n if complex_word: - require((h,v,R,ell) == (24,1760,26417,528),'Paired-cube local dimensions') + require((h,v,R,ell) == (22,1320,16011,440),'Paired-cube local dimensions') require(R == row['c']+row['q']-row['matched'],'Compatible carrier roles') - require(selected == {20:4840},'Selected rank20 gauges') + require(selected == {18:2970},'Selected rank18 gauges') for r,n in enumerate(row['remaining_internal_histogram']): H[r] += 3*n parts = ('source_data_histogram','target_data_histogram') @@ -72,6 +87,30 @@ def shared_profile(row,complex_word): deficit_per_vertex=W*m-mass,child_multiplicities=H,maxchild=max(H),edge_count=sum(H.values())) +def bitcube_profile(row): + """Shared-core profile of the paired-cube bit word, in the complex word's ledger format.""" + h,v,R,ell = (row[k] for k in ('h','v','R','loss')) + require((h,v,R,ell) == (24,1760,23368,528),'Paired-cube bit dimensions') + m,W,H = 3*h,2*v+R,Counter() + selected = {int(r):n for r,n in row['selected_rank_histogram'].items()} + require(selected == {20:5720} and sum(selected.values()) == row['selected_roles'],'Selected bit gauges') + for r,n in selected.items(): + H[3*r] += n + for name in ('remaining_internal_histogram','source_data_histogram','target_data_histogram'): + for r,n in row[name].items(): + H[int(r)] += 3*n + H[2] += 2*v + H = clean(H) + require(all(0 < r < m and n > 0 for r,n in H.items()),'Proper shared-core children') + mass = sum(r*n for r,n in H.items()) + require(H == clean(row['child_histogram']),'Saved complete child histogram') + require((m,W,mass,W*m-mass) == (row['m'],row['W_per_vertex'],row['rank_per_vertex'],row['deficit_per_vertex']), + 'Saved sharing dimensions and rank') + require(W*m-mass == 2*v-3*ell,'Shared-core telescoping deficit') + return dict(m=m,local_dimension=h,W_per_vertex=W,rank_per_vertex=mass,deficit_per_vertex=W*m-mass, + child_multiplicities=H,maxchild=max(H),edge_count=sum(H.values())) + + def exact_moment(p,saving): m,W = p['m'],p['W_per_vertex'] logs = {r:log_upper(Q(m,r)) for r in p['child_multiplicities']} @@ -82,7 +121,7 @@ def exact_moment(p,saving): def bit_certificate(row): - p = shared_profile(row,False) + p = bitcube_profile(row) exact = exact_moment(p,COARSE) m,W = p['m'],p['W_per_vertex'] fallback = 32*m*m @@ -98,8 +137,32 @@ def bit_certificate(row): row_stock='O(w log e) internally borrowed radix-q digits; retained stopped wrapper restores them') -def complex_certificate(row): - p = shared_profile(row,True) +def physical_profile(phys,row): + """Shared-core profile of the physical word (paired_cube_physical.py): descended frames, compensated reuse.""" + h,v,R,ell = (row[k] for k in ('h','v','R','loss')) + shared_profile(row,True) + require((phys['h'],phys['v'],phys['R'],phys['loss']) == (h,v,R,ell),'Physical word dimensions') + require(phys['physical_R'] == R-phys['pairs'] and phys['W_per_vertex'] == 2*v+phys['physical_R'],'Physical stock') + m,W,H = 3*h,phys['W_per_vertex'],Counter() + for name in ('local_histogram','source_data_histogram','target_data_histogram'): + for r,n in phys[name].items(): + H[int(r)] += 3*n + for d,n in phys['physical_gauge_histogram'].items(): + H[3*int(d)] += n + H[2] += 2*v + H = clean(H) + require(H == clean(phys['child_histogram']),'Saved physical child histogram') + require(all(0 < r < m and n > 0 for r,n in H.items()),'Proper shared-core children') + mass = sum(r*n for r,n in H.items()) + require((m,mass,W*m-mass) == (phys['m'],phys['rank_per_vertex'],phys['deficit_per_vertex']),'Saved physical rank') + require(W*m-mass == 2*v-3*ell,'Shared-core telescoping deficit') + return dict(m=m,local_dimension=h,W_per_vertex=W,rank_per_vertex=mass,deficit_per_vertex=W*m-mass, + child_multiplicities=H,maxchild=max(H),edge_count=sum(H.values()),physical_roles=phys['physical_R'], + reuse_pairs=phys['pairs'],late_reads=phys['late_pairs'],moved_operation_frames=phys['changed_operation_frames']) + + +def complex_certificate(row,phys): + p = physical_profile(phys,row) exact = exact_moment(p,AC) m = p['m'] n = m//2 @@ -148,9 +211,9 @@ def certificate(): def read(name): return json.loads((ROOT/'certificates'/name).read_text()) expected = read('paired-cube-input.json') - bit_row = reconstruct() + bit_row = bitcube_row() row = read('paired-cube-complex-input.json') - bit,phase = bit_certificate(bit_row),complex_certificate(row) + bit,phase = bit_certificate(bit_row),complex_certificate(row,checked_record()) bridge = finite_bridge(phase,bit,row) require(Q(read('copied-centers-network.json')['bit']['saving']) == OLD,'Retained ordinary leaf') result = assembly(ASSEMBLY_BIT,AC,bridge,KAPPA,beta=PHASE_STOP) @@ -170,6 +233,8 @@ def read(name): sources += sorted(p for folder in ('scripts/paired_cube','references/paired-cube') for p in (ROOT/folder).rglob('*') if p.is_file() and '__pycache__' not in p.parts) sources += sorted((ROOT/'notes').glob('paired-cube-*.tex')) + sources += sorted(p for p in (ROOT/'research/paired-cube-bit').rglob('*') + if p.is_file() and '__pycache__' not in p.parts) sources += [ROOT/p for p in ('scripts/three_stage_cover_network.py','scripts/structured_bulk_assembly.py', 'scripts/partial_gauge_bit.py','certificates/partial-gauge-bit-input.json', 'references/partial-gauge/pr97/SOURCE.json','certificates/copied-centers-network.json', @@ -187,7 +252,7 @@ def main(): p.add_argument('--output',type=Path,default=ROOT/'certificates/paired-cube-network.json') args = p.parse_args() args.output.write_text(json.dumps(js(certificate()),indent=2,sort_keys=True)+'\n') - print('PASS kappa=4609169/10000000000 = 4.609169e-4; both moments, shared cores, finite router and 47 strict constraints') + print('PASS kappa=5878747/10000000000 = 5.878747e-4; both moments, shared cores, finite router and 47 strict constraints') if __name__ == '__main__': diff --git a/scripts/paired_cube_physical.py b/scripts/paired_cube_physical.py new file mode 100644 index 0000000000..187bd31c59 --- /dev/null +++ b/scripts/paired_cube_physical.py @@ -0,0 +1,328 @@ +#!/usr/bin/env python3 +"""Physical frame descent and compensated birth-cut reuse on the selected paired-cube complex word. + +Prepared by eumemic with Anthropic Claude assistance. Apache-2.0. The signed word, frozen arcs, gauges, +chronology and shared-core accounting are icekylinx's PR144. Frame descent under PR130's general Clifford +frames and compensated birth-cut reuse follow PR131 and jamesyc's PR124; deadline (late) compensation reads +follow PR143. Frozen inputs: references/paired-cube/physical/{frames,pairs}.json. + +Checks, from the regenerated PR144 word and the frozen inputs only: + * every operation frame contains its node's value span, and every role chain + start -> operation frames -> root frame -> full space + is nested, with each recipient's chain spliced after its donor's last operation; + * every pair: donor ungauged, not a root, its last operation before the recipient's old-value read; the + recipient gauged and untouched before its read; donor last frame inside the recipient gauge; reads in + chronological order keep every target chain nested; + * the spliced positive-rank recount gives the shared-core profile with W = 2v + R - pairs and the unchanged + deficit 2v - 3 loss; + * an exact numeric replay mod 2^61-1 of the aliased signed word with arbitrary dirty scratch restores every + slot and adds x to y, for two seeds; omitting one recipient's read, reading a late recipient before its + donor's last operation, or flipping one signed coefficient is rejected. +Writes certificates/paired-cube-physical-input.json with --write; otherwise checks it. +""" +import argparse +from collections import Counter, defaultdict +import json +from pathlib import Path +import random +import sys +import tempfile + +from paired_cube.frames import basis, perp, contained + +ROOT = Path(__file__).resolve().parents[1] +REF = ROOT / 'references/paired-cube/physical' +OUTPUT = ROOT / 'certificates/paired-cube-physical-input.json' +P = (1 << 61) - 1 + + +def require(condition, message): + if not condition: + raise AssertionError(message) + + +def regenerated_word(): + from paired_cube_producer import regenerate + expected = json.loads((ROOT / 'certificates/paired-cube-complex-input.json').read_text()) + with tempfile.TemporaryDirectory(prefix='paired-cube-physical-') as d: + regenerate(expected, Path(d)) + return tuple(json.loads((Path(d) / name).read_text()) + for name in ('graph.json', 'frames.json', 'selection.json')) + (expected,) + + +def physical(g, witness, word, record, frames_in, pairs_in): + h, v, R = g['h'], g['v'], record['R'] + full = basis(1 << i for i in range(h)) + args = [None] + [None if a is None else (a[0] + 1, a[1] + 1) for a in g['args']] + inputs = g['inputs'] + roots = [dict(r, node=r['node'] + 1) for r in g['roots']] + ann = witness['annihilators'] + ops = [tuple(o) for o in word['ops']] + coef = [tuple(c) for c in word['opcoeff']] + phase1 = sorted(word['phase1']) + pset = set(phase1) + rest = [i for i in range(len(ops)) if i not in pset] + sources = {int(x): s for x, s in word['sources'].items()} + require(len(ops) == record['total_M_operations'], 'Operation count') + spans = [()] * len(args) + for x in range(1, len(args)): + spans[x] = (inputs[x - 1],) if args[x] is None else basis(spans[args[x][0]] + spans[args[x][1]]) + frames = [perp(tuple(ann[x]), h) for _, _, x in ops] # PR144's full backward-intersection frames + moved = set() + for i, F in frames_in: + require(i not in moved and basis(F) == tuple(F) != frames[i], 'Duplicate, noncanonical or unmoved frame') + frames[i] = tuple(F) + moved.add(i) + changed = len(moved) + for i, (_, _, x) in enumerate(ops): + require(contained(spans[x], frames[i]), 'Value span outside operation frame') + start = [()] * R + for x, s in sources.items(): + start[s] = (inputs[x - 1],) + gauge = {} + for z in word['selected']: + gauge[z['role']] = perp(tuple(z['annihilator']), h) + require(len(gauge[z['role']]) == z['rank'], 'Gauge rank') + start[z['role']] = gauge[z['role']] + rootframe, rootkind, rootann = {}, {}, {} + for r, s in zip(roots, word['rootroles']): + if r['kind'] == 'center': + U = spans[r['node']] + require(len(U) == h - 2, 'Centre frame rank') + A = perp(U, h) + else: + A = basis(inputs[t] for t in r['targets']) + U = perp(A, h) + require(s not in rootframe, 'Duplicate root role') + rootframe[s], rootkind[s], rootann[s] = U, r['kind'], A + role_ops = defaultdict(list) + for i, (a, b, _) in enumerate(ops): + require(a != b, 'Gate ports') + role_ops[a].append(i) + role_ops[b].append(i) + position = {i: k for k, i in enumerate(phase1 + rest)} + + def chain(s): + return [start[s]] + [frames[i] for i in role_ops[s]] + ([rootframe[s]] if s in rootframe else []) + [full] + + # Pairs and deadlines. + pairs = [(int(a), int(b)) for a, b, _ in pairs_in] + deadline = {int(b): (None if t is None else int(t)) for _, b, t in pairs_in} + donors = {a: b for a, b in pairs} + merge = {b: a for a, b in pairs} + require(len(donors) == len(merge) == len(pairs) and not set(donors) & set(merge), 'Pair aliases') + first = {s: l[0] for s, l in role_ops.items()} + for a, b in pairs: + require(a not in gauge and a not in rootframe, 'Donor kind') + require(b in gauge, 'Recipient kind') + last = role_ops[a][-1] + t = deadline[b] + if t is None: # read at the phase cut, after all of phase one + require(last in pset and first[b] not in pset, 'Early pair chronology') + else: # read immediately before operation t, the recipient's first operation + require(t == first[b] and t not in pset and position[last] < position[t], 'Late pair chronology') + require(contained(frames[last], gauge[b]), 'Donor frame outside recipient gauge') + # Chains, the spliced recount and the profile. + local = Counter() + for s in range(R): + seq = chain(s) + require(all(contained(A, B) for A, B in zip(seq, seq[1:])), 'Role chain nesting') + if s in merge: + continue + if s in sources.values(): + local[1] += 1 + if s in donors: + seq = seq[:-1] + chain(donors[s]) + dims = [len(F) for F in seq] + local.update(d1 - d0 for d0, d1 in zip(dims, dims[1:]) if d1 > d0) + for t in (s, donors.get(s)): + if t is not None and rootkind.get(t) == 'center': + local[len(rootframe[t])] += 1 + # Target chains along the actual read chronology (late reads move; sides last; inputs end). + cut = len(phase1) + when = {z['role']: (cut if deadline.get(z['role']) is None else position[deadline[z['role']]], k) + for k, z in enumerate(reversed(word['selected']))} + current = [full] * v + target = Counter() + for z in sorted(word['selected'], key=lambda z: when[z['role']]): + A = tuple(z['annihilator']) + for t in z['targets']: + require(contained(A, current[t]), 'Target chain order') + target[len(current[t]) - len(A)] += 1 + current[t] = A + for r, s in zip(roots, word['rootroles']): + if r['kind'] == 'side': + for t in r['targets']: + require(contained(rootann[s], current[t]), 'Side target order') + target[len(current[t]) - len(rootann[s])] += 1 + current[t] = rootann[s] + for t, q in enumerate(inputs): + require(contained((q,), current[t]), 'Target input') + target[len(current[t]) - 1] += 1 + require(dict(target) == {int(k): n for k, n in record['target_data_histogram'].items()}, 'Target histogram') + source = Counter({int(k): n for k, n in record['source_data_histogram'].items()}) + children = Counter({r: 3 * n for r, n in local.items() if r}) + children.update({r: 3 * n for r, n in source.items() if r}) + children.update({r: 3 * n for r, n in target.items() if r}) + tails = Counter(len(U) for s, U in gauge.items() if s not in merge) + children.update({3 * d: n for d, n in tails.items() if d}) + children[2] += 2 * v + m, W = 3 * h, 2 * v + R - len(pairs) + rank = sum(r * n for r, n in children.items()) + require(W * m - rank == 2 * v - 3 * record['loss'] == record['deficit_per_vertex'], 'Telescoping deficit') + require(all(0 < r < m for r in children), 'Proper children') + replay = scalar_replay(g, word, R, pairs, deadline, ops, coef, phase1, rest, sources, roots) + return dict(h=h, v=v, R=R, physical_R=R - len(pairs), pairs=len(pairs), + late_pairs=sum(t is not None for t in deadline.values()), changed_operation_frames=changed, + m=m, W_per_vertex=W, rank_per_vertex=rank, deficit_per_vertex=W * m - rank, loss=record['loss'], + local_histogram=dict(sorted(local.items())), physical_gauge_histogram=dict(sorted(tails.items())), + source_data_histogram=dict(sorted(source.items())), target_data_histogram=dict(sorted(target.items())), + child_histogram=dict(sorted(children.items())), scalar_replay=replay, + checks=dict(value_spans_inside_frames=True, role_chains_nested=True, pair_chronology=True, + donor_frames_inside_gauges=True, target_chains_nested_in_read_order=True, + telescoping_deficit=True)) + + +def scalar_replay(g, word, R, pairs, deadline, ops, coef, phase1, rest, sources, roots): + h, v, inputs, labels = g['h'], g['v'], g['inputs'], g['labels'] + inv = lambda a: pow(a % P, P - 2, P) + half, third, sixth = inv(2), inv(3), inv(6) + selected = [z['role'] for z in word['selected']] + deferred = set(selected) + cseed, dseed = {}, {} + for r, s in zip(roots, word['rootroles']): + if r['kind'] == 'center': + cseed[s] = [int(k == r['coordinate']) for k in range(h)] + else: + dseed[s] = {t: half if q == '1/2' else P - half for t, q in zip(r['targets'], r['coefficients'])} + cvec = [None] * R + dpart = [dict() for _ in range(R)] + for s, c in cseed.items(): + cvec[s] = list(c) + for s, d in dseed.items(): + dpart[s] = dict(d) + for (a, b, _), (ca, cb) in zip(reversed(ops), reversed(coef)): + if cvec[a] is not None: + cvec[b] = [(cb * u) % P for u in cvec[a]] if cvec[b] is None else \ + [(w + cb * u) % P for w, u in zip(cvec[b], cvec[a])] + for t, u in dpart[a].items(): + dpart[b][t] = (dpart[b].get(t, 0) + cb * u) % P + if ca != 1: + if cvec[a] is not None: + cvec[a] = [(ca * u) % P for u in cvec[a]] + dpart[a] = {t: (ca * u) % P for t, u in dpart[a].items()} + scatter = [[third if c in labels[t] else P - sixth for t in range(v)] for c in range(h)] + merge = {b: a for a, b in pairs} + late_at = defaultdict(list) + for b, t in deadline.items(): + if t is not None: + late_at[t].append(b) + leaf = {s: x for x, s in sources.items()} + cubes = [range(k, k + 8) for k in range(0, v, 8)] + + def run(seed, omit=None, early=None, flip=None): + rng = random.Random(seed) + live = sorted(set(range(R)) - set(merge)) + slot = {s: i for i, s in enumerate(live)} + alias = lambda s: slot[merge.get(s, s)] + x = [rng.randrange(P) for _ in range(v)] + z = [rng.randrange(P) for _ in live] + y0 = [rng.randrange(P) for _ in range(v)] + a, y, yc, chron = list(z), list(y0), [0] * h, [] + + def read(s, sign, seed_only=False): + val = a[alias(s)] + c = cseed.get(s) if seed_only else cvec[s] + d = dseed.get(s, {}) if seed_only else dpart[s] + if c is not None: + for k, u in enumerate(c): + yc[k] = (yc[k] + sign * u * val) % P + for t, u in d.items(): + y[t] = (y[t] + sign * u * val) % P + + def gate(i): + (dst, src, _), (ca, cb) = ops[i], coef[i] + cb = -cb if flip == i else cb + dd, ss = alias(dst), alias(src) + require(dd != ss, 'Aliased gate ports') + a[dd] = (ca * a[dd] + cb * a[ss]) % P + chron.append((dd, ss, ca, cb)) + for s in range(R): + if s not in deferred and s not in merge: + read(s, -1) + for s, xn in leaf.items(): + a[alias(s)] = (a[alias(s)] + x[xn - 1]) % P + chron.append((alias(s), None, xn - 1, None)) + for i in phase1: + gate(i) + for s in cseed: + read(s, +1, True) + for s in reversed(selected): + if s != omit and (deadline.get(s) is None or s == early): + read(s, -1) + for i in rest: + for s in late_at.get(i, ()): + if s != omit and s != early: + read(s, -1) + gate(i) + for s in dseed: + read(s, +1, True) + for cube in cubes: + qs = [inputs[t] for t in cube] + for i, t in enumerate(cube): + acc = sum(x[s] if (qs[i] ^ qs[j]).bit_count() == 6 else -x[s] if (qs[i] ^ qs[j]).bit_count() == 2 + else 0 for j, s in enumerate(cube)) + y[t] = (y[t] + acc * half) % P + for k in range(h): + if yc[k]: + for t in range(v): + y[t] = (y[t] + yc[k] * scatter[k][t]) % P + for dd, ss, ca, cb in reversed(chron): + if ss is None: + a[dd] = (a[dd] - x[ca]) % P + else: + a[dd] = (ca * (a[dd] - cb * a[ss])) % P + return a == z, all((y[t] - y0[t] - x[t]) % P == 0 for t in range(v)) + result = {'seed1': run(1), 'seed2': run(2)} + require(all(r == (True, True) for r in result.values()), 'Aliased dirty-scratch replay') + controls = {} + if pairs: + controls['omitted_recipient_read'] = run(3, omit=pairs[0][1]) + late = [b for b, t in deadline.items() if t is not None] + if late: + controls['late_read_before_donor_death'] = run(4, early=late[-1]) + controls['flipped_signed_operation'] = run(5, flip=next(i for i, c in enumerate(coef) if c[1] == -1)) + require(all(r[1] is False for r in controls.values()), 'Replay control accepted') + return dict(modulus='2^61-1', seeds_restored_and_y_plus_x=True, + rejected_controls=sorted(controls), slots=R - len(pairs)) + + +def checked_record(): + g, witness, word, record = regenerated_word() + frames_in = json.loads((REF / 'frames.json').read_text())['frames'] + pairs_in = json.loads((REF / 'pairs.json').read_text())['pairs'] + result = physical(g, witness, word, record, frames_in, pairs_in) + require(json.loads(OUTPUT.read_text()) == json.loads(json.dumps(result)), 'Frozen physical input differs') + return result + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument('--write', action='store_true') + a = parser.parse_args() + g, witness, word, record = regenerated_word() + frames_in = json.loads((REF / 'frames.json').read_text())['frames'] + pairs_in = json.loads((REF / 'pairs.json').read_text())['pairs'] + result = physical(g, witness, word, record, frames_in, pairs_in) + text = json.dumps(result, indent=2, sort_keys=True) + '\n' + if a.write: + OUTPUT.write_text(text) + else: + require(OUTPUT.read_text() == text, 'Frozen physical input differs') + print('PASS physical paired-cube word: %d operation frames moved, %d pairs (%d late), W %d, rank %d, deficit %d' + % (result['changed_operation_frames'], result['pairs'], result['late_pairs'], result['W_per_vertex'], + result['rank_per_vertex'], result['deficit_per_vertex']), flush=True) + + +if __name__ == '__main__': + main() diff --git a/scripts/paired_cube_producer.py b/scripts/paired_cube_producer.py index 3354975491..2ea477f898 100644 --- a/scripts/paired_cube_producer.py +++ b/scripts/paired_cube_producer.py @@ -13,7 +13,7 @@ from pathlib import Path import sys from paired_cube.graph import Graph -from paired_cube.modules import restricted_pairs, restricted_triples, all_but_one +from paired_cube.modules import restricted_pairs, restricted_triples, all_but_one, restricted_triples_from, restricted_pairs_from, pair_module_from, merge_outputs from paired_cube.frames import compile_graph from paired_cube.gauges import select from paired_cube.verify import verify @@ -33,7 +33,10 @@ def stage(message):print(message,file=sys.stderr,flush=True) frozen=reference/'matching-arcs.json' assert hashlib.sha256(frozen.read_bytes()).hexdigest()==pin['matching_arcs_sha256'] stage('Regenerating paired-cube signed DAG from inherited PR117 restrictions') - g=Graph(12).finish(restricted_triples(12),restricted_pairs(11),all_but_one(10)) + src=ROOT/'references/paired-cube/sources' + srcpin=json.loads((src/'SOURCE.json').read_text())['files'] + for name,digest in srcpin.items():assert hashlib.sha256((src/name).read_bytes()).hexdigest()==digest + _G=Graph(11);g=_G.finish(restricted_triples_from(src/'h20_g1.json.gz',[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 16]),pair_module_from(src/'pmod_C35_5.483127e-4.json',10),all_but_one(9));g=merge_outputs(g,_G,'w02') g['matching_frames']='coordinate' binding={k:g[k] for k in ('inputs','labels','args','signs','roots','centers')} assert hashlib.sha256(json.dumps(binding,separators=(',',':')).encode()).hexdigest()==pin['graph_sha256']