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{"basis_root_ids":[81,83,84,90,91,97,113],"blade":[[191,1],[223,-1],[247,-1],[254,-1]],"dimension":7,"frame_id":671,"norm_squared":4,"normalization_contents":[1,2,2,1,2,1,1],"outside_residual_gcds":{"2":56,"4":1},"perpendicular_frame":121,"projector_denominator":4,"root_members":63} + ] +} diff --git a/research/r420-exterior-frames/CLAIM.md b/research/r420-exterior-frames/CLAIM.md new file mode 100644 index 0000000000..2f23fe4292 --- /dev/null +++ b/research/r420-exterior-frames/CLAIM.md @@ -0,0 +1,82 @@ +# Sprint claim — independent verification + +- Assignment seed: **78**, generated once with Python `secrets.randbelow(100)` + (operating-system randomness). +- Base: `66849e1475c930fd7732b1a0067ffee7e1e95f60`. +- Track: independent verification or formalization. +- Claimed at: 2026-10-13 04:36 UTC (10:06 IST). +- Expected finish: 2026-10-13 05:31 UTC (11:01 IST). +- Exact object: the 672 rational frames in the selected PR543 R420 word, + `research/e8-bit-r420/data/bitE8mut-r420.json.gz`, source head + `2065fe4f72a914d907f1046603f5a36f17018535`. +- Hypothesis: an independent exterior-algebra implementation can certify the + frame ranks, orthogonal projections and root-line closures, and determine + exactly which odd primes (if any) invalidate those finite certificates. +- Method: integer exterior products and contraction identities; no imported + producer, row-reduction, Gram determinant, or prime-witness engine. Seed 78 + rotates the root order when selecting a spanning blade and selects adversarial + controls. All 672 frames and all 120 root memberships per frame are covered; + there is no adaptive search or early-success stopping rule. +- Deliverable: scoped mathematical note, self-contained exact checker, pinned + inputs, deterministic certificate and meaningful rejecting controls. No new + kappa or full multiplication theorem is claimed. +- Time allocation: one hour from 04:31 UTC; selection completed within the first + 20%; stop implementation by 05:16 UTC and reserve the remaining 15 minutes for + verification, overlap refresh and a single focused PR or patch handoff. + +## Selection and overlap + +Three narrow candidates were considered: + +1. Independent enclosure of the selected bit moment: already covered by #678; + #706 has also just appeared for the complex moments. +2. Direct amplitude identities for selected cold wrappers: #702 already covers + all selected four-dimensional wrappers by a different exact engine. +3. Exterior-algebra certificates for the R420 frame geometry: selected. #612 + formalizes frame-table boundaries and B-XOR interfaces; #661 optimizes fixed + B-frame paths. Neither supplies this exterior projector/prime-specialization + check. Newly opened #707 accelerates the existing Bareiss Gram engine on the + same frames; this work independently checks geometry with exterior products, + without using that engine or its adapter. It makes no performance claim. + +The accepted #626/#543 certificate and actual construction/review were read, +including the conditional compiler, hosting and analytic boundaries. The PR +ledger, research integration index and archived frame-search negatives were +consulted. This is verification of a fixed object, not reopening a closed search. +The initial queue contained 523 open PRs through #704; the immediate follow-up +was refreshed through #708 at 04:35 UTC. GitHub reports no pinned issues and +Discussions disabled; the only open issue is #650 requesting Discussions. +Accordingly the claim is recorded locally, without creating a reservation issue. + +Prepared for Ishan (@ishan-1729), with substantial OpenAI Codex assistance. + +## Owner-directed extension within the claimed object + +The owner requested ideas from `math_prime`, including Ghost Parametrization +and Ghost Properties. The bounded extension uses the same fixed R420 word: +test correctness of the target translation separately from restoration of +source/helper state, then price gifting its full-frame scalar cleanup while +retaining every frame endpoint, copy, child and hosting obligation. It does +not start a new supplier search. The owner subsequently requested multiple +GPT Astra subagents; seven bounded reviews cover frame geometry, independent +F2 replay, dirty-state compatibility, overlap, provenance, adversarial scalar +controls, and mathematical limits within the existing recursive interface. + +## Result + +Completed at the owner's wind-down request. No new exponent is claimed. +The exact ghost screen records rank-539 restoration debt, rank-120 source +readout drift and the helper-251 counterexample to unchanged second-call use. +Gifting all cleanup leaves the fixed recursive profile unchanged; the selected +complex supplier excludes the next recorded exponent grid tick under the +inherited framework. Ten focused tests, both receipt checks, the selected +record check and unchanged predecessor scalar/Gram checks pass. + +The final overlap review found concurrent #737 had submitted the same main +exterior geometry result first. We credit Sagar Thakrar and retain that part as +independent corroboration; the ghost return/carryover evidence is central. +#710's restored-cleanup cost observation and #731's different compensation +matrix are also credited. Reviewed heads are in README.md. + +Result: [reviewable contribution branch](https://github.com/ishan-1729/integer-mult-bounds/tree/research/r420-exterior-frames-seed78/research/r420-exterior-frames). +See README.md and VALIDATION.md for commands, scope, provenance and outcomes. diff --git a/research/r420-exterior-frames/GHOST-SCREEN.json b/research/r420-exterior-frames/GHOST-SCREEN.json new file mode 100644 index 0000000000..356ec14353 --- /dev/null +++ b/research/r420-exterior-frames/GHOST-SCREEN.json @@ -0,0 +1,52 @@ +{ + "assembly": { + "bit_lower": "413456747036699736067/250000000000000000000000", + "cap_minus_selected_kappa": "456684945111964070059/1956324622484976732153468750000000000000", + "complex_fixed_favorable_cap": "102387919519255428911/62602387919519255428911", + "complex_lower": "65528268492323474503/40000000000000000000000", + "complex_rejected_endpoint": "102387919519255428911/62500000000000000000000", + "kappa": "51110230636731/31250000000000000", + "kappa_with_optimistically_doubled_bit_rate": "51110230636731/31250000000000000", + "next_1e_minus_18_grid_tick_excluded": true, + "source_semantic_sha256": "a7c7786172a8e8f22dcd9426a2fb48ecb554122bb433c9f333ef75f10aa15708" + }, + "carryover_obstruction": { + "helper_to_source_rank_F2": 120, + "original_sources_all_zero": true, + "second_call_target": 0, + "second_call_target_error": 1, + "source_drift_rank_F2": 120, + "unit_helper_input_register": 251 + }, + "cleanup_gates": 2661, + "cleanup_scalar_xors": 2973, + "columns_replayed": 660, + "correct_target_rows_before_return": 120, + "debt_rows_sha256": "4e2c66610e6431bd353f63222c7e306e25af3e60cce1ff9ede4d56119c0d07eb", + "full_frame": 1, + "gifted_cleanup_child_profile": { + "1": 1320, + "2": 1440, + "4": 60, + "5": 120, + "8": 9 + }, + "helper_rows_with_debt": 420, + "ordinary_child_profile": { + "1": 1320, + "2": 1440, + "4": 60, + "5": 120, + "8": 9 + }, + "rank_mass": 5112, + "rejected_return_controls": [ + "no_cleanup", + "last_cleanup_gate_omitted" + ], + "restoration_debt_rank_F2": 539, + "scope": "Fixed word, frames, endpoint frames and global hosting; gift all scalar C work, retain every child/copy. No new construction or kappa.", + "source_rows_with_debt": 120, + "status": "PASS_R420_GHOST_RETURN_SCREEN", + "word_semantic_sha256": "e597e3885c0a8daa262afa5985ee2930443329126aece76d33159d81882bdcc2" +} diff --git a/research/r420-exterior-frames/GHOST-SCREEN.md b/research/r420-exterior-frames/GHOST-SCREEN.md new file mode 100644 index 0000000000..51f3fc69a2 --- /dev/null +++ b/research/r420-exterior-frames/GHOST-SCREEN.md @@ -0,0 +1,188 @@ +# A ghost-state return screen for the fixed R420 supplier + +**Scoped negative result, not a new construction or a general impossibility +theorem.** Allowing this word's source/helper state to remain temporarily +unrestored does not improve its paid recursive profile when its frames, +endpoint contract, copies, children and hosting stay fixed. Even gifting every +scalar cleanup operation leaves the profile unchanged. Moreover, the selected +assembly is already complex-bound, so improving the bit rate alone cannot +improve its recorded exponent on the existing 10^-18 output grid. + +## Hypothesis and exact return condition + +The motivating Ghost Parametrization idea is to enlarge the working state, +preserve an exact property, and separately establish return to the required +output state. Ishan's `math_prime` work suggested applying that distinction to +dirty auxiliary registers. This note defines the complete application here; +it does not import a theorem, algorithm, or text from that project. + +Let z comprise the 120 source and 420 helper registers, and y the 120 targets. +After the word's D, A, scatter and B sections, exact F2 replay has the form + +```text +(z, y) -> (U z, y + S z), +``` + +where S selects the original 120 sources. U has no target dependence. The +ghost property is correctness of the target translation; the return condition +is additionally `U=I` on arbitrary source and helper input, not merely on clean +helpers. Section C restores that condition. This is the scalar state contract; +the frame and compiled child contracts are additional requirements. + +The independent bit-packed replay covers every one of the 660 input columns. +Before C, all 120 targets are correct, but all 120 source rows and 420 helper +rows differ from the identity. `rank_F2(U-I)=539`. This rank diagnoses a failed +return condition; it is **not** a lower bound of 539 gates, copies, new helper +registers, or recursive children. With the full word every required row is +correct. Both omitted C and omitted last C gate are rejected. These controls +exercise the endpoint identity directly, independently of source hash checks. + +## Favorable cost before implementation + +All 2,661 C gates (2,973 scalar XORs) are at the full frame. All z registers +already have that same required final frame. Thus, if C's scalar XORs are +gifted at zero cost, each frame climb previously needed by C is still needed +by the final-frame contract. It moves to the endpoint with the same source +frame, destination frame, and rank. Earlier gates and nine retained helper +copies do not change. Exact replay checks this accounting: + +| Child rank | Ordinary count | Cleanup gifted | +|---:|---:|---:| +| 1 | 1,320 | 1,320 | +| 2 | 1,440 | 1,440 | +| 4 | 60 | 60 | +| 5 | 120 | 120 | +| 8 | 9 | 9 | + +Both rank masses are 5,112. Every source-access gate, helper, frame change, +retained copy and recursive child outside the gifted scalar cleanup remains +charged. In particular this does not treat the unrestored word as a valid +replacement supplier. + +With the selected nine-stage composition and whole-call hosting held fixed, +the inherited full profile is + +```text +H = {1:380160, 2:426624, 4:32640, 5:34560, + 6:15360, 7:23040, 8:2592, 42:3456}. +m=64, W=30720, sum(r H[r])=1956096. +``` + +These values are pinned in `research/e8-bit-r420/EXPECTED.json`. The cleanup +gift changes only the finite scalar overhead in that composition; it leaves +the child profile and inherited bad-class contribution unchanged. Hence the +same bit moment inequality determines the same rate. This is a price-first +rejection of the fixed-interface mechanism, not an unsuccessful large search. +Changing endpoint frames, copied information, or child interfaces is outside +this class and needs a new complete invoice and compatibility proof. + +More explicitly, with `E=sum(H[r])=918432`, the complete contraction moment is + +```text +Phi(c) = sum_r [r H[r]/(64*30720)] (64/r)^c + + [9567/1562500000000000] 64^c. +``` + +The second term is the inherited bad-class charge `32*m*E/(10^16*W)`. +The deficit is 9,984. Gifting C removes at most `288*2973=856224` scalar XORs +from the additive invoice, while leaving every term of Phi unchanged. The +pinned `assemble9.py` sends scalar additions to `unit`/`coefficient`, whereas +the moment depends only on H, m, W and E. This prices the existing complete +recursive profile; it supplies no recurrence for a new ghost compiler. + +## Why the same dirty state cannot feed an unchanged next call + +There is a concrete obstruction beyond the failed final return condition. +Write P for projection from z onto its original source coordinates. The exact +map above is `G(z,y)=(Uz,y+Pz)`, so two unchanged invocations yield + +```text +G^2(z,y) = (U^2 z, y + Pz + PUz). +``` + +For the fixed word, `rank(PU-P)=120`, and the helper-to-source block of U also +has rank 120. With all sources zero and only helper register **251** set to +one, `(Uz)_0=1`. The second call therefore adds one to target 0, although its +original source was zero. This is an exact counterexample to using the same +readout on carried dirty state; it is not a counterexample to all possible +ghost interfaces. Helper dirt can induce arbitrary source drift. + +The pinned nine-stage `global_k.py` requires every z row restored and the +canonical target shear in both orientations. Its retained-copy window also +requires the center to remain unchanged through erase. A new encoding E of +the ghost state would require subsequent children to read `P E^-1`, compatible +bridge and copy operations, and final reconstruction back to E=I. Those +operations and any frame changes or extra children must be priced. No such +construction is supplied here, and the already-favorable fixed-profile price +cannot improve the exponent. + +## The other supplier still binds + +The consumed assembly is the accepted #626 replay, not the historical kappa +in the #543 supplier's own receipt. Its exact lower endpoints are + +```text +b = 65528268492323474503/40000000000000000000000 +c = 413456747036699736067/250000000000000000000000 +kappa = 51110230636731/31250000000000000 +``` + +Here c is the bit rate and b the complex rate; `c>b`. The screen independently +implements the pinned seven-level arithmetic using exact fractions: + +```text +eta = beta = 10^-24 +s = min(c, (1-beta)b) - 10^-27 +a_0 = 384599/10^10 +a_(j+1) = (1-s)s + s a_j, j=0,...,6 +q = (1-2 eta)a_7 +margin = (1-eta)q/(1+q) +kappa = largest multiple of 10^-18 strictly below margin. +``` + +Replacing c by 2c reproduces exactly the same kappa. This is an intentionally +favorable arithmetic control, not a claim that a doubled bit rate is attainable. +The complex moment's next rejected endpoint in the pinned receipt is +`b_plus=b+10^-24`. Inheriting that moment certificate and its monotonicity, +even eliminating outer losses and taking the limiting iteration gives the +favorable ceiling `b_plus/(1+b_plus)`. Its excess over the selected kappa is +exactly + +```text +456684945111964070059/1956324622484976732153468750000000000000 +``` + +which is strictly between zero and 10^-18. This excludes the next recorded +grid tick with that complex supplier fixed, within the inherited moment and +outer-assembly framework. It does not exclude every smaller +real-number improvement or a different complex supplier, and does not claim +a universal exponent ceiling. + +## Reproduction and boundaries + +```sh +python3 -I -B research/r420-exterior-frames/ghost_screen.py +``` + +Expected: `PASS_R420_GHOST_RETURN_SCREEN: debt rank 539; 2973 gifted XORs; child profile unchanged; selected kappa unchanged; 2 rejected returns`. + +[GHOST-SCREEN.json](GHOST-SCREEN.json) contains the exact result, both profiles, +debt-row digest and fraction comparisons. [SOURCE.json](SOURCE.json) pins the +word and selected assembly. The implementation imports this package's exterior +geometry checker, but no producer, original scalar replay or moment engine. +The two ghost tests compare F2 rank against exhaustive image cardinality on +all 512 binary 3x3 row sets and check that target correctness does not satisfy +the restored endpoint. The geometry check has its own separate controls. + +The screen does not re-prove the selected moment enclosures or compiler, +common-label, mutable-basis, hosting, precision, tape or analytic assumptions. +The finite F2 replay is exact for this word; the fixed-profile conclusion is +the cost argument above. Neither is a formal proof of the full multiplication +theorem, nor a reason to abandon other ghost-state mechanisms. + +Concurrent predecessor credit: #710 already shows that shorter full-frame +scalar cleanup leaves the recursive terms unchanged; this screen extends the +favorable relaxation to all cleanup and tests the actual failed return. +#731's rank-120 helper-to-target compensation matrix is a different map from +the rank-120 helper-to-source drift certified here. #737 independently reached +the supporting exterior-frame result first. See README.md for reviewed heads. diff --git a/research/r420-exterior-frames/NOTICE.md b/research/r420-exterior-frames/NOTICE.md new file mode 100644 index 0000000000..e187f46291 --- /dev/null +++ b/research/r420-exterior-frames/NOTICE.md @@ -0,0 +1,53 @@ +# Attribution and license + +Prepared for Ishan (@ishan-1729), with substantial OpenAI Codex assistance in +research, implementation, mathematical exposition, tests and packaging. +GPT-6 Astra subagents provided independent geometry review, a separately +implemented scalar replay, and a bounded recursive-interface audit. Their +checks are computational/mathematical reviews, not proof-assistant proofs. +New files are contributed under this repository's Apache-2.0 license. + +The unchanged R420 word and its construction are due to eumemic, with +substantial Anthropic Claude assistance, in PR #543, source commit +`2065fe4f72a914d907f1046603f5a36f17018535`. Its own +[`NOTICE.md`](../e8-bit-r420/NOTICE.md) preserves the full predecessor chain, +including the D4 carrier skeleton, retained totals, hosting and pricing work. +No supplier source is vendored or relicensed by this package. + +The reference exporter and Gram/prime-witness checkers are credited to Dylan +Crow / utcorvusvolat-dotcom and their recorded predecessors; see +[`NOTICE.md`](../e8-bit-carrier-skeleton/NOTICE.md). Their implementation was +consulted for the exact obligation, but this geometry verifier uses exterior +products and contractions and imports none of those engines. The original +R420 scalar/nullspace checker is eumemic's independent checker. Concurrent +PR #707 by CharlesBethin optimizes the inherited Gram engine; our result is +independent geometry evidence, not a competing timing claim. + +The selected #626 complex construction and its accepted review are credited +to their authors and retained source notices in +[`community-k492-review.md`](../../docs/research/community-k492-review.md) and +[`complex-k492-owner-repair`](../complex-k492-owner-repair). Exact source pins +appear in [SOURCE.json](SOURCE.json). The cost screen inherits the selected +assembly's moment certificate and does not claim authorship of that bound. + +Concurrent exterior-algebra work by Sagar Thakrar / @thakrarsagar in +[PR #737](https://github.com/CrocSwap/integer-mult-bounds/pull/737) reached +the same main geometry result before this submission. The geometry code here +was independently developed; its inclusion is corroboration, with additional +all-pairs inclusion/complement coverage. The distinct result is the ghost +return screen. Credit also [PR #710](https://github.com/CrocSwap/integer-mult-bounds/pull/710) +for restored cleanup conjugation and unchanged child pricing, and Joshua +Snider's [PR #731](https://github.com/CrocSwap/integer-mult-bounds/pull/731) +for the different helper-to-target compensation map. None is a code dependency. + +The owner suggested Ghost Parametrization and Ghost Properties from their +`math_prime` research as motivation: distinguish an enlarged working state +and preserved property from exact return to the desired state. Only that +methodological idea is used; no private project text or code is reproduced, +and no result from that project is a dependency of this public certificate. +The operational definition and the bounded test are self-contained here. + +PR #612's formal boundary work, #661's fixed-order optimization, #678's bit +moment verification, and #702's cold-wrapper amplitude audit informed scope +selection. This package neither supersedes those contributions nor turns +their different scopes into alleged defects. diff --git a/research/r420-exterior-frames/README.md b/research/r420-exterior-frames/README.md new file mode 100644 index 0000000000..f0ed34da9e --- /dev/null +++ b/research/r420-exterior-frames/README.md @@ -0,0 +1,186 @@ +# R420 ghost-return obstruction and supporting frame certificates + +**Independent verification and a scoped negative result; no new κ.** +The [ghost-return screen](GHOST-SCREEN.md) tests a specific application of +Ghost Parametrization: retain temporarily dirty source/helper state and defer +its restoration. The first target output remains correct, but the restoration +debt has rank 539. Helper-to-source drift has rank 120; setting only helper +register 251 makes an unchanged second call corrupt target 0. Even gifting +all 2,973 cleanup XORs leaves the paid child profile unchanged under the +original frame endpoints. The selected assembly remains complex-bound. + +The supporting geometry audit was developed independently during the sprint. +Concurrent [PR #737](https://github.com/CrocSwap/integer-mult-bounds/pull/737) +by Sagar Thakrar submitted the same main exterior-algebra result first. We +credit that overlap; the distinct central contribution here is the concrete +ghost-return/carryover obstruction and its complete fixed-profile cost screen. +Our geometry receipt additionally enumerates every frame inclusion and +orthogonal complement, and remains supporting independent corroboration. + +All 672 frames of the selected #543 bit supplier have exact orthogonal +projectors with denominators dividing 4. Their dimensions, root-line +memberships, inclusions and orthogonal complements survive reduction modulo +every odd prime power. This is certified by integer exterior products, +without importing the supplier's elimination, determinant or prime-witness +code. The checked finite certificate and the specialization argument below +are not a proof-assistant formalization or a full multiplication theorem. + +## Reproduce + +From the repository root, Python 3.10+ and the standard library suffice: + +```sh +python3 -I -B research/r420-exterior-frames/ghost_screen.py +python3 -I -B research/r420-exterior-frames/verify.py +``` + +Expected: + +```text +PASS_R420_GHOST_RETURN_SCREEN: debt rank 539; 2973 gifted XORs; child profile unchanged; selected kappa unchanged; 2 rejected returns +PASS_R420_EXTERIOR_FRAMES: 672 frames; 80640 memberships; 28713 inclusions; 8 rejected controls; every odd prime +``` + +The additional screen and focused tests are: + +```sh +python3 -I -B research/r420-exterior-frames/ghost_screen.py +python3 -I -B research/r420-exterior-frames/test_verify.py +python3 -I -B research/r420-exterior-frames/test_ghost_screen.py +``` + +The screen prints `PASS_R420_GHOST_RETURN_SCREEN`; the test scripts pass eight +and two tests respectively. Optimized Python (`-O`) retains all mathematical +checks. The verifier compares every field against [CERTIFICATE.json](CERTIFICATE.json). +`--output /new/path.json` writes a fresh checked copy; it refuses to overwrite. +`--word /path/to/recompressed.json.gz` accepts another gzip container only when +its canonical decoded JSON matches the pinned word. Nothing is regenerated +in the supplier, `upstream/`, or selected-frontier files. + +## Pins and coverage + +- Base: `66849e1475c930fd7732b1a0067ffee7e1e95f60`. +- #543 supplier: `2065fe4f72a914d907f1046603f5a36f17018535`. +- Word: `research/e8-bit-r420/data/bitE8mut-r420.json.gz`; + canonical decoded SHA-256 + `e597e3885c0a8daa262afa5985ee2930443329126aece76d33159d81882bdcc2`. +- [SOURCE.json](SOURCE.json) records source hashes and Git blob IDs, including + the unchanged reference checkers and #626 assembly used by the cost screen. + The independent geometry checker consumes only the word. +- [CLAIM.md](CLAIM.md) records OS-random seed 78, the three candidate tasks, + scope, time budget and overlap review. Seed 78 rotates the candidate root + order for choosing bases; every frame and every root is still checked. + +There is no adaptive search, timeout-based conclusion, or sampled coverage. +The domain is exactly the pinned 672-frame table and its 120 integer labels: +80,640 membership obligations, comprising 14,400 members and 66,240 exclusions. +There are 28,713 ordered inclusions (including 672 equalities), 422,871 ordered +noninclusions, and 336 complementary pairs. Symmetric projector idempotence +and complement identities each check 43,008 entries. + +| Dimension | Frame count | +|---:|---:| +| 0 | 1 | +| 1 | 120 | +| 2 | 180 | +| 4 | 70 | +| 6 | 180 | +| 7 | 120 | +| 8 | 1 | + +## Certificate and specialization argument + +The form on the source labels is `dot/2`. Multiplying a nondegenerate form +by a nonzero scalar does not change its subspaces or orthogonal projector, +so work with the ordinary dot product until reduction; 2 is invertible at +every odd prime. + +For each frame, append the root vectors in seed-rotated order to an exterior +product. A zero product skips a dependent root. Divide each nonzero product +by the gcd of its integer coordinates and fix its sign. The resulting +primitive blade B, the selected root IDs and every removed gcd are in the +certificate. Masks encode increasing exterior basis indices. This calculation +uses alternating products and integer gcds, not a rational echelon basis. + +Put `S = ` and `N_ij = <ι_i B, ι_j B>`, where `ι_i` contracts with the +i-th coordinate vector. For a nonzero decomposable d-blade, the orthogonal +projector onto its span is `P=N/S`. One proof chooses a real orthonormal basis +of the span: the formula becomes the sum of its d rank-one projectors and is +independent of the chosen basis. This gives the rational identity as well. +For d=0 the contractions vanish; for d=8 the projector is the identity. +The checker independently checks symmetry, `N²=SN`, `trace(N)=dS`, and both +membership tests `B ∧ u=0` and `Nu=Su` on every source label u. + +All removed gcds are 1 or 2. If their product is c, the wedge of the selected +*original* roots is `±cB`. Cauchy–Binet therefore gives the original basis's +dot-Gram determinant `c²S`. The 672 values of S are 1 (28 frames), 4 (344), +16 (268), and 64 (32). Thus those Gram determinants are units modulo every +odd prime power. This also proves the selected roots remain a free basis of +a nondegenerate direct summand of the same dimension there: the inverse Gram +matrix gives a left inverse and the orthogonal projection. Dividing by the +normalization contents has introduced no hidden odd-prime restriction. + +For every excluded root u, the nonzero integer vector `Nu-Su` has gcd a power +of two (between 1 and 64). Consequently it cannot vanish modulo any odd prime, +or modulo any power of that prime. Every included root satisfies an exact +integer zero identity. Root membership is therefore identical after reduction. +Because each listed frame is generated by its listed roots, this preserves +*all* frame inclusions and noninclusions, not just the pairs counted by a +finite-field replay. The checker also finds each perpendicular frame and +checks `P_F + P_(F-perp) = I` entrywise; this identity reduces over the same rings. + +The reduced projectors have common denominators 1 (28 frames), 2 (100), or 4 +(544). These are address-label statements over odd-characteristic rings; +the bit **payload** XOR arithmetic is still over F2. No claim excludes F2 +payloads or proves the complete q-adic compiler/bad-class interface. + +## Controls and independence + +Eight controls run the mathematical predicates directly, bypassing source +hashes: remove a redundant root from the full frame; corrupt a projector's +diagonal; double its denominator; change its dimension; make a symmetric, +trace-preserving idempotence error; insert an A2 plane with an odd bad prime; +scale an in-frame generator by 3; and scale an outside root by 3. The latter +two isolate restrictions invisible to rational projector equality alone. +They are deliberate modified inputs, not alleged production defects. + +The actual A2 plane uses source roots 0, 73 and 78. Its primitive squared +blade norm is 3. It passes the rational closure/projector checks but has +nonzero radical `(0,0,0,2,2,0,0,1)` modulo 3. The full witness is in the +certificate. Thus **the conclusion does not cover arbitrary E8-root spans**. +Positive controls preserve the projector under generator rescaling and the +primitive blade under a different basis order. The tests additionally check +all 512 binary 3×3 exterior determinants against a signed-permutation oracle, +all 672 frames in unrotated order, and source/receipt corruption behavior. + +eumemic's #543 `xcheck.py` already checks rational closures by nullspaces. +Dylan Crow's exporter and Gram checker use echelon bases and the inherited +Bareiss/prime-witness engine. This is a separate implementation of related +obligations and supplies explicit projectors and all-odd-prime-power root +exclusions. Concurrent #707 accelerates that Bareiss engine on these same +frames; no code from #707 is used. #612 checks source boundaries and scalar +interfaces in Lean; #661 concerns fixed-order frame optimization. Those +scopes are credited and are not reclassified as unfinished by this audit. + +Submission overlap review also credits #710's shorter restored cleanup and +its unchanged-child-cost observation, and Joshua Snider's #731 compensation +kernel audit. The latter studies helper-to-target compensation K, whereas +this screen studies helper-to-source drift in U. Complete cleanup omission, +the unchanged-second-call counterexample and the inherited #626 grid ceiling +are the additional scope here. Reviewed heads: #737 +`b41e73169430641be18c988c659d2855a6a6218a`, #710 +`d97c8d20a8b885271a24c5a4e8446ac7f5697d5c`, #731 +`cf5ff54a358b400a7e69abe941d1d0ad0944b25e`. + +## Limits and credit + +This package proves no improved circuit, hosting schedule, optimum, or new +exponent. It does not certify fresh all-size compilation, common-label or +mutable-basis realization, full whole-call hosting, q-adic selector density, +precision, tape, exact recovery, or analytic transfer. Those remain inherited +from the selected construction. In particular the dyadic certificate does +**not** authorize deleting the bit bad-class surcharge. + +See [NOTICE.md](NOTICE.md) for predecessor attribution, the auxiliary-state +motivation, Apache-2.0 licensing, and substantial OpenAI Codex assistance. +Validation commands and their actual outcomes are in [VALIDATION.md](VALIDATION.md). diff --git a/research/r420-exterior-frames/SOURCE.json b/research/r420-exterior-frames/SOURCE.json new file mode 100644 index 0000000000..aab18a9db8 --- /dev/null +++ b/research/r420-exterior-frames/SOURCE.json @@ -0,0 +1,59 @@ +{ + "base_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60", + "supplier_commit": "2065fe4f72a914d907f1046603f5a36f17018535", + "word": { + "path": "research/e8-bit-r420/data/bitE8mut-r420.json.gz", + "semantic_sha256": "e597e3885c0a8daa262afa5985ee2930443329126aece76d33159d81882bdcc2" + }, + "source_pins": { + "research/e8-bit-r420/data/bitE8mut-r420.json.gz": { + "sha256": "9721a3819efd68307f66f00d84a952eb28eb7c3ad2ce79653ce295c98a764784", + "git_blob": "0c63ca62759dc63d302be6552aeeee1ede70350f" + }, + "research/e8-bit-r420/EXPECTED.json": { + "sha256": "06eccee9f6b77e140833df83407e7996dc948c17d49b0779ab171fbe463fd28d", + "git_blob": "b61d859826246e0c3dabbe096e57ad7015eadfd5" + }, + "research/e8-bit-r420/code/xcheck.py": { + "sha256": "5737d18a7826fd51e131035f2d7bebba2e2868af91dd7672a5f11208b036d424", + "git_blob": "fbff8607e4c61e18481f696b99dd38a32444e90d" + }, + "research/e8-bit-carrier-skeleton/code/bit/export_records.py": { + "sha256": "8a5999601f917dbb492bb9ae9a344611f18cc7cedc53dc3c76496a369e530eb3", + "git_blob": "3632d370486a02f5456f7643ced03f8c925b2350" + }, + "research/e8-bit-carrier-skeleton/code/bit/checkers/gram_frames.py": { + "sha256": "223a5a07731fc315a90a3e48177796babc155c1d9985c7ae5017eff6af2a0608", + "git_blob": "b740d47295c02692260d7cc01874a49fe48f78ab" + }, + "research/e8-bit-carrier-skeleton/code/pricing/prime_witnesses.py": { + "sha256": "98cd03bc9582e7bd0ab194c191ea8eea940539e510051443ab9fd3e572a364ab", + "git_blob": "7f16d930302ae879c562dd58e085d9747fcac077" + }, + "research/e8-bit-r420/NOTICE.md": { + "sha256": "12ba6e0845f222e9940b00f6c60d261db104246a8d3143da267682973479f811", + "git_blob": "cb68783d7d76c3fc13faa571429e9b067a0db3f7" + }, + "research/community-k492-audit/pr626/replay/ASSEMBLY.json": { + "sha256": "6127732e2fcdd470dab349c1c51d03cd9f0787ad0309b1b4a7c811a908028f4e", + "git_blob": "2e97b75bd8e74e8aa1b5fb83adcdc461aa07c398" + }, + "research/complex-k492-owner-repair/code/assemble.py": { + "sha256": "1b1a9024ac910dc9b9d9e30150684e1fa9cb228a4c1c275b75ab04b57fdef1dd", + "git_blob": "d1674e657ff046f9d55c917723002f270e05c42a" + }, + "research/e8-bit-r420/vendor/pr510/e8-r480-nine-stage/code/assemble9.py": { + "sha256": "4bf6768224a3fac81ccda26df24f3763f9a4fbc35ad62bef2888595cdae519fe", + "git_blob": "238226c568742dd5723b03eabae564a2eaa2bda9" + }, + "research/e8-bit-r420/vendor/pr510/e8-r480-nine-stage/vendor/e8-r715-eleven-stage-whole/code/global_k.py": { + "sha256": "6368196a393ac25bb2918880e4c4bd31ed733b762c27b6873ab391cbc0c5c3c8", + "git_blob": "3df12bd632f1fdba825ced57110a1fee82eb2857" + } + }, + "scope": "verify.py consumes only the pinned decoded R420 word. ghost_screen.py also consumes the pinned assembly as inherited arithmetic input. Remaining pins credit the unchanged reference/comparison sources. Regenerated gzip containers may differ if canonical decoded JSON agrees.", + "assembly": { + "path": "research/community-k492-audit/pr626/replay/ASSEMBLY.json", + "semantic_sha256": "a7c7786172a8e8f22dcd9426a2fb48ecb554122bb433c9f333ef75f10aa15708" + } +} diff --git a/research/r420-exterior-frames/VALIDATION.md b/research/r420-exterior-frames/VALIDATION.md new file mode 100644 index 0000000000..f9dcad2c40 --- /dev/null +++ b/research/r420-exterior-frames/VALIDATION.md @@ -0,0 +1,60 @@ +# Validation receipt + +Base: `66849e1475c930fd7732b1a0067ffee7e1e95f60`. +Environment: Ubuntu 24.04 WSL, Python 3.12.3, standard library only for this +package. Validation performed 2026-10-13. No selected source or `upstream/` +file was changed. Python 3.10 grammar compatibility was checked, but that +interpreter was not run. A scoped workflow was drafted but omitted because +the available GitHub token lacks workflow-upload scope. No remote CI success +is claimed in this receipt. + +Commands run from the repository root, all successful: + +```sh +python3 -I -B -O research/r420-exterior-frames/verify.py +python3 -I -B -O research/r420-exterior-frames/ghost_screen.py +python3 -I -B -O -m unittest discover -s research/r420-exterior-frames -p 'test_*.py' -v +make selected-record-check +python3 -I -B research/e8-bit-r420/code/xcheck.py research/e8-bit-r420/data/bitE8mut-r420.json.gz --json /tmp/r420-original-xcheck.json +python3 -I -B research/e8-bit-carrier-skeleton/code/bit/export_records.py research/e8-bit-r420/data/bitE8mut-r420.json.gz /tmp/r420-exterior-reference-records +python3 -I -B research/e8-bit-carrier-skeleton/code/bit/checkers/gram_frames.py /tmp/r420-exterior-reference-records --gram e8ham --h 8 --positive +``` + +The package runs ten tests, eight direct mathematical rejecting controls, and +two failed-return controls. It matches every field of both committed exact +receipts. Alternate root order, exact small determinant/rank oracles, +recompression acceptance, corrupted-source/receipt rejection and no-overwrite +behavior pass. Both main checkers also pass from `/tmp` with `-I -B -O`. + +The unchanged R420 checker passes all 660 symbolic columns and its H1–H5 +interface decomposition. The unchanged Gram checker reports 672 positive, +nondegenerate frames, valid prime witnesses, and primes `[2]`. The selected +record remains `51110230636731/31250000000000000`. All 11 source SHA-256/Git +blob pins and both semantic hashes match the recorded base and local inputs. + +Seven GPT-6 Astra subagents provided bounded AI-assisted review: + +1. A separate Bareiss-minor/Fraction inverse-Gram calculation matched all 672 + blades, basis determinants, projectors, denominators and 80,640 residuals. +2. A dense 660-by-660 F2 implementation independently confirmed restoration + debt rank 539, correct first target output, equal child profiles and all + seven selected assembly iterates. +3. A z-only transvection product found source drift rank 120 and the helper + 251 counterexample; the fixed child/copy interface and full moment were read. +4. An overlap review inspected #737, #710 and #731 at the heads in README.md. + The result is expressly narrowed and those concurrent authors credited. +5. A provenance/CI review checked pins, source notices, local links, action + pins (for the subsequently omitted workflow), isolation, optimized execution + and Python syntax compatibility. +6. An independent scalar-bit-list replay checked the literal two-call helper + 251 counterexample, every basis input's first target output, source drift + ranks and the positive full-cleanup two-call control. +7. An exact-fraction review checked the complete moment, finite-overhead + dependence, complex-bound ceiling and encoded-state obligations. + +These additional implementations were scratch review tools, not dependencies +of the submitted checker; the ten-test package above is the durable replay. +No independent human review, full theorem formalization or new exponent is +claimed. The full repository suite, selected cold replay and Lean build were +not rerun: this package is an additive fixed-input audit and changes no +generator, supplier, assembly or selected record. diff --git a/research/r420-exterior-frames/ghost_screen.py b/research/r420-exterior-frames/ghost_screen.py new file mode 100644 index 0000000000..eab43af6bd --- /dev/null +++ b/research/r420-exterior-frames/ghost_screen.py @@ -0,0 +1,204 @@ +#!/usr/bin/env python3 +"""Bounded auxiliary-state/return-marker screen on the selected R420 word. + +This is a favorable cost relaxation, not an executable improved supplier. +Prepared for Ishan (@ishan-1729) with substantial OpenAI Codex assistance. +SPDX-License-Identifier: Apache-2.0 +""" + +from collections import Counter +from copy import deepcopy +from fractions import Fraction as Q +import importlib.util +import json +from pathlib import Path +import sys + + +HERE = Path(__file__).resolve().parent +spec = importlib.util.spec_from_file_location("exterior_verifier", HERE / "verify.py") +v = importlib.util.module_from_spec(spec) +spec.loader.exec_module(v) + + +def rank_f2(rows): + pivots = {} + for row in rows: + while row: + pivot = row.bit_length() - 1 + if pivot not in pivots: + pivots[pivot] = row + break + row ^= pivots[pivot] + return len(pivots) + + +def replay(word, dimensions, cleanup=True): + n = 2 * word["v"] + word["R"] + current = word["start"][:] + values = [1 << r for r in range(n)] + profile, xors = Counter(), Counter() + + def climb(register, frame): + old = current[register] + if old != frame: + v.need(set(word["frames"][old]) <= set(word["frames"][frame]), "non-nested climb") + delta = dimensions[frame] - dimensions[old] + v.need(delta > 0, "nonpositive climb") + profile[delta] += 1 + current[register] = frame + + def gates(section): + for frame, dst, sources in word[section]: + for register in [dst] + sources: + climb(register, frame) + for source in sources: + values[dst] ^= values[source] + xors[section] += 1 + + gates("D") + gates("A") + # Retained helper copies are still paid and targets are still at frame zero. + v.need(all(dimensions[current[word["v"] + t]] == 0 for t in range(word["v"])), + "target left zero before scatter") + for register in word["ret"]: + profile[dimensions[current[register]]] += 1 + for target, retained_indices in enumerate(word["scatter"]): + for index in retained_indices: + values[word["v"] + target] ^= values[word["ret"][index]] + gates("B") + if cleanup: + gates("C") + # Keep the original endpoint contract even when the scalar cleanup is gifted. + for register, frame in enumerate(word["final"]): + climb(register, frame) + return values, v.counts(profile), v.counts(xors) + + +def restored(values, word): + ports = word["v"] + for r, value in enumerate(values): + expected = (1 << r) ^ (1 << (r - ports)) if ports <= r < 2 * ports else 1 << r + v.need(value == expected, "restoration/target identity") + + +def fixed_outer(bit, complex_rate): + # Literal seven-level/final-grid arithmetic of the pinned #626 assembly. + eta = beta = Q(1, 10**24) + s = min(bit, (1 - beta) * complex_rate) - Q(1, 10**27) + a = Q(384599, 10**10) + for _ in range(7): + a = (1 - s) * s + s * a + q = a * (1 - 2 * eta) + margin = (1 - eta) * q / (1 + q) + ticks = margin * 10**18 + return Q((ticks.numerator - 1) // ticks.denominator, 10**18) + + +def screen(word, frame_receipt, assembly): + dimensions = [r["dimension"] for r in frame_receipt["frames"]] + full = next(i for i, d in enumerate(dimensions) if d == 8) + v.need(all(g[0] == full for g in word["C"]), "cleanup must use full frame") + normal, profile, xors = replay(word, dimensions) + restored(normal, word) + ghost, relaxed_profile, _ = replay(word, dimensions, cleanup=False) + ports, n = word["v"], len(normal) + z = list(range(ports)) + list(range(2 * ports, n)) + target_mask = sum(1 << r for r in range(ports, 2 * ports)) + v.need(all(ghost[r] & target_mask == 0 for r in z), "target feeds a ghost register") + target_ok = all(ghost[r] == normal[r] for r in range(ports, 2 * ports)) + v.need(target_ok and profile == relaxed_profile, "favorable relaxation changed target/profile") + debt = [ghost[r] ^ (1 << r) for r in z] + v.need(all(row for row in debt), "expected every source/helper row to carry debt") + helper_mask = sum(1 << r for r in range(2 * ports, n)) + source_drift_rank = rank_f2(debt[:ports]) + helper_source_rank = rank_f2([ghost[r] & helper_mask for r in range(ports)]) + v.need(source_drift_rank == helper_source_rank == ports, "source drift is not surjective") + witness = next(r for r in range(2 * ports, n) if (ghost[0] >> r) & 1) + # Applying the exact ghost map twice gives y + Pz + PUz. A restored + # canonical shear applied twice would give y. Test an explicit helper input. + second_target_zero = (1 << ports) ^ (1 << 0) ^ ghost[0] + v.need((second_target_zero >> witness) & 1, "second-call witness lost") + rejected = [] + for name, values in [("no_cleanup", ghost)]: + try: + restored(values, word) + except v.Reject: + rejected.append(name) + else: + raise v.Reject("incorrect ghost return accepted") + mutant = deepcopy(word) + mutant["C"].pop() + try: + restored(replay(mutant, dimensions)[0], word) + except v.Reject: + rejected.append("last_cleanup_gate_omitted") + else: + raise v.Reject("omitted cleanup accepted") + + b, c, kappa = (Q(assembly[k]) for k in ("complex_lower", "bit_lower", "kappa")) + v.need(assembly["binding"] == "complex" and c > b, "complex must bind") + v.need(fixed_outer(c, b) == fixed_outer(2 * c, b) == kappa, "fixed assembly comparison") + # Inherited rejected moment endpoint; this screen does not re-prove that moment. + upper_b = b + Q(1, 10**24) + endpoints = assembly["moments"] + v.need(any(r["side"] == "complex" and not r["accept"] and Q(r["endpoint"]) == upper_b + for r in endpoints), "complex upper endpoint not present") + optimistic_cap = upper_b / (1 + upper_b) + v.need(kappa < optimistic_cap < kappa + Q(1, 10**18), "next-grid favorable ceiling") + return { + "status": "PASS_R420_GHOST_RETURN_SCREEN", + "scope": "Fixed word, frames, endpoint frames and global hosting; gift all scalar C work, retain every child/copy. No new construction or kappa.", + "word_semantic_sha256": v.digest(word), + "full_frame": full, + "cleanup_gates": len(word["C"]), + "cleanup_scalar_xors": xors["C"], + "ordinary_child_profile": profile, + "gifted_cleanup_child_profile": relaxed_profile, + "rank_mass": sum(int(r) * count for r, count in profile.items()), + "source_rows_with_debt": sum(ghost[r] != 1 << r for r in range(ports)), + "helper_rows_with_debt": sum(ghost[r] != 1 << r for r in range(2 * ports, n)), + "restoration_debt_rank_F2": rank_f2(debt), + "debt_rows_sha256": v.digest(debt), + "carryover_obstruction": { + "source_drift_rank_F2": source_drift_rank, + "helper_to_source_rank_F2": helper_source_rank, + "unit_helper_input_register": witness, + "original_sources_all_zero": True, + "second_call_target": 0, + "second_call_target_error": 1, + }, + "correct_target_rows_before_return": ports, + "columns_replayed": n, + "rejected_return_controls": rejected, + "assembly": { + "source_semantic_sha256": v.digest(assembly), + "bit_lower": str(c), "complex_lower": str(b), "kappa": str(kappa), + "kappa_with_optimistically_doubled_bit_rate": str(fixed_outer(2 * c, b)), + "complex_rejected_endpoint": str(upper_b), + "complex_fixed_favorable_cap": str(optimistic_cap), + "cap_minus_selected_kappa": str(optimistic_cap - kappa), + "next_1e_minus_18_grid_tick_excluded": True, + }, + } + + +def main(): + source = json.loads((HERE / "SOURCE.json").read_text()) + word = v.load_word(v.ROOT / source["word"]["path"], source) + frame_receipt = v.build_receipt(word, source) + v.need(frame_receipt == json.loads((HERE / "CERTIFICATE.json").read_text()), "frame certificate mismatch") + assembly = json.loads((v.ROOT / source["assembly"]["path"]).read_text()) + v.need(v.digest(assembly) == source["assembly"]["semantic_sha256"], "assembly source mismatch") + result = screen(word, frame_receipt, assembly) + v.need(result == json.loads((HERE / "GHOST-SCREEN.json").read_text()), "ghost screen mismatch") + print("PASS_R420_GHOST_RETURN_SCREEN: debt rank 539; 2973 gifted XORs; " + "child profile unchanged; selected kappa unchanged; 2 rejected returns") + + +if __name__ == "__main__": + try: + main() + except (v.Reject, OSError, ValueError, KeyError, TypeError) as error: + print("REJECT:", error, file=sys.stderr) + sys.exit(1) diff --git a/research/r420-exterior-frames/test_ghost_screen.py b/research/r420-exterior-frames/test_ghost_screen.py new file mode 100644 index 0000000000..491977e96a --- /dev/null +++ b/research/r420-exterior-frames/test_ghost_screen.py @@ -0,0 +1,37 @@ +"""Exact return-marker and cost-screen controls. SPDX-License-Identifier: Apache-2.0.""" + +import importlib.util +from itertools import product +import json +from pathlib import Path +import unittest + +HERE = Path(__file__).resolve().parent +spec = importlib.util.spec_from_file_location("ghost_screen", HERE / "ghost_screen.py") +g = importlib.util.module_from_spec(spec) +spec.loader.exec_module(g) + + +class GhostTests(unittest.TestCase): + def test_rank_against_exhaustive_image_cardinality(self): + for rows in product(range(8), repeat=3): + image = {0} + for row in rows: + image |= {x ^ row for x in image} + self.assertEqual(1 << g.rank_f2(rows), len(image)) + + def test_complete_screen_and_failure_of_target_only_return(self): + source = json.loads((HERE / "SOURCE.json").read_text()) + word = g.v.load_word(g.v.ROOT / source["word"]["path"], source) + frames = g.v.build_receipt(word, source) + assembly = json.loads((g.v.ROOT / source["assembly"]["path"]).read_text()) + result = g.screen(word, frames, assembly) + self.assertEqual(result, json.loads((HERE / "GHOST-SCREEN.json").read_text())) + values, _, _ = g.replay(word, [r["dimension"] for r in frames["frames"]], cleanup=False) + self.assertTrue(all(values[120 + t] == (1 << (120 + t)) ^ (1 << t) for t in range(120))) + with self.assertRaisesRegex(g.v.Reject, "restoration/target identity"): + g.restored(values, word) + + +if __name__ == "__main__": + unittest.main(verbosity=2) diff --git a/research/r420-exterior-frames/test_verify.py b/research/r420-exterior-frames/test_verify.py new file mode 100644 index 0000000000..49ed8f56ae --- /dev/null +++ b/research/r420-exterior-frames/test_verify.py @@ -0,0 +1,109 @@ +"""Focused algebra and failure-detection checks; Python standard library only. + +Prepared for Ishan (@ishan-1729) with substantial OpenAI Codex assistance. +SPDX-License-Identifier: Apache-2.0 +""" + +import gzip +import importlib.util +from itertools import permutations, product +import json +from pathlib import Path +import subprocess +import sys +from tempfile import TemporaryDirectory +import unittest + + +HERE = Path(__file__).resolve().parent +spec = importlib.util.spec_from_file_location("exterior_verifier", HERE / "verify.py") +v = importlib.util.module_from_spec(spec) +spec.loader.exec_module(v) + + +class ExteriorTests(unittest.TestCase): + @classmethod + def setUpClass(cls): + cls.source = json.loads((HERE / "SOURCE.json").read_text()) + cls.word_path = v.ROOT / cls.source["word"]["path"] + cls.word = v.load_word(cls.word_path, cls.source) + cls.receipt = v.build_receipt(cls.word, cls.source) + + def test_three_dimensional_leibniz_oracle(self): + # All 512 binary 3x3 matrices, including singular and negative determinants. + # The oracle explicitly sums signed permutations; no exterior operations. + terms = [(p, (-1) ** sum(p[i] > p[j] for i in range(3) for j in range(i + 1, 3))) + for p in permutations(range(3))] + for values in product((0, 1), repeat=9): + matrix = [values[i:i + 3] for i in (0, 3, 6)] + expected = sum(sign * matrix[0][p[0]] * matrix[1][p[1]] * matrix[2][p[2]] + for p, sign in terms) + blade = {0: 1} + for row in matrix: + blade = v.wedge(blade, row) + self.assertEqual(blade.get(7, 0), expected) + + def test_projectors_on_nonorthogonal_examples(self): + # Span((1,1,0), (0,1,1)) has normal (1,-1,1), hence P=I-n*n^T/3. + blade = v.wedge(v.wedge({0: 1}, [1, 1, 0]), [0, 1, 1]) + matrix = v.projection_numerator(blade, 3) + self.assertEqual(matrix, [[2, 1, -1], [1, 2, 1], [-1, 1, 2]]) + v.check_projector(matrix, 3, 2) + self.assertEqual(v.projection_numerator({0: 1}, 3), [[0] * 3 for _ in range(3)]) + self.assertEqual(v.projection_numerator({7: 1}, 3), [[int(i == j) for j in range(3)] for i in range(3)]) + + def test_all_frames_basis_order_invariant(self): + for ids, expected in zip(self.word["frames"], self.receipt["frames"]): + alternate, _ = v.frame_certificate(self.word["lines"], ids, seed=0) + self.assertEqual(alternate["blade"], expected["blade"]) + self.assertEqual(alternate["norm_squared"], expected["norm_squared"]) + self.assertEqual(alternate["projector_denominator"], expected["projector_denominator"]) + v.check_dyadic(alternate) + + def test_mathematical_controls(self): + self.assertEqual(len(self.receipt["controls"]["rejected"]), 8) + a2 = self.receipt["controls"]["A2_control"] + self.assertEqual(a2["root_ids"], [0, 73, 78]) + self.assertEqual(a2["norm_squared"], 3) + self.assertTrue(any(a2["radical_mod_3"])) + + def test_receipt_matches_all_fields(self): + self.assertEqual(self.receipt, json.loads((HERE / "CERTIFICATE.json").read_text())) + + def test_recompressed_input_and_corruption(self): + with TemporaryDirectory(prefix="r420 exterior ") as tmp: + path = Path(tmp) / "word.json.gz" + path.write_bytes(gzip.compress(json.dumps(self.word, indent=1).encode(), compresslevel=1, mtime=78)) + self.assertEqual(v.load_word(path, self.source), self.word) + corrupted = json.loads(json.dumps(self.word)) + corrupted["frames"][1].pop() + path.write_bytes(gzip.compress(json.dumps(corrupted).encode(), mtime=0)) + with self.assertRaisesRegex(v.Reject, "word semantic SHA-256 mismatch"): + v.load_word(path, self.source) + + def test_cli_rejects_tampered_certificate_without_output(self): + with TemporaryDirectory(prefix="r420 exterior ") as tmp: + tmp = Path(tmp) + bad = json.loads(json.dumps(self.receipt)) + bad["frames"][78]["norm_squared"] += 1 + certificate = tmp / "bad certificate.json" + certificate.write_text(json.dumps(bad)) + output = tmp / "output.json" + result = subprocess.run([sys.executable, "-I", "-O", "-B", str(HERE / "verify.py"), + "--check", str(certificate), "--output", str(output)], + cwd=tmp, text=True, capture_output=True, timeout=60) + self.assertEqual(result.returncode, 1) + self.assertIn("certificate mismatch", result.stderr) + self.assertFalse(output.exists()) + + def test_output_reuse_is_rejected_and_preserved(self): + with TemporaryDirectory() as tmp: + output = Path(tmp) / "existing.json" + output.write_text("preserve me") + with self.assertRaisesRegex(v.Reject, "output already exists"): + v.main(["--output", str(output)]) + self.assertEqual(output.read_text(), "preserve me") + + +if __name__ == "__main__": + unittest.main(verbosity=2) diff --git a/research/r420-exterior-frames/verify.py b/research/r420-exterior-frames/verify.py new file mode 100644 index 0000000000..3bba8e0ca6 --- /dev/null +++ b/research/r420-exterior-frames/verify.py @@ -0,0 +1,342 @@ +#!/usr/bin/env python3 +"""Independent exterior certificates for the pinned R420 frame geometry. + +Standard library only. No producer, elimination, determinant or prime-witness +code is imported. See README.md for the specialization argument and scope. +Prepared for Ishan (@ishan-1729) with substantial OpenAI Codex assistance. +SPDX-License-Identifier: Apache-2.0 +""" + +import argparse +from collections import Counter, defaultdict +from copy import deepcopy +from functools import reduce +import gzip +import hashlib +import json +from math import gcd +from pathlib import Path +import sys + + +HERE = Path(__file__).resolve().parent +ROOT = HERE.parents[1] +SEED = 78 +H = 8 + + +class Reject(ValueError): + """A failed mathematical obligation or source/receipt binding.""" + + +def need(condition, message): + # Deliberately not an assert: the checker also runs under python -O. + if not condition: + raise Reject(message) + + +def canonical(value): + return json.dumps(value, sort_keys=True, separators=(",", ":")).encode() + + +def digest(value): + return hashlib.sha256(canonical(value)).hexdigest() + + +def counts(counter): + return {str(k): v for k, v in sorted(counter.items())} + + +def power_of_two(n): + return type(n) is int and n > 0 and n & (n - 1) == 0 + + +def wedge(blade, vector): + """Append a vector; bit masks encode increasing exterior basis indices.""" + result = defaultdict(int) + for mask, coefficient in blade.items(): + for i, value in enumerate(vector): + if value and not (mask >> i) & 1: + sign = -1 if (mask >> (i + 1)).bit_count() % 2 else 1 + result[mask | (1 << i)] += sign * coefficient * value + return {mask: x for mask, x in result.items() if x} + + +def spanning_blade(lines, ids, seed=SEED): + """Greedy independent roots, with primitive integer normalization.""" + blade, selected, contents = {0: 1}, [], [] + for root in sorted(ids, key=lambda i: (i - seed) % len(lines)): + extended = wedge(blade, lines[root]) + if extended: + content = reduce(gcd, extended.values(), 0) + sign = 1 if extended[min(extended)] > 0 else -1 + blade = {mask: sign * x // content for mask, x in extended.items()} + selected.append(root) + contents.append(content) + return blade, selected, contents + + +def projection_numerator(blade, h=H): + """N_ij = ; P=N/||B||^2.""" + contractions = [] + for i in range(h): + contractions.append({ + mask ^ (1 << i): x * (-1 if (mask & ((1 << i) - 1)).bit_count() % 2 else 1) + for mask, x in blade.items() if (mask >> i) & 1 + }) + return [[sum(x * contractions[j].get(mask, 0) + for mask, x in contractions[i].items()) + for j in range(h)] for i in range(h)] + + +def check_projector(matrix, norm, dimension): + h = len(matrix) + need(type(norm) is int and norm > 0, "nonpositive blade norm") + need(all(len(row) == h and all(type(x) is int for x in row) for row in matrix), + "projector shape or entry type") + need(all(matrix[i][j] == matrix[j][i] for i in range(h) for j in range(h)), + "projector symmetry") + need(sum(matrix[i][i] for i in range(h)) == dimension * norm, "projector trace") + need(all(sum(matrix[i][k] * matrix[k][j] for k in range(h)) == norm * matrix[i][j] + for i in range(h) for j in range(h)), "projector idempotence") + + +def frame_certificate(lines, ids, seed=SEED): + """Rational frame obligations, before imposing any prime restrictions.""" + h = len(lines[0]) + need(all(type(i) is int and 0 <= i < len(lines) for i in ids), "root index") + need(len(ids) == len(set(ids)), "duplicate root in frame") + blade, selected, contents = spanning_blade(lines, ids, seed) + dimension = len(selected) + need(all(mask.bit_count() == dimension for mask in blade), "blade degree") + norm = sum(x * x for x in blade.values()) + matrix = projection_numerator(blade, h) + check_projector(matrix, norm, dimension) + outside = Counter() + members = set(ids) + for k, vector in enumerate(lines): + residual = [sum(matrix[i][j] * vector[j] for j in range(h)) - norm * vector[i] + for i in range(h)] + divisor = reduce(gcd, residual) + # Wedge membership and contraction membership are independently evaluated. + need((divisor == 0) == (not wedge(blade, vector)), "wedge/projector membership") + need((divisor == 0) == (k in members), "root closure") + if divisor: + outside[divisor] += 1 + denominator = norm // reduce(gcd, (x for row in matrix for x in row), norm) + record = { + "dimension": dimension, + "basis_root_ids": selected, + "normalization_contents": contents, + "blade": [[mask, x] for mask, x in sorted(blade.items())], + "norm_squared": norm, + "projector_denominator": denominator, + "root_members": len(ids), + "outside_residual_gcds": counts(outside), + } + return record, matrix + + +def check_dyadic(record): + need(power_of_two(record["norm_squared"]), "blade norm has an odd prime factor") + need(all(power_of_two(n) for n in record["normalization_contents"]), + "basis normalization has an odd prime factor") + need(all(power_of_two(int(n)) for n in record["outside_residual_gcds"]), + "outside residual has an odd prime factor") + + +def validate_labels(word): + need((word["h"], word["v"], word["R"], word["form"]) == (8, 120, 420, "/2"), + "R420 header") + lines = word["lines"] + need(len(lines) == 120, "root count") + need(all(len(v) == H and all(type(x) is int for x in v) for v in lines), "root shape") + need(all(sum(x * x for x in v) == 4 for v in lines), "root norm") + need(all(abs(sum(x * y for x, y in zip(a, b))) <= 2 + for i, a in enumerate(lines) for b in lines[i + 1:]), "distinct root lines") + need(all(sum(x * y for x, y in zip(a, b)) % 2 == 0 for a in lines for b in lines), + "integral E8 form") + need(len(word["frames"]) == 672, "frame count") + need(len({tuple(sorted(ids)) for ids in word["frames"]}) == 672, "duplicate frames") + + +def geometry(word): + validate_labels(word) + lines, frames = word["lines"], word["frames"] + records, matrices = [], [] + for frame_id, ids in enumerate(frames): + record, matrix = frame_certificate(lines, ids) + check_dyadic(record) + record["frame_id"] = frame_id + records.append(record) + matrices.append(matrix) + + lookup = {tuple(sorted(ids)): i for i, ids in enumerate(frames)} + for frame_id, ids in enumerate(frames): + perpendicular = tuple(i for i, v in enumerate(lines) + if all(sum(x * y for x, y in zip(v, lines[j])) == 0 for j in ids)) + need(perpendicular in lookup, "perpendicular frame absent") + other = lookup[perpendicular] + a, b = records[frame_id], records[other] + need(a["dimension"] + b["dimension"] == H, "complement dimension") + s, t = a["norm_squared"], b["norm_squared"] + need(all(t * matrices[frame_id][i][j] + s * matrices[other][i][j] == s * t * (i == j) + for i in range(H) for j in range(H)), "complement projectors") + a["perpendicular_frame"] = other + + sets = [set(ids) for ids in frames] + inclusions = 0 + for i, a in enumerate(sets): + for j, b in enumerate(sets): + if a <= b: + need(i == j or records[i]["dimension"] < records[j]["dimension"], + "proper inclusion dimension") + inclusions += 1 + summary = { + "frames": len(records), + "root_membership_obligations": len(records) * len(lines), + "inside_memberships": sum(len(ids) for ids in frames), + "outside_memberships": sum(len(lines) - len(ids) for ids in frames), + "ordered_inclusions_including_equalities": inclusions, + "ordered_noninclusions": len(frames) ** 2 - inclusions, + "perpendicular_pairs": len(records) // 2, + "projector_entry_identities": len(records) * H * H, + "complement_entry_identities": len(records) * H * H, + "dimensions": counts(Counter(r["dimension"] for r in records)), + "primitive_blade_norms": counts(Counter(r["norm_squared"] for r in records)), + "projector_denominators": counts(Counter(r["projector_denominator"] for r in records)), + "normalization_contents": counts(Counter(n for r in records for n in r["normalization_contents"])), + "outside_residual_gcds": counts(sum((Counter({int(k): v for k, v in r["outside_residual_gcds"].items()}) + for r in records), Counter())), + } + return records, matrices, summary + + +def negative_controls(word, records, matrices): + """Exercise mathematical predicates directly, bypassing all input hashes.""" + lines = word["lines"] + rejected = [] + + def reject(name, expected_reason, action): + try: + action() + except Reject as error: + need(str(error) == expected_reason, name + ": unexpected rejection: " + str(error)) + rejected.append({"name": name, "reason": str(error)}) + else: + raise Reject("negative control accepted: " + name) + + # The remaining 119 lines still span all eight dimensions. + reject("missing_redundant_root", "root closure", + lambda: frame_certificate(lines, [i for i in range(120) if i != SEED])) + selected_frame = next(i for i, ids in enumerate(word["frames"]) if ids == [SEED]) + record, matrix = records[selected_frame], matrices[selected_frame] + norm = record["norm_squared"] + broken = deepcopy(matrix) + broken[0][0] += norm + reject("projector_diagonal_changed", "projector trace", + lambda: check_projector(broken, norm, 1)) + reject("projector_denominator_doubled", "projector trace", + lambda: check_projector(matrix, 2 * norm, 1)) + reject("dimension_changed", "projector trace", + lambda: check_projector(matrix, norm, 2)) + # A trace-preserving symmetric perturbation must reach the idempotence guard. + broken2 = deepcopy(matrix) + broken2[0][0] += norm + broken2[1][1] -= norm + reject("trace_preserving_projector_corruption", "projector idempotence", + lambda: check_projector(broken2, norm, 1)) + + # An actual rational E8 A2 plane: legal over Q, degenerate at p=3. + u = SEED + v = next(i for i in range(120) if abs(sum(x * y for x, y in zip(lines[u], lines[i]))) == 2) + blade = wedge(wedge({0: 1}, lines[u]), lines[v]) + a2_ids = [i for i, root in enumerate(lines) if not wedge(blade, root)] + a2, _ = frame_certificate(lines, a2_ids) + reject("A2_plane_at_prime_3", "blade norm has an odd prime factor", lambda: check_dyadic(a2)) + sign = sum(x * y for x, y in zip(lines[u], lines[v])) // 2 + radical = [(x + sign * y) % 3 for x, y in zip(lines[u], lines[v])] + need(any(radical) and all(sum(x * y for x, y in zip(radical, lines[i])) % 3 == 0 for i in a2_ids), + "A2 radical witness") + + # Rational projectors alone do not detect nonunit changes of integral generators. + scaled = deepcopy(lines) + scaled[u] = [3 * x for x in scaled[u]] + scaled_frame, scaled_matrix = frame_certificate(scaled, [u]) + need(scaled_matrix == matrix and scaled_frame["norm_squared"] == norm, + "scaled-basis positive rational control") + reject("basis_scaled_by_3", "basis normalization has an odd prime factor", + lambda: check_dyadic(scaled_frame)) + scaled = deepcopy(lines) + scaled[v] = [3 * x for x in scaled[v]] + outside_frame, _ = frame_certificate(scaled, [u]) + reject("outside_root_scaled_by_3", "outside residual has an odd prime factor", + lambda: check_dyadic(outside_frame)) + + # A second traversal order must recover the same primitive oriented blade. + alternate, alternate_matrix = frame_certificate(lines, a2_ids, seed=(SEED + 1) % 120) + need(alternate["blade"] == a2["blade"], "basis-order invariance") + return { + "rejected": rejected, + "A2_control": {"root_ids": a2_ids, "chosen_pair": [u, v], "norm_squared": a2["norm_squared"], + "radical_mod_3": radical, "basis_coefficients_mod_3": [1, sign % 3], + "rational_dimension": 2, "rational_closure_passes": True}, + "positive_controls": ["scaled basis has identical rational projector", "basis-order invariance"], + } + + +def load_word(path, source): + with gzip.open(path, "rt", encoding="utf-8") as handle: + word = json.load(handle) + need(digest(word) == source["word"]["semantic_sha256"], "word semantic SHA-256 mismatch") + return word + + +def build_receipt(word, source): + records, matrices, summary = geometry(word) + controls = negative_controls(word, records, matrices) + return { + "status": "PASS_R420_EXTERIOR_FRAMES", + "schema": "r420-exterior-frames/1", + "assignment_seed": SEED, + "base_commit": source["base_commit"], + "supplier_commit": source["supplier_commit"], + "word_semantic_sha256": digest(word), + "geometry_sha256": digest({k: word[k] for k in ("lines", "frames")}), + "summary": summary, + "controls": controls, + "frames": records, + } + + +def main(argv=None): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--word", type=Path, help="Alternate gzip container for the same pinned word") + parser.add_argument("--check", type=Path, default=HERE / "CERTIFICATE.json") + parser.add_argument("--output", type=Path, help="Write computed certificate to a NEW file after checking") + args = parser.parse_args(argv) + if args.output: + need(not args.output.exists(), "output already exists") + source = json.loads((HERE / "SOURCE.json").read_text(encoding="utf-8")) + word = load_word(args.word or ROOT / source["word"]["path"], source) + receipt = build_receipt(word, source) + expected = json.loads(args.check.read_text(encoding="utf-8")) + need(receipt == expected, "certificate mismatch") + if args.output: + with args.output.open("x", encoding="utf-8") as handle: + json.dump(receipt, handle, sort_keys=True, indent=2) + handle.write("\n") + s = receipt["summary"] + print(f"PASS_R420_EXTERIOR_FRAMES: {s['frames']} frames; " + f"{s['root_membership_obligations']} memberships; " + f"{s['ordered_inclusions_including_equalities']} inclusions; " + f"{len(receipt['controls']['rejected'])} rejected controls; every odd prime") + return 0 + + +if __name__ == "__main__": + try: + sys.exit(main()) + except (Reject, OSError, ValueError, KeyError, TypeError) as error: + print("REJECT:", error, file=sys.stderr) + sys.exit(1)