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+ "word_payload_sha256": "e597e3885c0a8daa262afa5985ee2930443329126aece76d33159d81882bdcc2" +} diff --git a/research/r420-exterior-audit/CLAIM.md b/research/r420-exterior-audit/CLAIM.md new file mode 100644 index 0000000000..21c2cb0588 --- /dev/null +++ b/research/r420-exterior-audit/CLAIM.md @@ -0,0 +1,48 @@ +# R420 exterior frame audit + +Status: completed on 2026-10-13. GitHub returned no pinned issues; the claim +was recorded locally before implementation. + +- Track: independent verification; OS-random assignment seed 83. +- Base: `66849e1475c930fd7732b1a0067ffee7e1e95f60`. +- Object: every rational frame of eumemic's selected PR543 R420 bit word, + including its closed root-line set and the nested moves used by that word. +- Method: primitive exterior coordinates and contraction projectors over exact + integers, with seed-83 input ordering. No import of the predecessor rank, + Hermite-normal-form, determinant or geometry routines. +- Deliverable: standard-library verifier, exact receipt, mathematical explanation, + source pins and negative controls. No new kappa or full theorem claim. +- Expected finish: 2026-10-13 05:25 UTC. Final 12 minutes reserved for checks + and handoff. + +Three candidates were considered. PR543 moment verification overlaps #678; +selected cold-wrapper verification overlaps #702 and #712. This frame audit +instead supplies a separate algebraic method for the finite geometry obligation. +The initial scan covered 535 open PRs through #716. #707 accelerates the existing +Gram checker; #709 caches the existing geometry checker; #612 formalizes literal +boundary and scalar interfaces. Their results are credited but are not code +dependencies. No exterior-coordinate projector audit was found in that scan. + +Acceptance requires exact checks for all frames and all distinct ordered frame +transitions in D/A/scatter/B/C/finalization, dyadic projector denominators, +closed line sets, expected source/target endpoints, and rejection of deliberately +broken geometry. Scalar XOR semantics, hosted invoices, analytic bounds and +all-size realization remain outside this contribution. Existing sources, +`upstream/` and the selected frontier remain unchanged. + +Prepared by Sagar Thakrar with substantial OpenAI Codex assistance, Apache-2.0. + +## Result update + +All 672 frames passed the independent exterior audit, including 2,460 distinct +nested pairs, 66,240 nonincidences and dyadic basis/Gram witnesses. The complete +certificate reproduces with 258 positive oracles and 11 rejected controls. +The selected kappa is unchanged. The final overlap scan covered 549 open PRs +through #730, followed by the new #731/#732 heads immediately before submission. +No duplicate rational-projector audit was found. Verification and its limits +are recorded in the README. No issue or PR belonging to another contributor +was modified. + +The reviewable result is on +[the contribution branch](https://github.com/thakrarsagar/integer-mult-bounds/tree/research/r420-exterior-frame-audit/research/r420-exterior-audit). +One focused upstream PR will carry this branch; no merge is requested. diff --git a/research/r420-exterior-audit/README.md b/research/r420-exterior-audit/README.md new file mode 100644 index 0000000000..3355c0db31 --- /dev/null +++ b/research/r420-exterior-audit/README.md @@ -0,0 +1,203 @@ +# Independent exterior-algebra audit of the R420 frames + +**Independent finite verification; no new kappa.** All 672 rational frames in +eumemic's selected PR543 bit word have orthogonal projectors with common +denominator 1, 2 or 4. An exterior-algebra implementation independently checks +their root-line closure and all 2,460 distinct nested pairs used by the word. +It also supplies exact witnesses that the listed root spans and incidences +survive reduction over every odd prime-power ring. + +This corroborates the predecessor's finite geometry result by a different +method. It does not remove the other prime restrictions or the conditional +physical, recursive and analytic interfaces of the multiplication construction. +No defect in the selected word was found. + +## Reproduce + +From the repository root, with Python 3.9 or later and its standard library: + +```sh +python3 -I -B research/r420-exterior-audit/verify.py +``` + +Expected output: + +```text +PASS_R420_EXTERIOR_GEOMETRY: 672 frames; 2460 nested pairs; 258 positive oracles; 11 rejected controls +``` + +The command checks pinned inputs, recomputes every witness, runs the controls, +and compares the complete result with [CERTIFICATE.json](CERTIFICATE.json). +It does not write files. `--emit` prints the full regenerated certificate. +Only the decompressed word hash determines word equality, so recompressing +identical payload bytes is allowed. The original container hash is provenance. + +## Exact result + +| Quantity | Count | +|---|---:| +| Frames of ranks 0, 1, 2, 4, 6, 7, 8 | 1, 120, 180, 70, 180, 120, 1 | +| Projector denominator 1 | 28 | +| Projector denominator 2 | 100 | +| Projector denominator 4 | 544 | +| Primitive exterior squared norm 1, 4, 16, 64 | 28, 344, 268, 32 | +| Persistent-register moves in A, B, C | 1,649, 840, 451 | +| Distinct ordered nested frame pairs | 2,460 | +| Nonincident frame/root pairs checked | 66,240 | +| Retained copy frames | 9, each rank 8 | + +There are no persistent-register moves in D, scatter or finalization. The +2,940 move occurrences agree with the predecessor's 29,874-record export after +subtracting 26,916 binary additions and 18 COPY/ERASE records. This audit checks +the retained frames and zero target frames at scatter, not the COPY/ERASE +operator or its restoration semantics. + +For each frame, the certificate records a root basis, its exterior content, +the primitive exterior coordinates, their squared norm, the reduced common +projector denominator and a histogram of nonzero residual contents. Frame +records use the original frame-table order. A coordinate mask denotes the +increasing wedge of the corresponding axes. The stream record lists every +distinct ordered pair rather than only a digest or a sample. + +## Method and finite implication + +The word uses eight integer coordinates with inner product `dot/2`. Multiplying +the inner product by a nonzero scalar does not change an orthogonal projector. +All exterior norms and contractions below use the ordinary dot product. +The implementation imports no producer, rational elimination, Hermite reduction, +determinant, Gram checker or geometry cache. It uses integer wedge products and +contractions instead. + +For a frame of rank r, append its listed root vectors to an exterior product, +skipping a vector exactly when its wedge is zero. Divide each nonzero result by +its coefficient gcd and normalize its sign. Root indices are ordered by +`(index - 83) mod 120`; reversing that order must give the same normalized +exterior vector w. This cross-checks two basis orders on every frame. The product c +of the discarded positive gcds is retained, so the selected basis B satisfies +`wedge(B) = +/- c w`. The empty basis has w=1 and c=1. + +Let `D = sum_I w_I^2`. For each coordinate i, contract w with the i-th coordinate +vector, obtaining `a_i`. Set `N_ij = `. Then `P = N/D` is the orthogonal +projector onto the frame. To see this, choose an orthonormal basis u_1,...,u_r +of its real span and write `w = alpha u_1 wedge ... wedge u_r`. The r products +obtained by omitting one u_k are orthonormal. Their contraction coefficients +therefore give `N_ij = alpha^2 sum_k u_ki u_kj` and `D = alpha^2`. +The rank-zero case gives N=0 and D=1. This argument establishes the identity; +the verifier checks its finite instances using exact integers. + +The code reduces N and D by their common gcd, giving P=n/d, and checks +symmetry, `n*n = d*n`, and `trace(n) = r*d`. A listed root v lies in the frame +exactly when `n*v = d*v`. Testing all 120 roots for every frame proves closure +relative to this finite label set. A move from P=a/ad to Q=b/bd is nested +exactly when `b*a = bd*a`; the stream additionally requires a strict rank +increase for a changed frame. The source starts, target hyperplanes, helper +endpoints, D frames and retained-copy frames are checked separately. + +Two extra witnesses justify reduction at every odd prime, without sampling +primes or assuming that positive rational determinants suffice: + +1. Every c and D is a power of two. Cauchy-Binet gives + `det(B B^T) = c^2 D`, so the Gram determinant for `dot/2` is + `c^2 D / 2^r`. It is a unit over `Z[1/2]` and over every `Z/p^e` with p odd + and e >= 1. Thus the actual listed basis spans a nondegenerate free + rank-r direct summand there. Tracking c matters: primitive normalization + alone would conceal an odd index in the supplied basis. +2. For each nonincident root, the gcd of the coordinates of `n*v-d*v` is a + power of two. At every odd p at least one residual coordinate is a unit + modulo p. A nonincident listed root therefore remains nonincident modulo + p^e. Incident roots remain incident by the exact zero identity. + +Since d is a unit at every odd p, the exact projector and nesting identities +also reduce there. In particular nested frames retain their ranks and split +orthogonally: for nested P and Q, Q-P is the complementary idempotent inside Q. +These statements concern this frame table and these 120 labels. They do not +assert that it exhausts all modular roots or establish any all-size compiler. + +## Controls and verification boundary + +The 258 positive oracles cover all 256 coordinate subspaces, an oblique line, +and a mixed-sign plane with an explicit projector. Eleven negative controls +reject a corrupted projector, an isotropic line at prime 3, a closed A2 frame, +a nondecomposable exterior vector, an odd-index basis, a new incidence modulo 3, +an omitted line, a duplicate line, a wrong target hyperplane, an illegal +phase-B descent and a nonzero D frame. + +The A2 control uses the word's actual root indices `[0,16,23]`. Its rational +root-line set is closed, but its primitive exterior squared norm and projector +denominator are both 3. It demonstrates why rational closure and positive +Gram determinant alone would not justify the odd-prime conclusion. It is an +alternative frame, not a counterexample to the selected construction. The +odd-index and extra-incidence controls use synthetic vectors to isolate those +two logically separate obligations. + +The finite domain is exactly the pinned 672 frames, all 80,640 frame/root +incidence tests, and all 2,940 persistent moves in the frozen stream. It is +exhaustive within that domain, with no search stopping heuristic. The assignment +seed changes the greedy basis order, not the covered domain. + +This checker does not independently prove that the labels generate the E8 +lattice, replay scalar XOR semantics, reconstruct hosted parents, price calls, +or validate coefficients, tapes, rounding or analytic transfer. Those remain +the predecessor's checks and inherited assumptions. This is an executable +finite certificate with a written algebraic argument, not a Lean proof or a +formal verification of integer multiplication. + +## Provenance and credit + +[SOURCE.json](SOURCE.json) pins the base commit +`66849e1475c930fd7732b1a0067ffee7e1e95f60`, PR543 source +`2065fe4f72a914d907f1046603f5a36f17018535`, reference files and word payload. +At that base, the [selected result](../../certificates/selected-result.json) +is #626 with #543, conditional kappa `51110230636731/31250000000000000`. +The [maintainer review](../../docs/research/community-k492-review.md) explains +the construction and inherited obligations. This package changes none of them. + +Credit belongs to eumemic for the R420 word and its independent checker, +Dylan Crow for the E8 word/export and earlier Gram checks, and the supplier +lineage credited in the [PR543 README](../e8-bit-r420/README.md), including +DreamingOfClouds, Zhihao Chen and the shared-label work. Chafik Boukhalfa's #626 +uses this bit supplier with the complex construction reviewed by Douglas +Colkitt. No predecessor code is copied or imported by the new verifier. + +The [local claim](CLAIM.md) records the three candidates and overlap check. +Concurrent #707 and #709 accelerate existing geometry checks; #612 formalizes +boundary and scalar interfaces. This contribution instead reconstructs finite +geometry using exterior algebra and records the odd-prime incidence witnesses. +Those concurrent implementations are not dependencies. + +The submission refresh covered 549 open PRs through #730. #718 studies split +target-read schedules, #726 studies A2 gather neighborhoods, and #729 audits +complex helper lifetimes. None supplies this exterior-coordinate frame audit. +The A2 control here is an independently derived prime-three obstruction, not +a gather construction or a claimed census result. +The final pre-submission metadata check also covered #731's binary compensation +kernel and #732's star punctures; neither studies these rational projectors. + +## Local validation + +The reproduction command passed on macOS arm64 with CPython 3.9.6 and 3.14.8, +including 3.9.6 with `-O`. The selected-record integrity check and the original +PR543 `xcheck.py` also passed. The latter independently retains its complete +660-column scalar replay and interleaved-interface checks. + +A second AI-assisted implementation used rational Gram-Schmidt projectors and +Gaussian-elimination minors, with the listed root order rather than the seeded +order. It matched every frame's primitive exterior coordinates, norm, all 64 +projector entries and residual-content histogram, and every used transition. +It also checked every emitted basis content by maximal minors. Its implementation +was a separate review experiment, not an additional committed checker. + +Six in-memory implementation mutations were detected: wrong wedge signs, +omitted closure, omitted basis content, omitted residual-prime check, omitted +idempotence and reversed nesting. Recompressed word bytes reproduced the entire +certificate; changed payload bytes and a changed certificate were rejected. +Local Markdown links and staged whitespace were checked. + +The repository-wide suite, selected cold replay, native cohort replay, Linux CI +and Lean were not rerun. This isolated package does not change their inputs or +claim their obligations as newly verified. + +Prepared by **Sagar Thakrar / @thakrarsagar**, with substantial **OpenAI Codex +assistance** in implementation, mathematical checking and documentation. +New material is under Apache-2.0. Independent implementations here were also +AI-assisted; no independent human review is claimed. diff --git a/research/r420-exterior-audit/SOURCE.json b/research/r420-exterior-audit/SOURCE.json new file mode 100644 index 0000000000..27becc7785 --- /dev/null +++ b/research/r420-exterior-audit/SOURCE.json @@ -0,0 +1,24 @@ +{ + "base_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60", + "assignment_seed": 83, + "supplier_pr": 543, + "supplier_commit": "2065fe4f72a914d907f1046603f5a36f17018535", + "consuming_pr": 626, + "consuming_commit": "660de893d27bb7abd1d5b01b1640207528a5c851", + "word": "research/e8-bit-r420/data/bitE8mut-r420.json.gz", + "committed_container_sha256": "9721a3819efd68307f66f00d84a952eb28eb7c3ad2ce79653ce295c98a764784", + "word_payload_sha256": "e597e3885c0a8daa262afa5985ee2930443329126aece76d33159d81882bdcc2", + "reference_sha256": { + "research/e8-bit-r420/README.md": "9807be57c6adf26b30e4f54e2f8160075bbc05091f1a6d157e43d2a4620c1c9f", + "research/e8-bit-r420/code/xcheck.py": "5737d18a7826fd51e131035f2d7bebba2e2868af91dd7672a5f11208b036d424", + "research/e8-bit-carrier-skeleton/code/bit/export_records.py": "8a5999601f917dbb492bb9ae9a344611f18cc7cedc53dc3c76496a369e530eb3" + }, + "selected_at_base": { + "kappa": "51110230636731/31250000000000000", + "certificate_sha256": "b3bf402f31a69bde0629936a9b569bed48281568d7b16d5fba120b5311ea1f53", + "review_sha256": "3b731d30e80010b15408a289b0b32192700885474c8b36d48ee3320a7a8c5c47" + }, + "author": "Sagar Thakrar (@thakrarsagar)", + "assistance": "Substantial OpenAI Codex assistance in implementation, mathematical review and documentation", + "license": "Apache-2.0" +} diff --git a/research/r420-exterior-audit/verify.py b/research/r420-exterior-audit/verify.py new file mode 100644 index 0000000000..76b21f7365 --- /dev/null +++ b/research/r420-exterior-audit/verify.py @@ -0,0 +1,320 @@ +#!/usr/bin/env python3 +"""Independent finite R420 geometry audit. Sagar Thakrar / OpenAI Codex. + +Apache-2.0. Uses no predecessor implementation and no third-party packages. +""" + +import argparse +from collections import Counter +import copy +import gzip +import hashlib +import json +from math import gcd +from pathlib import Path + + +HERE = Path(__file__).resolve().parent +ROOT = HERE.parents[1] +SEED = 83 +H = 8 + + +class Rejected(ValueError): + pass + + +def require(condition, message): + if not condition: + raise Rejected(message) + + +def dyadic(n): + return n > 0 and n & (n - 1) == 0 + + +def wedge(w, vector): + """Append one vector; masks label the increasing exterior basis.""" + out = Counter() + for mask, value in w.items(): + for j, x in enumerate(vector): + if x and not mask & (1 << j): + sign = -1 if bin(mask >> (j + 1)).count("1") % 2 else 1 + out[mask | (1 << j)] += sign * value * x + return {mask: value for mask, value in out.items() if value} + + +def exterior_basis(lines, indices): + w = {0: 1} + selected = [] + content = 1 + for i in indices: + candidate = wedge(w, lines[i]) + if not candidate: + continue + divisor = gcd(*candidate.values()) + content *= divisor + if candidate[min(candidate)] < 0: + divisor = -divisor + w = {mask: value // divisor for mask, value in candidate.items()} + selected.append(i) + return w, selected, content + + +def projector(w): + contractions = [] + for j in range(H): + contractions.append({ + mask ^ (1 << j): value * (-1 if bin(mask & ((1 << j) - 1)).count("1") % 2 else 1) + for mask, value in w.items() if mask & (1 << j) + }) + norm = sum(value * value for value in w.values()) + numerator = [[sum(value * contractions[j].get(mask, 0) + for mask, value in contractions[i].items()) + for j in range(H)] for i in range(H)] + divisor = gcd(norm, *(value for row in numerator for value in row)) + return [[value // divisor for value in row] for row in numerator], norm // divisor, norm + + +def product(a, b): + return [[sum(a[i][k] * b[k][j] for k in range(H)) + for j in range(H)] for i in range(H)] + + +def apply(a, vector): + return [sum(x * y for x, y in zip(row, vector)) for row in a] + + +def check_projector(n, d, rank): + require(dyadic(d), "non-dyadic projector denominator") + require(all(n[i][j] == n[j][i] for i in range(H) for j in range(H)), + "projector not symmetric") + require(product(n, n) == [[d * x for x in row] for row in n], + "projector not idempotent") + require(sum(n[i][i] for i in range(H)) == rank * d, "projector trace") + + +def check_frame(lines, indices): + require(len(indices) == len(set(indices)) and + all(type(i) is int and 0 <= i < len(lines) for i in indices), + "frame line indices") + ordered = sorted(indices, key=lambda i: (i - SEED) % len(lines)) + w, basis, content = exterior_basis(lines, ordered) + n, d, norm = projector(w) + rank = len(basis) + check_projector(n, d, rank) + require(dyadic(content) and dyadic(norm), "non-dyadic basis Gram witness") + inside = [] + residual_contents = Counter() + for i, line in enumerate(lines): + residual = [a - d * x for a, x in zip(apply(n, line), line)] + divisor = gcd(*residual) + if divisor == 0: + inside.append(i) + else: + require(dyadic(divisor), "root incidence can change at an odd prime") + residual_contents[str(divisor)] += 1 + require(inside == sorted(indices), "frame not closed") + # Changing every greedy choice must preserve the primitive exterior line. + reverse, _, _ = exterior_basis(lines, reversed(ordered)) + require(reverse == w, "exterior coordinates depend on basis order") + receipt = {"rank": rank, "basis": basis, "basis_content": content, + "primitive_norm_squared": norm, "projector_denominator": d, + "exterior": sorted([mask, value] for mask, value in w.items()), + "nonincident_residual_contents": dict(sorted(residual_contents.items()))} + return n, d, receipt + + +def geometry(word): + require((word["h"], word["v"], word["R"], word["form"]) == + (8, 120, 420, "/2"), "R420 header") + lines = word["lines"] + require(len(lines) == 120 and all(len(line) == H and + all(type(x) is int for x in line) for line in lines), "label dimensions") + require(all(sum(x * x for x in line) == 4 for line in lines), "label norm") + require(all(sum(x * y for x, y in zip(lines[i], lines[j])) in (-2, 0, 2) + for i in range(120) for j in range(i)), "label inner product") + frames = [check_frame(lines, indices) for indices in word["frames"]] + zero = [i for i, (_, _, rec) in enumerate(frames) if rec["rank"] == 0] + full = [i for i, (_, _, rec) in enumerate(frames) if rec["rank"] == H] + require(len(zero) == len(full) == 1, "unique zero and full frames") + return frames, zero[0], full[0] + + +def check_stream(word, frames, zero, full): + v, size = 120, 660 + require(len(word["start"]) == len(word["final"]) == size, "endpoint lengths") + require(all(type(f) is int and 0 <= f < len(frames) + for f in word["start"] + word["final"]), "endpoint frame index") + for r, f in enumerate(word["start"]): + if r < v: + n, d, rec = frames[f] + require(rec["rank"] == 1 and apply(n, word["lines"][r]) == + [d * x for x in word["lines"][r]], "source start line") + else: + require(f == zero, "non-source start frame") + for r, f in enumerate(word["final"]): + if v <= r < 2 * v: + n, _, rec = frames[f] + require(rec["rank"] == 7 and apply(n, word["lines"][r - v]) == [0] * H, + "target final hyperplane") + else: + require(f == full, "source/helper final frame") + cur = word["start"][:] + edges = set() + moves = Counter() + + def move(register, frame, phase): + old = cur[register] + if old == frame: + return + a, _, ar = frames[old] + b, bd, br = frames[frame] + if (old, frame) not in edges: + require(ar["rank"] < br["rank"] and + product(b, a) == [[bd * x for x in row] for row in a], + "non-nested frame move") + edges.add((old, frame)) + moves[phase] += 1 + cur[register] = frame + + retained_ranks = [] + for phase in ("D", "A", "scatter", "B", "C"): + if phase == "scatter": + require(all(cur[r] == zero for r in range(v, 2 * v)), "scatter target frame") + require(len(word["ret"]) == 9 and len(set(word["ret"])) == 9 and + all(type(r) is int and 2 * v <= r < size for r in word["ret"]), + "retained helper indices") + retained_ranks = [frames[cur[r]][2]["rank"] for r in word["ret"]] + require(retained_ranks == [8] * 9, "retained copy ranks") + continue + for f, destination, sources in word[phase]: + registers = [destination] + sources + require(type(f) is int and 0 <= f < len(frames), "gate frame index") + require(len(registers) == len(set(registers)) and + all(type(r) is int and 0 <= r < size for r in registers), "gate registers") + if phase == "D": + require(f == zero, "D frame") + for r in registers: + move(r, f, phase) + for r, f in enumerate(word["final"]): + move(r, f, "final") + return {"moves_by_phase": dict(sorted(moves.items())), + "distinct_nested_pairs": sorted([a, b] for a, b in edges), + "retained_copy_ranks": retained_ranks} + + +def controls(word, frames, zero, full): + rejected = [] + + def reject(name, expected, operation): + try: + operation() + except Rejected as error: + require(str(error) == expected, "wrong control rejection: " + name + ": " + str(error)) + rejected.append(name) + else: + raise Rejected("control accepted: " + name) + + positive = 0 + axes = [[int(i == j) for j in range(H)] for i in range(H)] + for mask in range(1 << H): + chosen = [i for i in range(H) if mask & (1 << i)] + w, _, _ = exterior_basis(axes, chosen) + n, d, _ = projector(w) + require(d == 1 and n == [[int(i == j and i in chosen) for j in range(H)] + for i in range(H)], "coordinate-subspace oracle") + positive += 1 + oblique = [[1, 1, 0, 0, 0, 0, 0, 0]] + w, _, _ = exterior_basis(oblique, [0]) + n, d, _ = projector(w) + require(d == 2 and n == [[int(i < 2 and j < 2) for j in range(H)] + for i in range(H)], "oblique-line oracle") + positive += 1 + mixed = [[1, 0, 1, 0, 0, 0, 0, 0], [0, 1, 0, 1, 0, 0, 0, 0]] + w, _, _ = exterior_basis(mixed, [0, 1]) + mixed_n, mixed_d, norm = projector(w) + require(w == {3: 1, 9: 1, 6: -1, 12: 1} and norm == 4 and mixed_d == 2 and + mixed_n == [[int(i < 4 and j < 4 and i % 2 == j % 2) + for j in range(H)] for i in range(H)], "mixed-sign plane oracle") + positive += 1 + corrupt = copy.deepcopy(n) + corrupt[0][0] += 1 + reject("corrupted projector", "projector not idempotent", lambda: check_projector(corrupt, d, 1)) + w, _, _ = exterior_basis([[1, 1, 1, 0, 0, 0, 0, 0]], [0]) + bad_n, bad_d, _ = projector(w) + require(bad_d == 3, "prime-three control denominator") + reject("rational line singular at prime 3", "non-dyadic projector denominator", + lambda: check_projector(bad_n, bad_d, 1)) + a2 = [0, 16, 23] + w, _, _ = exterior_basis(word["lines"], a2) + a2_n, a2_d, a2_norm = projector(w) + require(a2_norm == a2_d == 3 and + [i for i, line in enumerate(word["lines"]) if apply(a2_n, line) == + [a2_d * x for x in line]] == a2, "closed A2 control") + reject("closed A2 frame singular at prime 3", "non-dyadic projector denominator", + lambda: check_frame(word["lines"], a2)) + bad_n, bad_d, _ = projector({3: 1, 12: 1}) + reject("nondecomposable exterior two-vector", "projector not idempotent", + lambda: check_projector(bad_n, bad_d, 2)) + reject("odd-index basis with integral projector", "non-dyadic basis Gram witness", + lambda: check_frame([[3, 0, 0, 0, 0, 0, 0, 0]], [0])) + reject("extra incidence after reduction modulo 3", "root incidence can change at an odd prime", + lambda: check_frame([[1, 0, 0, 0, 0, 0, 0, 0], + [1, 3, 0, 0, 0, 0, 0, 0]], [0])) + reject("omitted line from full frame", "frame not closed", + lambda: check_frame(word["lines"], word["frames"][full][1:])) + reject("duplicate line", "frame line indices", + lambda: check_frame(word["lines"], [0, 0])) + changed = copy.deepcopy(word) + changed["final"][120] = word["final"][121] + reject("wrong target hyperplane", "target final hyperplane", + lambda: check_stream(changed, frames, zero, full)) + changed = copy.deepcopy(word) + changed["B"][0][0] = zero + reject("phase-B descent to zero", "non-nested frame move", + lambda: check_stream(changed, frames, zero, full)) + changed = copy.deepcopy(word) + changed["D"][0][0] = full + reject("D moved off zero", "D frame", lambda: check_stream(changed, frames, zero, full)) + return {"positive_oracles": positive, "rejected": rejected} + + +def certificate(): + source = json.loads((HERE / "SOURCE.json").read_text()) + for name, digest in source["reference_sha256"].items(): + require(hashlib.sha256((ROOT / name).read_bytes()).hexdigest() == digest, + "reference source drift: " + name) + payload = gzip.decompress((ROOT / source["word"]).read_bytes()) + require(hashlib.sha256(payload).hexdigest() == source["word_payload_sha256"], "word payload drift") + word = json.loads(payload) + frames, zero, full = geometry(word) + stream = check_stream(word, frames, zero, full) + return {"status": "PASS_R420_EXTERIOR_GEOMETRY", "seed": SEED, + "word_payload_sha256": source["word_payload_sha256"], + "frame_count": len(frames), + "rank_histogram": dict(sorted(Counter(str(rec["rank"]) for _, _, rec in frames).items())), + "denominator_histogram": dict(sorted(Counter(str(d) for _, d, _ in frames).items())), + "nonincident_pairs": sum(sum(rec["nonincident_residual_contents"].values()) + for _, _, rec in frames), + "frames": [rec for _, _, rec in frames], + "stream": stream, "controls": controls(word, frames, zero, full)} + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--emit", action="store_true", help="print the regenerated certificate to stdout") + args = parser.parse_args() + result = certificate() + if args.emit: + print(json.dumps(result, sort_keys=True, separators=(",", ":"))) + else: + require(result == json.loads((HERE / "CERTIFICATE.json").read_text()), "certificate mismatch") + print(result["status"] + ": " + str(result["frame_count"]) + " frames; " + + str(len(result["stream"]["distinct_nested_pairs"])) + " nested pairs; " + + str(result["controls"]["positive_oracles"]) + " positive oracles; " + + str(len(result["controls"]["rejected"])) + " rejected controls") + + +if __name__ == "__main__": + main()