diff --git a/research/tensor-e8-copy-floor/CLAIM.json b/research/tensor-e8-copy-floor/CLAIM.json new file mode 100644 index 0000000000..a92f6cd956 --- /dev/null +++ b/research/tensor-e8-copy-floor/CLAIM.json @@ -0,0 +1,45 @@ +{ + "track": "Structural decoder research", + "assignment_seed": 43, + "randomness": "secrets.randbelow(100), OS randomness; drawn once at session start", + "base_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60", + "started_at": "2026-10-13T04:45:49.912942+00:00", + "claimed_at": "2026-10-13T04:58:13.000008+00:00", + "expected_finish": "2026-10-13T05:45:49Z", + "verification_reserved_minutes": 15, + "coordination": "No pinned issues; only open issue #650 asks for Discussions. Local claim per sprint instructions.", + "object": "The fixed binary full-E8 star decoder matrix under tensor powers; arbitrary mixing among the minimal number of scalar totals.", + "hypothesis": "Nonseparable changes of total basis across tensor factors might lower copied-information cost; test tensor stability of full rational reader spans and price the exact floor.", + "method": "Exact independent E8 root generation and all 511 base reader spans, written tensor-support induction, rank-minimal factorization argument; seed-selected nonsingular binary basis mixtures at depth 2 as finite controls.", + "expected_deliverable": "One scoped theorem/negative result with portable standard-library certificate checker, meaningful negative controls, source pins and no kappa claim.", + "candidates": [ + { + "task": "Higher-order hypercube stars", + "decision": "deferred; #231/#669 already cover the main parity and puncture obstruction" + }, + { + "task": "Nonseparable tensor-basis decoder mixing", + "decision": "selected; #620 establishes only the depth-one reader-span floor and #682 audits the canonical paid tensor rate, so test arbitrary rank-minimal total basis changes" + }, + { + "task": "Mixed-size paired-port stars", + "decision": "deferred; broad new geometry and complete correction circuits would not fit the bounded session" + } + ], + "seed_diversification": "43 modulo 3 selects tensor mixing in the recorded candidate order; 43 also seeds a nonseparable depth-two factorization control, rather than repeating a prior geometry search.", + "stopping_rule": "Prove or refute tensor-support stability and check the finite base within first 45 minutes; do not broaden to higher-rank decoder search or claim an implemented supplier.", + "research_completed_at": "2026-10-13T05:11:07Z", + "orientation_note": "Local claim recorded 12 minutes 23 seconds after session start; the fetch, worktree and network authorization delays slightly exceeded the 12-minute orientation allocation. Research completed with more than half of the 60-minute budget remaining for verification and handoff.", + "status": "complete: scoped negative result", + "result": "For the fixed binary full-E8 matrix B, every rank-minimal factorization of B tensor-powered d has copied-information rank sum 72^d. Also excludes overcomplete decompositions with both factors constrained to the tensor code; no new kappa.", + "result_url": "https://github.com/CrocSwap/integer-mult-bounds/tree/research/decoder-seed43/research/tensor-e8-copy-floor", + "verification": "511 exact base minors, one seeded 81-total nonseparable depth-two factorization checked entrywise, 162 full-rank tensor supports, exhaustive 3+15 toy words, 8 mathematical negative controls, 2 packaging controls, independent determinant recheck, Python 3.13.0 and 3.14.6 including optimized mode; selected-record integrity passed.", + "scope_exclusions": [ + "Different decoder matrices", + "Unrestricted overcomplete factorizations", + "Punctures/relabelings", + "Dynamic interactions between delivery trees", + "Complete physical supplier and multiplication theorem" + ], + "final_overlap": "581 open PRs; #757 inspected and credited. No pinned coordination issue; main unchanged." +} diff --git a/research/tensor-e8-copy-floor/README.md b/research/tensor-e8-copy-floor/README.md new file mode 100644 index 0000000000..be2c80b2d4 --- /dev/null +++ b/research/tensor-e8-copy-floor/README.md @@ -0,0 +1,263 @@ +# Tensor E8 decoder mixing has copy-rank floor 72^d + +**Scoped negative result; no new kappa.** For the fixed full-E8 binary decoder +matrix `B`, every factorization of `B` tensor-powered `d` times into the minimum +`9^d` scalar totals has copied-information rank sum exactly **`72^d`**. This +includes arbitrary nonsymmetric, nonseparable changes of total basis across +the tensor factors. Such changes cannot improve the canonical tensor decoder's +copy-rank budget. The conclusion holds at every integer depth `d >= 1` by the +proof below, conditional only on the explicitly stated copy-tree model when +interpreted as a physical cost bound. + +The new step is tensor stability of the full-span support property, followed +by its application to arbitrary minimal factorizations. The depth-one floor +is predecessor work: GamingPuzzled's #620 and Chi Zhao's #631. This package +also supplies a portable exact check of that finite premise, without the +NumPy floating rank routine used by #620's `geo.py`. + +## Reproduce + +From the repository root, with Python 3.10 or later and no external packages: + +```sh +python3 -B research/tensor-e8-copy-floor/verify.py +``` + +Expected: + +```text +PASS_TENSOR_E8_COPY_FLOOR: 511 exact base minors; 162 tensor supports; 8 negative controls; cost 72^d +``` + +Each run checks source hashes and compares the complete regenerated result +against `RESULTS.json`. To write a separate receipt, append +`--write /tmp/tensor-e8-copy-floor.json`. Checks remain active under `python3 -O`. +The default command writes nothing. The receipt's 511 exact minors can also +be checked independently using just integer determinants. + +## Object and cost model + +Let `u_i` be the 120 rational E8 root-line representatives in dimension eight, +in the norm-four Hamming coordinates used by the repository. Define over F2 + +```text +B_ij = 1 + (u_i dot u_j)/2 (mod 2), C = columnspace_F2(B). +``` + +The checker generates the eight coordinate lines and the 112 signed affine +plane lines of F2^3. It binds their set and all 14,400 binary pairings to the +pinned #531 prototype. Thus its ordering is local, but the geometry and binary +matrix agree with the predecessor. `B` is symmetric, has diagonal one and +binary rank nine. Every nonzero `c in C` has rational support span + +```text +span_Q {u_i : c_i = 1} = Q^8. (F) +``` + +There are exactly 511 such words: 255 of weight 56, 255 of weight 64 and one +of weight 120. These finite facts are checked exhaustively, with a nonzero +8-by-8 integer minor for each word. The rational rank calculation and the +fraction-free determinant calculation are separate algorithms. + +At depth `d`, ports are tuples `I = (i_1,...,i_d)`, physical lines are +`u_I = u_i1 tensor ... tensor u_id` in `V_d = (Q^8) tensor-powered d`, and +the **fixed matrix under study** is `B_d = B tensor-powered d` over F2. +Its binary rank is `r_d = 9^d`. This is the rank of this matrix, not a +minimum over every decoder compatible with the product geometry. + +The matrix has the required algebraic support for orthogonal corrections: +its diagonal is one, and for distinct ports with nonzero physical inner +product, every factor inner product is nonzero and at least one factor is +an off-diagonal E8 pairing. That factor's entry of `B` is zero, so the +entry of `B_d` is zero. Hence `I + B_d` is supported on orthogonal pairs. +This observation supplies no correction circuit or paid schedule. + +A scalar-total decomposition is + +```text +B_d = sum_j a_j b_j^T (over F2). +``` + +Total `j` gathers sources `P_j = supp(b_j)` and is delivered to readers +`Q_j = supp(a_j)`. Set + +```text +beta_j = rank_Q(pairing(span_Q u[P_j], span_Q u[Q_j])), +C_pair = sum_j beta_j. +``` + +Theorem C of DreamingOfClouds' [#446 proof, section 4](https://github.com/CrocSwap/integer-mult-bounds/blob/309482849b6ddf34db998f53612d34c19f6e3233/research/format-obstructions/PROOF.md) +is the physical-model dependency: a retained total gathered without drops, +then delivered through its own copy tree to readers at orthogonal frames, +pays at least `beta_j` in drops. This package does not re-prove that theorem +or establish its applicability to a different compiler. Per-total costs are +added under that model. Dynamic mixing between different totals during their +delivery is outside this claim. + +At the base commit the review ledger marks #446 **unreviewed**, and #620 +**partially reviewed**. The algebraic statement (1) does not depend on their +approval or solver results: its premise is independently checked here and +its linear algebra proof is below. Its physical drop-cost interpretation +explicitly inherits #446's model and written theorem. + +## Proof at every tensor depth + +**Tensor-support lemma.** Let binary codes `A <= F2^S`, `D <= F2^T` have +rational physical labels `v_s in V`, `w_t in W`. Suppose every nonzero word +in each code has support spanning its entire physical space. Then every +nonzero word of `A tensor D` has physical support spanning `V tensor W`. + +View a word `z` as a binary matrix `(z_st)`. Each row lies in `D`, so a +nonzero row at `s` contributes support vectors spanning `v_s tensor W`. +Choose any nonzero column. It is a nonzero word of `A`; its support spans +`V` and is contained in the set of nonzero rows of `z`. Summing the spaces +`v_s tensor W` for those rows therefore gives all of `V tensor W`. This +argument allows arbitrary sums of pure tensors; it makes no separability +assumption about `z`. Induction and (F) prove the property for `C^tensor d`. + +**Minimal-factorization lemma.** If an F2 matrix `M` has rank `r` and +`M = A D^T` with exactly `r` columns in each factor, then every column of +`A` lies in `columnspace(M)` and every column of `D` lies in +`columnspace(M^T)`. Indeed `rank(A) = rank(D) = r` and +`columnspace(M) <= columnspace(A)`; their dimensions agree. The transpose +gives the other statement. All columns are nonzero. This proof does not +require the factors to be equal or the matrix to be positive definite. + +Apply this to `B_d`. Its row and column spaces are both `C^tensor d`, +and its rank is `9^d` by multiplicativity of matrix rank over a field. +The tensor-support lemma says every source support and every reader support +of any minimal factorization spans the entire `8^d`-dimensional rational +space. Its Euclidean tensor pairing is nondegenerate. Hence every +`beta_j = 8^d`, and + +```text +C_pair = 9^d * 8^d = 72^d. (1) +``` + +An explicit rank factorization of `B`, tensored `d` times, attains (1) as +a pairing-rank objective. Dropping each retained total from the full frame +to zero costs `8^d`, so it also attains the per-total copy-tree drop bound +when those root frames are available. This is not a full supplier schedule: +source access, gathering, restoration, helpers, inter-stage work and recursive +calls still require their own charges and legal interfaces. + +There is also an overcomplete corollary **with a code constraint**. If both +factors of every nonzero summand are required to belong to `C^tensor d`, +every term still costs `8^d` and at least `9^d` terms are required by rank. +Thus no such overcomplete label decoder beats (1). General overcomplete +factorizations need not obey the code constraint and are not covered. + +## Optimistic budget and stopping rule + +The proposed mechanism was to reduce copy expense by rebasing several tensor +units jointly. Its best possible copied-information budget is already the +canonical one: + +| Depth d | Ports | Minimum totals | Rank per total | Copy-rank sum | +|---:|---:|---:|---:|---:| +| 1 | 120 | 9 | 8 | 72 | +| 2 | 14,400 | 81 | 64 | 5,184 | +| 3 | 1,728,000 | 729 | 512 | 373,248 | + +The normalized copy cost is `(3/5)^d`. The result preserves the existing +tensor scaling; it only rules out improving that expense through the stated +basis mixing. Even granting all basis changes, source accesses, extra helpers, +restorations and recursive work for free cannot make `C_pair` smaller. A real +implementation must charge all those operations and cannot obtain a positive +copy saving from this mechanism. We therefore stopped before building a +supplier. No inference about a better helper schedule or a different stage +structure follows. + +The selected #626/#543 assembly at the base is complex-bound, with complex +lower endpoint `65528268492323474503/40000000000000000000000` and bit lower +endpoint `413456747036699736067/250000000000000000000000`. A hypothetical +bit-only copy improvement would still require checking that binding constraint; +this result supplies no improvement on either side. + +## Finite evidence and negative controls + +The verifier covers: + +- All 120 base roots, all base binary pairings, rank nine, and all 511 nonzero + base-code supports, each with a committed exact determinant witness. +- One depth-two factorization changed by 243 elementary invertible basis + operations drawn deterministically with seed 43. The other factor receives + the inverse-transpose changes. It checks all **207,360,000 binary entries** + of the resulting matrix identity via complete bit-packed row equality, and + all 162 producer/reader supports using nonzero 64-dimensional minors modulo + three. A nonzero modular minor proves the rational minor is nonzero. +- Every nonzero word in a separate `[3,2]` toy code at depths one and two + (3 and 15 words). Its rational labels are `(1,0), (0,1), (1,1)`. +- Eight mandatory negative controls: corrupt root norm, corrupt diagonal, + omitted independent total, repeated minor row, one-root puncture, an + unrestricted overcomplete factor outside the code, degenerate physical + pairing, and a missing inverse-transpose update. + +The one-root puncture is a real boundary example: remove root zero from the +support of its own `B` row. The remaining 63 roots have rational rank seven. +Thus the full-span hypothesis cannot simply be reused for punctured units. +Likewise `B = sum_i e_i (row_i B)` is a valid 120-total factorization whose +`e_i` factors are outside `C` and have rank-one physical supports. This +invalidates extending the per-term rank-eight assertion to unrestricted +overcomplete factorizations; it is not a cheaper decoder. + +The finite depth-two run is a control, not exhaustive coverage of all changes +of basis. Universal coverage comes from the two written lemmas. This is not +a proof-assistant formalization or a proof of integer multiplication. + +## Provenance, scope and handoff + +Exact base: `66849e1475c930fd7732b1a0067ffee7e1e95f60`. Assignment seed **43** +was drawn once using OS randomness; `CLAIM.json` records task selection and the +local claim. No pinned coordination issue existed. `SOURCES.json` pins source +bytes and predecessor/overlap heads. No repository generator, selected record, +or `upstream/` file is changed. + +Predecessors and concurrent work retain credit: + +- DreamingOfClouds' #446 supplies the per-total copy-tree rank lower bound, + following Jacob Sussman's frame model and the E8 work of Dylan Crow and the + community contributors credited there. +- Joel Pulikkan / GamingPuzzled's #620 identifies the full-E8 depth-one floor. + Chi Zhao's #631 supplies exact puncture thresholds and the reader census. + The depth-one result is independently corroborated here, not newly claimed. +- icekylinx's #531 package supplies the pinned Hamming-coordinate prototype; + eumemic's #543 is the selected bit supplier. The selected complex construction + and allocation remain credited to their #622/#623/#626 predecessors. +- Artus' #647 and Scott Woolsey Biggart's #682 concern a separate paid complex + RAM tensor provider and its rate. They are overlap checks, not dependencies + or constructions ruled out by this binary decoder theorem. #669, #703, + #715 and #717 also have different geometry or factorization scopes. +- Russell Seligmann's concurrent #757 gives a solver-free copied-rank + certificate for the actual P111 fixtures and corrects a root-order mismatch. + Its punctured single-unit scope differs from this full-unit tensor result. + Here root coordinates and all prototype pairings are matched explicitly; + no predecessor's numeric puncture list is used. + +Outstanding possibilities include changing `B_d`, puncturing/relabeling, +unrestricted overcomplete factorizations, and interacting delivery programs +that do not have the per-total copy-tree discipline. None is searched here. +All inherited all-size, compiler, hosting, restoration, precision/tape and +analytic obligations of the selected multiplication result remain unchanged. + +Prepared for Douglas R. Colkitt with substantial **OpenAI Codex assistance** +in the mathematics, implementation, verification and writing. No independent +human review is claimed. GitHub submission uses the authenticated contributor +account **@Mister-Meeseeks**. New files are licensed under the repository's +Apache-2.0 license; no predecessor implementation is vendored or imported. + +## Validation receipt + +Local runs passed on Python **3.14.6** (normal, optimized and isolated modes) +and **3.13.0**, with the same complete JSON receipt. A separate implementation +generated affine planes as four-element subsets with XOR zero, and recomputed +all 511 committed determinants by Fraction elimination without importing +`verify.py`. Two additional packaging controls rejected an incorrect source +hash and a mutated determinant receipt. Selected-record integrity and whitespace +checks passed. Final overlap inspection covered 581 open PRs; main and the +selected frontier were unchanged. + +The repository-wide `make verify` and selected supplier cold replay were not +run: this contribution changes only its new research package, and its affected +checks are the command above and selected-record integrity. 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"corrupt_decoder_diagonal", + "omit_independent_total", + "duplicate_minor_row", + "apply_full_span_claim_after_one_puncture", + "extend_to_unrestricted_overcomplete_factors", + "drop_nondegeneracy_hypothesis", + "omit_inverse_transpose_update" + ], + "scope": "Exact base premises and finite controls; all-depth theorem is a written proof.", + "status": "PASS_TENSOR_E8_COPY_FLOOR", + "tensor_control": { + "basis_changes": 243, + "copy_rank_sum": 5184, + "full_rank_supports": 162, + "matrix_entries_checked": 207360000, + "minor_indices_sha256": "a40fed3d92b2dfd49f81165dfec972f8c74df1cd7ffda14904f0a56cda11864f", + "minor_prime": 3, + "nonseparable_factors": 153, + "operations_sha256": "49e6ad8875c585ec391fb16beb78e0ebe22fcfafb63ea43a7db4b1b99ac6e48c", + "seed": 43, + "support_dimension": 64, + "totals": 81 + }, + "theorem_prices": [ + { + "copy_per_port": "3/5", + "copy_rank_sum": 72, + "depth": 1, + "ports": 120, + "rank_per_copy": 8, + "totals": 9 + }, + { + "copy_per_port": "9/25", + 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"prototype": "research/direct-delivery504-global/inputs/hamming/PROTOTYPE.json", + "local_source_sha256": { + "research/direct-delivery504-global/inputs/hamming/PROTOTYPE.json": "327d2a868043e846381aa58a33053c24425e8cad522ff401b10ec8af1d649a3f", + "research/e8-decoder-floors-and-cold14-notes/s2-decoder/geo.py": "73f9c4e4fb7d424fcba45eb5da36bdab694b10020341924b7ebfe7276868716a", + "research/e8-decoder-floors-and-cold14-notes/s2-decoder/e8units.py": "3134443d8327d0bf2a8a66c26327cf5268cec6b7ffd85dde750b7e197ce63ef1", + "research/e8-decoder-floors-and-cold14-notes/s2-decoder/REPORT.md": "16c53bf8afd9f91eca02b2af9c9f5f09385823fc705dd1a48153508cfdff28b6", + "certificates/selected-result.json": "b3bf402f31a69bde0629936a9b569bed48281568d7b16d5fba120b5311ea1f53", + "docs/research/community-k492-review.md": "3b731d30e80010b15408a289b0b32192700885474c8b36d48ee3320a7a8c5c47", + "research/community-k492-audit/pr626/replay/ASSEMBLY.json": "6127732e2fcdd470dab349c1c51d03cd9f0787ad0309b1b4a7c811a908028f4e" + }, + "dependencies": { + "446": { + "head": "309482849b6ddf34db998f53612d34c19f6e3233", + "role": "Theorem C copied-information/drop-pricing model; written dependency, no predecessor solver replay", + "proof_sha256": "7a8c3bab633c08e7d96297dec9f9e7ed68d6bd0e076b73310c7d6b08a16df017" + }, + "620": { + "head": "e9cf03bbdf7094b5a83f6dbd1d74bfd77a05fb87", + "role": "Depth-one reader-span floor; imported geometry context pinned locally" + }, + "631": { + "head": "fb893b5ee1317c2da3b297e6fe78eca0857d1a76", + "role": "Exact depth-one reader-weight census and puncture boundary; comparison only", + "readme_sha256": "281f706bf29d1f91dd64ba096f9d66d622f49e9d82d0ff306fb9374f77ba7333" + } + }, + "overlap": { + "682": { + "head": "cfc7130e0e371395f4cf2f96c122c240cdbeab97", + "scope": "Paid-profile asymptotics for a complex RAM provider; different object from fixed binary tensor decoder basis mixing" + }, + "647": { + "head": "956bb581e7f97328e1b53baa5aca3d2490f2f607", + "scope": "Complex RAM provider, not a prerequisite" + }, + "669": { + "head": "308f1e6ec3bb0b6ec0584cf7bfe219addc219b1e", + "scope": "Paired-hypercube star parity and punctures" + }, + "703": { + "head": "358383e2f1b0f292d4d6eb897c06fd8558a4f5e6", + "scope": "Quaternionic Hurwitz single-unit asymmetric cost" + }, + "715": { + "head": "d8ed5c6cf2781d6efa0062a2636e1a2644f96c85", + "scope": "Minimum-puncture E8 mixed-port saturation" + }, + "717": { + "head": "67e9e846d9e8814b2dd01133251bb83fafa8f5b6", + "scope": "Complex P112 and word-format limits, not binary tensor code support stability" + }, + "757": { + "head": "e2ec7d7e719031e8f8b8012a4ab5b20faed074ce", + "scope": "Russell Seligmann: solver-free asymmetric copied-rank floor for actual P111 punctures and source-order correction. Different from full-P120 tensor-code mixing; our prototype binding uses coordinates rather than numeric deletion indices." + } + }, + "initial_open_pr_count": 551, + "selected_conditional_kappa": "51110230636731/31250000000000000", + "final_overlap_open_pr_count": 581, + "final_overlap_observed_at": "2026-10-13T05:15:12.262543+00:00" +} diff --git a/research/tensor-e8-copy-floor/verify.py b/research/tensor-e8-copy-floor/verify.py new file mode 100644 index 0000000000..888e931d5d --- /dev/null +++ b/research/tensor-e8-copy-floor/verify.py @@ -0,0 +1,332 @@ +#!/usr/bin/env python3 +"""Exact finite premises for the tensor E8 copy-floor note. Standard library only. + +No producer code, floating point, SAT solver, or precomputed rank is imported. +The all-depth argument is in README.md, not inferred from the finite controls. +""" + +import argparse +from collections import Counter +from fractions import Fraction +import hashlib +from itertools import product +import json +from pathlib import Path +import random + +HERE = Path(__file__).resolve().parent +ROOT = HERE.parents[1] + + +def require(condition, message): + if not condition: + raise ValueError(message) + + +def digest(value): + return hashlib.sha256(json.dumps(value, sort_keys=True, separators=(",", ":")).encode()).hexdigest() + + +def bits(word): + while word: + low = word & -word + yield low.bit_length() - 1 + word ^= low + + +def dot(a, b): + return sum(x * y for x, y in zip(a, b)) + + +def roots(): + """Coordinate roots and signed affine planes of F_2^3, modulo sign.""" + out = [tuple(2 * (i == j) for j in range(8)) for i in range(8)] + planes = sorted({tuple(x for x in range(8) if (a & x).bit_count() % 2 == b) + for a in range(1, 8) for b in range(2)}) + for plane in planes: + for signs in product((1, -1), repeat=3): + row = [0] * 8 + for i, sign in zip(plane, (1,) + signs): + row[i] = sign + out.append(tuple(row)) + require(len(out) == len(set(out)) == 120, "root count") + require(all(dot(r, r) == 4 for r in out), "root norm") + return out + + +def binary_basis(rows): + echelon = {} + original = [] + for row in rows: + value = row + for pivot in sorted(echelon, reverse=True): + if (value >> pivot) & 1: + value ^= echelon[pivot] + if value: + echelon[value.bit_length() - 1] = value + original.append(row) + return original + + +def coordinates(rows, basis): + echelon = {} + for i, row in enumerate(basis): + value, coeff = row, 1 << i + for pivot in sorted(echelon, reverse=True): + if (value >> pivot) & 1: + v, c = echelon[pivot] + value, coeff = value ^ v, coeff ^ c + require(value != 0, "dependent binary basis") + echelon[value.bit_length() - 1] = (value, coeff) + out = [] + for row in rows: + value, coeff = row, 0 + for pivot in sorted(echelon, reverse=True): + if (value >> pivot) & 1: + v, c = echelon[pivot] + value, coeff = value ^ v, coeff ^ c + require(value == 0, "word outside code") + out.append(coeff) + return out + + +def combine(mask, vectors): + result = 0 + for i in bits(mask): + result ^= vectors[i] + return result + + +def q_basis_indices(rows, indices, dimension): + """Fraction elimination; returns an actual independent subset.""" + echelon, chosen = {}, [] + for i in indices: + row = list(map(Fraction, rows[i])) + for pivot, old in sorted(echelon.items()): + factor = row[pivot] + if factor: + row = [x - factor * y for x, y in zip(row, old)] + pivot = next((j for j, x in enumerate(row) if x), None) + if pivot is not None: + factor = row[pivot] + echelon[pivot] = [x / factor for x in row] + chosen.append(i) + if len(chosen) == dimension: + break + return chosen + + +def determinant(rows): + """Independent fraction-free Bareiss determinant, including row swaps.""" + a = [list(row) for row in rows] + n, previous, sign = len(a), 1, 1 + require(all(len(row) == n for row in a), "square determinant") + for k in range(n - 1): + pivot = next((i for i in range(k, n) if a[i][k]), None) + if pivot is None: + return 0 + if pivot != k: + a[k], a[pivot] = a[pivot], a[k] + sign = -sign + for i in range(k + 1, n): + for j in range(k + 1, n): + numerator = a[k][k] * a[i][j] - a[i][k] * a[k][j] + require(numerator % previous == 0, "inexact Bareiss division") + a[i][j] = numerator // previous + a[i][k] = 0 + previous = a[k][k] + return sign * a[-1][-1] + + +def kron_vector(a, b): + return tuple(x * y for x in a for y in b) + + +def kron_support(a, b, size): + return sum(b << (i * size) for i in bits(a)) + + +def mod_basis_indices(rows, indices, dimension, prime=3): + """A nonzero minor modulo 3 also certifies full rational rank.""" + echelon, chosen = {}, [] + for i in indices: + row = [x % prime for x in rows[i]] + for pivot, old in sorted(echelon.items()): + factor = row[pivot] + if factor: + row = [(x - factor * y) % prime for x, y in zip(row, old)] + pivot = next((j for j, x in enumerate(row) if x), None) + if pivot is not None: + inv = pow(row[pivot], -1, prime) + echelon[pivot] = [(x * inv) % prime for x in row] + chosen.append(i) + if len(chosen) == dimension: + break + return chosen + + +def tensor_basis_control(matrix, base_basis, physical): + """One seeded nonseparable 81-total factorization; every matrix entry checked.""" + n, rank = len(matrix), len(base_basis) + left_rows = coordinates(matrix, base_basis) + left = [sum(1 << i for i, row in enumerate(left_rows) if (row >> k) & 1) + for k in range(rank)] + lcols = [kron_support(a, b, n) for a in left for b in left] + rcols = [kron_support(a, b, n) for a in base_basis for b in base_basis] + lcoeff = [1 << i for i in range(rank * rank)] + rcoeff = lcoeff.copy() + rng = random.Random(43) + operations = [tuple(rng.sample(range(rank * rank), 2)) for _ in range(3 * rank * rank)] + for dst, src in operations: + lcols[dst] ^= lcols[src] + lcoeff[dst] ^= lcoeff[src] + rcols[src] ^= rcols[dst] + rcoeff[src] ^= rcoeff[dst] + require(len(binary_basis(lcoeff)) == len(binary_basis(rcoeff)) == 81, + "singular tensor basis") + entangled = sum(len(binary_basis([(c >> (9 * j)) & 511 for j in range(9)])) > 1 + for c in lcoeff + rcoeff) + require(entangled > 0, "tensor control did not mix factors") + for i, a in enumerate(matrix): + for j, b in enumerate(matrix): + t = i * n + j + actual = combine(sum(1 << k for k, col in enumerate(lcols) if (col >> t) & 1), rcols) + require(actual == kron_support(a, b, n), "tensor decoder identity") + proots = [kron_vector(a, b) for a in physical for b in physical] + witnesses = [] + for support in lcols + rcols: + witness = mod_basis_indices(proots, bits(support), 64) + require(len(witness) == 64, "tensor support rank") + witnesses.append(witness) + return { + "seed": 43, "basis_changes": len(operations), "totals": 81, + "nonseparable_factors": entangled, "full_rank_supports": len(witnesses), + "support_dimension": 64, "minor_prime": 3, + "matrix_entries_checked": n ** 4, "copy_rank_sum": 5184, + "operations_sha256": digest(operations), "minor_indices_sha256": digest(witnesses), + } + + +def toy_tensor_controls(): + # Every nonzero word of this [3,2] code selects two independent Q^2 vectors. + code = [0b101, 0b110] + vectors = [(1, 0), (0, 1), (1, 1)] + receipt = [] + for depth in (1, 2): + basis, labels, size = code, vectors, 3 + for _ in range(1, depth): + basis = [kron_support(a, b, 3) for a in basis for b in code] + labels = [kron_vector(a, b) for a in labels for b in vectors] + size *= 3 + for mask in range(1, 1 << len(basis)): + support = combine(mask, basis) + require(len(q_basis_indices(labels, bits(support), 2 ** depth)) == 2 ** depth, + "toy tensor premise") + receipt.append({"depth": depth, "all_nonzero_words": (1 << len(basis)) - 1, + "physical_dimension": 2 ** depth, "ports": size}) + return receipt + + +def negative_controls(physical, matrix, base_basis, witnesses): + passed = [] + + def rejects(name, f): + try: + f() + except ValueError: + passed.append(name) + else: + raise ValueError("negative control accepted: " + name) + + bad = [list(row) for row in physical] + bad[0][0] += 1 + rejects("corrupt_root_norm", lambda: require(all(dot(r, r) == 4 for r in bad), "root norm")) + mutant = matrix.copy() + mutant[0] ^= 1 + rejects("corrupt_decoder_diagonal", lambda: require(all((r >> i) & 1 for i, r in enumerate(mutant)), "diagonal")) + rejects("omit_independent_total", lambda: coordinates(matrix, base_basis[:-1])) + bad_minor = witnesses[0]["minor_indices"].copy() + bad_minor[-1] = bad_minor[0] + rejects("duplicate_minor_row", lambda: require(determinant([physical[i] for i in bad_minor]) != 0, "minor")) + # A single deleted root breaks the full-span premise: its B row now + # selects exactly the 63 roots orthogonal to it, which have rank seven. + reader = matrix[0] & ~1 + rank = len(q_basis_indices(physical, bits(reader), 8)) + require(rank == 7, "puncture control rank") + rejects("apply_full_span_claim_after_one_puncture", lambda: require(rank == 8, "puncture")) + # Extra factors need not lie in the code. B = sum_i e_i (row_i B) + # is an explicit valid 120-total factorization with rank-one source supports. + outside = next(1 << i for i in range(120) if len(binary_basis(base_basis + [1 << i])) == 10) + rejects("extend_to_unrestricted_overcomplete_factors", lambda: coordinates([outside], base_basis)) + # Degenerate physical pairing: full spans alone do not force full pairing rank. + gram = [[sum(a[k] * b[k] for k in range(7)) for b in physical[:8]] for a in physical[:8]] + rejects("drop_nondegeneracy_hypothesis", lambda: require(determinant(gram) != 0, "degenerate form")) + # Tensor identities require inverse-transpose changes on the other side. + left = [0b01, 0b11] + wrong_right = [0b01, 0b10] + rejects("omit_inverse_transpose_update", lambda: require( + [combine(sum(1 << k for k, c in enumerate(left) if (c >> i) & 1), wrong_right) + for i in range(2)] == [1, 2], "factorization")) + return passed + + +def verify(): + sources = json.loads((HERE / "SOURCES.json").read_text()) + for path, sha in sources["local_source_sha256"].items(): + require(hashlib.sha256((ROOT / path).read_bytes()).hexdigest() == sha, "source hash: " + path) + physical = roots() + prototype = json.loads((ROOT / sources["prototype"]).read_text()) + require(set(physical) == set(map(tuple, prototype["root_lines"])), "prototype root set") + matrix = [] + for a in physical: + require(all(dot(a, b) % 2 == 0 for b in physical), "integral Gram matrix") + matrix.append(sum(((1 + dot(a, b) // 2) % 2) << i for i, b in enumerate(physical))) + require(all((row >> i) & 1 for i, row in enumerate(matrix)), "decoder diagonal") + port_by_root = dict(zip(map(tuple, prototype["root_lines"]), prototype["binary_ports"])) + labels = [port_by_root[r] for r in physical] + require(all(((matrix[i] >> j) & 1) == ((labels[i] & labels[j]).bit_count() % 2) + for i in range(120) for j in range(120)), "prototype binary pairing") + base_basis = binary_basis(matrix) + require(len(base_basis) == 9, "binary decoder rank") + witnesses, histogram = [], Counter() + for coeff in range(1, 512): + support = combine(coeff, base_basis) + chosen = q_basis_indices(physical, bits(support), 8) + require(len(chosen) == 8, "base reader span") + det = determinant([physical[i] for i in chosen]) + require(det != 0, "base exact minor") + histogram[support.bit_count()] += 1 + witnesses.append({"coefficient": coeff, "support_hex": hex(support), + "minor_indices": chosen, "determinant": det}) + require(dict(histogram) == {56: 255, 64: 255, 120: 1}, "reader weight census") + return { + "status": "PASS_TENSOR_E8_COPY_FLOOR", "base_commit": sources["base_commit"], + "scope": "Exact base premises and finite controls; all-depth theorem is a written proof.", + "base": {"ports": 120, "binary_rank": 9, "physical_dimension": 8, + "nonzero_codewords": 511, "weight_histogram": dict(sorted(histogram.items())), + "roots_sha256": digest(physical), "binary_matrix_sha256": digest(matrix), + "basis_rows_hex": list(map(hex, base_basis)), "minors": witnesses}, + "tensor_control": tensor_basis_control(matrix, base_basis, physical), + "exhaustive_toy_controls": toy_tensor_controls(), + "negative_controls": negative_controls(physical, matrix, base_basis, witnesses), + "theorem_prices": [{"depth": d, "ports": 120 ** d, "totals": 9 ** d, + "rank_per_copy": 8 ** d, "copy_rank_sum": 72 ** d, + "copy_per_port": str(Fraction(3, 5) ** d)} for d in range(1, 7)], + } + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--write", type=Path, help="write a new receipt instead of comparing the committed receipt") + args = parser.parse_args() + result = verify() + encoded = json.dumps(result, indent=2, sort_keys=True) + "\n" + if args.write: + args.write.write_text(encoded) + else: + require(encoded == (HERE / "RESULTS.json").read_text(), "committed receipt mismatch") + print("PASS_TENSOR_E8_COPY_FLOOR: 511 exact base minors; 162 tensor supports; 8 negative controls; cost 72^d") + + +if __name__ == "__main__": + main()