[petalas]

Summary

Prove a sharp 1/4 maximum coefficient-error floor for contractive rank-seven approximation of the P104 Gram channel. This is a scoped obstruction with exact finite witnesses, without a new kappa or physical supplier.

K = R R^T / 2
K_D = R D R^T / 2, D real symmetric, 0 <= D <= I, rank D <= 7
min_D max_ij |K_ij - (K_D)_ij| = 1/4

Every omitted half-spin root q gives an attaining dyadic projector D=I-qq^T/2. The lower bound uses only the retained D8 roots and covers every admissible real D. For some input B e_i, the output bias is at least |B|/4, so increasing arithmetic precision alone cannot make this mode deletion harmless.

The proof and verifier were produced by Nick Petalas. Credit Matarener's PR6166 for the P104 D8 coordinate identification and omitted-root signs, cited without re-verification; Rohan Arun PR1972, Chafik Boukhalfa PR1991 and their retained root/frame/native lineage for the Gram channel. All P104/helper/compiler/BIT predecessor credit is retained. Root geometry, projection and positive-semidefinite order are classical; no literature-wide novelty claim is made.

Evidence

The exact finite replay checks sixteen new rank-seven projectors, all 173,056 entries of their error matrices, 6,560 nonzero integer-direction fixtures and three rejecting controls. A changed-receipt control also rejects under optimized Python. The written proof supplies the all-real-D quantifier.

Normal and optimized warning-strict checks and both registered VERIFY commands pass locally. The individually timed commands take about 0.17 seconds each, below their 120-second limits. Environment: CPython 3.12.15, macOS 27.0.1, arm64; standard library only. Syntax inspection, package metadata/pins, exact committed bytes and source-history preflight pass. Official export integrity and a Git-free, read-only optimized replay pass; all nine exported files retain their bytes and modes.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/p104-contractive-gram-deflation/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/p104-contractive-gram-deflation/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/p104-contractive-gram-deflation
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/p104-contractive-gram-deflation --check-source-history

Expected markers: PASS_P104_CONTRACTIVE_GRAM_DEFLATION, PASS_REGISTERED_RESEARCH_PACKAGE, PASS_RESEARCH_COMMIT_BYTES. A clean CPython 3.12 may replace the local interpreter path. RESULT.json is generated and binds all seven proof/input/metadata files.

Canonical main is 3f0152a32eeebac0df11946dbd50cabe5a7f7839, priority revision 8. The coverage-law-change card explicitly changes to an original approximate-decoder question. The external coordinate identification is pinned by PR6166 head 096938513138868f0ed27639061f1a5054900f96 and path in MISSING.json; its factual signs are retained locally. The generic source-history pass covers declared canonical provenance, not scientific validation or ancestry of that external head. No pending Git object is needed for replay.

Limitations and unrun obligations

This only covers real symmetric positive contractive filters and maximum entry error on the complete Gram block. Unrestricted rank-seven matrices, noncontractive/nonsymmetric filters, masked accuracy, changed side corrections, source encodings and nonlinear consumers remain open. A different word could correct the residual. No native, dirty-restoration, physical caller, full signed invoice, precision/tape, compiler, BIT, outer, Lean or selected cold replay is supplied. Independent human and formal proof-kernel review are unclaimed. The selected frontier is unchanged.

Open next questions

• Can an exact noncontractive or nonsymmetric rank-seven factor beat 1/4? Start with rational factors and certify every error entry; physical realization remains a separate obligation.
• Can a changed side correction absorb the explicit rank-one residual with a winning complete invoice? Start from one omitted-root projector and an actual scalar word.

Related open work

• Prove stable joint-bank Gram spectral obstruction (no new κ) #6181 studies the complete P104 Gram stages under a common Gaussian-dyadic invertible code, including joint-bank mixing and restored scratch. Its endpoint correction-rank obstruction differs from this real contractive rank-seven coefficient-error floor; no common numerical optimum is computed.
• Prove functional decoder pairings cost at least v over every field #6178 bounds exact decoder costs for functional binary pairings. Its support and lift hypotheses differ from this positive contractive approximation model; no decoder-cost optimum is computed here.
• P104 carrier code: the length-8 decoding code is unique (E8 root code, min weight 49) #6166 supplies the cited P104 coordinates and studies ternary code uniqueness. This studies real contractive approximation and does not replay its code certificate.
• Prove exact two-channel P104 support 72 over F3 and Q (no new κ) #6173 uses the same cited PR6166 root code and computes joint support 72 over F3 and Q. This theorem concerns real contractive Gram approximation and computes no competing support invariant.
• Conditional κ .003128386 (+2.582e-6 over #6030): owned registers 770, 771 and 772 descend through the frames of helpers 705, 706 and 731, which no longer relay them; helper 732 reads 771 at frame 117 #6144 changes owned-register relay descent. This changes no word, lifetime or paid profile.
• Conditional κ = 0.003128645 (+2.59e-7 over #6144): retain an operand to remove a root stop #6167 retains an operand to remove a root stop. This supplies no competing complete kappa computation and does not exclude that changed word.
• No new κ: exact multi-gate resynthesis removes 117 P104 events #6169 resynthesizes scalar events while keeping the invoice. This concerns mathematical Gram approximation with its residual explicit.
• Add P104 designer toolkit v2: exact pricing and legality, #6030 rebuild with fold/hoist/retain, generators, sub-circuit tools (no kappa claim) #6170 supplies P104 design and pricing tools. This standalone certificate invokes no word optimizer or legality oracle.
• Prove an all-width tensor-frame reduction under fixed boundaries (no new κ) #6171 reduces tensor frames under fixed boundaries. This is a real coefficient-error problem, with no tensor-moment optimum claimed.
• Prove sharp signed tensor gap with variable target boundaries (no new κ) #6172 studies variable productive-boundary signed moments. This introduces no variable productive targets or negative credits.
• Add exact P104 ADD-coefficient fault localization (no new κ) #6175 localizes ADD coefficient faults. This deliberately changes the mathematical Gram channel rather than diagnosing a source word.
• Conditional κ 0.00309871 from a prepared P104 split helper #5606 prepares a split helper. This supplies no helper-preparation word or invoice change.
• Two successor hand-offs on the P104 word: conditional κ 0.0030428242 #5553 changes helper handoffs. This does not study or recheck those lifetimes.

The completion refresh inspected this account's open PRs created since the advance CLAIM, including #6181 and #6173. None computes the same contractive coefficient-error optimum. No additional computation was run to validate or compare predecessor results.

Live main README, merged-PR, open-PR, Discussion and local-lock searches were checked for overlap. Their coverage does not certify literature priority or unpublished work. The original three-contact CLAIM was withdrawn under the seven-day exclusion; the delivered scope follows the explicit scope update.

CI notes

Both hosted checks passed at exact head eaf77790466f34b86ce72080e3ec690e26dfacab in run 40881430320: plan and replay (research/p104-contractive-gram-deflation). The checks were read once; run metadata confirms the same head and completed success.

Hosted replay certifies execution of both registered normal and optimized checks at that commit. All recorded local mathematical, metadata/pin, committed-byte/source-history and read-only export checks also pass. The P104 coordinate identification remains a cited premise, and independent human and formal proof-kernel acceptance remain unclaimed.

Merge danger

Door: two-way. Blast radius: isolated package. Eight new files; no shared code, frozen sources or selected-frontier edits.