Scoped theorem: sharp excess-rank budgets for exact overcomplete E8 decoders

[bbMisk]

For an exact all-pairs E8 decoder with m terms, the two factors' ranks outside the live root code satisfy s+t <= m-8. This package proves the complete feasible region, constructs every feasible pair with every term active, and derives the full rank-one residual form for two-sided ten-term decoders. It also proves the symmetric-factor distinction: real symmetric decoders stay in the root code; complex symmetric decoders can have residual rank up to floor((m-8)/2), subject to ambient dimension.

The proof applies standard rank algebra and isotropic cancellation. The contribution is a sharp decoder-specific constraint and parameterization extending joaqcoo's #1224 projection equations and nine-term dichotomy. It does not claim priority for the known existence of a two-sided ten-term decoder.

Only research/decoder-excess-rank/ is added. PROOF.md is the universal written argument. The exact P112 witness has copy bill 82, rising to 90 under an identity-preserving orthogonal term gauge; this records why the parameterization cannot be treated as a cost-preserving search normalization.

Verification

python3.12 -B scripts/research_package.py --package research/decoder-excess-rank

Passed locally on macOS 27.0.1 arm64 / Python 3.12.13 in about 59 seconds:

• 168 general and 32 symmetric complex examples, every feasible residual-rank choice for 8–14 terms on P112/P120; all terms active; 2,694,400 exact identity entries.
• Gaussian-rational and separate integer-realification rank computations; four extracted ten-term residual forms; ten mathematical/source controls.
• A standalone integer checker, importing neither the primary checker nor predecessor code, verifies the stored symmetric witness, its nonzero rank-one residual, ordinary-versus-Hermitian distinction and both copy invoices.
• Six regression groups, including a 256-matrix determinant oracle, each ten-term obligation, witness/source corruptions, and optimized read-only relocation. Regression groups also passed under optimized Python 3.14.6.
• Package metadata/source pins, selected-result integrity and staged whitespace checks passed.

These are implementations by the same assisted author. Independent human review, Lean formalization, selected-construction cold replay and native supplier admission are unclaimed. Hosted CI is separate.

Scope, sources and coordination

Started from main ee055dc44815497927ef618fc985b93c2dcbda2a; submission base 0a201997097b8a89b638f3a2afcb35db4184c8b2, contribution head 652264fa1f70820139b2730363a0056b7fadc689. Guidance, catalogue, selected record and live inputs were unchanged by the intervening main update. SOURCES.json retains exact Git/blob/SHA-256 bindings; no frozen source or selected frontier changes.

The revision-3 complex-rank-shape-decoder card is explicitly marked changed scope to this algebraic side result. Residual rank is not recursive child rank. No cheaper decoder, complete physical invoice, 5% construction-gate success or new kappa is claimed. The README and TASK identify the remaining support-aware ten-term P112 search and its blockers.

Final overlap refresh covered 741 open PRs and 267 discussion records with complete reported pagination and no warnings, plus local notes/ledger. This does not establish novelty. Claim and final handoff.

Credit joaqcoo/#1224, woklerx/#1077, themilkman216/#968, Philipp/mightyOil/#1197 (including the prior gauge/copy-cost warning), Sagar Thakrar/#1327, DreamingOfClouds/#717/#446, Douglas Colkitt/#1240 and Arham Tahir/#1605. Exact overlapping heads and evidence boundaries are retained in the package.

Prepared by Hanson Sun (@bbMisk) with substantial OpenAI Codex assistance in proof, implementation, verification and writing. No subagents. Apache-2.0.

[Mister-Meeseeks]

Batch review at 652264fa1f70820139b2730363a0056b7fadc689.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Sharp residual-rank region s+t <= m-8 for exact overcomplete all-pairs decoders, constructive active-term attainment, complete two-sided ten-term form and real/complex symmetric distinction. Residual ranks are not COPY costs; finite examples corroborate the written proof.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.