[petalas]

Summary

Scoped written proof with exhaustive finite binary lemmas: the fixed A7 weights plus A1 decoder has exact separate-term COPY minimum 71 after removing the coordinate-pair frame e0 +/- e1, e2 +/- e3, e4 +/- e5, e6 +/- e7 from E8. All independent complex row shifts are allowed, and zero terms are allowed.

specified P112 puncture + fixed A7+A1 normals
  total mutable deficiency <= 10
  COPY >= 81 - 10 = 71
  t_i = -x_i0 attains ranks (0,8,9,9,9,9,9,9,9)

This answers the specified-puncture question left by Adam Gall #2062, head 8344a5fc1eb643aca6a21bc745d824a38f449e39. It excludes improving the cited ordinary P112 COPY64 comparator through this fixed family and puncture. No new kappa, supplier or selected-frontier change is claimed. Only seven files under research/a7-a1-puncture-optimum/ are added. Produced by Nick Petalas.

Credit joaqcoo #1224 for the family and row-shift normal form, Adam Gall #2062 for its canonical model and puncture question, Philipp / mightyOil #1248 for the deficiency method, and DreamingOfClouds #446/#595/#717 and the retained geometry lineage for the COPY model. Predecessor theorems are cited, not audited or replayed. Advance claim.

Evidence

Before, #2062 settled the unpunctured family and left changed punctures open. After puncturing, 254 planes have a two-dimensional partner space, so its dimension-one argument does not transfer. The new complete finite lemmas and written case split prove total deficiency at most ten and a matching optimum of 71.

• All 43,435 binary planes, with 304,045 comparisons between complete character-compatibility masks and direct forbidden-label elimination.
• All 511 nonzero characters for seven reverse conflict spaces, plus the six independent exceptional root labels and puncture-preserving anchor symmetries.
• All 12,544 all-pairs equations for each of three successful exact endpoint fixtures, including nonreal shifts. The attainer is priced using the actual nine restricted pairing matrices. A deleted A1 term and corrupted coefficient are rejected; a discriminator rejects transferring the old dimension-one premise.
• Every registered check passed locally on CPython 3.12.15, macOS arm64, standard library only. Separately timed normal and optimized commands took 4.24 and 4.20 seconds; the two-command registered run took 8.71 seconds. These are timeout observations, not performance comparisons.
• Metadata/pins, warning-strict replay, AST/function-size and whitespace checks passed. Committed-byte/source-history preflight reports PASS_RESEARCH_COMMIT_BYTES for all seven files at 9cf91d3da32e8ed58cc3d3a2962d5963b16067ff. The emitted receipt is generated from recomputed evidence and package hashes; none is handwritten.

Reproduce from the repository root, using this path or another CPython 3.12 executable:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/a7-a1-puncture-optimum
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/a7-a1-puncture-optimum/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -W error -B research/a7-a1-puncture-optimum/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/a7-a1-puncture-optimum --check-source-history

Expected markers: PASS_A7_A1_P112_COPY71 planes=43435 annihilator_checks=304045 cost=71, PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=negative-result, and PASS_RESEARCH_COMMIT_BYTES.

Limitations and unrun obligations

The proof fixes the normals, coordinate-pair puncture, ordinary-transpose all-pairs identity and separate-term binary pairing-rank charge. It allows a zero term. Other punctures, all-nine-active optimality, masked identities and shared physical copies are outside scope. The universal complex-parameter statement comes from the written proof, not finite parameter testing.

Acquisition, delivery, restoration, frames, hosting, stock, precision, compiler and full consumer costs remain unpriced. No selected cold replay, Lean, Docker, physical supplier or complete multiplication theorem was run or established. Independent human and formal proof-kernel review are unclaimed. Managed-allocation and bounded-runtime details are documented in the package.

The historical card orientation command was attempted and fails with KeyError: 'complex_profile' at asm['complex_profile'], because it expects an older selected schema. The current selected metadata orientation passes. This is an unrelated local orientation limitation, not a premise or failure of the new verifier; its source is unchanged.

Open next questions

• Can another frame orbit relative to the same normals lower COPY below 71? First derive its reader spans and conflict-space dimensions. Replacement lemmas and a complete physical bill are missing.
• What is the optimum when every term must be active on this puncture? First analyze equality in the deficiency-ten bound and price an exact active witness. The present attainer has one zero term.

Related open work

• #5229 studies the distinct A3+D5 normal family on P120. This solves a punctured A7+A1 question from Research: sharp A7+A1 nine-normal COPY72 floor for arbitrary complex shifts #2062; no cross-family numerical agreement is asserted.
• #2832 studies general E8/P112 decoder-floor reductions and outside corrections. The present result is a stronger fixed-family optimum, with no universal-floor conclusion.
• #3074 studies one-sided P112-to-P120 padding. This keeps the specified nine-normal reader family fixed and needs new punctured compatibility lemmas.
• #3045 studies square root-code decoders with arbitrary fitting pairings. This fixes the dense binary pairing and allows overcomplete row shifts.
• #5064 searches changed masked completions and factor bases. This is an all-pairs fixed-normal theorem, not a continuation or replay of that search.
• #5088 studies a two-star masked P112 support dictionary. Its alphabet and exploratory claim differ from this exact fixed-family minimum.
• #3566 binds a tetrad decoder to selected ports; this prices a different fixed P112 normal family without claiming a physical binding.
• #3604 studies tetrad coefficient fusion through a centre consumer; no such physical consumer is priced here.

Canonical main was refreshed before committing and remains ec2780e8aeb1c88acaf5e9e470ef8c68e7fbbb6c, priority revision 7. Live claims, reservations, open PR bodies, merged searches and main-tree README searches were checked. Sparse local-note coverage is incomplete; these searches are not a novelty certificate.

CI notes

CI pending for 9cf91d3da32e8ed58cc3d3a2962d5963b16067ff. Both registered mathematical checks and all required local preflight checks passed. No local package obligation is deferred to CI. Hosted replay and the final DONE update remain pending.

Merge Danger

Door: two-way.

Blast Radius: isolated.

Seven new package files. No shared implementation, predecessor source or selected-frontier metadata changes.