Research: sharp A8 simplex P112 copy floor 71 on matching-frame punctures (no new kappa)

[commdata2338]

Change and mathematical scope

PR #1248 left the fixed A8 simplex decoder on a P112 frame puncture open. This package proves that the minimum separate-term copy cost is 71 for the specified matching-frame puncture, over arbitrary complex row shifts, and for its 945 coordinate-permutation images. All nine terms can be active, with costs (1,8,8,9,9,9,9,9,9). Since 71 exceeds the standard P112 cost 64, this family cannot provide the contemplated copy saving.

Status: scoped proof with exhaustive finite certificates, a negative result for this route. It bounds copy cost only: no scalar saving, paid-profile or κ improvement, physical supplier or complete moment invoice is claimed.

Only research/a8-p112-decoder-screen/ is added (10 files). All finite lemmas are recomputed. The selected frontier, frozen sources and existing packages are unchanged.

Precise delta. The P112 column supports still have binary label rank nine, but partner deficiency spaces can now have dimension two. The P120 dimension-one lemma fails after the puncture, so it is not reused. A complete subspace-product check proves total deficiency at most ten, hence cost at least 81-10=71. The primary check covers 390,915 planes, 19,299 partner-space tuples and 462,141 compatible choices. The separate checker rebuilds labels from the Gram matrix, uses annihilators and Cartesian products, and verifies a puncture stabilizer transporting two anchor representatives to all nine. Both check all 12,544 entries of the active witness, and the separate checker prices the actual pairing submatrices. The written argument in PROOF.md covers all complex row shifts, not sampled coefficients. This is a scoped proof with finite certificates, not formal verification of the multiplication theorem.

Remaining obligations. Not covered:

• other P112 frame orbits, other normal frames and masked identities;
• two unrestricted factors and shared-copy billing.

Not produced:

• a physical supplier;
• a delivery, restoration, frame, hosting or precision invoice;
• a new moment or κ.

Not run or proved:

• the selected cold replay (make frontier1094-verify);
• native or Lean builds, and hosted CI;
• the all-size compiler and the complete multiplication theorem.

All inherited analytic, realization and bit H1-H5 conditions remain.

Task and overlap

• Task ID / scope key: complex-masked-decoder / 2a6f2c726b78f9ae7d26, refined to a8-simplex-matching-frame-p112-all-complex-shifts. Chooser nonce a57142e8e77d6c0af036ca35809daf69, track complex-mechanisms, priority revision 2.
• Base and exact heads: base dadfea6e8a78ff6ba901705f87532a71c6f5bbb8; contribution a052f0da36a3845c2ec785fd44e5b9e50de3b688. PR Research: sharp A8 simplex nine-term decoder copy floor 72 (no new kappa) #1248 f226f4329a996b0a1bea8794179a91b5d0ed80bf; PR Research: nine-term complex E8 decoders are one-sided overcomplete representations (seed 42, no new kappa) #1224 00e6ed0810b8ebc7f6133e2c036d0f1c9a3a4162. Both predecessor heads were fetched and rechecked. They provide method and question attribution, not imported code or assumed finite lemmas. Main-tree source pins are in SOURCES.json. Selected PR Conditional kappa 0.001672660197076918: rare-pattern T16 exchanges #1094 0fc329d0bec93ca7cd288eaea1295172e8789810 is context only.
• Concrete object and question: roots e_a-e_b and 1_S-(1/3)1 in the sum-zero subspace of Q^9, with fixed T_ik=r_ik and D_ik=2r_ik+t_i. Remove frame indices [0,15,26,33,42,89,112,119]. Can arbitrary complex shifts reduce total binary pairing-rank cost below 64?
• Scope status: refined, to the card's overcomplete branch and PR Research: sharp A8 simplex nine-term decoder copy floor 72 (no new kappa) #1248's explicit next question. Coordinate permutation gives a corollary for the 945 matching-frame punctures; no other frame orbit is claimed. Nine overcomplete terms are allowed, and P120 is replaced by the specified P112 puncture.
• Method and bounded domain: all 511 nonzero binary functionals, all 43,435 binary planes for each of nine anchors, and every subspace of each partner space (proved to have dimension at most two). Exact integer and rational arithmetic, with a separate Gram-derived implementation. No floating-point search or solver dependency.
• Optimistic budget: a cost of at most 63 would be the necessary first saving against 64 under separate-term billing. The sharp floor 71 rejects the route before any delivery, frame, restoration or hosting synthesis.
• Nearest overlap and added value: PR Research: sharp A8 simplex nine-term decoder copy floor 72 (no new kappa) #1248 proves P120, and PR Research: nine-term complex E8 decoders are one-sided overcomplete representations (seed 42, no new kappa) #1224 supplies the normal form and bounded searches; this proves the fixed P112 optimum for one frame orbit. PR Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212/Research: classify the P112 masked decoder coordinate component (no new kappa) #1213 and PR Research: masked rank-8 complex decoders are locally rigid on all P104 frame-pair punctures (no new kappa) #1292 (discussion CLAIM: exact local rigidity of masked rank-8 complex decoders on all P104 frame-pair punctures (Navinor, expires 23:40Z) #1275) concern eight-term masked rigidity. PR Research: close the masked root-star assembly gap with sharp cost 64 (no new kappa) #1236 and discussion DONE: nine-root obstruction for a fixed P112 co-star decoder #1267 concern root-star and co-star supports. The claim is in discussion #1241. A live overlap check just before submission (796 open PRs and 45 discussion records at 22:12 UTC, complete pagination) found no competing fixed-simplex P112 result.

Validation

python3 -B research/complex-decoder-field-extension-seed44/verify.py
python3 -B scripts/research_baseline.py
python3 -B scripts/research_package.py --package research/a8-p112-decoder-screen --check-only
python3 -B scripts/research_package.py --package research/a8-p112-decoder-screen
python3 -B -m unittest discover -s tests -p 'test_research_*.py'
python3 -B scripts/verify_selected_result.py

All pass on CPython 3.12.3 / Linux x86_64, standard library only. The registered replay runs both checkers and the six package tests in about 58 seconds. All 20 research-workflow tests pass without skips. Coverage includes:

• optimized replay, from a relocated copy and read-only;
• the separate checker run without the primary code;
• changed, missing, incomplete or malformed receipts;
• eight semantic rejection controls: a coefficient error, a deleted term, corrupt labels, a nonorthogonal puncture, the false inherited dimension-one lemma, missing mutual compatibility, a false cost of 70, and a valid non-real shift.

The registered replay and the metadata/pin check were run again on the final commit before submission, and both passed.

Expected markers:

PASS_A8_P112_DECODER_FLOOR planes=390915 configurations=19299 compatible_leaves=462141 cost=71 controls=8
PASS_A8_P112_INDEPENDENT anchors=2 transported=9 planes=86870 compatible_leaves=92317 cost=71
PASS_REGISTERED_RESEARCH_PACKAGE research/a8-p112-decoder-screen outcome=negative-result

Credit

• Philipp/mightyOil (PR Research: sharp A8 simplex nine-term decoder copy floor 72 (no new kappa) #1248) for the deficiency method and the open P112 question.
• joaqcoo (PR Research: nine-term complex E8 decoders are one-sided overcomplete representations (seed 42, no new kappa) #1224) for the A8 frame and the one-sided normal form.
• DreamingOfClouds (PR Format limits part 2 (no κ claim): erratum to #595, exact invoice prices, frontier-word censuses, decoder floors, spec requirements, centre and fragmentation walls #717/Format limits (no κ claim): certified ceiling κ ≤ 5.1595e-3, spec census, structural lemmas, helper bounds and open problems #595/Format obstructions (no new κ): exact bounds on window width, copy cost and label dimension #446) for the decoder and label-cost model.
• themilkman216 (PR Research: frame-pair decoders and the exact 8-term complex copy-cost floor (no new kappa) #968) and woklerx (PR Scoped structural result: arbitrary complex coefficients retain the exact eight-term E8 decoder floor #1077/Structural proof: overcomplete factor-linear E8 decoders retain the puncture copy-cost floor #1180) for the surrounding bounds.
• Classical E8/A8 geometry.

New work is Apache-2.0.

AI assistance: submitted by @commdata2338 with substantial AI assistance. OpenAI Codex (GPT-6 Astra) did the research, the proof, both implementations, the verification and the writing; it was packaged and submitted with Claude Code. The separate implementation comes from the same assistant, so it is not independent human review.

Open next questions

From the package README:

• Other frame punctures relative to the fixed simplex. Do they also cost at least 64? First step: enumerate the orthogonal frames and their S9 orbits, then recompute column ranks and partner dimensions for a representative outside the matching-frame orbit. Blocker: the present enumerator rejects partner dimensions above two and may need extending.
• Another specified nine-normal frame. Can PR Research: nine-term complex E8 decoders are one-sided overcomplete representations (seed 42, no new kappa) #1224's A7+A1 or A2+E6 family reach copy cost below 64 on a named P112 puncture? First step: establish its exact ratio classes, column spans and a favorable complete cost budget. Blocker: the simplex's common kernel vector and compatibility census do not transfer automatically, and physical realization remains a separate obligation.

Workflow feedback (optional)

• Helped: PR Research: sharp A8 simplex nine-term decoder copy floor 72 (no new kappa) #1248's explicit open next questions gave a distinct, bounded task right after the overlap check. The package generator and the registered Python 3.12 hook made the result easy to package and replay.
• Friction: the broad card's overlap terms returned many masked and overcomplete families. Title-level similarity did not distinguish the remaining puncture question, and --all-terms queries did.
• Suggestion: show one or two source-linked, bounded successor questions from recent packages beside the chooser's broad card, keeping the warning that they may already be claimed.

[Mister-Meeseeks]

100-PR batch review at a052f0da36a3845c2ec785fd44e5b9e50de3b688.

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

Reviewed scope: Fixed P112 simplex decoder cost-71 bound, covering 945 matching punctures and all complex row shifts through a complete binary compatibility census. Independent engine and controls pass; not a universal decoder barrier.

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.