Research: sharp COPY69 for the A2+E6 nine-normal P112 decoder

[adamgall]

Result

Sharp COPY69 for PR1224's fixed A2-weight + E6 root-basis nine-normal decoder on standard P112, with arbitrary independent complex per-root kernel shifts and ordinary transpose. The all-nine-active rational witness has term costs (1,8,8,8,8,9,9,9,9) and differs from the dropped-first-term base only by t_45=-4.

This resolves the named A2+E6 follow-up explicitly left in PR1304 and PR1612. It is not a finite coefficient-box extension, the A8-simplex result, retained-coordinate sparse-lift PR1325, or the partition-gauge PR1962/1988 family.

Proof and evidence

Six forced terms cost 52. The three variable column supports have full binary label rank, so their cost is 27 - sum(kernel dimensions). A complete 512-character compatibility census shows that any kernel of dimension at least three forces both other nonzero kernels onto a unique exceptional one-dimensional pair. With the separately handled zero-kernel cases, the total deficiency is at most ten, hence variable cost at least 17. The all-active witness attains 69.

The package contains the written proof, exact finite certificate, independent geometry/character-basis implementation, actual support-pair charges, 12,544 all-pairs rational identities per checker, semantic negative controls and full chooser/source/replay metadata.

The complete optimistic conditional fixed-noncentre screen already fails at the elementary COPY65 bound. COPY69 sharpens its root to [1452819253108,1452819253109]/10^15, even granting fractional rank-nine centres and zero added interface overhead. The independent integer-power bound gives moment >=15238506806272258653/15232000000000000000 > 1 at a=1/625.

No new physical supplier, selected frontier or kappa is claimed. The theorem fixes one normal basis and puncture, all-pairs exactness and separate term charges. Other normals, masks, shared physical copies and changed noncentre work remain outside scope.

Validation

CPython 3.12.12, darwin arm64, standard library only:

uv run --python 3.12 --no-project python -I -B research/a2-e6-nine-normal-floor/verify.py --write
uv run --python 3.12 --no-project python scripts/research_package.py --package research/a2-e6-nine-normal-floor

Observed: primary certificate PASS; independent geometry/basis and integer-power replay PASS; six semantic test groups PASS; optimized -O replay PASS; PASS_REGISTERED_RESEARCH_PACKAGE. Parent independently reviewed the proof and replayed all four registered commands before publication. No selected cold replay, Lean/kernel acceptance, independent human review or hosted-CI success is asserted.

Provenance and scope

Only research/a2-e6-nine-normal-floor/ is added. Base e112d71c8d7bb32435debcc54df932593dd8223a; independent seed 814c825e12629ac65e009a4beab63757, complex-mechanisms / complex-rank-shape-decoder. Explicit changed scope: the named support-family obstruction rather than a physical synthesis success. Full original draw is preserved in SELECTION.json.

Credit joaqcoo / PR1224 for the family and normal form; Philipp/mightyOil / PR1248, commdata2338 / PR1304, and Hanson Sun/bbMisk / PR1612 for the deficiency approach and handoff; DreamingOfClouds / PR717/595/446 for the decoder/COPY model; Joel Pulikkan/GamingPuzzled / PR620 for coordinate COPY64; themilkman216 / PR968 and woklerx / PR1077 for surrounding bounds; Sagar Thakrar / PR1012 and Douglas Colkitt / PR1240 for coverage and ceiling accounting. Frozen inputs remain unchanged and hash-pinned.

Coordination claim. Final cached overlap and targeted live merged-head/search checks are recorded in TASK.json, including coverage limits.

Prepared for Adam Gall with substantial OpenAI Codex assistance in research, proof, implementation, checking and writing. Apache-2.0.

Research claim and completion record

Status: completed. The claim record is preserved here so this PR is self-contained. Historical coordination links in the committed research metadata refer to this record; the separate coordination discussion is being removed.

DONE — seed 814c825e12629ac65e009a4beab63757. PR #2061 #2061 at eed2398 proves sharp separate COPY69 for PR1224 fixed A2-weight + E6 root-basis nine-normal decoder on standard P112, allowing arbitrary independent complex row shifts. All nine terms active: (1,8,8,8,8,9,9,9,9), t45=-4 only. Complete binary compatibility gives deficiency sum <=10 and variable cost17 plus forced52. Conditional fixed-noncentre optimistic budget fails even with all centre mass at rank9 and zero additional overhead; no new supplier or kappa. CPython3.12 registered replay passed primary/independent checks,12544 identities per checker,6 test groups and optimized replay; parent independently reviewed/replayed before publication. Base e112d71. Credit joaqcoo PR1224, mightyOil PR1248, commdata2338 PR1304, bbMisk PR1612, decoder and coverage/ceiling lineage in package. No hosted-CI success, proof-assistant or independent human review claimed.