Research: close the masked root-star assembly gap with sharp cost 64 (no new kappa)

[DavidAriasC]

PR #1226 leaves open whether nine cheap trimmed-star atoms can assemble into a masked P112 decoder below copy cost 64. This package closes that question: the exact minimum for its entire root-star support family is 64, for any finite term count and arbitrary real or complex coefficients. The existing coordinate decoder attains the bound.

The new constraint is simultaneous frame coverage. Every live orthogonal eight-frame needs eight rank-one contributions. Nine cost-seven atoms would need pairwise disjoint missed-frame sets; selecting one live normal from each produces nine pairwise nonorthogonal E8 lines. A short primitive-integer-dependence parity argument rules those out. Eight terms cannot include a cost-seven atom, and ten already cost at least 70. This strengthens #1226's correct averaged bound 784/13 without relying on pending eight-term rigidity proofs.

The supports must be exactly C(alpha) or their self-trimmed versions N(alpha) on the specified P112. Arbitrary free cost-zero corrections are allowed. Deeper trims, arbitrary subsets, non-star supports, other punctures and shared-copy pricing remain outside the theorem. No new kappa, physical word or supplier admission is claimed.

Only research/masked-root-star-assembly/ is added. Start with its README.md and PROOF.md. The base is current main 49b3157d1d89adcef16df8be98445451672be5ba; the selected record, upstream and frozen packages are unchanged. The package opts into the existing bounded research-package replay hook.

Validation

python3 -I -B research/masked-root-star-assembly/verify.py
python3 -I -B research/masked-root-star-assembly/independent.py
python3 -I -B research/masked-root-star-assembly/test_package.py
python3 scripts/research_package.py --package research/masked-root-star-assembly --check-only

All passed on macOS arm64 / CPython 3.14.7. Both separately written implementations rebuild all 57,600 support-pair ranks and 1,260 frames, cover all 3,304 live orthogonal pairs, and verify the sharp coordinate identity. The primary checker uses bitset pairing ranks and complete clique enumeration; the other uses dense binary elimination, integral lattice coordinates and a quadratic-form obstruction. Neither imports predecessor code. Thirteen semantic controls and seven package tests pass, including optimized isolated relocation, read-only replay and changed/missing receipt rejection. Selected-record integrity also passes. A separate AI review reproduced the core finite facts and checked the proof and implementations.

Hosted CI, the Python 3.12 generic replay, Lean and the full selected-supplier cold replay were not run; all-size/compiler, realization, hosting, precision/tape, analytic and bit-interface obligations remain inherited. Full commands and limits are in VALIDATION.md.

Scope, overlap and credit

Task-selector card: complex-masked-decoder, nonce b0886c49d1a831e2034b53c13c05bae4, refined to #1226's explicit assembly gap. The final overlap refresh covered all 838 open PRs, including drafts, and all discussion bodies/comments/replies (12 indexed records). Discussion #1210 records the predecessor's released claim and open assembly question. Heads, hashes, coverage limits and open next questions are included.

Credit Aaron Wang / Navinor for #1226's support family, census and question; DreamingOfClouds for #717/#595's masked model and label costs; Joel Pulikkan / GamingPuzzled and the retained lineage for the coordinate construction; themilkman216 and woklerx for #968/#1077's all-pairs context. #1212/#1213's eight-term masked rigidity, #1191's one-sided root-linear theorem, and #1180/#1224's all-pairs overcomplete work are complementary and are not proof dependencies.

Prepared for David Arias with substantial OpenAI Codex assistance in research, proof, both implementations, testing, review and writing. Separate AI review is not independent human review. Apache-2.0.

[Mister-Meeseeks]

100-PR batch review at ee8437bd9879272094e20c024cd34a53c4e84cd8.

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

Reviewed scope: Sharp64 copied-information floor for actual full/self-trimmed P112 root-star supports at any term count: simultaneous frame misses exclude nine cost7 atoms. Retains PR1226 relaxation and coordinate-witness attribution; arbitrary sub-supports excluded.

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.