Research: uniform proper-star ranks and all-target puncture thresholds

[ellipsi2]

Research theorem / scoped negative result; no new kappa.

This proves a uniform rank formula for global proper stars on signed k-subset ports, and the sharp algebraic all-target puncture threshold when k is a power of two. Granting the global sum adds one dimension exactly when k is a power of two and p >= 2k-1. In the 1,120-port (8,4) geometry, nonempty stars can retain at most 447 simultaneously decodable ports (673 deletions); granting the empty star permits 448 (672 deletions). Both limits have literal attaining decoders.

The proof separates sign monomials and applies the integral unsigned inclusion basis of Ghorbani et al., arXiv:0709.3144v1. PR #747 already supplies the finite (6,4) all-target threshold85 and information-set argument; those are credited corroboration. This adds the uniform formula and identifies a boundary invisible in that smaller geometry. PR #732's correction-absorption mechanism and #669/#231's paired-star lineage are credited. No predecessor executable code is imported.

Only research/paired-star-rank/ is added. Exact base: 66849e1475c930fd7732b1a0067ffee7e1e95f60. Source heads, inspected-byte hashes, OS-random seed46, candidate selection and the local sprint claim are included. No pinned coordination issue existed. Accepted kappa and upstream/ remain unchanged.

python3 -I -B research/paired-star-rank/verify.py
python3 -I -O -B research/paired-star-rank/test_verify.py

Expected: PASS_PAIRED_STAR_RANK: 36 cases; (8,4) ranks 447/448; all-target deletions 673/672; 8 negative controls; kappa unchanged, followed by four passing test groups.

Validation passed on Windows Python3.11.15 and local Linux Python3.12.3, including optimized isolated mode, byte-identical regeneration, relocated minimal-input replay, modified-input rejection and preservation of prior output on failure. The separate verifier performs no elimination: 2,706 explicit unit decoders prove the lower bounds and 3,714 full-family spanning identities prove the upper bounds. Selected-record/source integrity and whitespace checks pass. The unchanged broad supplier suites were not repeated; no hosted CI or formal multiplication-proof pass is claimed.

The result concerns unrestricted scalar star identities, not preparation cost, dirty restoration, legal frames, hosting or a paid recursive supplier. More retained ports are impossible within the stated decoder class even with zero implementation cost; the attaining information sets have no priced construction. Inherited compiler, precision/tape/recovery and analytic assumptions remain.

Prepared by ellipsi2 (@ellipsi2) with substantial OpenAI Codex assistance, including one scoped AI proof review, under Apache-2.0. No independent human review is claimed. Pre-submission overlap refresh covered 586 open PRs through #767 and found no duplicate uniform-rank result.