Research (no κ claim): Gaussian E8 c = 20 exact certificates; local-rewrite census on #4088's T360 words

[GamingPuzzled]

No κ claim. This continues #620's Gaussian-E8 notes (§13–§16, merged) and adds one negative census on the current T360 frontier words.

1. Gaussian E8, c = 20 — exact certificates (research/e8-gaussian-c20-exact/)

A Gaussian-E8 decoder (60 ℚ(i)-lines, F2 pattern) beats real E8 only if its Theorem-C cost is ≤ 42. §15–§16 exclude c ≤ 19.

cost cap	cases	status
≤ 41	20	all excluded
≤ 42	21	16 excluded; open: 5, 6, 17, 18, 23

• So no 20-total Gaussian decoder costs ≤ 41.
• At cost 42, the five open cases sit within 1 of the counting threshold. They may be counting-feasible, in which case only the F2 pattern (e.g. §16's 2×2 support lemma) can close them.
• Method: an exact branch-and-bound (ebb4.py). HiGHS solves the LPs, every bound is recomputed from dyadic-rounded duals in integer arithmetic, probing and reduced-cost fixing are recorded as certified steps, and symmetry is reduced with each case's stabilizer.
• Checking: each certificate is re-checked without floating point by ecert.py.
• Reproduce: sh verify.sh /fresh/dir rebuilds every table from scratch and reports verified 35 / 35 certificates EXCLUDED (exact), as tested locally from a clean directory.
• Case 22's certificate (349 MB) exceeds GitHub's limit. Its SHA-256, our exact check and the deterministic regeneration command are included.

2. Local-rewrite census on #4088's T360 words (research/t360-local-rewrite-census/)

Searched on the #4088 head (72d7efc1), with a full frame re-solve after every candidate:

• receiver sharing (Conditional κ = 0.001730970419199163702738 (+1.614e-7 over #4076): receiver sharing on the staged T360 supplier #4085's identity, dyadic ratios, clusters of up to 8);
• source sharing;
• commuting-gate retiming.

About 20,000 candidates across the A, B and restore stages gave 0 strict savings, and the current frames are already optimal. The only payable receiver pair, 672/676 via 544, is the one #4085 already uses. Further 8→7 staged entries are capped by owner sink capacity (#4076's LP, #4091's ceiling).

Credits:

• Gaussian line: 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, Research notes (no κ claim): E8 decoder floors are tight; extension-ring, triple and Lorentzian units lose after helpers; cold-end, dispatch and fusion-search negatives #620, Scoped negative result: Gaussian E8 copy floor 40 excludes three stages #738, Scoped negative result: Gaussian E8 planar and incidence decoder obstructions #767, Research: Gaussian E8 weighted maximal-clique ceiling 40 and independent replay #770, Research: certified format ceilings for Eisenstein/Gaussian/Hurwitz E8 copy floors (no κ claim) #801, Research: Gaussian E8 all-clique certificate ceiling is 40 (no new kappa) #868, Research: Gaussian nonlinear message copy floor 40b under explicit width pricing #888, Scoped proof: Gaussian whole-hyperplane reciprocity costs at least 180 #2066.
• T360 census: Improve conditional κ with a mixed fourth staged entry (+8.696e-9) #4088 (1337isnot1337), Conditional κ = 0.001730970419199163702738 (+1.614e-7 over #4076): receiver sharing on the staged T360 supplier #4085 (zxv; Nick Petalas's fan-out search), Conditional κ = 0.001730809033576552081734 (+1.458e-7 over #3945): three staged helper entries and an LP-exact T360 fill #4076 and Raise conditional staged544 κ by filling idle owner slots #3945 (rohanarun), Research: census of staged entries on PR3734's other helpers; none found that pays (no new κ) #3949, Prove the sharp pooled optimum for fourth staging #4091 (RossRichardson).

AI disclosure: prepared by Joel Pulikkan (GamingPuzzled) with Claude Opus 5.5 (Anthropic) via Claude Code. Apache-2.0.

🤖 Generated with Claude Code