[ikeboy]

κ = 4498144/10^11 ≈ 4.498144×10⁻⁵ > 2⁻¹⁵, conditional on OpenAI's base theorem and the retained analytic and fixed-tape interfaces. That is 9.2146906% above #48 (411862541/10^13), which it builds on. It is also 7.8378235% above the highest open claim at the time of writing, #51 (4171211781/10^14); #49 and #50 claim 4.123863984e-5 and 4.129418696e-5. The exact bit saving is 4498347/10^11.

Change and mathematical scope

This builds on #48 (b7194b8), so the branch also carries the #43, #46 and #48 commits; only the last commit is new. Everything except the scalar producer is #48's research/copied-fixed pipeline, with constants recomputed. The new files are in research/skip-strips/.

What changes. Every vertex's leave-one-out strip sums use a skip-prefix layout:

s_j = A_{j−2} + (v_{j−1} + B_{j+1}) instead of s_j = A_{j−1} + B_{j+1}

In the ordinary layout, the two consumers of each prefix A_i are A_{i+1} and s_{i+1}. Their envelopes are incomparable, so the second use always costs a fresh role. With the skip layout, the consumers of A_i are A_{i+1} and s_{i+2}, and cover(A_{i+1}) ⊆ cover(s_{i+2}). Likewise, the new node v_{i−1} + B_{i+1} sits inside B_{i−1}. The inherited carrier matching can join these nested pairs.

• Layout details. The top level runs the layout on the reversed value list. Deeper levels run it on values sorted by (support size, ±support), Reordered exclusion sums: conditional κ = 4.11862541e-5 #48's idea. Totals follow Reordered exclusion sums: conditional κ = 4.11862541e-5 #48's rules.
• Matching. It is Weighted carrier matching: conditional κ = 4.100629254e-5 > 2^-15 #44's weighted optimizer, pinned in links-{23,25}.uses and regenerable with optimize_matching.py.

	#48	this PR
additions (h=23 / 25)	37,098 / 49,119	40,329 / 53,475
carrier links (h=23 / 25)	6,002 / 7,715	12,719 / 16,991
roles R (h=23 / 25)	36,432 / 48,329	32,946 / 43,409
physical wires W	177,530,859	160,799,739

Hypotheses. No hypothesis or interface changes; the bound uses the same inherited framework as #48. Two points of scope:

• This result leans more heavily on the carrier-continuation rule than Reordered exclusion sums: conditional κ = 4.11862541e-5 #48 does. Without carrier matching, the new DAG has more roles than Reordered exclusion sums: conditional κ = 4.11862541e-5 #48's (45,665 vs 42,434 at h=23).
• Both fixed bases and copied reversed corners: conditional κ = 3.918734894e-5 > 2^-15 #40 and later, including Reordered exclusion sums: conditional κ = 4.11862541e-5 #48, are outside the maintainer audit of Fixed middle basis and copied reversed corners: conditional κ = 3.886675852e-5 > 2^-15 #39 (docs/research/community-final-audit.md). This PR inherits that status.

Validation

Finite checks. make skip-strips-verify checks:

• scalar exactness: every output is the exact leave-two-out sum or retained total;
• the original-envelope labels;
• pinned-link legality: every link is replayed in profiles.cpp and recounted independently;
• the fixed I+J profiles: 61,461 / 82,606 distinct transition matrices, 5,642 / 7,434 of them checked under several primes (CRT), with 0 disagreements;
• the full physical timeline and every dirty-scratch basis vector, in both orientations;
• all 4,073,300 data pairs, with exact rational recovery of the ten primary-prime failures;
• the exact moment, plus exact lower bounds showing that the Copied retained centers and two-stage assembly for conditional kappa = 3.84569e-5 #36 baseline and the Both fixed bases and copied reversed corners: conditional κ = 3.918734894e-5 > 2^-15 #40–Weighted carrier matching: conditional κ = 4.100629254e-5 > 2^-15 #44 and Weighted matching and exact data recovery: conditional κ = 4.10062945e-5 #46–Reordered exclusion sums: conditional κ = 4.11862541e-5 #48 networks fail at the new bit saving;
• all 47 strict assembly conditions and 7 margins.

tests/test_skip_strips.py adds 20 tests, including leave-one-out exactness, the cover-nesting identity and negative controls.

An earlier version of the same change, applied on top of #43, was audited independently from clean copies. The audit re-ran the full producer check and recomputed every transition profile in exact rational arithmetic, with no modular step. It found no correctness issue. The stacked version here has not had that separate audit.

Proof arguments. As in #41, #47 and #48, which also changed the producer DAG, the new DAG enters the inherited proofs only through properties and quantities that the finite checks above recompute: disjoint additions, common points, exact outputs, admissible carrier links, the physical timeline and the transition profiles. The cover-nesting identity explains why the matching finds more links; correctness does not depend on it. The inherited written obligations are not re-proved here: the base theorem, the residual compiler and fixed-tape recursion, the physical meaning of carrier continuations and copied centers, and the analytic, routing and eventual-threshold interfaces. This is not a formal verification, and no global optimality or speedup is claimed.

Attribution

The skip-prefix layout is new here, by Avi Eisenberg with Anthropic Claude (Claude Opus 5.5) assistance. The rest is inherited from:

• Chafik Boukhalfa: Changed graphs and fixed bases: conditional κ = 4.09953e-5 > 2^-15 #43, Weighted matching and exact data recovery: conditional κ = 4.10062945e-5 #46, Reordered exclusion sums: conditional κ = 4.11862541e-5 #48;
• Rohan Arun: Weighted carrier matching: conditional κ = 4.100629254e-5 > 2^-15 #44, plus Copied centers and reversed corners for conditional κ = 3.850771033e-5 > 2^-15 #37, Fixed middle basis and copied reversed corners: conditional κ = 3.886675852e-5 > 2^-15 #39, Both fixed bases and copied reversed corners: conditional κ = 3.918734894e-5 > 2^-15 #40, RaD producers with fixed reversed geometry: conditional κ = 4.099494519e-5 > 2^-15 #42, Hill-climbed producers for a conditional κ = 4.105106623e-5 bound #47;
• RaD / hipotures: Alternating pair-block networks for conditional κ = 4.034168124 × 10⁻⁵ > 2⁻¹⁵ #41;
• Dominik Scholz, James Chang, icekylinx, Zhihao Chen, Paureel, Swapnil Jain, Douglas Colkitt and OpenAI.

Full credits are in research/skip-strips/PROOF.md and NOTICE.

🤖 Generated with Claude Code

https://claude.ai/code/session_013eMahk2wJVfY7JKgp81K6y