[ikeboy]

κ = 4986133/10^11 ≈ 4.986133×10⁻⁵ > 2⁻¹⁵, conditional on OpenAI's base theorem and the retained analytic and fixed-tape interfaces. This is 4.651% above #60 (4764513337/10^14), the highest open claim at the time of writing, 6.782% above #57 (4669442391/10^14) and 10.849% above #53. It does not use #57's compiler or #54's clones. The exact bit saving is 4986382/10^11.

Stacked with #57's unchanged joint frame compiler, the same graph gives κ = 5101691/10^11 ≈ 5.101691×10⁻⁵, 7.077% above #60 and 9.257% above #57. That second witness also depends on #57's compiler argument.

Change and mathematical scope

This builds on #53 (3ffd402). The branch also carries #57's commit (cd350f7, by eumemic) unchanged, for the stacked witness. Only the scalar producer DAG changes. The new files are in research/pair-assembly/.

A role is a physical scratch wire: R = c + q − μ, where μ counts retained carrier links. A link needs the donor's original envelope inside the target's: the target's common core inside the donor's, and the donor's cover inside the target's. So R counts the additions that cannot continue a carrier, plus q. Both changes raise μ.

1. Interval strips. Each vertex needs every leave-one-out sum of its k strip values. #53 used s_j = A_(j−2) + (v_(j−1) + B_(j+1)). We build every cyclic interval instead, each from an interval one item shorter: I(a, m) = I(a, m−1) + v_(a+m−1), and s_j = I(j+1, k−1). Every interval shorter than k−1 can continue into I(a−1, m+1), which consumes the same last item and contains it. This costs k(k−2)+1 additions instead of about 4k, but only the k outputs and the total stay unmatched. In a model of one strip, an exact integer program over every leave-one-out circuit shows k−1 unmatched additions is the minimum for k ≤ 8. Skip-prefix leaves 2k−3.

2. Core-aware pair assembly. For groups (a, a2) and (b, b2), with F = far[i,j] and strip outputs Sa, Sa2, Sb, Sb2:

F+Sa, F+Sa2, F+Sb, F+Sb2
Y1 = (F+Sb)+Sa,       Y2 = (F+Sa)+Sb2
Y3 = (F+Sa2)+e(a,b2), Y4 = (F+Sb2)+e(a,b)
out[a,b]  = e(a2,b2)+Y1,  out[a,b2]  = e(a2,b)+Y2
out[a2,b] = Y3+Sb,        out[a2,b2] = Y4+Sa2

Each F+S node now continues a carrier into a later consumer of its strip operand with the same core and a larger cover. #53's assembly leaves all 10 additions per group pair unmatched; this one leaves 8. An exact integer program over every assembly of the nine pieces, with their actual cores and covers, shows 8 is optimal. Cores matter here: Sa + e(a2,b2) has the two-point core {c, a2}, so a one-point-core donor cannot continue into it.

	#53	this PR	stacked with #57's compiler
additions (h=23 / 25)	40,329 / 53,475	62,294 / 86,595	—
carrier links (h=23 / 25)	12,719 / 16,991	38,911 / 55,844	—
roles R (h=23 / 25)	32,946 / 43,409	28,719 / 37,676	27,918 / 36,586
physical wires W	160,799,739	140,924,496	137,151,806
κ	4498144/10^11	4986133/10^11	5101691/10^11

Hypotheses. No hypothesis or interface changes for the main witness; it uses the same inherited framework as #53. The stacked witness adds #57's compiler mechanism. #40 and later PRs, including #53 and #57, are outside the maintainer audit of #39 (docs/research/community-final-audit.md); this PR has the same status.

Validation

Finite checks, main witness (make pair-assembly-verify, all passed locally):

• scalar exactness on dense global supports: every output is the exact leave-two-out sum or retained total;
• the original-envelope labels;
• pinned-link legality: every link replayed in profiles.cpp and recounted independently;
• fixed I+J profiles: 37,728 / 51,927 distinct transition matrices, 6,026 / 7,981 CRT-checked, 0 disagreements;
• the full physical timeline and every dirty basis vector, in both orientations, at h = 23 and 25;
• all 4,073,300 data pairs, with exact rational recovery of the ten primary-prime failures;
• the exact moment at 4986382/10^11 and failure of the next grid value; exact lower bounds excluding the Joint frame compiler: conditional κ = 4.669442391e-5 #57, Paid clones on skip-prefix strips: conditional κ = 4.529040672e-5 #54, Skip-prefix strips: conditional κ = 4498144/10^11 > 2^-15 #53 and Both fixed bases and copied reversed corners: conditional κ = 3.918734894e-5 > 2^-15 #40–Reordered exclusion sums: conditional κ = 4.11862541e-5 #48 networks and the Copied retained centers and two-stage assembly for conditional kappa = 3.84569e-5 #36 baseline;
• all 47 strict assembly conditions and 7 margins at κ = 4986133/10^11, and failure at the next κ grid value;
• 22 tests in tests/test_pair_assembly.py, including exact leave-one-out for every layout, interval nesting, the symbolic assembly identities and negative controls.

Finite checks, stacked witness (research/pair-assembly/frame/frame_verify.py, adapted from #57's verifier):

• both XOR words replayed independently on every input and dirty basis vector, in both orientations;
• every frame transition reconstructed from the actual XOR word;
• all fixed-basis profiles with bounded-minor and CRT checks;
• the exact moment at 5101952/10^11, rejection of the next grid value, exclusion of Skip-prefix strips: conditional κ = 4498144/10^11 > 2^-15 #53's and Joint frame compiler: conditional κ = 4.669442391e-5 #57's networks, all 47 strict conditions and 7 margins, and rejection of the next κ grid value.

Proof arguments. As in #41, #47, #48 and #53, the new DAG enters the inherited proofs only through properties that the finite checks recompute: disjoint additions, common points, exact outputs, admissible carrier links, the physical timeline and the transition profiles. The integer programs explain why the links exist; correctness does not depend on them. The inherited written obligations are not re-proved here. This is not a formal verification, and no global optimality or speedup is claimed.

The checks ran on Windows with a static g++ build. The inherited geometry loader imports the Unix-only resource module and strips its setrlimit calls, so the local run used an empty stub module for that import. A Linux make pair-assembly-verify needs no stub.

Attribution

The interval strips and core-aware pair assembly are new here, by Avi Eisenberg with Anthropic Claude (Claude Opus 5.5) assistance. The rest is inherited from:

• eumemic: Joint frame compiler: conditional κ = 4.669442391e-5 #57's joint frame compiler, used unchanged for the stacked witness;
• 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/pair-assembly/PROOF.md and NOTICE.

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