Joint frame compiler: conditional κ = 4.669442391e-5

[eumemic]

Conditional exponent saving: κ = 4.669442391e-5.

The PR #53 skip-prefix graph can use fewer physical auxiliary roles when each equal-frame region is compiled as one invertible binary map and retired signal components are reclaimed. The resulting conditional witness is:

κ ≈ 4669442391/10¹⁴ > 2⁻¹⁵ (exactly 4669442391/100000000000000).

This is 2.6551% above #56, 2.8283% above #55, 3.1000% above #54, and 3.8082% above #53, comparing their stated conditional exponent savings, not measured runtimes. #56 appeared during submission; its draft reports independent exact recertification still pending. No global optimality claim is made.

Quantity	PR #53	This PR
h23 auxiliary roles	32,946	31,416
h25 auxiliary roles	43,409	41,264
Physical wires W	160,799,739	153,481,944

The compiler groups scalar nodes with the same original rational envelope, synthesizes joint reversible linear maps, matches compatible retained signals to later uses, and clears retired input-dependent signals with paid XOR gates. Arbitrary old dirty components are retained and canceled by the full compute/scatter/uncompute word. Designated output roles stay distinct; copied centers and endpoint corrections remain charged.

The actual fixed-I+J transition profiles give bit saving 116741511/2500000000000. The 4,073,300 data pairs, rank deficit 1,846,900, maximum child width 529, complex branch, and balanced assembly interfaces remain inherited.

Reproduce

make skip-frame-verify
make verify

Start with scripts/experiments/skip_frame_README.txt, skip_frame_compiler.py, and certificates/skip-frame-kappa.json. The README includes the finite mechanism and remaining proof dependencies.

Focused verification passed on this publication branch: complete input/dirty basis in both orientations; independent reconstruction of every frame transition from the serialized XOR word; all 60,989 / 81,960 distinct fixed-basis matrices, including 6,373 / 8,518 CRT checks with zero disagreements; exact rational recurrence; all 47 strict assembly inequalities, seven margins, and eventual cutoff arithmetic. Both producer words were separately regenerated and matched the saved words byte for byte after decompression.

Full repository make verify passed at construction commit cd350f76c9bc01489ec83568bded532cb69be938: 246 integrated unit tests, the retained 12/10/17/5-test focused suites, all 20 historical patch checks, the complete new auxiliary replay, and inherited PR48/49 producer/data checks covering all 4,073,300 data pairs, including the ten exact primary-prime recoveries. The tracked worktree was clean afterward. Validation receipt; log SHA256 e7d271e0db1530e2c26a157b114ad46cacc19c5e476eca6e37e909b2e766bf33. The subsequent commit only adds this receipt. The focused command alone inherits the unchanged data calculation; the full suite replayed it. No Lean build is claimed.

Dependencies and credit

This is a focused commit on current main 64ed4cb, with hash-pinned copies of the required #48 and #53 sources. It does not carry the old #15 branch or unrelated scratch experiments.

• Avi Eisenberg / ikeboy, with Anthropic Claude assistance: Skip-prefix strips: conditional κ = 4498144/10^11 > 2^-15 #53's skip-prefix graph.
• Chafik Boukhalfa, with OpenAI Codex assistance: 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 original-envelope pipeline and exact data recovery.
• Rohan Arun, with Anthropic Claude assistance: weighted matching and fixed-basis/assembly components; RaD / hipotures: alternating producer work.
• icekylinx, Dominik Scholz, James Chang, Zhihao Chen, Aurel Prosz / Paureel, Swapnil Jain, eumemic, Douglas Colkitt, OpenAI, David Harvey, Joris van der Hoeven, and the earlier contributors retained in the source notices.
• This joint frame compiler and integration were prepared by eumemic with substantial OpenAI Codex assistance. Original LICENSE and NOTICE files are retained.

The result remains conditional on the inherited all-size residual compiler, fixed finite-alphabet multitape, analytic, routing, and recovery interfaces. Finite checks are not formal verification or external acceptance of the complete multiplication theorem. No manuscript or PDF is added.