Research: cap T74 scalar and handoff frame refits at b < 0.0016972

[jeremydiviney]

The leading PR2322/2374 T74 words leave a concrete question after owner optimization: can changing their intermediate frames, including the entry/exit charts of fixed logical handoffs, materially improve the primitive saving? This package proves b < 0.0016972 for that broader model, with arbitrary fractional whole-helper packing on the unchanged T74 stock. No higher κ is claimed.

The remaining headroom above PR2374's primitive endpoint is less than 5.05e-9. Scalar gates, coefficients, order, logical handoff maps/positions, helper endpoints, fixed prefix/cut frames and parent stock remain hypotheses; changing those can escape the result. This is a scoped lower-cost certificate, not new native/global/all-size or outer-theorem admission.

The new contribution is complete frame minimization for these newer borrowed words:

• Propagate transported subspace bounds and enumerate all intermediate GF(2)^9 subspaces, not just the input's frame catalogue.
• Contract 1,434 free nodes per provider using concavity, preserving a true-cost optimum. Then exact min-sum variable elimination resolves all 25 remaining components per provider; its largest table has 1,953,125 entries.
• Use directed rational powers, downward integer prices and bounded NumPy int64 tables. Both providers' rounded optima equal their source-profile bills. The proof accounts for rounding; it does not claim exact real-valued optimality.
• Check all 1,163 whole-helper packing-dual inequalities. The resulting moment lower endpoint exceeds one by about 1.17422517e-8 at b=4243/2500000.

The frame computation takes about one second for both providers on the author's machine, where exploratory general CP-SAT runs had left gaps after minutes. Variable elimination and concavity are standard methods; the contribution is their source-bound application and verifier. New candidate words need new pins and billing, but can reuse the small elimination engine.

Overlap: Concurrent #2402 (Sagar Thakrar, head 2e2cb4f) gives the tighter owner-only cap b<0.001697195759 with local frames fixed, and prices hypothetical circuit moves. This package's slightly looser cap additionally quantifies over all permitted scalar/handoff frame refits. #1995/#2013 concern the older selected scalar word. #2360/#2387 concern the different PR2339 400-helper inventory. These are credited; no priority for their packing relaxation or a new algorithm is claimed. At the original submission, main was 0fdd7d8, priority revision 4; selected result unchanged. These historical inputs remain the scope of this certificate. Live PR/Discussion overlap and relevant heads were refreshed, including #2402 and PR2374's text-only follow-up 5449da6. Both latest PR2374 word JSONs were compared in full with the bundled PR2322 inputs.

Validation: Registered CPython 3.12.3 / NumPy 2.5.4 replay passed. Optimized -O execution, exhaustive 81-assignment cyclic comparison, complete GF(2)^3 lattice control, improving contraction, fixed-anchor/unbalanced-node checks, undercharged bill/singular transport/overstated dual rejection, read-only standalone export, source/receipt tampering and no-clobber controls passed. Fresh archives of the original commit fafb673 and the current source-loader fix 626ce8e passed outside Git; all committed immutable source bytes match their pins. No optimizer or network is needed for certificate replay.

Verifier follow-up (commit 626ce8e): The arithmetic loader now compiles the exact bytes authenticated by the input manifest, preventing a preexisting .pyc from substituting different code. Four regression controls cover timestamp and unchecked-hash caches under normal and optimized Python; all eight export controls and the registered mathematical replay pass. Vendored sources, inputs, dual and expected receipt are unchanged, as is the cap. Credit to ywz65's general runner work in #2553 and David Arias C.'s related loader control in #2766. The companion fix for the merged width-cap package is #2788. This correction does not extend the theorem to newer words.

python3 scripts/research_package.py --package research/t74-variable-frame-cap --check-only
python3.12 scripts/research_package.py --package research/t74-variable-frame-cap

Credits and source notices retain Vlado Vojdanovski/PR2322, Jeremy Kothe's T74 line, Cinna Davis/PR2226, Rohan Arun, WildIdeas, Ali Romman/PR1129 and the incorporated pattern-dual audit, Sagar Thakrar/PR2339/2387/2402, and the earlier frame-bound lineage. Prepared by Jeremy Diviney with substantial OpenAI Codex assistance; no independent human review claimed. Open next questions are in the README.