[hexapode]

Raise the conditional multiplication saving from #4535’s 0.003020165 to 0.00302016597, an absolute gain of 9.7e-10. This combines Ryanson Jonathan’s smaller outer slack in #4564 with the unchanged #4532/#4470 critical-root supplier, funding a new fixed word at N=2^(10^14). It does not reuse #4564’s smaller rank at the new exponent.

New material is isolated in research/critical-slack-multiplication/. This branch is stacked on #4535 at a6d7237; predecessor changes are retained dependencies, not new work here. The native paid profile and raw root are unchanged. Unlike the same-kappa rank refinements, this retunes the uniform supplier and complete outer point to obtain a strictly larger conditional exponent.

Validation: the registered source-bound replay passes the critical-root certificate, fresh finite envelopes and absorption (margin >418/15), all54 outer inequalities, four functional guards, and four rejection controls. A separate Git-free export replay under isolated optimized Python also passes. Unchanged native/BIT sources retain the pinned #4535 replay coverage. The new expanded word’s physical, precision, stock and fixed-tape constants are specified anew by finite procedures; they are not numerically materialized. All inherited physical and transfer assumptions remain explicit. No unconditional theorem, practical runtime gain, formal verification, hosted-CI pass or selected-frontier change is claimed.

Reproduce with Python3.12:

python3.12 -B scripts/research_package.py --package research/critical-slack-multiplication

Credit Ryanson Jonathan (#4563/#4564), Dillon (#4532), zxv, Jacob Alan Sussman, Cinna Davis and all #4535/native/BIT/transfer contributors. #4565’s newer majorant retuning is acknowledged but not required. Prepared for hexapode with substantial OpenAI Codex assistance; Apache-2.0. Canonical main checked at f626db5; open PR headline refresh through #4566 found no newer whole-kappa claim.