[hexapode]

Combining #4532's critical-root compiler with #4470's unchanged native supplier raises the conditional whole multiplication saving from κ = 0.00302015 to κ = 0.003020165 (+0.000000015).

The new contribution is research/critical-root-multiplication/: a finite-template argument, exact source-bound certificate and new complete outer parameter point. The productive width (10,928) and complete paid invoice (64,620 jobs) are unchanged. At sufficient frozen local rank N = 2^(10^12), the compiler yields uniform complex saving 0.0030201656; all 54 strict outer predicates and four functional guards pass. Ordinary BIT saving remains 0.0034.

Dependency: this branch is stacked on #4532 at 644607ba59ebe401ac7ecd627ae7f331f924d1b0; its files are unchanged predecessor work. Canonical main checked at f9f85c1e4fa1e955ff39f10fca447e3e2a4e4adc. New work commit: cca1112d6f8e462e8df92bffa0b939b8fb73d1b4.

Validation: complete registered dependency/native/new-arithmetic replay passed, including both 632-column arbitrary-input orientations and 423 physical frame pairs. Final metadata/source pins, isolated optimized arithmetic with rejection controls, committed-byte and source-history checks pass. VALIDATION.md records the exact scope and final additions. Hosted CI is not asserted; predecessor reports existing upstream planning failures.

The proof uses fresh finite bounds for owned storage, routing, precision, stock and fixed-tape execution. The enormous word is algorithmically specified, not materialized. Generic finite-template closure and all inherited native/physical/analytic/recovery contracts remain conditional premises; finite arithmetic does not prove them. No overall numerical input threshold, practical speedup, formal acceptance or change to the selected record is claimed.

Credit Dillon (#4532), zxv (#4483 and earlier compiler work), Jacob Alan Sussman, Ryanson Jonathan, Cinna Davis and #4470's native lineage, Dugongue (#4479), and retained dependency authors. Prepared for hexapode with substantial OpenAI Codex assistance. Apache-2.0. The package records source pins and open next questions.

Hosted update: run 39721672717 stopped during onboarding tests with four failures in test_research_current_status.py, before selecting/running research packages. All four reproduce locally; the test, status page, CI navigation and selected record are unchanged from canonical main. This is not a hosted scientific replay pass. Separately, the 77-file committed export, archive check and extracted isolated arithmetic replay passed at the published head.

Independent follow-up: #4536 reproduces its checked arithmetic, while excluding compiler/physical proofs. Commit a6d7237f697185e857090e174c23c1cb22c6563c explicitly labels the distinct evaluation points of margin_lower and cost_margin_lower. All pre-existing numerical fields are identical; metadata, isolated optimized arithmetic and committed-byte/source-history checks pass. No κ or proof change.