[mihir-s-05]

The R437 numerical wrapper in #4176 uses the sufficient stock K^D and explicitly leaves descendant path admission open. This package isolates the actual caller's row-cardinality component: with A = 28416 |GL64(F_q)| q^(4096(w-1)), the literal quotient and subsequent 9475 divisibility guard need 9475·A^D, rather than (9475·A)^D. The ratio is exactly 9475^(D-1) for positive depth. This is a sharper sufficient condition, not a defect in the conservative predecessor.

The written induction proves sufficiency for every path with child width at most 42 floor(n/64), and necessity within that complete rank-ceiling path model. A separate arithmetic certificate anchors each descendant to its supplied root, preserves original-root field/precision and checks the exact row quotient. A forged descendant with twice the correct rows passes local divisibility but is rejected by the path checker.

Only research/r437-descendant-stock/ is added. The frozen numerical wrapper remains unchanged and intentionally rejects the smaller stock at depth >1. No new κ, supplier admission, selected-frontier change or practical runtime claim.

Validation on exact head ca336bd72aff54e025a8ddbb1f3c8b756d9b9631, Windows/Python 3.12.0:

• Registered ordinary and optimized replays pass: 323,232 width transitions, 25,620 abstract guard cases, 14 internal nodes through the literal source guard logic, and 17 rejection controls.
• Source metadata/pins and committed-byte preflight pass for 14 files.
• The changed-package planner selects only research/r437-descendant-stock; no shared replay controls or predecessor files are changed.
• Hosted CI run 39810033114 passed on this exact head, including onboarding, planning, the registered replay and committed-input immutability.
• The committed export passes integrity checking. Ordinary and optimized Git-free replay pass; deliberate source and receipt corruption are rejected, and all 15 exported files are unchanged after restoration.
• Reproduction: python -B scripts/research_package.py --package research/r437-descendant-stock.

The source AST runs through an explicitly disclosed integer Count/Word adapter; binary tapes and recursive payloads were not executed. Actual Program/request authentication, root-prime/precision admission, additional stock predicates, physical costs and complete reserve admission remain open. The minimum claim is for the stated guard model, not all possible production executions. Selected orientation stopped on a sparse omitted checkpoint; no baseline or selected cold-replay pass is claimed. The optional full-history check was not run in this shallow checkout.

Started from and refreshed canonical main b58ad9ef4d4e861b0eaa5dba41742bdc072dca51, priority revision 7. Final overlap snapshot covers 451 open PRs and 3,613 Discussion records with complete pagination; local notes and historical/code-diff coverage remain partial. Canonical investigation and handoff. This complements #4584's pivot audit and is separate from #4612's selected BIT stock audit.

Credit Chris R / Clear1nsight (#4067/#4176), unpaidmoon (#4147), Jose Manuel / jmbarrancoml, David Arias, Nick Petalas, ywz65, and the retained R437/hosting/source lineage. Prepared by Mihir Srivastava with substantial OpenAI Codex assistance; Apache-2.0. Author-checked proof and separately written arithmetic, not independent human or formal-kernel verification.