DONE: three reciprocal guards and uniform induction (PR1638)

[ToikY]

CLAIM — adversarial-audit / uniform-transfer-audit, priority revision 3. Chooser nonce b43a43f25c1b77e3be01ef6fdcf22b71, scope key 603f341b99f590f17a76; base main 0a20199. Refined domain: Section 09 reciprocal only, R_t(z)=2^-t floor(2^t z), normalized real c in [1,2), seed 5/8 and t=6 warm-up, capped doubling to every integer p>=1. Question: do the actual directed signs suffice with three fractional guard bits rather than the retained five? Prove the signed residual implication with quantified hypotheses; exact finite cells and deliberate-failure controls only corroborate it. Expected artifact: isolated research/reciprocal-directed-guards/ proof and standard-library certificate/replay. Stop by 05:07 UTC for verification; session ends 05:22 UTC. Advisory claim expires 2026-10-14T06:30:00Z.

Nearest predecessor: Hanson Sun (@bbMisk), PR1528 head f43caa5, bounded p=1..8 Newton prefix audit and its explicit all-precision question. I credit that boundary and add a directed-rounding guard refinement; no priority claim, no imported pending executable. OpenAI manuscript / classical Brent reduction and Douglas Colkitt retain credit. Section 09 is a downstream corollary, not a multiplication dependency: this is a scoped side result with no kappa gain, root-stage change, physical-tape proof or selected-frontier change.

Overlap at entry: 741 open PR bodies with complete pagination plus 347 local package records; fresh start-here/handoff/active board reads and the GitHub Newton discussion search. Nearest broader parking audit PR1379 already closes the tempting locality target, so I will not duplicate it. No matching guard refinement found in this bounded text coverage; Discussions are not fully paginated by the local helper. Prepared for @ToikY with substantial OpenAI Codex assistance, including a bounded independent mathematical check.

[ToikY]

DONE / HANDOFF — PR1638, head394b90ee7fdaedd1f666a78a442127be3da17810, adds only research/reciprocal-directed-guards/ (11 files, one commit). Claim released.

The written all-real/all-positive-precision theorem proves three reciprocal guards suffice, retaining the six-bit warm-up and all incoming iterate bits. Exact lower-guard witnesses prove sharpness for propagation from every arbitrary admitted invariant state, not optimality of the actual seeded routine. Section09 is a downstream corollary; no kappa or selected-supplier claim.

Evidence: CPython3.12.14/macOSarm64 registered replay passes in13.989s; paired rational/integer traces on16,320 common cells, separate integer census81,920 cells throughp10 with774,784 endpoint checks, eight semantic/fault/relocation groups,20 repository workflow tests and selected baseline bindings. Separate static Codex review found no actionable issue; no human/kernel review is claimed. At04:44:48UTC all61 hosted checks were queued. No selected cold replay or root/tape/native integration was run.

Smallest follow-up: derive a stronger reachable-state invariant to decide whether the seeded two-guard routine works. Generic-state failures and additional finite no-failure cells cannot settle that question. Kernel formalization and complete root/recovery caller binding remain separate obligations. Credit Hanson Sun/@bbMisk PR1528 headf43caa53c5cc9717a8a04fe5a1538e31adeef313, the OpenAI manuscript/classical Brent reductions, and Douglas Colkitt. Prepared for @ToikY with substantial OpenAI Codex assistance. Sources, overlap limitations, commands and open questions are in TASK/SOURCES/VALIDATION and the README.