Research audit: signed prefixes give 18/21-bit local bounds for selected PR1094

[bitplane]

Change and mathematical scope

Conditional precision-interface audit; no new κ. The unchanged selected PR1094 R480 forward/backward words have exact signed scalar/Horner prefix maxima 131088 / 1154896, giving 18 / 21 bits instead of the retained positive-recurrence 52 / 68 bits. Exact squared inequalities cover partially stored real/imaginary components within the same bit bounds. Under the inherited global execution/precision hypotheses, substituting only that local term gives 397 → 350 sufficient bits. The production 512-bit payload, paid profile and selected result are unchanged.

The written norm corollary applies the credited Lemma G conjugation theorem at every prefix. Exact examples show why equal labels, unitary label maps and exact recipes are required; endpoint correctness alone is insufficient. A source-specific witness also shows that omitting coefficient scratch underestimates the forward maximum: 131088 > 131072. This is not a defect in the original conservative bound or a new fixed-point rounding theorem.

Only research/complex-signed-prefix-audit/ is added. Complete native/Clifford/physical provider admission, recursive child internals, compiler/common-label/hosting realization, roundoff/tape/exact recovery, analytic transfer and Lean remain outside this fresh replay.

Task and overlap

Base dadfea6e8a78ff6ba901705f87532a71c6f5bbb8; selected PR1094 0fc329d0bec93ca7cd288eaea1295172e8789810, parent PR1087 589d1cedf5ef5f231182f4ec86408616311a8af0. Fresh OS nonce 9e66acdf5b86cc4b02a0d212138d7b3e, adversarial-audit / uniform-transfer-audit, scope key ce7411fb7cf6f3460bfc, priority revision 2. Scope is refined to a quantified common-frame precision implication and its exact selected-source application. Claim/result: #1260.

The initial replica-norm idea was dropped after finding #821. This result does not repeat or compose its argument. #1184 accelerates the older positive-envelope checker; #734 changes coefficient circuits while retaining those envelopes; #745 supplies reverse endpoint/private-clear checks; #1255 audits chart phases. Their exact heads and distinctions are in README/OVERLAP.json. Fresh complete pagination covered 868 open PRs and 30 Discussions with bodies/comments. Concurrent #1263 changes helper gauges, #1277 audits ancestry, and #1278 bounds another recursive word; none is imported as a dependency.

Validation

python3.12 scripts/research_package.py --package research/complex-signed-prefix-audit
python3 -I -O -B research/complex-signed-prefix-audit/verify.py
python3 -I -O -B research/complex-signed-prefix-audit/controls.py

All registered commands pass on Linux CPython 3.12.3; optimized 3.14.4 replay agrees. Evidence includes:

• All 39,386 scalar prefix rows checked by sparse rational and dense integer engines, both 704-column endpoints and 960 wrapper unit rows.
• Separate reverse functional checks and whole-word primitive executions attain both peaks, with 198766 / 198814 real-component primitives matching the source.
• 216 exact unitary-frame words, 2,376 cuts, 57,024 matrix-entry comparisons, four mathematical boundary examples and eight selected-word/bound controls.
• The entire unchanged original local driver, assertions enabled, both orientations and 25 inherited controls; its complete regenerated receipt equals the archive.
• Six package/kernel tests including read-only isolated optimized relocation, source/pin corruption and an understated receipt; selected-record integrity, metadata/pins, links and whitespace checks pass.

The signed receipt is 17c82644f920abd7b9a776481efec1740f0ae75b99c53bf331e3e1442d328300. Exact sources, witnesses, proof, commands and unrun obligations are in the package. Hosted CI is reported separately from these local passes.

Credit

Gareth Davidson (@bitplane), with substantial OpenAI Codex assistance; no subagents or independent human review. Credit Zhihao Chen/PR425, DreamingOfClouds/PR484 and the Lemma G/A/HPE formalization authors for the inherited conjugation framework; chafreaky/PR458 and Dugongue/PR471 for whole-helper/normalization work; Rohan Arun, Jeremy Kothe, Hao Yan, Josie Elliston, Douglas Colkitt and the retained gauge44/Clifford/compiler lineage for the selected provider and audits. Reverse functional, integer-row and Horner techniques retain the PR745/734/1137/1156/1184 predecessors' credit. Apache-2.0. No priority claim for the standard norm corollary.

Open next questions

README/TASK give two bounded follow-ups: formalize the Hilbert-prefix/partial-store corollaries alongside Lemma G; or integrate the signed recount into one production provider with complete native/global admission. The latter still needs the retained internal compiler/precision obligations before changing any reserve or κ.

[Mister-Meeseeks]

100-PR batch review at 851e5232f7c9e105fb6e8f3a7be7a070e7524012.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Signed complex-prefix audit, Horner preparation and inverse-input bounds, with independent source binding and controls. Conditional fractional-width improvement leaves the charged 512 guard unchanged; no new multiplication bound.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.