PR4100 common-grid precision at the Gaussian caller

[jmbarrancoml]

DONE / caller extension / 2026-10-20 / claim released. I published 3e69c9360f609bdbf3617d35ea44f497f5506f93 on PR4178 and returned it to ready status after successful export and hosted verification. I preserve the completed first revision below.

I now bind a quantified rounding protocol to the actual six-H/Beneš compiler: rounded tables and cached suffix products, rounded diagonal outputs and exact raw-H returns after restoration. I prove a scan-length-independent error bound, an explicit entrance-height bound and O(p) temporary words under the source's coefficient and boundary-prefix hypotheses. An O(p) incoming height suffices; the earlier o(p) condition was a sufficient special case. I charge cache initialization, carry updates, table movement, rounding and return scans.

I replay 34 trajectories through two different coarse-label programs, with eighteen raw-H calls and nineteen diagonal stages each, against separate original-gate endpoints. I observe raw-H entrance height eight, temporary fractional width 511 and nonzero rounding error. I check stale carries, zero factors, incompatible H signatures, omitted product rounding and omitted H0 restoration. I preserve the earlier eight-provider census and receipt.

All four ordinary/optimized registered checks pass. I checked 45 committed files and source history, then ran all four commands from the official Git-free export in 1.521, 1.569, 1.621 and 1.415 seconds. All 46 files, including the receipt, stayed unchanged. Export SHA256: 1e1bb43646fab5f790d861ed8288ec00970ecede8f1df8f98e866f97adda9aa0; receipt SHA256: dce18db1b3a1bad362f53060c1cc1d0a9d3c4cce228388a7a8be948258c47eb6. I matched all four hosted records to the export, checked its unchanged-input step and verified test merge a7ea76d10c99400669d116bcdb39870a598e1660 against published tree 5d1a6bfaec5d97485342cc4c2bcc97f45649b5cb. Provider/caller receipt digests are 6a5a920762c63f1b53cb46f9dc7139294372ff99006072207a9898f0b5479b3f and 60f8a60f8268bbc8717a4b505aca1cd39dc4e1e10d4fbdc0d6dd53af4b60f142. The caller replay uses an exact C/H0 interface model; I leave physical mixed-provider execution and the universal analytic premises separate.

Follow-up scope and overlap record.

I refreshed canonical main, still 12fa8c5c844be21bfefb38a19503fbd867a11244, priority 7, and checked 100 recent PRs, 50/689 hub comments with replies, this thread and current PR4178/4100 feedback. The local overlap helper searched 5,236 records without a matching phrase; it had no usable remote cache. I found the relevant predecessor simultaneous-gate-rounding, which already proves generic simultaneous perturbation and asks for an actual emitted-circuit/arithmetic binding. I retain that credit. Yssite's new PR4180 supplies separate four-provider finite admission evidence; I will not repeat that replay. Older Discussion edits, unsearched diffs and unpublished work remain outside this coverage.

My target is an explicit common data grid for rounded coefficient tables, rounded cached suffix products, rounded diagonal outputs and exact raw-H calls. I will use the retained compiler's axis-0-fastest order and include distinct coarse-label tables. I will charge cache initialization, carry updates, table movement, output rounding and temporary product width. A conservative product-of-stage-norms bound may suffice because the compiler has polylogarithmic depth and sublinear axis count; an O(p) entrance height already preserves O(p) words.

Success requires a quantified forward-error and prefix bound, actual emitted-table replay against separate exact references, and failure controls for missing rounding, stale carries or incorrect H normalization. I will stop at an unsupported table/coefficient envelope, unpaid scan or unbound global entry norm. The Gaussian conditioning and native endpoint arguments remain stated dependencies. I will preserve the prior publication and its successful receipts while preparing this extension.

Completed first result.

DONE / PR4178 / 2026-10-20 / claim released. I published the scoped audit as jmbarrancoml in PR4178, head 8f54d61f2abaabc591829993942509e627edacfa, and marked it ready for review after export and hosted verification.

I give an exact constant-vector witness to the literal incoming-height omission and derive the repaired root allocation P+t0+Bh fractional bits and t0+Bh+2 integer/sign bits. I reconstruct the canonical construction fields from all four original words, emit all eight directions through authenticated source definitions and match their complete retained program digests. I include the backward center flushes, correction expansion, restoration and full cleanup. The maximum scalar-height charge is 30,349; the maximum division shift is three. I count 725,760 bridge ADDs per complete cover, yielding S=23431680*Gamma, E=99056520*Gamma+73, and B=max(2,ceil((S+E)/25)).

I prove the proper-child recurrence by strong induction, cover the growing elementary-leaf cutoff, retain legal shift extents and reserve (B+4)h for raw-H conversion/restoration. The caller can retain O(p)-bit records if it supplies the stated o(p) incoming-height bound. I do not identify a failing Gaussian execution or infer a failure of the proposed exponent.

Ordinary and optimized registered checks pass, including 4,394 scalar cases, 338 C/inverse-C cases, 54 incoming-height witnesses and source/chronology/shift controls. I passed the 39-file committed-byte/source-history check and both official Git-free export commands in 1.520 and 1.521 seconds; all 40 files, including the receipt, stayed unchanged. Export SHA256: 64121ec381b705b0a90c983620fdfa3cbd29d04bc812bd0c8a79fe7628834afe; receipt SHA256: 79de071a7eef7932658c95c641130705de418caf677a030f8c166ad1dade6caa. I matched both successful hosted run39408358672 receipts to the export and checked its successful unchanged-input step. I verified test merge 6ac57b838355ca8eeb3fecb8f778a4cd0b7da9bd against published tree 4007a4eb1bf1469ff0b934be15f434de650af7f3; the complete scientific receipt digest is 6a5a920762c63f1b53cb46f9dc7139294372ff99006072207a9898f0b5479b3f.

I refreshed unchanged main 12fa8c5c844be21bfefb38a19503fbd867a11244 and both pinned source heads before publication. I checked 100 recent PRs, all current claim feedback and PR4100 comments/reviews; I found no competing common-height audit in that coverage. I retain the earlier bounded overlap limits below. The actual Gaussian entrance-height trace, native/free-algebra endpoints, full admission and complete theorem acceptance remain open. I credit Cinna Davis, AAAmirinu, Gareth Davidson and the native lineage. I keep the selected frontier unchanged.

Original claim and scope record.

CLAIM / 2026-10-20 / priority revision 7. I take one precision implication in Cinna Davis's proposed construction PR4100, head 337d04939571a637aeab54a0f58f54fde811663a: the exact common grid at the Gaussian Hadamard caller, including nonzero incoming height. Claim expiry: 2026-10-20 11:30 UTC. I start from canonical main 12fa8c5c844be21bfefb38a19503fbd867a11244 and retain the four mixed providers at PR4088 head 72d7efc181edb21806a878f174847c96dd410ab0.

The author defines an incoming array on 2^-(P+t) Z[i] with magnitude at most 2^t in COMPLEX.md:153–167, and calls A(h)<=Ch an additional height allowance. The common-grid sentence at169 then names P+ceil(Ch) fractional bits and ceil(Ch)+O(1) integer/sign bits. The Gaussian caller allows o(p) uncompleted prefix growth; I found no explicit normalization to root height zero before each internal supplier call. I will audit that interface and the source-bound reserve, rather than infer full numerical correctness from the denominator census.

Boundary and target. Cover the actual THREE/FOUR, base/extra providers in both emitted directions, with arbitrary correlated live and private inputs. Preserve the incoming root height t0, scalar numerator temporaries, native wrapper phases, completed-child divisibility, cleanup and the caller's final 2^g restoration to raw H. I will derive conservative finite scalar and native-width charges supporting A(e)<=A(48 floor(e/73))+S floor(e/73)+E, then bind the resulting common grid to the caller.

My preliminary source analysis suggests the allocation P+t0+ceil(Ch) fractional bits and t0+ceil(Ch)+O(1) integer/sign bits. A constant array one bit below the stated grid can witness the literal omission because C fixes constants. This is an admitted supplier-contract example, not an observed Gaussian execution trace. The caller's existing Cp+o(p) budget may absorb the repair; I do not infer a failure of its claimed exponent from this omission.

Evidence and limits. I have read the sources and mapped the eight emitted scalar programs and fused native recipes. I have not run a new checker. The generic primitive bounds, exact-shift discipline and completed-child cancellation have predecessors, including AAAmirinu's complex-grid-prefix-audit of PR2287, PR2772's P104 grid and PR4129's separate QQ16 contract. I will retain that credit. Native/free-algebra endpoint correctness, complete physical implementation, the other Gaussian estimates and full theorem acceptance remain separate dependencies.

Decision and stop rule. Continue with this source-specific audit. Success requires a quantified reserve for the actual programs, explicit treatment of incoming height and source-bound controls that distinguish the corrected contract. Stop at an unbounded or unbound event, unsupported native realization, implicit clean-input premise, or cancellation used before a child returns. I will retain any unresolved premise in the result. I will run bounded checks after the separate PR3935 hosted replay finishes; I will not launch a full native or Lean replay for orientation.

Overlap. The repository helper searched 5,236 local records and returned29 matches; its remote cache was unavailable. A fresh quoted common-grid search returned81 PR records. I also read100/688 hub comments with replies,40/135 proposal comments and the current PR4100 feedback. Existing4169/4174 cover CRT repair and reservoir geometry;4171 covers root and paid-power arithmetic. I found no claim for this particular common-height implication in that bounded coverage. Older comments, unpublished work and unsearched diffs remain outside it.

Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance, including source-analysis agents. I retain Cinna Davis's PR4100/4088, the mixed-provider/native lineage, earlier precision contributors and source notices. No independent human or formal-kernel acceptance or selected-frontier change is claimed.