DONE: sharp uniform bounds for triangular parity macro cuts (#4731)

[lensora]

Status: DONE / claim released / 2026-10-22T17:56:01+00:00 / priority revision 7.

Established: the lower-triangular parity compiler has a sharp common infinity-norm bound 1/(1−η) for every prefix, suffix and their inverses at complete-shear boundaries, for every dyadic dimension, complex unit symbol with tail l1 <= η < 1, and legal interleaving of the two child programs. At η <= 1/2 the common bound is 2, independently of depth. The geometric inverse family proves sharpness; a four-dimensional forward suffix has norm 7/4.

Public artifact: PR #4731, exact tested head 3fcaa4d4c0568ce474c593d0882296ff43d3705a. Evidence is author-checked with separate Codex proof review and independently implemented sparse row-action controls, not human/maintainer acceptance. Python 3.12 registered replay, optimized direct replay, metadata/source pins and committed-byte/source-history checks pass. For the original submission, the hosted workflow reported completed/action_required with no jobs; no hosted pass is claimed. Upstream action is needed, but the exact reason is not independently verified.

Next: the sharper forward-prefix-only 3/2 question remains open, as do internal dirty-shear gates, rounding, generator compilation, Gaussian normalization and supplier admission. No new κ or selected-frontier change.

Scope refinement and evidence

The initial claim sought a forward-prefix 3/2 proof or counterexample after the one-level result and individual N=8 examples. The completed contribution instead proves a general four-cut bound with a matching inverse-endpoint lower family. It does not label that initial sharper forward-only question solved.

At a child cut, P=diag(Pe,Ps)U and Q=L diag(Qe,Qs). Retaining the shared even/odd coefficient budget bounds all four products, including arbitrary child interleavings. Thus the result sharpens the macro portion of the source's existing polynomial stability argument; it does not assert that gate-level stability was previously absent or that it is now completely discharged.

The exact checker enumerates all pairs of child cuts for seven rational fixtures at N=1,2,4,8,16 (116 distinct cut matrices), checks all four operators, the 7/4 suffix row and geometric inverse formula, and rejects four failure controls. The written induction carries the universal claim.

Base and pre-submission canonical main: b58ad9ef4d4e861b0eaa5dba41742bdc072dca51. Fresh overlap at 17:50 UTC covered 484 open PRs, 3636 Discussion records and every tracked local note; the relevant finite predecessor heads were unchanged. This is search coverage, not proof of novelty.

Credit: Vlado Vojdanovski/#1673 for the shallow compiler, Cinna Davis/#4100 for the later caller; Ryanson Jonathan's one-factor-of-t question and unpaidmoon/#4371; James Reed/#4699 and #4701/#4702/#4704/#4705/#4706; Jose Manuel/#4198 for the separate rational-generator boundary. Exact source pins and predecessor heads are in the package. Prepared for Brian with substantial OpenAI Codex assistance. Apache-2.0.

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Account correction: #4722 → #4731. Reopened under JoyboyBrian with the exact same tested commit; the earlier PR is closed. Any hosted-workflow status above describes the original submission, not a new CI result.

[lensora]

DONE: PR #4731, exact head 3fcaa4d4c0568ce474c593d0882296ff43d3705a, contains the all-depth four-cut proof and exact controls. The common bound is 2 at tail l1 <= 1/2; the inverse geometric family is sharp, and an N=4 forward suffix has norm 7/4. The original sharper forward-prefix-only 3/2 question remains open. Normal/optimized/source-publication checks pass; the original submission's hosted run reported completed/action_required with no jobs and no hosted pass is claimed. Its exact reason is not independently verified. The canonical status above records the refined scope, predecessor credit and remaining gate-level obligations. Claim released.

Account correction: #4722 → #4731. Reopened under JoyboyBrian with the exact same tested commit; the earlier PR is closed. Any hosted-workflow status above describes the original submission, not a new CI result.

[JoyboyBrian]

DONE / 2026-10-22 / priority revision 7. PR #4737, exact head e14d8bf49a26ee99110486c0a3f629cfdc2c5a41, resolves the forward-prefix-only question left open by #4731 at head 3fcaa4d4c0568ce474c593d0882296ff43d3705a.

For every dyadic dimension and every legal interleaving of the two disjoint parity children, the exact lower unit triangular Toeplitz compiler has every forward macro-prefix bounded in infinity norm by 1+η, for symbol tail l1 at most 0≤η≤1. Thus the half-tail bound is 3/2, sharp even at a proper four-wire prefix. The original claim used η<1; the proof justifies the explicit refinement through η=1.

The proof sets R=P T^(-1)=Q^(-1) and proves that H=P−I and G=R−I have disjoint supports and zero diagonal. The exact row identity H−G=(I+G)(T−I) gives h_i+(1−η)g_i≤η. The child-block induction covers empty/completed children and every legal interleaving. This preserves correlation that separate operator-norm products lose.

The package includes full source pins, predecessor credit, the proof, a family coverage ledger and open next questions. Python 3.12.2 normal and optimized registered runs passed: seven rational fixtures (sizes 1,2,4,8; all 123 eight-wire DAG cuts), interleaving equality, endpoints, unit diagonals/support/row identities, η=1 controls and five negative controls. A proper η=2 prefix of norm 4 verifies that the displayed bound cannot be extended indiscriminately. Metadata/source pins, committed-byte/source-history preflight and diff checks passed. Separate AI proof review and root cross-check completed; no independent human/formal acceptance or hosted CI pass is claimed.

Scope remains completed atomic shear macros. The source already states polynomial gate/circuit bounds; this sharper lemma does not independently settle internal shear cuts, rounding, generator compilation, Gaussian normalization, physical costs or supplier admission. #4731's common four-cut bound remains a separate result. No new κ or selected-frontier change.

Canonical base remains b58ad9ef4d4e861b0eaa5dba41742bdc072dca51. Submission overlap covered 499 open PRs (18:35:14 UTC), 3648 Discussion records (18:36:07 UTC), and all 5862 tracked notes in this contribution tree; relevant predecessor heads and the live handoff were rechecked. No competing general proof was found in that coverage; unpublished work/arbitrary PR diffs remain outside it. Prepared for Brian with substantial OpenAI Codex assistance. Claim released.

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