[jwr404]

Summary

One parity split of the triangular Toeplitz compiler has infinity norm at most 3/2 when the off-diagonal coefficient l1 is at most 1/2. This is the prefix-norm step left open by research/one-factor-of-t. It branches from fetched main and does not replay that package.

Let T be lower triangular Toeplitz with constant term 1 and tail l1 at most η ≤ 1/2. Write the symbol as e(z^2) + z o(z^2), and let a and b be the coefficient l1 norms of e − 1 and of o. Then a + b ≤ η. The Neumann bound ‖e^{−1}‖_1 ≤ 1/(1 − a) and the shared budget give:

• the even child satisfies ‖E − I‖_∞ ≤ a ≤ η;
• k = o/e satisfies ‖k‖_1 ≤ b/(1 − a) ≤ η, so each shear has infinity norm at most 1 + η;
• the Schur child satisfies ‖S − I‖_∞ ≤ a + b ≤ η.

At η = 1/2 every factor is at most 3/2. That is sharper than the descendant cap η/(1 − η) = 1 in SHALLOW-TOEPLITZ.md. The pure odd symbol (1, 1/2, 0, 0) meets 3/2 on the completed map and on the right-shear prefix. The three-factor product identity holds in integers over denominator 4. Replacing that coefficient 1/2 by 1 makes the tail l1 equal to 1 and raises the same prefix from 3/2 to 2. A perturbation of total l1 at most ε keeps the 3/2 cap only when η + ε ≤ 1/2.

This is one prefix. It does not prove a second nested split, the O(log^2 N) compiler, the generator-shear compiler, the completed Gaussian stage, or that the dyadic p^100 normalizer absorbs the prefix. No new κ. The selected record is 60403/20000000.

Test plan

• python3.12 -B scripts/research_package.py --package research/triangular-parity-prefix
• Prints PASS_TRIANGULAR_PARITY_PREFIX and outcome=completed
• Maintainer replay was not run

Base b58ad9ef4d4e861b0eaa5dba41742bdc072dca51, priority revision 7. Upstream fetches at 2026-10-22T16:21:18Z, 2026-10-22T16:31:19Z, 2026-10-22T16:33:00Z, and the pre-submit fetch at 2026-10-22T16:33:49Z were that commit. The pinned blobs of certificates/selected-result.json (c35482cd9035843b15a634cc8276a7e1e40e8395), GAUSSIAN.md (adfb28c93e6b06178c901caaac39ede41a9e541b), and vendor/SHALLOW-TOEPLITZ.md (bc4349b2daea548ffa09e1f38a31f1df86b0b7be) were unchanged. κ remains 60403/20000000.

Prepared by James Reed with substantial Cursor Grok 4.7 assistance.