[lensora]

Superseded by PR #4731, reopened under the intended contributor account, JoyboyBrian. The tested commit and research contents are unchanged.

The triangular Toeplitz parity compiler's one-level bounds do not directly control an unfinished child composed with its ancestor shear. This package proves a sharp, depth-independent common infinity-norm bound of 1/(1−η) for every prefix, suffix and their inverses at complete-shear boundaries, for every dyadic dimension and every legal interleaving of the disjoint child programs. At tail l1 ≤ 1/2, all four bounds are at most 2. This includes the source's parallel child schedule.

The proof simultaneously inducts on all four cut operators while retaining the shared even/odd coefficient budget. The geometric inverse family proves the common constant is sharp. An exact N=4 suffix of norm 7/4 shows why a one-level 3/2 cap cannot cover all composed cuts. The sharper forward-prefix-only 3/2 question remains open.

This sharpens macro-boundary conditioning within the polynomial stability claim already present in SHALLOW-TOEPLITZ.md; it does not replace the internal dirty-shear gate, rounding, full Gaussian, normalization or supplier-admission obligations. No new κ or selected-frontier change.

Validation:

• Python 3.12 registered metadata/source-pin validation and package replay pass.
• Seven exact rational fixtures at N=1,2,4,8,16: 116 distinct cut matrices, including all pairs of child cuts for each fixture; all four operators checked.
• Exact suffix witness and geometric inverse checks; four failure controls; direct replay also passes under Python -O.
• Committed-byte and declared-source-history publication preflight pass.
• A second Codex agent independently reviewed the proof and checker and used a separate sparse row-action implementation for three interleaving schedules. Root agent reviewed the four block identities and inequalities. These are AI checks, not independent human or maintainer acceptance.

Credit: Vlado Vojdanovski/#1673 for the shallow compiler and Cinna Davis/#4100 for its later caller; Jose Manuel/#4198 for the separate rational-generator work; Ryanson Jonathan's one-factor-of-t question and unpaidmoon's #4371; James Reed's #4699 one-level budget and #4701/#4702/#4704/#4705/#4706 finite examples. Exact predecessor heads and source hashes are retained in the package. No unmerged executable dependency is required.

Canonical investigation: #4715 . Base: b58ad9ef4d4e861b0eaa5dba41742bdc072dca51, priority revision 7. Prepared for Brian with substantial OpenAI Codex assistance; Apache-2.0.