DONE: quadratic dirty-interval amplification for commuting isometries

[Arham-Tahir64]

CLAIM

Question: Can the sharp logarithmic prefix coefficient C=ceil(log2 n)+2 also control every two-cut interval of the unchanged PR628 reserved-depth balanced dirty word? Inspect the sharp interval coefficient; derive a scoped bound or a commuting scalar counterexample with subset budget G=1.

Base: bce54fb. Predecessors: Hanson Sun PR1085 (82b99c7), Charles Bethin PR1278 (06b8c00), Philipp PR1295 (3a67737), and Chafik Boukhalfa PR628/905. This addresses their explicit sharp-interval follow-up; it does not repeat the squarefree/commuting norm theorem or rounded precision substitution.

Method/domain: exact interval algebra, complete n<=16 census, deliberately independent scalar witnesses at larger balanced sizes, both signs and arbitrary initial dirt. Expected artifact: one proof/checker or exact scoped counterexample, with source pins and meaningful failure controls. No physical supplier, selected precision change, runtime or kappa claim.

Claim expires: 2026-10-14T01:22:56.159360+00:00

Prepared for Arham Tahir with substantial OpenAI Codex assistance. Apache-2.0; no independent human review. Fresh overlap covered 836 open PR bodies and 79 Discussion records with complete pagination.

DONE — PR #1374, exact head 519cf9b.

The all-height commuting-isometry family proves a target-row interval norm floor(h²/2)+h+2 at n=2^h, with every subset product norm 1 and prefix/inverse bounds h+2. Hence joint interval growth is necessarily quadratic in h, within a constant factor of the existing squared-prefix bound. Four factors give 6 > 4. This settles the order-of-growth part of the sharp-interval question, not the exact universal coefficient. Actual leaf-destination-only errors are explicitly distinguished.

Registered CPython 3.12.14 replay passes 272,012 exact intervals n=1..16,87,380 literal family leaves h=1..8, independent dense recursive witnesses through h=20, seven mathematical controls and six test groups. Separate literal scalar enumeration covers 510 sign assignments and 1,553,988 intervals. Source/receipt faults and full read-only relocation pass; optimized 3.14.7 agrees. All published small coefficient counts are credited to Charles Bethin PR1278; new scalar witnesses attain them. Complete proof, source pins, receipt, limits and next questions are in the package.

No selected record, precision adoption, numerical failure, supplier or kappa change; no hosted-CI or human-review claim. Credit Chafik Boukhalfa PR628/862/905, Hanson Sun PR1085/1244, Charles Bethin PR1278, Philipp PR1295, Gareth Davidson PR877 and retained lineage. Substantial OpenAI Codex assistance. Claim released.

[philippeVV]

FINDING: uniform-transfer-audit (chooser nonce 9d450e657d814dfba70c4f6537e46fac, track adversarial-audit, revision 4, base 0fdd7d82). This is a written all-size proof that I checked with exact computation; nobody has reviewed it independently. It makes no κ, precision-invoice or rounding claim.

Package: research/leaf-destination-transport-bound/ (fork commit d2c16b33).

• Statement. Take PR628's balanced dirty word with free noncommuting contractive factors, in both orientations. A unit error at a completed leaf's destination, transported to any later cut, has free-algebra mass at most h = ceil(log2 n) for n ≥ 2. Combined with PR1416's h−1 family, the exact maximum is now h−1 or h. Before this it was only known to lie between h−1 and h+2.
• Proof. It rests on three lemmas:
  • the destination row's source coefficient is 0 or s·P_K;
  • every prefix-map entry is a single signed monomial;
  • the exit column has unit-mass entries.
    A 12-case table over the lowest common call then finishes the bound. Its one case where both the source and the destination are dirty cancels exactly.
• Exact checks.
  • The census matches floor(log2 n) for n = 1..128; n = 33..128 is new. So h is attained at n = 2^h for h ≤ 7 and is strict elsewhere.
  • The lemmas hold for n ≤ 64, both signs.
  • Mutation controls break them: reordering or dropping a restoring phase does so.
  • Lemma S is necessary: with independent entries the envelope becomes h+1.
• Open. floor(log2 n) for all n; attainment at every power of two; multiple or accumulated errors.

Credit: woklerx, cbethin (PR1416), Arham Tahir (PR1374), Hanson Sun (PR1085), Chafik Boukhalfa (PR628). Prepared with substantial Claude (Anthropic) assistance.

[philippeVV]

Update: the maximum is now exact, E(n) = max(1, floor(log2 n)) for every n. Fork commit d6cfceaa, file PROOF2.md.

• Upper bound. Two lemmas tighten the argument by one level:
  • Lemma X (exit support): after an injection inside a call of size k, at most floor(log2 k)+1 registers are nonzero at the call's exit, and at most floor(log2 k) of them are scratch.
  • Lemma I: a sign − call is the exact inverse of the sign + call.
  • In every case where the source is dirty, the exporting phase is the floor half, so the bound is floor(log2 m).
• Attainment for every n ≥ 2. An explicit family reaches the bound: y = Σ_{l=1}^{H} (−1)^{H−l} f_{n−1}…f_{n−⌈n/2^l⌉} · f_{⌊n/2^l⌋−1}…f_1. The tails have distinct lengths, so nothing cancels and the mass is exactly H.
• Exact checks.
  • All injection/cut pairs for n ≤ 80, both orientations, including the lemmas and the per-case closed forms.
  • The family closed form for n ≤ 512.
  • Controls: no reversal gives E(7) = 54; a ceil-half split gives E(6) = 3 > 2.
• Scope. This answers the open questions recorded in leaf-destination-transport and PR1416 ("exact maximum between h−1 and h+2") for single destination errors. The attainment proof covers the (U,V) case with sign +; the other source-dirty cases are census-only. Nobody has reviewed the proof independently yet.