[ywz65]

Change and mathematical scope

Supply the simultaneous-error step in PR1673's shallow Toeplitz precision argument. Exact prefix and inverse-prefix bounds imply explicit bounds for every jointly perturbed prefix, its inverse, and an arithmetic trajectory. The interaction-coordinate proof retains noncommuting order and does not assume the rounded circuit already has the exact circuit's prefix bound.

For K two-wire gates, exact prefix/inverse norm at most B>=1, unit input and target p>=0, P=p+ceil(log2(8KB^3)) fractional bits suffice under the stated coefficient and local arithmetic error contracts. This preserves the source's asymptotic guards in the near-identity, SPD and grouped cases. The exact compiler prefix envelope and arithmetic implementation remain hypotheses; no new all-size compiler, supplier, physical reversibility or kappa claim.

Task and overlap

The fresh draw 5c2f3ef5e7f8813b17cd63e079a43faf offered independent-moment and formal-interface-lemma. This refines the latter through PR2578's explicit outer-precision handoff. PR2044's reused review and PR2578's rectangular/union checks are credited; PR2683 is an unrelated T74 retention audit. Main base b5ea56e, priority five, remained unchanged at final fetch. The three exact source notes are vendored from PR1673 head 85ddaa3 and authenticated before replay. TASK records cached coverage, targeted REST and the claim. Standard product-perturbation mathematics is not claimed as new.

Validation

python3.12 -I -B scripts/research_package.py --package research/simultaneous-gate-rounding passes. Exact rational checks cover 27 circuits, 279 prefixes and 756 rounded trajectory steps; direct products and a separate interaction recurrence agree. Five counterexamples cover missing higher-order terms, an endpoint-only norm bound, rho=1 singularity, omitted arithmetic noise and changed noncommuting order. Four Git-free public-CLI test groups pass with read-only replay, three reference corruptions, changed/Boolean/float receipt controls, overwrite refusal and actual interaction-index/precision-guard mutations. Source compiled circuits, hosted CI and the selected cold multiplication replay are not claimed.

Credit and open next questions

Vlado Vojdanovski and the original PR1673/PR2044/PR2578 contributors retain their circuit, review and transfer credit. Prepared for ywz65 with substantial OpenAI Codex assistance. Next: bind a concrete emitted tile's K, prefix/inverse envelope and actual local arithmetic error; this note does not infer machine precision from unspecified big-O constants.

Independent root review found no remaining proof/code issue. Finite controls use the induced infinity norm; the source-facing complex two-wire consequence uses Euclidean norm, as explicitly distinguished in the note.