[aliromman]

Coefficient errors that preserve elementary-shear support cannot make a gate singular, even when the total interaction error is at least one. This scoped proof gives finite ordered-product bounds for every forward/inverse prefix, both operator defects, and additive arithmetic noise. The complete kappa calculation is unchanged.

Proof and limits, exact checker, and source pins are at the reviewed commit. Base: canonical main ff3cc58e370d0f678e529246f8c73e8094e18fcf, priority revision 7. The claim records the overlap scope.

For exact gates I+a_i N_i and changed gates I+(a_i+e_i)N_i, with N_i^2=0, each interaction error is also square-zero. Both its forward and inverse factors have norm at most 1+||D_i||. Thus P_j=product_(i<=j)(1+||D_i||) controls the ordered products without the general-error condition sum ||D_i||<1. Structural zeros and diagonal ones must be retained. This is a sufficient bound, not a sharp bound or a claim that rounded grid instructions are reversible.

CPython 3.12.14 registered normal/optimized replay passed 24 circuits, 192 prefixes, 192 noisy steps, 16 circuits with interaction sum at least one, and five controls. Direct normal/optimized CPython 3.14.7 runs also passed. The checker computes literal inverse matrices by rational Gaussian elimination. Committed-byte comparison and direct base ancestry passed. The optional full-history preflight is unavailable because this checkout is shallow.

python3 -B scripts/research_package.py --package research/shear-structure-perturbation

Expected markers: PASS_SHEAR_PERTURBATION in both modes and PASS_REGISTERED_RESEARCH_PACKAGE. Hosted planning and the registered package replay passed on this exact head in run 39983657784. No selected cold replay, native replay, compiler build, Lean build, or Docker run was used. A production coefficient policy, prefix envelope, arithmetic kernel, overflow bound, physical costs, and supplier/outer admission remain separate obligations.

Credit ywz65's PR #2701 for the interaction-coordinate and general noise contract, Vlado Vojdanovski's PR #1673 for the circuit/precision context, and Nick Petalas's PR #5153 for the distinct endpoint-norm refinement. Classical nilpotent inverse and product estimates are retained. Prepared by Ali Romman with substantial OpenAI Codex assistance; Apache-2.0. No independent human or proof-kernel acceptance is claimed.