[aliromman]

This enabling proof gives the physical inverse-coded pair an explicit budget for every listed returned residual. The same guard P=p+ceil(9f/2)+5 now covers native and Gaussian errors, under stated provider contracts. It also proves the exact real-component terminal norm 26*2^f, improving a product-norm bound by a factor two at odd widths. The complete paid Gaussian profile and κ are unchanged.

Proof, receipt, source pins and checker. Base: canonical main ff3cc58e370d0f678e529246f8c73e8094e18fcf, priority 7.

The scope is every positive fiber width, with exact copies and signed monomial wrappers. Each residual is defined on its actual computed input; no rounded operator is assumed linear. With S=Q^-1 C Q:

Delta X = -DQ C e0 + bX
Delta Y = Q^-1 bX + S bY + Q^-1 C e1 + S et
          + Q^-1 C eQ + Q^-1 eC + eL.

The proof defines every site. The two actual inverse-kernel residuals satisfy C e=-13e. If each is at most two grid units and each of the other six residual vectors is at most one, the stated guard gives output error strictly below 2^-p. The earlier PR #5322 treated native and Gaussian calls as exact; it supplies the credited inverse-only cancellation and error shape.

Use CPython 3.12 and the standard library:

python3.12 -B scripts/research_package.py --package research/physical-pair-full-residuals
python3.12 -B scripts/check_research_commit.py --package research/physical-pair-full-residuals

Both registered Python 3.12.13 modes passed, with warnings as errors and -O. Expected verifier marker: PASS_FULL_PAIR_RESIDUAL_TRANSFER. Checks include 63 chronological cases in seven actual Walsh fibers, all 5,824 terminal-block premises, 3,072 literal row/column checks, 12 source-kernel calls, 500 guard widths, and omitted-residual/order controls. The written proof covers all widths. Committed-byte and export checks, both direct exported modes, and git diff --check passed. Full-history validation was not run in the shallow checkout. Provider accuracy, a complete physical caller, Lean, native/compiler and selected replay were not run.

Input ranges, provider accuracy and their internal precision costs remain explicit obligations. The unshared boundary cost is unchanged. This does not claim a cheaper complete caller or alter the active native-history construction. Scoped claim.

@JoyboyBrian’s PR #5069 supplies the pair and PR #5035 supplies the finite kernel. @dillona’s PR #4933 supplies the normalizers and four-point terminal expansion. Those inputs are pinned at the base. Retained native, controller and compiler credit remains. New all-site theorem and checks: Ali Romman (@aliromman) with substantial OpenAI Codex assistance. Apache-2.0; author-checked, not independent acceptance.