[petalas]

Summary

Scoped obstruction and construction proof with finite exact corroboration. For N>=2 arbitrary integer coefficients, an exact coefficient sum plus fixed binary-linear checks permits uniform recovery from coordinate errors at most R exactly as follows:

R < 1/2        recover without checks
1/2 <= R < 1   recover iff rank_F2([H; all-ones]) = N
R >= 1         impossible even with all coordinate parities

The middle regime requires exactly N-1 checks, attained by the first N-1 coordinate parities. Every rank-deficient augmented sketch has an explicit equal-sum zero/one collision at error 1/2. This closes an original possible precision-saving route for the unrestricted coefficient interface. It adds only research/sum-parity-recovery/; no new kappa or selected-frontier change is claimed.

The contribution was produced by Nick Petalas. Standard binary linear algebra, coding and lattice separation are background, with no literature-priority claim. The selected review and zxv's PR6062 supply analytic-recovery context rather than a theorem being audited. The P104/native/helper/compiler/outer lineage retains its credit. Advance CLAIM.

Evidence

• Before: a small fixed parity sketch and the exact sum are a possible proposed replacement for the strict half-unit precision requirement. After: the written all-input proof forces N-1 checks and gives an attaining decoder; unit-radius recovery is impossible.
• The independent finite coefficient-observation oracle enumerates all 3,287 binary row spaces through N=6. Exactly 67 admit recovery; each of the other 3,220 supplies an explicit ambiguity. The decoder passes 5,704 closed-boundary and 1,089 sub-half-unit fixtures; 66 omitted-check controls, 63 unit-radius ambiguities, 20 invalid-contract rejections and a 4,001-bit integer fixture pass.
• Warning-strict normal and optimized direct runs pass on CPython 3.12.15, macOS arm64, in about 0.25 seconds each. Every registered VERIFY command passes; the complete registered replay takes about 0.75 seconds, well below its 120-second command limits. These are replay-budget observations.
• Publication commit 64a58e7bce47a1745aa74918485afd657bc5f871 passes committed bytes, modes and declared source ancestry for all seven package files. Official export integrity and declared-input coverage pass. Both read-only Git-free direct modes pass with empty PATH; all eight exported files, including the export manifest, retain their bytes and modes. Generated JSON receipts bind all seven package files. Syntax/AST, function-length and whitespace checks pass.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/sum-parity-recovery/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/sum-parity-recovery/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/sum-parity-recovery
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/sum-parity-recovery --check-source-history --check-modes

Any CPython 3.12 executable may replace the local path. Expected markers are PASS_SUM_PARITY_RECOVERY, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The verifier writes nothing. PROOF.md supplies the universal argument; finite checks corroborate it.

Limitations and unrun obligations

The coefficient vector is unrestricted, the error bound is uniform and worst-case, and the parity matrix is fixed before the input/observation. The mathematical theorem has no dimension or magnitude cutoff. The executable decoder accepts N=2..64, at most 64 rows, and 4096-bit input integers or rational numerators/denominators.

This is not a lower bound for actual convolution coefficient sets or multiplication. The exact sum is granted for the obstruction; its physical acquisition and the parity/decoding/restoration/tape bill remain unpriced. Sparse or probabilistic errors, adaptive/nonlinear checks, odd moduli and other exact measurements can change the answer. No actual convolution caller, numerical supplier, selected cold replay, Lean, Docker, independent human review or formal proof-kernel acceptance is supplied. The five selected assumptions remain.

Open next questions

• Can an actual completed numerical caller guarantee sparse coefficient-error supports? Prove that restriction before choosing and pricing exact checks. A uniform norm bound is insufficient.
• Can a fully paid odd-modulus interface improve the precision budget? First specify its coefficient domain, exact residue acquisition, returned error, storage and restoration.

Related open work

No open PR studying this unrestricted exact-sum/fixed-parity coefficient-recovery quantity was identified in the refreshed open corpus, targeted live searches and current reservations. Coverage is bounded and excludes unpublished work and arbitrary diffs. PR6062 is merged context. This account's merged PR6182 studies a deterministic rounded native inverse with coordinate tags, a different operator and error model that is not used or extended here. No extra computations were run to check predecessors or compare their results.

The completion refresh inspected this account's twelve other open PRs created since the advance CLAIM at 2026-10-29T16:17:14Z: #6192, #6193, #6194 and #6196 through #6204. They concern cyclotomic operators, cyclic reversal, projection moves, XOR snapshots, P104 birth costs, E8 spans, draft creation, cubic-reader repair, ordered-shear primes, cache trees, and verifier performance. None studies this unrestricted exact-sum/fixed-parity coefficient-recovery quantity or computes a competing value for it. No additional computations were run to check their results.

CI notes

Both hosted checks passed at exact head 64a58e7bce47a1745aa74918485afd657bc5f871 in run 40911486584: plan and replay (research/sum-parity-recovery). The completed checks were read once, and the run's successful conclusion and exact head were confirmed.

Hosted replay certifies the registered normal and optimized package checks. The local mathematical, publication-byte, source-history and read-only export checks listed above also passed. Independent human and formal proof-kernel acceptance remain unclaimed; the stated physical and full-theorem obligations remain open. The delivered scope matches the advance CLAIM.

Merge danger

Door: two-way. Blast radius: isolated research package. Shared code, frozen sources and the selected frontier are unchanged.