[petalas]

Summary

Prove a sharp intermediate-growth bound for exact restoring words of signed unit row ADDs. Claim type: scoped written proof with exact finite corroboration. For every finite number of arbitrary dirty rows n>=2 and every at-most-L gate budget:

maximum absolute coefficient = F_(floor(L/2)+1)
maximum absolute row sum     = F_(floor(L/2)+2)

At a cut k of a length-ell identity word, the upper bounds instead use min(k,ell-k). Alternating positive ADDs on two rows followed by their literal inverse attain both global bounds. The sharp input-magnitude cap and signed storage width follow. The new step bounds the same prefix through its restoring suffix. Fibonacci majorization and elementary row operations are classical; no global literature-priority claim is made.

Only research/restored-unit-shear-growth/ is added. The result is produced by Nick Petalas. Independent human and formal proof-kernel review are unclaimed. No predecessor result is audited or replayed as the contribution, and no new kappa or selected-frontier change is claimed.

Evidence

The unrestricted one-way estimate grows with all executed gates. The new restoration-aware estimate uses the smaller distance to either endpoint, and its two-row witness attains equality.

Local CPython 3.12.15 on macOS 27.0.1 arm64 passes:

• All 358,834 signed-unit matrix words in the two declared domains, including 3,266 exact identity loops, with every cut checked.
• A separate literal-cube oracle on 28,082 words; all common identity counts and maximum row norms agree. It imports no matrix-model code.
• 387 even-budget attaining loops through budget 256, 128 odd-budget attainers, 231 literal magnitude/width guards and 12 contract/semantic controls.
• Both registered warning-strict normal and optimized replays. Separately timed identical direct commands take 4.26 s and 3.87 s, respectively. These are timeout checks, not a speedup claim.
• Direct scientific/oracle checks, syntax/function-size inspection, staged whitespace and exact committed-byte/source-history/mode checks. The verifier emits a generated receipt bound to every package file's SHA256. No static linter or type checker was installed.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/restored-unit-shear-growth
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/restored-unit-shear-growth/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/restored-unit-shear-growth/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/restored-unit-shear-growth/oracle.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/restored-unit-shear-growth --check-source-history --check-modes

Another CPython 3.12 executable may replace the local path. Expected markers are PASS_RESTORED_UNIT_SHEAR_GROWTH, PASS_RESTORED_SHEAR_LITERAL_CUBE, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Replays need only the standard library and write no checkout files.

Canonical main was refreshed before submission and remains 7c49fe3, priority revision 8, selected PR6030. Advance claim. Live claims, local reservations, open/account PRs, main object/formula searches and merged searches were checked. Cached card coverage included 375 open PRs and 6,298 Discussion records with complete remote pagination. Arbitrary historical diffs and unpublished work remain outside coverage.

Limitations and unrun obligations

The theorem requires an exact full identity endpoint and permits only distinct-row ADDs with coefficient +1 or -1. It excludes scales, other coefficients, native operations, source insertions and rounding. The universal proof is written; the implementation's bounds of eight rows and 256 gates do not restrict the theorem. No assertion is made about every exact word length or sharpness at each prescribed cut.

No selected caller, physical tape implementation, rounded cleanup, error bound, coefficient preparation or arithmetic-temporary evaluation order is admitted. All five selected framework assumptions remain. No predecessor scientific suite, selected cold replay, Lean or Docker build ran. No local proof obligation is deferred solely to CI.

Open next questions

• Find a sharp endpoint-dependent envelope for one specified nonidentity matrix, beginning with the inverse suffix followed by that endpoint. No such classification or selected-word binding is supplied.
• Specify rounding points and an increment-error model, then test whether the exact inverse-prefix bound sharpens error propagation. Exact restoration does not imply rounded restoration or dirt-independent increments.

Related open work

• PR6407 prices rank units in four pure bank-shear stages with a swap endpoint. This result bounds integer intermediate magnitude under full identity restoration and a gate-count budget; it computes no bank-swap tariff.

• PR6402 gives square-zero ordinary tensor endpoint error and precision choices. This result bounds exact scalar prefix magnitudes and supplies no tensor-error or rounded-cleanup claim.

• PR6418 gives adaptive rounding bias for a reused seed. This result has exact arithmetic and no random seed, and compares no rounding-bias number.

• PR6429 bounds arithmetic temporaries in a once-rounded rational gate. This result bounds exact row states in restoring signed-unit loops; neither result supplies the other's temporary or rounding contract.

• PR6427 minimizes ADD counts for fixed-weight parity delivery with a restored anchor. This result instead maximizes intermediate magnitude under a full identity endpoint and an at-most gate budget; no parity-tour tariff is compared.

• PR6426 bounds cyclic inverse row norms as polynomial degree grows. This result bounds row norms of integer shear prefixes by restoring-word length; no common matrix family or inverse-norm value is computed.

The completion refresh inspected all eight account open PRs created since the advance CLAIM, including this PR. The seven others are PR6425 through PR6430 and PR6432. No competing value for the restoring signed-unit identity-word envelope or shared-instance numerical disagreement was found. No additional scientific computation checked another contribution.

CI notes

Both hosted checks, plan and replay (research/restored-unit-shear-growth), passed at exact head b0cb9c60ac4c237ba498ce71ed8cf5597fb6062a in run 41537320120. The completed checks were read once with gh pr checks 6431 --repo CrocSwap/integer-mult-bounds. Separate run metadata confirms completed status, success and the matching head. Planning took 3m34s and package replay took 1m41s.

Hosted replay certifies execution of both registered warning-strict normal and optimized commands. Local scientific, literal-oracle, committed-byte/source-history/mode and syntax checks also passed. No failed or waived check remains. Independent human and formal proof-kernel acceptance, selected-caller binding and the listed physical/rounded obligations remain unclaimed.

Merge Danger

Door: two-way. Blast radius: isolated. Eight new files in one self-contained package; no shared code, frozen predecessor, selected record or maintainer ledger changes.