[petalas]

Summary

Construct a once-rounded two-coordinate Gaussian rational gate with a smaller sufficient arithmetic-word width. Claim type: scoped arithmetic construction and written proof with exact finite corroboration. For signed B-bit input grid numerators and coefficient-component height at most C, with B>=2 and C>=1:

- cited generic temporary allowance: 2B + 8C + 32
+ constructed sufficient width:      B + 4C + 2

The integer kernel uses common-denominator accumulation, shift/add multiplication and binary long division, with one final nearest rounding per real output and ties toward positive infinity. It saves all four real inputs before writing outputs. No multiplication/division machine instruction or intermediate rounding is assumed. This answers the arithmetic Open next question in Armin Pourshafeie / PR5240, without auditing its rounded-layer theorem.

The contribution was produced by Nick Petalas. Credit apoursh for the explicit question; Jose Manuel / jmbarrancoml for the retained executor, phase factory and temporary/coefficients contracts; Vlado Vojdanovski, Chafik Boukhalfa, Brian / JoyboyBrian, James Reed, Cinna Davis and the unchanged numerical/compiler lineage. Common-denominator arithmetic and binary long multiplication/division are classical. The new step is this explicit once-rounded kernel and its smaller sufficient width. Advance CLAIM.

Evidence

• Before, the source uses the conservative 2B+8C+32 cap and PR5240 leaves the concrete bounded arithmetic kernel open. After, PROOF.md bounds every product, accumulator, division remainder and quotient, and both signed tie cases by B+4C+2 bits.
• Exact independent Fraction comparisons pass 50,625 four-term rows and 3,600 two-term rows. Binary division agrees with divmod on 31,744 cases. Fifty extreme rows reach data/coefficient widths 2048/256. Eight semantic checks and seventeen invalid-domain/overflow controls pass.
• The actual retained phase factory produces a component-height-15 butterfly. All 768 complete-gate fixtures agree with its saved-input rounded executor, across three fractional grids. At B=18,C=15, the new cap is 80 bits; the cited formula evaluated at those same parameters is 188. This is not a claim that a full caller allocated an 188-bit word.
• Warning-strict direct normal/optimized checks and both registered commands pass locally. Timed direct checks take 2.337 and 2.338 seconds; the final registered pair takes 4.421 seconds, within the 120-second per-command caps. CPython 3.12.15, Darwin 27.0.0 arm64, standard library only.
• Actual negative-tie and premature-input-overwrite mutations reject, as does changed retained source content. Committed bytes, modes and source history pass for all twelve package/input files at head 08f0609cc997da02944cf5517eba1aa954665050.
• Official committed export integrity passes. Normal and optimized read-only Git-free export replay pass with PATH=/nonexistent, identical receipts and all thirteen exported file hashes/modes unchanged. Receipts are generated from checks and file hashes; none is handwritten or committed.

From the repository root:

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

Any CPython 3.12 path can replace the local interpreter. Expected markers are PASS_INTEGER_ROUNDED_GATE, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Exact source pins and the originating card are retained in SOURCES.json and TASK.json.

Limitations

The all-width result is a written proof, not finite extrapolation or a proof-kernel theorem. The bound is sufficient, not proved optimal. The reference accepts B<=2048 and C<=256. Its checked-word observer is not a physical machine emitter. Outputs fit the arithmetic width; storing them back into B-bit data words still needs the caller's amplitude bound.

No total-live-storage, runtime, bit-tape, kappa or selected-frontier improvement is claimed. Coefficient preparation and its own temporary storage, reversible cleanup, instruction encoding, physical movement, complete Full allocation, supplier admission and all five selected framework assumptions remain open. No selected cold replay, native/Lean/Docker build or independent human/formal proof-kernel acceptance is claimed. No scientific check was left solely to CI.

Open next questions

Can one complete Full allocation use the kernel with a fully priced physical arithmetic and ownership contract? Start with one emitted two-wire call and its actual input/output amplitude bounds, then count snapshots, output staging, live words, bit operations, cleanup and tape movement. The physical backend and complete caller invoice remain missing. No further optimization is reserved.

Related open work

The completion refresh inspected all fourteen account open PRs created since the CLAIM, including this PR. The other thirteen are PR6419 through PR6432, excluding PR6429. None computes this once-rounded rational-gate temporary-width bound. No extra computation checked another contribution's number. The checked main baseline is 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision eight. Sparse local notes and unpublished work remain coverage limits.

• PR6431 bounds intermediate integer magnitude and signed storage width in restoring unit-ADD words. This PR bounds arithmetic temporaries for a once-rounded rational matrix gate. The contracts differ and no common numerical instance is computed.

Merged PR5240 poses the arithmetic question; merged PR4205 supplies the retained program and generic allowance. Merged PR5940 bounds conditioned matching-cut operator norms and explicitly excludes a multiply/add/round kernel. Merged unpaidmoon / PR4123 supplies a sharper specialized two-point H bound with toward-zero return; this contribution neither audits nor improves that special-case formula. No extra computation checked another contribution's number.

CI notes

Both hosted checks passed at exact contribution head 08f0609cc997da02944cf5517eba1aa954665050 in run 41537260744: plan in 4m28s and registered package replay in 1m29s. Completed checks were confirmed once; run metadata confirms successful completion and the matching head. The replay runs both registered warning-strict normal and optimized commands.

All registered scientific commands, metadata/source pins, committed-byte/source-history/mode preflight and read-only Git-free replay also passed locally. Hosted success covers the declared finite checks and source bindings; it does not establish independent human or formal proof-kernel acceptance of the quantified theorem.

Merge Danger

Door: two-way. Blast radius: isolated. Seven new package files; no shared implementation, predecessor source, pinned evidence or selected record changes.