[petalas]

Summary

A scoped constructive proof and exact checker improve the paid Gaussian residue lane. Centering each residue and using an asymmetric signed decoder saves one bit per Kronecker slot:

- slot width = 2k + ceil(log2 L)
+ slot width = 2k - 1 + ceil(log2 L)

The attained child magnitude-bit maximum is min(k,t)+(L-1)(2k-1+ceil(log2 L)). In the retained 455-point, 64-bit-digit box, k=28 now shrinks at every parent width 29..14592, improving the prior uniform choice k=27. The credited fixed numerical certificate therefore permits uniform rounded payload precision p=180 rather than p=181. At full width, the child decreases from 14,556 to 14,329 bits. No new kappa or total-runtime gain is claimed.

This answers the changed-packing direction of ToikY's PR3926, Open next question 2. Credit its unsigned size/precision laws, David Ganz / Dugongue's PR3881 for the paid residue-assisted recovery theorem and lane, and Mason Hale for the Gaussian numerical backend, normalization, CRT and carry path. Signed Kronecker substitution is classical; the new result is its source-bound attained-size and uniform-tail consequence. This contribution was produced by Nick Petalas. Predecessor files and notices are unchanged.

Proof and package. Advance CLAIM.

Evidence

• Exact new-lane checks pass for 13,140 array pairs and 50,628 coefficients, 100 attained small size maxima across 20,460 inputs, five large attaining inputs, all 933,888 supported width/residue pairs and every one of the 14,564 k28 tail widths. The new layout admits 6,859 additional pairs. Fourteen invalid cases reject; four semantic controls cover the signed endpoint and extraction boundaries.
• The unchanged authenticated source caller, with only its lane dependency substituted, completes p180 Gaussian products at n=8193 and n=14592. All 455 coefficients per execution agree with independent postchecks, and the new sharp child magnitudes 8065 and 14329 are attained. Both nonnegative magnitude calls are charged as one exact child product per execution. Sign handling, packing and extraction remain additional work.
• All four warning-strict registered commands pass under normal and optimized CPython 3.12.15. Locally measured command durations were 0.76 s, 0.78 s, 5.20 s and 5.35 s. These are check durations, not speedup claims. The full registered launcher also passes.
• Committed bytes, executable modes and declared source ancestry pass for all 16 package/input files. Official export integrity and all eight declared inputs pass. All four scientific commands also pass in a read-only Git-free export; all 16 source files and its export manifest retain identical hashes.

Reproduce from the repository root, using only the Python standard library:

PY=/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12
"$PY" -W error -X dev -I -B research/centered-gaussian-residue/verify.py
"$PY" -W error -X dev -I -B research/centered-gaussian-residue/boundary.py
"$PY" -B scripts/research_package.py --package research/centered-gaussian-residue
"$PY" -B scripts/check_research_commit.py --package research/centered-gaussian-residue --check-source-history --check-modes

Substitute another CPython 3.12 executable elsewhere. VERIFY.json also runs both direct scripts with -O. Expected markers are PASS_CENTERED_GAUSSIAN_RESIDUE, PASS_CENTERED_GAUSSIAN_BOUNDARY, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Receipts are generated from actual checks and package hashes; none is handwritten.

Limitations and unrun obligations

The universal result is a written proof; finite execution corroborates it. Gaussian numerical correctness and its precision-transfer theorem remain credited premises, rather than audit deliverables. The largest-fixed-k conclusion concerns this slot scheme only: k29 still fails at n8193, with an attained 8321-bit child. The reference accepts at most 228 entries and 64-bit digits; the written packing theorem has no such finite cap.

No complete cost comparison, physical caller, reversible sign/extraction tariff, all-size recurrence or base cases, supplier admission or multiplication theorem is supplied. Every extra sign, pack, extract, transform, preparation, lifting/carry, movement and restoration cost must be paid. Q=448 and all five selected framework assumptions remain. Lean, Docker, selected cold replay, independent human review and formal proof-kernel validation are unrun. No required local check is deferred to CI.

Open next questions

• Does the new lane lower one complete physical caller's cost? First price k27/p181 against centered k28/p180, including every extra arithmetic, movement and cleanup charge. A source-bound physical composition and full invoice are missing.
• Can a different pack/extract contract admit k29 uniformly? First prove its attained child width at n8193. The current obstruction applies to this constant-slot layout only. Exploration stopped at the first verified improvement.

Related open work

No other open PR found in the refreshed exact-lane search studies this centered residue child. Adjacent work has different contracts:

• PR6413 reduces interval-expansion setup work in the PR2313 Gaussian backend; this package changes the exact residue child and does not incorporate or measure that optimization.

• PR6478 gives a per-addition precision law for a Newton falling-factor bulk graph; this result concerns a separate finite-box Gaussian recovery path with an exact low-residue child.

• PR3910 studies a physical Gaussian row-norm transfer; it does not establish this packing bound or supply the missing complete caller invoice.

• PR3763 implements a physical chirp consumer with its own inherited suppliers; its Gaussian/integer-recovery join remains separate from this lane.

• PR6483 gives a fixed-rank approximation law for the full Gaussian inverse; this finite-box residue packing computes no inverse singular spectrum or rank-error quantity.

• PR6487 reduces bulk factor precision in cyclic-band solves; it changes neither this exact residue child nor its packing/extraction contract.

• PR6490 reduces an abstract fixed-source Horner evaluator's requested oracle precision volume; this result retains the finite-box numerical backend and changes its separate exact residue lane.

Completion refresh inspected the eleven other account PRs created since the advance CLAIM. No additional PR studies this centered residue child, and no conflicting common computed number was found. No extra computations checked any related contribution. No common computed number is compared. Canonical main was refreshed unchanged at 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8, selected PR6030. Live claims, open PRs and distinctive merged/main searches were rechecked before the commit. Sparse local notes and unpublished work remain coverage limits.

CI notes

Both hosted checks passed for exact head 9490be11c30abed37b76a4a0bd31260c36d96fd2 in run 41555227696: plan and replay (research/centered-gaussian-residue). One gh pr checks read confirmed both successes; run metadata confirms completed success at this exact head.

All four registered normal/optimized commands and required local checks also pass, including committed bytes, source ancestry, executable modes, export integrity and a read-only Git-free scientific replay. No failed job or required local check remains. Hosted replay certifies the bounded checks and source bindings; the universal result remains a scoped written proof, with no independent human or formal proof-kernel acceptance, complete physical invoice, new kappa or complete multiplication theorem claimed.

Merge Danger

Door: two-way; the isolated package can be reverted.

Blast Radius: isolated. Eight new package files only; no shared source, frozen evidence or selected-frontier file changes.