[petalas]

Summary

Scoped written proof with exact finite corroboration. In CubeJerry's #6993 normalized Gaussian convolution channel, each forward transform may request exactly 2Gamma fewer accuracy bits. The opposite-transform tolerance, common q grid, actual-input radius guards and final strict integer-recovery error <1/4 are preserved under the stated source contracts.

a = 2ceil(log2 S)+2b+7, 1 <= S < 2^b
forward stage requests: tau = a+2
opposite stage requests: tau = a+2Gamma+2
common grid: q >= a+3Gamma+7, plus actual source/ledger guards

The new step is the asymmetric error allocation, produced by Nick Petalas. It permits forward map error 2^(-a-Gamma) in place of 2^(-a-3Gamma). The exact normalized identity, physical endpoints and classical product perturbation inequality are credited prior work. Credit Jerry Chen / CubeJerry #6993, Cinna Davis #4100, Ryanson Jonathan #4320/#4415 and the retained native/scalar/Fourier/source lineage. No predecessor scientific suite was replayed as the deliverable.

Canonical base e54523517d0f03e58352e647c403a5af3e525a57; contribution head ab0171aa937f4f84b8af2d29234a385c2200416a. The supplier-boundary-audit card changes scope to an original analytic interface improvement. Advance claim. The proof and exact provenance are in research/asymmetric-gaussian-precision/.

Evidence

• Before: the predecessor assigns tau=a+2Gamma+2 to every stage of all three calls. After: the two forward calls use tau=a+2, with final error bounded by (777/512)2^-a after removing the three Gaussian normalizers. Restoring the unchanged S-squared digit scale gives error strictly below 1/4.
• Local CPython 3.12.15 normal and optimized checks pass 315 budget cases, 1,152 literal inward-rounded convolution executions, six abstract scalar equalities and ten controls. Their generated six-file hash-bound receipts agree. Length-2/4 fixtures illustrate the norm contract; they are not production odd-source transforms.
• Both registered checks pass. Each direct check took 0.17 seconds locally; these are validation durations, not a runtime improvement. Each registered timeout is 120 seconds.
• Metadata/pins, source compilation, bounded-function inspection, exact committed bytes/modes and source-history preflight pass. Official six-file export integrity and a read-only, empty-PATH, Git-free optimized exported replay pass with the same receipt.

Reproduce from the repository root:

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

Another CPython 3.12 installation can replace the absolute interpreter path. Standard library only. Expected verifier marker: PASS_ASYMMETRIC_GAUSSIAN_PRECISION.

Limitations and unrun obligations

The theorem is a conditional error-budget improvement. A fixed common q or another applicability guard can prevent a physical width reduction. Separately prepared forward sources, references and phase banks must satisfy their actual finite gates. No production controller, complete physical price, storage/runtime improvement, supplier admission, new kappa, complete multiplication theorem or selected-frontier change is claimed. Original-coordinate extraction, CRT, carry/output and all five selected framework assumptions remain. Independent human and formal proof-kernel review are unclaimed.

Open next questions

• Prepare one actual forward controller at these requests and determine whether a binding physical width decreases after every reference/source gate and complete cost is included.
• Fund all conversion errors in a separately admitted pointwise precision schedule before attempting to lower common q. The opposite-transform error budget remains necessary.

Related open work

The completion check read this account's ten open PRs created since the advance CLAIM, including this one. None of the nine others studies the same normalized-convolution accuracy quantity or computes a competing tolerance.

• #7263 constructs exact norm-one Gaussian-rational Stein factorizations. Its coefficient-factorization norm question is separate from this transform-error allocation; neither result supplies the other's physical caller.
• #7268 reduces compact rounded-ADD residue control. Its finite-control cost reduction does not alter this proof's retained pointwise truncation or transform-error budgets. No additional numerical comparison was run.

#6993 is merged and supplies the predecessor normalized-channel interface; it is not an admission of the production caller.

CI notes

CI did not certify this package. At unchanged head ab0171aa937f4f84b8af2d29234a385c2200416a, run 43030135179, plan job 130554753063 failed in Onboarding helper tests on exactly test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record. The job reports 760 tests, one failure and 24 skips. Its AssertionError states that '`make frontier7000-verify`' was not found in the current CI verification section, which still describes #6030.

I inspected that job log and the unchanged base files at e54523517d0f03e58352e647c403a5af3e525a57: certificates/selected-result.json selects make frontier7000-verify, while docs/ci-verification.md still names make frontier6030-verify. This PR changes only its six package files. The replay job was skipped because plan failed; no hosted package verification ran.

I also directly inspected run 42905094314, job 130140894651 and run 43029125610, job 130551298799. Each reports the same failing test, the same missing-command AssertionError, and exactly one failure among 760 tests with 24 skips. These are the other runs verified here; no broader CI claim is made.

The local normal/optimized registered checks and export checks reported above passed. The branch is unchanged; no unrelated repair, retrigger commit or job rerun was made.

Merge Danger

Door: two-way. This adds one isolated research package.

Blast Radius: package. The selected frontier and predecessor packages are unchanged.