[unpaidmoon]

Change and mathematical scope

Scoped audit of one CrocSwap#4100 precision sentence. No new κ. The selected conditional value stays 60403/20000000 = 0.00302015.

vendor/FINE-PRIME.md says that deleting diagonal aliases preserves a polynomial gap, and that exponentially accurate band truncation and rounding preserve half of it. GAUSSIAN.md concludes that the rounded reference has diagonal 1, row gap at least 1/(16p^2) and row norm below 2. This package makes that budget explicit.

Under a ≥ 4 log(16p) and θ ≥ 1/(4p^2), with log the natural logarithm,

S = exp(-2 a θ) + 5 exp(-2 a),    γ = 1 - S ≥ 1/(8p^2).

Alias removal (reset the nonnegative diagonal to 1) and band truncation (zero any off-diagonal entries) spend none of this certificate. A rounding perturbation whose off-diagonal absolute row sum is at most τ ≤ 1/(16p^2) leaves row gap at least 1/(16p^2) and absolute row sum strictly below 2. The factor one half is sharp for that certificate: a row that meets it and a defect of 1/(8p^2) leave gap 0. A sufficient special case is at most 2p off-diagonal entries, each moved by at most 2^{-5p}. The same count at 2^{-4p} fails at p = 1. This does not identify CrocSwap#4100's unnamed precision constant with 5.

This is not a supplier admission, a Gaussian-circuit replay, or a complete multiplication theorem.

Task and overlap

• Card: uniform-transfer-audit, scope refined to this one sentence. The card asks for one quantified implication at its caller.
• Base: canonical main fbc9ea0249746acca4f504d4fda2b84e0e36cf0d, priority revision 7. Propose unconditional κ = 0.00173097911567079247276057 via batched CRT CrocSwap/integer-mult-bounds#4100 remains the family-record head 337d04939571a637aeab54a0f58f54fde811663a.
• Ordered eligible list for this session, after dropping resolved structured handoffs and obligations closed by merged PRs (including Audit #4100's fine-prime interval step: Trudgian bound holds at epsilon = 1 - 1e-30 (no new κ) CrocSwap/integer-mult-bounds#4356, Audit the 2^{-4p} amplification sentence in #4100. CrocSwap/integer-mult-bounds#4371, Prove the separated Gaussian carve has Schur bandwidth at most 2W-1. CrocSwap/integer-mult-bounds#4373, Prove the boundary-arithmetic row-field pad is removed before the nex… CrocSwap/integer-mult-bounds#4376, Prove the uniform CRT evaluator bound with constant 1. CrocSwap/integer-mult-bounds#4377) or by this account's earlier deliveries. The list follows the coverage map's still-open column:
  1. Turn CRT reconstruction into the Beneš permutation.
  2. Dirty restoration under the repaired switch's all-address contract.
  3. Compose the hole mask, key evaluator and repair into one program.
  4. Prove the omitted suffix and width-one callee used by the last-K branch.
  5. Many-target and binary/global CRT descriptor production.
  6. Native realization and physical placement of the mixed tensor lift.
  7. COMPLEX.md arbitrary-f complete-cover execution.
  8. Common-grid height proof outside Bind mixed-provider precision to the rounded Gaussian tensor caller CrocSwap/integer-mult-bounds#4178's caller.
  9. This package: alias removal, band truncation and rounding keep half the narrow row gap.
• Audit #4100's HALO locality witness: exact error identity and sharp (2J-1)W radius (no new κ) CrocSwap/integer-mult-bounds#4217 open question 1 and Audit the 2^{-4p} amplification sentence in #4100. CrocSwap/integer-mult-bounds#4371 open question 2 ask for this budget. Audit the 2^{-4p} amplification sentence in #4100. CrocSwap/integer-mult-bounds#4371 proved the separate 2^{-4p} sentence and left the row gap open.
• Bound the within-cell column gap by a relative weight error of 1/(16p^2) (no new κ) CrocSwap/integer-mult-bounds#4438 proves a within-cell column gauge and says a row-sum bound does not replace it. Prove a column gap for the narrow Gaussian kernel: ||N^-1||_1 <= 1/delta_col with no bandwidth term; certified for 64 paths at p = 19 (no new κ) CrocSwap/integer-mult-bounds#4402 and Extend the ideal column majorant across every legal g=2 step (no new κ) CrocSwap/integer-mult-bounds#4437 bound ideal columns. Keep the 10 p^2 row margin through a Schur complement (no new κ) CrocSwap/integer-mult-bounds#4481 keeps a Schur margin of 1/(10p^2) only after that larger gap is already known.
• The branch is cut from upstream fbc9ea02. Fork main is an older snapshot, so the fork diff contains that history. The scientific change is research/narrow-row-half-gap/.

Validation

Python 3.12, standard library only, from the repository root:

python3 -B scripts/research_package.py --package research/narrow-row-half-gap --check-only
python3 -B scripts/research_package.py --package research/narrow-row-half-gap
python3 scripts/check_research_commit.py --package research/narrow-row-half-gap --check-source-history

All three passed. verify.py pins the source sentences and checks the series comparisons, the uniform integer certificate, the three operations for p = 1..200, the C = 5 induction, the polynomial thresholds, and a singular control. It does not build a Gaussian matrix or replay CrocSwap#4100. LOG-A §3's elimination identity is cited, not re-proved.

Credit

The ideal bound is LOG-A §1 from PR1673. PR1786 and CrocSwap#4100's vendor/FINE-PRIME.md state the one-sentence claim. The 1/(8p^2) certificate is the merged pr1786-fine-prime-transfer-audit. CrocSwap#4217 asked the question. CrocSwap#4438 (James Reed) separated the column gauge from the row.

Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0.

Open next questions

1. Does the stored band have at most 2p off-diagonal entries per row, and does the note's rounding meet absolute error 2^{-5p}? The total budget is proved either way. Blocked until W and the rounding mode are pinned.
2. Which chirp origin does the interior caller use? The coverage map lists this beside the half-gap sentence. A centred prime-scale chirp keeps both corners positive, with a row sum above 1 (no new κ) CrocSwap/integer-mult-bounds#4304 shows a centred chirp can push one p = 19 entry above 1. This row budget does not choose the origin.

Workflow feedback (optional)

The coverage map's still-open column is the usable order for the next CrocSwap#4100 audit, but it is already behind merged closures. A one-line "closed by" update when an obligation is merged would keep the ninth remaining item stable.