[unpaidmoon]

Change and mathematical scope

Scoped audit of one sentence in Cinna Davis's #4100 PROOF.md. No new κ. #4100 is not accepted, and the selected frontier stays 0.001672660197076918.

PROOF.md says that polynomially many full-array norm changes cost at most 2^{O(p)} because T=2^{Θ(p)}, and that a sufficiently large fixed C then makes every absolute source-transform coefficient error smaller than 2^{-4p}.

With the size section's own parameters (b=ceil(log2 n), p=6b, T the unique power of two in [4n/b, 8n/b)), every n≥4 satisfies

b − floor(log2 b) + 1 ≤ log2 T ≤ b + 2 − floor(log2 b).

Both ends are attained (n=5 and n=4). So log2 T ≤ p/6 + 2 − floor(log2 b). Charging p factors of T costs 2^{Θ(p^2)} bits: at n=2^b the exponent is exactly 14p, 60p, 250p, 1016p, 4086p for b=16, 64, 256, 1024, 4096. That product reading does not determine C.

The one-factor reading does. One factor of T, times p^{10}, is below 2^{-4p} at C=5 for every bucket b≥13, and at C=8 for every b≥2. Six factors leave no room at C=5. If the displayed row gap δ≥1/(16p^2) holds and one inverse is charged in the infinity norm, the factor is at most 16p^2, and C=6 works for every b≥2.

Task and overlap

• Base: canonical main 12fa8c5c844be21bfefb38a19503fbd867a11244. Priority revision 7.
• Card: uniform-transfer-audit, scope refined to PROOF.md's 2^{-4p} amplification sentence. The card asks for one quantified Propose unconditional κ = 0.00173097911567079247276057 via batched CRT #4100 implication; this is that sentence and not CRT repair, the free-algebra lift, or the row gap.
• Propose unconditional κ = 0.00173097911567079247276057 via batched CRT #4100 head 337d04939571a637aeab54a0f58f54fde811663a.
• Nearest work: Audit #4100 PROOF.md's transform and exact-recovery chain: all identities hold (no new κ) #4320 audited the algebraic composition and left this sentence as open question 1, saying the row gap blocked a constant. Audit #4100's cached chirp odometer: O(p) head travel per record uniformly in d (no new κ) #4325 audited chirp head travel. Map independent audit coverage of #4100's obligations (navigation; no new κ) #4311 lists this sentence as unaudited obligation 2. The exponent lemma and the product-reading failure do not use the row gap. The 16p^2 / 512p^5 rows remain conditional on it.
• Package: research/proof-explicit-4p-error.
• Current public draft, same commit: Audit #4100's 2^{-4p} sentence: log2 T is p/6, and a polynomial number of full-array factors is not 2^{O(p)} unpaidmoon/integer-mult-bounds#63

Validation

Python 3.12, standard library only. verify.py pins the source sentences and checks:

• the exponent for every n from 4 through 200000, and both endpoints of every bucket through b=4096;
• the product quotients above;
• the conditional C table by exact powers of two;
• failure controls: a one-bit-tighter upper bound is false at n=4; a 10p product claim is false at b=64; C=5 never absorbs six dimension factors on 2≤b<80.

python3 scripts/research_package.py --package research/proof-explicit-4p-error --check-only
python3.12 scripts/research_package.py --package research/proof-explicit-4p-error
python3 scripts/check_research_commit.py --package research/proof-explicit-4p-error

Local results: PASS_PACKAGE_METADATA_AND_PINS, PASS_REGISTERED_RESEARCH_PACKAGE outcome completed, PASS_RESEARCH_COMMIT_BYTES, and PASS_EXPLICIT_4P_DIMENSION.

Not run: a proof kernel, the Gaussian circuit, and any count of how many factors of T one real coefficient path contains. Finite checks corroborate the size identities; they do not discharge #4100.

Credit

The construction and the 2^{-4p} sentence are Cinna Davis's #4100, with OpenAI Codex assistance. #4320 and #4311 are Ryanson Jonathan's, with Claude Code assistance. Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0.

Open next questions

1. Count the factors of T on one coefficient of the three normalized transforms, the pointwise product and the final M scale. That integer is the m the table prices. The normalized H0/2 butterfly may contribute none. Nothing in the size section blocks the count; a Gaussian inverse on the same path still needs the row gap before its norm is 16p^2.
2. Make δ≥1/(16p^2) effective after alias removal, truncation and rounding (Audit #4100's HALO locality witness: exact error identity and sharp (2J-1)W radius (no new κ) #4217 open question 1). Until that holds, the inverse rows stay conditional.

Workflow feedback (optional)

The #4311 coverage map made this obligation easy to separate from #4320's finished identities. No separate feedback post.