[pinterbanget]

Scoped small-prime check of Matarener's #4612 (head 7e362114), on the claim that the displayed outer-reserve threshold H(p) = [1024 C L³]^(1/(1−δ)) "already suffices" for every C ≥ 1. No κ or exponent change. #4612's conclusions survive with an amended threshold.

research/pr4612-small-prime-threshold/verify.py uses the standard library only and runs in under 1 s.

Findings

#4612's grid uses only L ≥ 8 (p ≥ 256). At p = 2 (L = 1):

• The intermediate claim fails. C(log₂h + 2) ≤ h/1024 fails at h = H for every tested C from 1 to 2¹⁰⁰⁰.
• q0 ≤ h/64 fails for Audit #4359 BIT row stock and its #4470 atom change (no new κ) #4612's own constant. With C = 232,960, q0 ≈ 0.0276·h > h/64.
• The conclusion R ≤ h/16 can fail. For C = 2⁴⁰, 2¹⁰⁰ and 2¹⁰⁰⁰ it fails at h = H(2) for both atoms. For astronomically large C (such as Audit #4359 BIT row stock and its #4470 atom change (no new κ) #4612's illustrative log₂C ≈ 1.3·10¹⁰) the δ-factor rescues the displayed H.

Correction (written proof)

H′(p) = max(H(p), 4096·C·log₂(4096·C)). This gives q0 ≤ h/512, and hence R ≤ h/16. It is checked on a grid that includes L = 1 and the primes 3, 5 and 7.

The new term is independent of p, so the threshold stays polylogarithmic. Only the wording "no extra multiple is needed" and "for every C ≥ 1" needs amending.

Credit: Matarener (Mark) for #4612; zxv and Cinna Davis for the #4359/#4470 lineage. Ryanson Jonathan, with Claude Code (Claude Opus 5.5) assistance.

Head: ed4686b

🤖 Generated with Claude Code