[jwr404]

Change and mathematical scope

Scoped comparison. No new κ. The selected conditional exponent stays 60403/20000000, and PR #4100 is not accepted.

PR #4646 reaches r (2/B + β) ≤ p 2^{-G} / 6400 when L ≤ p. On the admitted layout p = 2, ell = 2, L = 65536, G = 256, and r = r_max = 2, the exact sharp ratio is

C_* = 2 + 2^{-15871} - 2^{-16126}.

This is the C in r (2/B + β) = C p 2^{-G}. It lies strictly between 2 and 3, so C = 2 fails and C = 3 holds. The integer 3 replaces 1/6400 for this one inequality.

The published window term has exact ratio 2 + 2^{-15869} and the same integer threshold. The production plan r_max = 5, p = 9240576 still satisfies 5 · 3 · 6400 ≤ p, so 1/6400 is not retuned there.

Task and overlap

• Base: canonical main 624de9dc66858af0945e5f8a0942488d62280d43, fetched 2026-10-22T23:47:42Z, priority revision 7.
• Card: formal-interface-lemma, scope refined. This is the open question of PR Record one #4174 production plan with L = p/6 (no new κ) #4913.
• Predecessor: PR Audit CRT reservoir placement and split-axis coordinate descriptors #4174 admits the layout. PR Compare the sharp compact-switch window with CRT.md's repair room. #4646 supplies the 1/6400 line. PR Audit #4100 CRT.md's sparse exception repair tariff at Θ(p)-bit records: O(V) dense, o(V) sparse at every p (no new κ) #4328 supplies the L ≤ p hypothesis.
• Package: research/crt-reservoir-length-constant/.
• Overlap at 2026-10-22T23:47:39Z: this package, PR Record one #4174 production plan with L = p/6 (no new κ) #4913's open question, and the original 1/6400 proof. No other evaluation of this ratio.

Validation

Python 3.12, standard library only:

python3.12 -B scripts/research_package.py --package research/crt-reservoir-length-constant

Observed PASS_CRT_RESERVOIR_LENGTH_CONSTANT layout_p=2 layout_L=65536 r=2 G=256 sharp_between=2,3 integer_C=3, then PASS_REGISTERED_RESEARCH_PACKAGE research/crt-reservoir-length-constant outcome=completed.

Not run: the 104,864-plan census, the repair tariff, the E product, Lean, and Docker.

Credit

Hiroki KUMAZAKI, PR #4174, for the constructor. PR #4646 for the 1/6400 line. PR #4328 for the L ≤ p hypothesis. Cinna Davis, PR #4100, for the CRT parameters.

Prepared by James Reed with substantial Cursor Grok 4.7 assistance. Apache-2.0. No independent human review and no formal-kernel acceptance are claimed.

Open next questions

1. For every layout the Audit CRT reservoir placement and split-axis coordinate descriptors #4174 constructor accepts, is C = 3 r_max / p sufficient at r = r_max? First step: show that every accepted layout has G ≥ 256, so 2 + 2^{1-62G} < 3.