[unpaidmoon]

Change and mathematical scope

Scoped comparison. No new κ. The selected conditional exponent is unchanged, and PR #4100 is not accepted.

CRT.md §4 charges a batch at r [2/B + 8B/2^K]. The window summand is the fraction of K−1-bit windows outside the published margin 2B ≤ W < 2^m − 2B. The isolated post-increment step admits 0 ≤ W ≤ 2^m − B + 1, and the bad-window count on that predicate is exactly B − 2. The replacement term is

β = 2(B − 2) / 2^K = ((B − 2) / (4B)) · (8B / 2^K),

which is strictly less than one quarter of the published window term for every B > 2, and is zero at B = 2.

Under CRT.md's parameters, with ell ≥ 1, p ≥ 2, L ≤ p and r ≤ floor(L/(100G)),

r (2/B + β) < p 2^{-G} / 6400.

The same ceiling holds for the published window term. The displayed line 2^{-G} E(p, ell) ≤ C 2^{-60 ell} (p+2)^{-48} does not contain 8B/2^K, so it is unchanged. Replacing the window term does not shrink the scalar room CRT.md uses to put bad-key work in O(V).

This is a comparison of those two displayed scalars. It does not extend the simultaneous carry argument from the 2B margin to the extra windows, and it does not replay the tape repair.

Task and overlap

• Base: canonical main fbc9ea0249746acca4f504d4fda2b84e0e36cf0d, priority revision 7. Selected κ stays 0.00302015 (Conditional kappa = 0.00302015 (+0.6304% over #4447): joint helper boundaries #4470).
• Card: formal-interface-lemma, scope refined. Item 7 of the priority-table order in docs/research/current-priorities.md, then the catalogue tasks not already in that table. The compact-switch contract audit is already Audit the compact-switch coefficient and guard contract. #4321. This is its open question 2, which is a different obligation from the live-digit identity.
• Predecessor: Cinna Davis, PR Propose unconditional κ = 0.00173097911567079247276057 via batched CRT #4100, head retained as 337d04939571a637aeab54a0f58f54fde811663a in the family record. The sharp window is PR Audit the compact-switch coefficient and guard contract. #4321. The published density identity and the L ≤ p hypothesis are 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. PR Decide #4100 CRT.md's guard premise: mod-K coloring needs (K-1)-bit guards; vacuous on #4174's production plans (no new κ) #4329 and PR Prove the uniform CRT evaluator bound with constant 1. #4377 are not rerun.
• Package: research/compact-switch-sharp-density/.

Nearest work, and why this is different:

• Audit the compact-switch coefficient and guard contract. #4321 proves the live-digit lemma and leaves the density recomputation open.
• 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 proves δ_r = 1 − [(1−1/B)^2 (1 − 8B/2^K)]^r for the published margin. It does not substitute the sharp window.
• Decide #4100 CRT.md's guard premise: mod-K coloring needs (K-1)-bit guards; vacuous on #4174's production plans (no new κ) #4329 decides the mod-K guard length. It does not recompute 8B/2^K.
• Prove the uniform CRT evaluator bound with constant 1. #4377 bounds the uniform evaluator. The room line here is CRT.md's displayed comparison, not that evaluator constant.

Validation

Python 3.12, standard library only:

python3.12 -B scripts/research_package.py --package research/compact-switch-sharp-density

Observed PASS_COMPACT_SWITCH_SHARP_DENSITY counts=7 rooms=7 rows=15 positive_r=6 gaps=4 controls=2 sample_ell=1 sample_p=1048576 sample_G=1408 sample_r=7, then PASS_REGISTERED_RESEARCH_PACKAGE research/compact-switch-sharp-density outcome=completed.

The proof is research/compact-switch-sharp-density/PROOF.md. The executable re-derives the bad-window count by running the isolated step, checks the integer ratio, the room identity, and the C = 1/6400 bound. Two controls reject a forged 4B count and a room line that drops E.

Not run: Lean, Docker, the selected cold replay, simultaneous two-target macros, and the tape repair.

Credit

Cinna Davis, PR #4100, for CRT.md and check_batched_crt.py. PR #4321 for the live-digit lemma and the sharp window. PR #4328 for the published density identity and the L ≤ p absorption hypothesis.

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

Open next questions

1. Does one two-target macro with disjoint guards restore both controls on a window that is sharp-good but outside the published 2B margin? First step: run the live macro at W = 0 and at W = 2^m − B + 1 with K-separated positions. Blocker: the twelve paid H-bit exchanges are not replayed here.
2. Is L ≤ p true for a real Audit CRT reservoir placement and split-axis coordinate descriptors #4174 placement, or only a sufficient hypothesis inherited from 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? First step: read one production plan's address length against its p. Blocker: reservoir placement is not an input here.