Weak mixing and neighborhood shadowing without small periodic sets

Jonathan Meddaugh and Elyssa Stephens, Shadowing with small diameter sets of bounded cardinality, arXiv:2509.04312v1, Question 5.6:

Is there a weakly mixing system which has the neighborhood N-shadowing property (N>1) that does not have dense small periodic sets?

Use the intended nonempty meaning of a small periodic set: a nonempty compact set B with f^p(B)=B for a positive natural p. The printed definition omits nonemptiness. The phase space is compact metric and the map continuous; the source does not require a homeomorphism.

Ordinary shadowing and weak mixing yield dense small periodic sets in the source's cited setting. The question asks whether the initial small diameter of a finite tracing set forces the same conclusion. The example separates that finite tracing property from every nonempty periodic-set neighborhood.

The full intended mathematical existence conclusion has a native Lean proof. No homeomorphism version, coherent single tracer or all-branches tracing statement is asserted. The optional period-collapse and superlinear-length generalization in the ordinary source are outside this endpoint.

The full-scope four-slot information-escape audit is unfinished under issue #26047. The Reg mirrors preserve full telescope positive-law attempts; they are not Contract.Registration values or declared_validated evidence. Enrolled source reconstruction, realization bridges, whole-family variation and sensitivity, actual observational dependence and lawful readout binding remain missing.

Let X consist of binary sequences on the natural numbers such that for every s>=0 and p>=1 some j<=p^2+2 satisfies x(s+jp)=0. Equip X with the genuine first-disagreement metric d(x,y)=2^(-firstDiff(x,y)), and shift left once. Each fixed progression condition is clopen. Their intersection is closed in the compact binary product. The zero point lies in X, and prepending zero gives a legal preimage of every point, so the shift is onto.

Allowed prefixes of lengths a,b can be joined by zeros at every position n>=a+max(a,b). A forbidden progression that crosses the gap must have step larger than either prefix and thus at most two supported terms. This is impossible for a forbidden length at least three. This proves mixing and, by simultaneous hitting of two open rectangles, weak mixing on X times X.

For an accuracy epsilon, first fix R>=1 with 2^(-(R+1))<epsilon. Put M=32(R+1), alpha=sqrt(2)/M, P=256M(M+3), D=max(R,P^3+2P) and delta=2^(-D). For an arbitrary infinite delta-pseudo-orbit z, its diagonal c(n)=z(n)(0) agrees with z(i) through D+1 after shifting by i. Hence all progression constraints with p<=P already hold in c.

The lower bound ||k alpha||>=1/[M(M+3)k], combined with a short Dirichlet return q<=16 and a signed mesh with explicit integer lifts, makes every quarter-circle interval hit within Pp steps, uniformly in the starting phase and positive p. Delete c on phases [1/8,3/8] to form a, and on [5/8,7/8] to form b. Both branches preserve the already present small-step zeros. For p>P, each forbidden progression meets each deletion mask by index Pp<p^2+3, so both whole branches lie in X.

The initial R-window avoids both masks, while every later R-window avoids at least one: R alpha<1/8 and the two quarter masks are separated across integer boundaries. Thus the same nonempty set A={a,b}, of cardinality at most two, has initial diameter less than epsilon and satisfies infDist(z(i),sigma^i(A))<epsilon for every natural i. The choice of branch may change with i; no all-time image-diameter bound is imposed.

Finally U=[1] contains 10-infinity and is clopen. If a nonempty B subset U satisfied sigma^p(B) subset B for p>0, any b in B would have b(jp)=1 for every natural j, contradicting its p-progression constraint. This excludes nonempty compact periodic sets in a nonempty open set.

The existence conclusion would fail if the specified carrier were not a nonempty compact metric system, product transitivity failed, or the same fixed N=2 did not satisfy the original infinite pseudo-orbit quantifiers. In particular, conditional mask existence, a finite time horizon, a time-dependent tracing set or omission of its initial diameter would not answer the question. The periodic-set obstruction must use a nonempty open U and explicitly nonempty B; the empty set supplies no obstruction.

The full endpoint is D5/S1/Dynamics/Shadowing/ProgressionExample.quadratic_progression_example. Its supporting modules are ProgressionSubshift, QuadraticRotationHitting and ProgressionTwoShadowing in the same directory, with corresponding Blueprint Scribe sources. Native canonical scoped elaboration has succeeded for the full endpoint. Formal independent scientific review, freezing and required CI are separate acceptance obligations.

The construction and explicit constants are attributed to the supplied completed Pro proposal and its isolated ordinary mathematical review. Native Lean implementation does not inherit a scientific or novelty verdict from that ordinary review.

The construction proves the intended Question 5.6 existence conclusion with the single fixed N=2. It also proves the stronger obstruction to every nonempty forward power-invariant subset of [1], without compactness or image equality. This is one original question; the arithmetic and topology lemmas do not count as additional solved originals. The hierarchy Questions 5.1 and 5.2 and language-characterization Questions 5.4 and 5.5 remain outside this result.

Source correspondence uses the explicitly nonempty interpretation and a genuine compatible symbolic metric, rather than the degenerate printed window-infimum expression. The current arXiv metadata lookup returns v1 and the primary v1 text still asks Question 5.6. Bounded repository, Mathlib, GitHub code and arXiv searches do not establish exhaustive exclusion of a prior answer, novelty, worldwide priority or unique status. No official acceptance, merged publication or complete escape audit is asserted here.