slug: meddaugh-stephens-2025-neighborhood-two-shadowing
bibkey: meddaugh2025neighborhoodshadowing
doi: null
url: https://arxiv.org/abs/2509.04312v1
triage: theorem
motivation_gids:
  - D5/S1/Dynamics/Shadowing/PowersTwoReturnDistances.powers_two_return_distance_example

Neighborhood two-shadowing without quasi-finite type

Problem

Jonathan Meddaugh and Elyssa Stephens, Shadowing with small diameter sets of bounded cardinality, arXiv:2509.04312v1, Question 5.3:

Are there subshifts with neighborhood 2-shadowing which are not a subshift of quasi-finite type?

Use neighborhood shadowing from Definition 2.1 and exact-length quasi-finite type from Definition 3.5, with their full quantifiers as retained in the source note.

Motivation

The explicit example separates the finite tracing-set property with an initial diameter bound from the exact-length language gluing property. The general theorem unbounded_return_neighborhood_two supplies a reusable positive result for every return-distance set containing one and arbitrarily large positive distances.

Route

Let $X_D\subseteq \{0,1{\}}^{\mathbb{Z}}$ consist of configurations whose consecutive ones have distances in $D=\{2^j:j\ge 0\}$, and put $\sigma (x)(k)=x(k+1)$. The zero-run lengths are $2^j-1$, rather than $2^j$. The space includes finite supports, singleton supports and the configuration with no ones. Its defining local exclusion is closed and invariant under integer shifts, so it is a nonempty compact subshift.

For each $\epsilon >0$, choose a signed window radius $r$ controlling the compatible product metric, an allowed distance $h>2r$, and $\delta >0$ forcing pseudo-orbit overlap on radius $2h$. The signed diagonal extends the negative-time part using the exact orbit of the initial point. Retain its ones from $-r$ onward. Short consecutive gaps are legal; a forbidden gap $(p,q)$ longer than $2h$ is repaired by inserting ones from $p+h$ to $q-h$. New consecutive distances are $h$ or $1$, both allowed. This gives one tracer; truncating the original support to $[-r,r]$ gives the other. They agree initially on this window. At every natural time, a window containing an original one agrees with the repaired tracer, while a window with no original one agrees with the truncated tracer. Their fixed set $A$ is nonempty, has at most two points and initial diameter less than $\epsilon$, and satisfies $\mathrm{infDist}(x_i,{\sigma }^i(A))<\epsilon$ for every $i$.

For a proposed QFT threshold $M$, take $n=\max (M,1)$ and $H=2^{n+2}$. The nonempty words $u=10^H$, $w=0^n$ and $v=1$ belong to the language, as do $uw$ and $wv$. A replacement $z$ of length exactly $n$ with $uzv$ in the language would force the first return after the leading one to lie in $(H,H+n+1]$. This interval lies strictly between the consecutive powers $H$ and $2H$, so no such return is allowed.

Falsifier

The proposed source-level answer would fail if some $\epsilon >0$ allowed, for every $\delta >0$, a natural-time $\delta$-pseudo-orbit in $X_D$ with no fixed nonempty finite $A\subseteq X_D$ satisfying $|A|\le 2$, $\mathrm{diam}(A)<\epsilon$ and $\mathrm{dist}(x_i,{\sigma }^i(A))<\epsilon$ for every $i\ge 0$. The displayed QFT obstruction would fail if, for some proposed threshold $M$, its words admitted a language word $z$ of length exactly $n=\max (M,1)$ with $uzv$ in the language. A mismatch with the original Definitions 2.1 or 3.5 would defeat the claimed source correspondence. These are failure criteria for the inference, not claims that they occur.

Evidence

The endpoint is D5/S1/Dynamics/Shadowing/PowersTwoReturnDistances.powers_two_return_distance_example. It states nonemptiness and compactness of the actual binary two-sided subshift, continuity of its shift, original neighborhood two-shadowing, and negation of the original exact-length QFT predicate. Its three supporting modules are ReturnDistanceSupport, ReturnDistanceSubshift and ReturnDistanceShadowing in the same D5 directory. The corresponding Blueprint document gives the statement and consumer exposition.

The encoded metric is Mathlib’s genuine compatible countable-product metric, not the degenerate printed infimum over disagreeing windows. The proof uses the product uniformity directly. It preserves natural pseudo-orbit time, integer configuration coordinates, a fixed nonempty finite tracing set and exact bridge lengths.

The four-slot information-escape audit is unfinished under issue #22664. The 23 typed positive-law checks are not registrations: source reconstruction, whole-family variation and sensitivity, observational dependence, and enrolled readout binding remain missing. They are not declared_validated evidence.

Gap

The encoded example answers the stated existence question affirmatively. The primary version still lists the question; bounded independent search found no exact prior answer within its searched scope. This is not an exhaustive novelty or global-priority claim. The source note records the inspected Definition 1.5, Theorem 3 and Lemma 16 in the author-linked 17-page manuscript of Carrasco-Olivera et al. (2022). Those passages provide no direct answer to Question 5.3; the printed supremum set distance, two-sided time and absence of a separate initial diameter bound prevent identifying their literal definition with Definition 2.1 here. The journal version and Dastjerdi–Hosseini (2010) remain unread, and Dastjerdi–Jangjoo arXiv:1108.3242 is withdrawn. Ordinary $N$-shadowing, density shadowing and entropy QFT do not replace the two definitions used here.

Triage

The mathematical endpoint concerns Question 5.3 only. It does not settle the full Question 5.2 hierarchy or assert nonsoficity. Kernel verification certifies the encoded statements; correspondence to the scholarly question remains a source-reading judgement. No official acceptance is claimed.

ASSUMED-UNVERIFIED

Source correspondence remains a source-reading judgement using the genuine compatible product metric, not equality with the degenerate printed infimum. Literature exclusions are bounded: the inspected passages of the author-linked Carrasco-Olivera et al. manuscript do not supply a direct answer to Question 5.3, and their literal definition is not identified with Definition 2.1. Its journal version and the full text of Dastjerdi–Hosseini (2010) remain unread; no exhaustive full-text exclusion is claimed. The audit under issue #22664 remains unfinished; the 23 positive-law checks are not declared_validated. No exhaustive novelty, global priority or official acceptance is claimed.