Quadratic Progression Two-Shadowing

Abstract

The same two genuine infinite configurations trace every natural-time pseudo-orbit, with small initial diameter.

Put ell(p)=p^2+3. The hereditary carrier forbids every all-one progression of that length and positive difference p. Given epsilon>0 choose R>=1 with 2^(-(R+1))<epsilon, then M=32(R+1), alpha=sqrt(2)/M, P=256 M(M+3), D=max(R,P^3+2P) and delta=2^(-D). These parameters are fixed before the arbitrary infinite natural-time pseudo-orbit z. The arithmetic-progression construction and explicit constants for Meddaugh–Stephens Question 5.6 are attributed to Pro.

Read c(n)=z(n)(0). Strict first-disagreement errors force edge equality through D. The existing MetricTracing.diagonal_splice_overlap gives c(i+j)=z(i)(j) for all i and j<=D+1. For 1<=p<=P, the span p(p^2+2)<=P^3+2P fits in that overlap, so c satisfies every such progression constraint at every starting coordinate. The proof does not assume c itself is in the carrier.

Theorem 1.1 (Quarter masks stay separated across wrap-around).

$$\begin{array}{r}\forall u,v\in \mathbb{R},\\ \mathrm{fract}(u)\in [\frac{1}{8},\frac{3}{8}]\Rightarrow \\ \mathrm{fract}(v)\in [\frac{5}{8},\frac{7}{8}]\Rightarrow \frac{1}{8}\le |v-u|\end{array}$$

Proof. Machine-checked in Lean as D5/S1/Dynamics/Shadowing/ProgressionTwoShadowing.quarter_masks_separated (✓ std3). ∎

Source. Repository-derived.

Commentary.

If the real phase difference were less than 1/8, the integer difference floor(v)-floor(u) would be strictly between -1 and 0. This contradiction handles every integer crossing. Since R alpha<1/8, no window [i,i+R] can meet both masks.

Let E0 be the natural coordinates whose alpha phases lie in [1/8,3/8], and E1 those in [5/8,7/8]. Define a by deleting c on E0 and b by deleting c on E1. For p<=P, an existing zero stays zero after deletion. For p>P, uniform quarter hitting puts a deletion in each forbidden support by index Pp<p^2+3. Thus both whole infinite branches satisfy every constraint. Their initial phases lie in [0,1/8), so both agree initially with c. Every later R-window misses at least one deletion color, so that branch agrees throughout with the corresponding pseudo-orbit window.

Theorem 1.2 (The original fixed two-point tracing contract).

$$\begin{array}{r}\mathrm{NeighborhoodTwoShadowing}(\mathrm{progressionShift}(\mathrm{quadraticProgressionLength}()))\end{array}$$

Proof. Machine-checked in Lean as D5/S1/Dynamics/Shadowing/ProgressionTwoShadowing.quadratic_progression_neighborhood_two_shadowing (✓ std3). ∎

Source. Repository-derived.

Commentary.

For every epsilon>0 choose delta>0 before the entire pseudo-orbit. The two constructed points are in X and have distance at most 2^(-(R+1))<epsilon. At every natural i, at least one shifted point is within the same strict epsilon of z(i). The existing neighborhood_two_of_pair packages the same nonempty finite set A={a,b}, card(A)<=2, initial diameter<epsilon and infDist(z(i),sigma^i(A))<epsilon. The minimizer may switch with i. No all-branches tracing, coherent single tracer or all-time diameter bound is claimed.

References

• Truth anchor: D5/S1/Dynamics/Shadowing/ProgressionTwoShadowing.quadratic_progression_neighborhood_two_shadowing
• Truth anchor: D5/S1/Dynamics/Shadowing/ProgressionTwoShadowing.quarter_masks_separated
• Dependency: D5/S1/Dynamics/Shadowing/ProgressionSubshift
• Dependency: D5/S1/Dynamics/Shadowing/QuadraticRotationHitting