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The regular Toeplitz and strictly ergodic branch is refuted by one base

Abstract

The regular Toeplitz and strictly ergodic branch is refuted by one base.

Definition 1.1 (Literal regular and strictly ergodic hypothesis).

Lean statement: D5/S1/Dynamics/Symbolic/PermutativeLifts/BalancedRegularRefutation.claim

Formalization. D5/S1/Dynamics/Symbolic/PermutativeLifts/BalancedRegularRefutation.claim (✓ std3).

Source. Repository-derived.

Acknowledgement. Z. Kang; T. Jäger (2026). On lifts of strictly ergodic subshifts by permutative sliding block codes. DOI: 10.48550/arXiv.2609.01140. URL: https://arxiv.org/abs/2609.01140v1.

Commentary.

The claim quantifies over all nonempty minimal binary subshifts having exactly one invariant probability measure and generated by an aperiodic Toeplitz point with a genuine essential period structure. The structure has positive nonempty essential periods, strict growth, divisibility and coverage of every signed coordinate. Its actual normalized periodic-skeleton densities converge to one. The conclusion asks for any allowed length and bipermutative local rule whose entire inverse image is minimal. This necessary condition also applies to a proposed strictly ergodic entire lift.

Theorem 1.2 (Literal negation using the same balanced base).

Lean statement: D5/S1/Dynamics/Symbolic/PermutativeLifts/BalancedRegularRefutation.result

Proof. Machine-checked in Lean as D5/S1/Dynamics/Symbolic/PermutativeLifts/BalancedRegularRefutation.result (✓ std3). ∎

Resolves. Problems/kang-jager-2026-universal-permutative-lift-refutation (refuted) by D5/S1/Dynamics/Symbolic/PermutativeLifts/BalancedRegularRefutation.result.

Source. Repository-derived.

Acknowledgement. Z. Kang; T. Jäger (2026). On lifts of strictly ergodic subshifts by permutative sliding block codes. DOI: 10.48550/arXiv.2609.01140. URL: https://arxiv.org/abs/2609.01140v1.

Commentary.

One fixed balanced binary subshift discharges every hypothesis: its integer orbits are dense, its invariant probability is unique, and one actual regular aperiodic Toeplitz point generates exactly this subshift. For every length k+2 and every bipermutative rule, the full inverse image splits over this same base with actual prefix fibre Bool^(k+1), so it is nonminimal. Thus no entire lift is strictly ergodic either. This strengthens the already delivered general minimal refutation of the same Question 1.10 and uses the same problem slug; it is not a second solved problem. Priority remains unestablished, and the coboundary mechanism has prior literature.

References