Question 1.10: regular Toeplitz and strictly ergodic strengthening

The cited v1 asks whether a given minimal/strictly ergodic (Toeplitz) subshift admits a permutative sliding block code of length ℓ≥2 whose entire inverse image is minimal/strictly ergodic. This dossier includes the delivered general minimal refutation and its strengthening to a regular Toeplitz, strictly ergodic base. Both concern the same numbered external question.

The original UniversalRefutation.claim quantifies over all finite discrete alphabets A with at least two letters and all nonempty minimal two-sided subshifts. Its conclusion asks for some k≥0 and h:A^(k+2)→A, bijective in both endpoints with all remaining letters fixed, whose whole inverse image is minimal. UniversalRefutation.result : Not claim is the general-branch refutation retained as a preliminary result.

The strengthened BalancedRegularRefutation.claim already restricts the input to binary subshifts that are nonempty and minimal, have exactly one invariant probability measure, and are generated by one aperiodic Toeplitz point with a genuine essential period structure. The structure has positive nonempty essential periods, strict growth, divisibility into the next period, and coverage of every signed coordinate; its actual normalized periodic skeleton densities tend to one. The conclusion still allows every length k+2 and every bipermutative local rule, and asks for minimality of the whole inverse image. BalancedRegularRefutation.result : ¬ claim is its literal negation. Since strict ergodicity implies minimality, it also rules out a strictly ergodic whole lift. A binary counterexample suffices for the original finite-alphabet question.

The original finite extension has the actual fibre A^(ℓ−1). Group holonomy over a regular group action cannot silently replace it. The construction trivializes every finite letter-permutation cocycle on one fixed infinite minimal binary base. The balanced version of this construction supplies unique ergodicity and actual regular Toeplitzness on the same base.

Tier 1: an explicitly numbered external question, preregistered before the formal probe in issue 24726. The inspected public sources included the original v1, the author-hosted paper, author publication lists, arXiv version/author searches, and adjacent finite-extension and profinite/S-adic papers. No exact uniform construction was found in that scope. Citation coverage is incomplete and announced works in preparation could not be checked. Priority remains unestablished. The coboundary/invariant-subset mechanism itself has prior literature.

The original factorial-word base and UniversalRefutation.result remain valid. The strengthened counterexample uses the balanced-word subshift, constructed as the intersection of compact two-sided tiling subshifts. Recognizable marked boundaries give unique continuous tilings, uniform recurrence and minimality, and exclude every positive global period.

Sparse disagreements between the two balanced words give a vanishing error for every finite-window event. Existence of an invariant probability and uniqueness on measurable cylinders establish unique ergodicity on actual subshift points. An aligned compact-limit point has periodic coordinates, is aperiodic and generates exactly this same base. Its actual periodic skeleton densities tend to one; extracting least skeleton periods gives strictly increasing essential periods with divisibility and coverage of all signed coordinates. Density is preserved under this extraction.

For any finite Q and two permutations, a high word level has identity holonomy on both block types. Prefix holonomy gives a continuous transfer β(Sy)=π(y₀)β(y). Independently, BipermutativeLift reconstructs every two-sided preimage from its coded point and actual length-(k+1) prefix. Together these homeomorphisms split the whole lift as Y×Bool^(k+1), with shift acting on Y and fixing the transformed fibre label. The invariant label is β(y)⁻¹(sourcePrefix x), not the raw prefix. Distinct labels give nonempty clopen invariant minimal fibres, exactly 2^(k+1) distinct orbit closures.

The same regular Toeplitz, strictly ergodic binary base works for every k and every permitted code. No length is fixed in advance and no component is substituted for the entire lift.

The construction would fail if the fixed base were empty or nonminimal; if unique ergodicity concerned different points or a different base; if the period structure lacked actual essentiality, divisibility, strict growth, coverage or normalized density convergence; if the transfer identity failed for some finite permutation action; or if reconstruction missed a point of the whole inverse image. An exact prior resolution would defeat a novelty claim without changing the formal statement.

The delivered general-branch modules and their Reg mirror compiled on Lean 4.35.0 with only Classical.choice, Quot.sound and propext, and have canonical frozen state fragments and emitted Blueprint mirrors. The new regular/strict strengthening requires its current scoped kernel report, actual Scribe consumption and required checks; the original evidence does not certify the strengthened statement. Public source-bound escape audit registrations remain unfinished under issue 24726.

The general minimal branch is refuted by the delivered factorial base. The strengthened literal target uses one fixed regular Toeplitz, strictly ergodic balanced binary base and proves that every permitted full lift is nonminimal. This is a strengthening of the same problem, with one problem slug and no second problem count. Priority remains unestablished.

The Library note locates the prior coboundary mechanism. The formal argument reuses pinned Mathlib compactness, symbolic dynamics, finite groups, measure theory and minimal-action APIs. Literature coverage and priority are not established by compilation. Current kernel, Scribe, CI, freezing and delivery status must be read from their actual evidence and checks.