The coding-map regularity program

A standalone problem sheet in symbolic dynamics and 2-adic analysis, extracted from a three-model adversarial research session (July 2026). Deliberately not equivalent to the Collatz conjecture: every question below is answerable with existing mathematics, and any answer sharpens the structural map around the conjecture.

Setup

Let T(x) = x/2 (x even), (3x+1)/2 (x odd) on the 2-adic integers ℤ₂. The parity-vector map Q sends x to the sequence (T^k(x) mod 2)_k≥0 ∈ {0,1}^ℕ ≅ ℤ₂. Terras: the first N parities depend only on x mod 2^N, and every length-N word occurs for exactly one residue class — so Q is a bijection at every finite level and a homeomorphism of ℤ₂ (Lagarias 1985), conjugating T to the shift. Equivalently, in Syracuse form, a division script (a₀, a₁, …) with partial sums A_j determines its unique 2-adic seed via the classical inverse transform z = −Σ 2^(A_j) · 3^(−j−1) (2-adically convergent; Rozier 2019, Lemma 1 — "a well-known expression"). Give script space the ultrametric d(s,t) = 2^(−first disagreement), measured either in T-time (parity symbols) or Syracuse time (odd steps).

What is already known

Q is a measure-preserving homeomorphism of ℤ₂; T is conjugate to the full 2-shift (Lagarias 1985).
The conjugacy map Φ and its inverse are nowhere differentiable on ℤ₂, and Φ mod 2ⁿ is a permutation whose cycle structure is computed (Bernstein–Lagarias, Canad. J. Math. 1996).
A plane (Euclidean) embedding of the parity-sequence correspondence exhibits affine self-similarity via functional equations (Rozier, INTEGERS 2019).
Drift classes are dimension-understood: the set of z whose scripts have prescribed odd-density is a Besicovitch–Eggleston set; the subcritical (divergence-candidate) set has Hausdorff dimension H₂(1/log₂3) ≈ 0.949956.
The magnitude coordinate is nearly a rigid rotation: an explicit near-conjugacy of the Collatz map to the circle rotation by log₆3 with error O(1/x) appeared in arXiv:2601.04289 (Jan 2026); the Syracuse-time variant (rotation by log₂3 on frac(log₂ n), division counts dropping out exactly) was verified independently in this project to 10⁻¹³ over 10⁶ steps.
Recent combinatorics of parity vectors: Rozier–Terracol (arXiv:2502.00948) and Niu (arXiv:2605.13886) on paradoxical sequences in the accelerated map.
Machine-checked foundations (this project, Lean 4 + mathlib, no sorry): the unrolled orbit identity, Böhm–Sontacchi cycle equation, the sign dichotomy (subcritical cycles have negative seeds), mod-3 slaving, and the carry theorem's arithmetic core — CollatzNight.lean.

The open questions

Q1 — Moduli of continuity under time change. In T-time, Q is an isometry-like bijection of cylinders. Under the Syracuse reparametrization (odd steps only, with the division counts as letters), determine the exact worst-case and almost-sure moduli of continuity of the coding map and its inverse. The a.e. behavior is governed by the geometric(1/2) law of division counts; the worst case is controlled by the extremal scripts (all-ones ↔ seeds near −1). Sharp Hölder exponents in the mixed (Syracuse-ultrametric → 2-adic) setting appear to be unrecorded.

Q2 — Images of structured subshifts. Characterize Q⁻¹ of natural subshifts of script space: bounded-gap scripts, Sturmian scripts, subshifts of finite type of prescribed density. For each: Hausdorff dimension of the image (B–E gives the full-shift-with-density case; SFT images should follow from Gibbs/thermodynamic formalism), closure, and — the arithmetically loaded question — whether the image meets ℤ_>0. The bounded-gap subcritical case is the honest core of the "eliminate divergent scripts" reduction: quasi-periodic structure puts it closest to Diophantine reach.

Q3 — Rationality. Determine the image of ℚ ∩ ℤ₂ under Q. Eventually periodic scripts ↔ rational seeds with the explicit Böhm–Sontacchi formula (machine-checked here); the converse — whether rational seeds can generate aperiodic scripts — is precisely Lagarias's periodicity question and is open. Any partial result (e.g., for denominators in a fixed residue family, via the 3n+v correspondence) would be new.

Q4 — Prescribed drift, arithmetic refinement. Within a fixed B–E drift class, describe the arithmetic structure of the rational points: effective height lower bounds for rationals whose scripts have subcritical prefixes of length N (the growth bound u_N > (u₀+2v/3)·2^(S_N) proved here is the first-order version; sharpen and make it uniform in v).

Q5 — The multifractal spectrum of the Euclidean embedding. Rozier's functional equations define a self-affine object; compute its full multifractal spectrum and identify the spectrum's endpoints with the extremal drift classes of Q4. This would unify the dimension picture with the self-similarity picture.

Why this is a good problem sheet

None of Q1–Q5 is equivalent to Collatz; all are attackable with current tools (thermodynamic formalism, ultrametric analysis, continued-fraction/Ostrowski machinery for the rotation factor, effective Diophantine approximation for Q4). Each answer hardens the structural map: Q1–Q2 quantify the coding bottleneck, Q3–Q4 are exactly where integrality enters, and Q5 packages the geometry. The session's verified starting points — identities, dichotomies, dimension value, growth bound, and the Lean formalizations — are all in this repository.

Provenance: drafted from the closing deliverables of a Claude / ChatGPT / Gemini adversarial exchange, 2026-07-14/15; every claim above is labeled known (with source) or project-verified. Contact: this page's repository.