math.NT — Number Theory
TRANSCENDENCE OF THE 3x+ 1 CONJUGACY MAP ON STURMIAN WORDS
Let T be the 3x + 1 map on the 2-adic integers and let Φ : ℤ₂ → ℤ₂ be the Bernstein–Lagarias conjugacy map, which sends a parity vector to the unique 2-adic integer realising it. We prove that Φ(cγ), Φ(1cγ), Φ(0cγ) and Φ of every shift of these words are transcendental for every irrational slope γ < γ* = (3 + √5)/(4 log₂ 3) = 0.8258977…, with no hypothesis whatever on the partial quotients of γ; in particular at the resonance slope γ = log₃ 2. The mechanism is a bridge: by the Bernstein–Lagarias isometry, the 2-adic approximation depth of a periodic shadow equals a longest common prefix, so the Diophantine behaviour of Φ is governed by the repetition structure of the parity word, at a height cost A(γ) = max(1, γ log₂ 3) that we control at every level. Quantitatively ω₁⁽²⁾(Φ(cγ)) ≥ ice(cγ)/A(γ) − 1, where ice is the initial critical exponent of Berthé–Holton–Zamboni; their floor ice(cγ) ≥ 1 + φ then beats the Ridout threshold ω₁⁽²⁾ ≤ 1 by a margin bounded away from zero. The result is not a corollary of the digit-based transcendence criteria: the 2-adic Hensel digits of Φ(c_{log₃ 2}) are measured to have full complexity 2ⁿ, against the parity word’s n + 1, and the shadow denominators 2^ℓ − 3^k are not S-units for any fixed finite S. We also record the Liouville alternative for unbounded partial quotients and a finite effective irrationality measure for bounded ones at slopes γ ≤ log₃ 2.