Tilted 3-adic Syracuse laws: absolute continuity, dimension and the limits of L²

Let g1, g2, … be independent with P(g = k) = θ(1 − θ)k−1 for k ≥ 1, and let ρ = Σj≥1 3j−12−(g1+⋯+gj) in the 3-adic integers. The law μθ of ρ is a 3-adic self-similar measure for the infinitely many maps x ↦ 2−g(1 + 3x), each contracting by exactly 1/3; at θ = 1/2 it is Tao's Syracuse random variable. We prove that μθ has a density with finite relative entropy with respect to Haar measure at θ = 1/4 and for every θ ∈ [1/2, 0.550024]. At θ = 1/4 the information gained by refining the r-th ternary digit is O(r−27/25), and the density lies in L41/20, with L² norm below 15/2; this last statement rests on a reviewed written proof with an exact finite computation and is not formalized. At θ = 1/2 the information is O(r−A) for every A. The mechanism is a carry-aware chi-square inequality for ternary digits together with a logarithmic averaging argument, and it applies to every law of independent positive gaps that satisfies an explicit scalar criterion. We also prove that the entropy dimension of μθ equals min(1, H/log 3) for every θ, where H is the entropy of one gap, so that μθ is singular for θ > 0.6090898; that μθ has no square-integrable density for any θ ≥ 1/2, and that laws with a fixed total are not L²-flat for θ > 1/2; and that Tao's polylogarithmic Fourier decay, which holds for every θ < log 2/log 3, fails at resonant frequencies for fixed totals of mean gap in (1, 1.0642). As an application, for every c < 7.503325 every large octave [X, 2X) contains at least Xe integers n whose Collatz orbit reaches 1 after at least c log n steps of the shortcut map, with no numerical census. Several scalar inequalities are verified by exact rational arithmetic, and one finite table by an exact integer computation; the programs and outputs are in the supplement (tilted-supplement.zip). Companion papers: Extremal stopping-time coefficients for the Collatz map; The stochastic 3x+1 model is a theorem: Brownian structure of Collatz orbits.