A Phase Law for Mixed-Base Tetration: Logarithmic Tails, Cocycle Structure, and Singular Asymptotics

We study the mixed-base height map obtained by applying the base-b superlogarithm to the base-d tetration function. The tetration functions are regular Paulsen–Cowgill/Kneser branches, particularly for real bases above the classical threshold e^(1/e). Along integer heights, exact Abel peeling produces diagonal logarithmic constants and hence a constant asymptotic shift. For real heights, however, numerical phase profiles for bases 2, e, and 10 show that the difference between the mixed-base height and the original height does not approach a single constant; instead, it approaches a small periodic function of the fractional height. The main theorem proves a conditional phase law. Under a regular real-logarithmic channel and a uniform peeling-growth estimate, the iterated logarithmic tails converge uniformly on the phase circle. Consequently, the mixed-base height difference decomposes asymptotically into a mean shift, a centered 1-periodic phase function, and a vanishing error term. We also prove an exact groupoid identity for mixed-base transformations, derive the associated nonlinear phase cocycle, establish exact inverse mean antisymmetry under a smooth circle-diffeomorphism assumption, and record the logarithmic singular expansion of real tetration near height −2. The results distinguish exact algebraic identities, conditional analytic theorems for the regular branch, and numerical phenomena including Fourier dominance and an approximate rank-one inner-base law.