Fine Structure of Mixed-Base Tetration Phases: Certified Evaluation, a Universal Mode-Decay Law, and a Boundary Essential Singularity

For regular tetration branches with real bases above the classical threshold e^(1/e), the mixed-base phase is a small 1-periodic circle map whose existence, groupoid structure, and continuously differentiable diffeomorphism property were established in the companion papers. This paper investigates the fine structure of these phases through a combination of mathematical proof and a high-precision numerical measurement campaign, and develops them into a certified computational primitive for base changes in tetration. On the theoretical side, we prove an exact two-level entry formula and an a posteriori error certificate for the logarithmic peel ladder, so that every phase evaluation carries a proven error bound. We establish a regularity dictionary relating branch smoothness to Fourier-mode decay, together with an explicit seam-jump formula calibrated on continuously differentiable branches. We also develop a first-order perturbation theory for small base mismatch, in which the derivative of the phase at equal bases is determined by a response kernel satisfying a linear cohomological equation. This explains the observed amplitude law for the Fourier modes. An exact hub decomposition of the mean shifts is derived, with second- and third-order defect formulas numerically validated to approximately 10^−15. Finally, a parabolic passage-time argument yields an edge law near the classical base threshold, with the predicted coefficient agreeing with measurement to within 2.3 percent. On the empirical side, Fourier modes of the regular phases are measured up to mode 300 after identifying and quantifying a systematic limitation of the underlying Kneser computation engine. The effective decay profile is universal across fourteen measured base pairs to three or four decimal places. A direct investigation of the complexified phase identifies an isolated essential singularity per period on the boundary of the analytic strip, at a pair-universal height of approximately 0.055 and with pair-universal strength; only its horizontal position depends on the base pair. Steepest-descent asymptotics connect the singularity data to the observed four-parameter decay law and yield mode-count estimates for high-precision base-change tables. We conclude with a hybrid representation combining a mean-shift hub, a Fourier head with a universal decay curve, and a certified logarithmic peel ladder, in which cross-base identities serve as proof residuals rather than heuristic checks.