A Completed-Cosh Thawing-Scalar Diagnostic with a Candidate Two-Measure Parent Mechanism

Abstract

This exploratory theory-and-numerics manuscript studies a canonical thawing-scalar dark-energy diagnostic generated by a completed-cosh potential with a finite zero-energy endpoint at Phi = 2. Displaced initial conditions above the endpoint produce non-phantom thawing behavior. For the fiducial initial condition Phi_i = 4 at N = ln(a) = -8, with the matter normalization chosen so that Omega_Phi(0) = 0.685, the background solution gives Phi_0 approximately 3.734 and CPL summary values (w_0, w_a) approximately (-0.874, -0.214). This places the branch in the non-phantom thawing quadrant and makes it a reproducible target for dynamical-dark-energy comparison, although not a statistical likelihood fit. The manuscript also identifies a candidate two-measure parent construction in which the same completed potential arises from the effective form V_eff = (V_1 + M)^2/(4 V_2), rather than being inserted directly. In the minimal Palatini version, the Einstein-frame kinetic map preserves the Fisher-Rao kinetic coefficient, so the same canonical scalar background equations are recovered. The parent construction remains conditional. The decisive unresolved issues are safe matter coupling, fifth-force avoidance, endpoint continuation through the measure-degeneracy surface, radiative stability of the measure sector, displacement selection, Boltzmann-level observables, and direct likelihood comparison.

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