Derivation of the SFT Action and Field Equation - Working Note on the Minimal Geometric Action

This working note proposes a minimal geometric action for Specular Field Theory (SFT), a geometric framework in which time is treated as a physical field with its own geometry. Starting from the minimal geometric structure of the specular sheet, we formulate the action for a real scalar coherence field Phi_s on a temporal metric, the associated potential V(Phi_s) = lambdaPhi_s^4 + v^2Phi_s^2, and an interface coupling singular at the informational horizon (A_SFT = 1/2) with coupling scale g = Lambda_0. We show how the operational field equation can be motivated as a working ansatz in the weak-coupling regime. The note also outlines a thermodynamic framework for the specular sheet, investigating the projection of the entanglement contribution E_ent. It models the probability density |psi|^2 via a Gibbs-like distribution governed by an entropic temperature T_eff, phenomenologically linked to the curvature of the potential (T_eff proportional to 1/|V''(Phi_s)|). The divergence of this temperature at the bifurcation point predicts a maximum of the Shannon entropy and a consistent bulk-boundary transition. The transition parameters--the critical curvature A_c and width Delta--are calibrated from circuit data (A_c = 1.051, Delta = 0.034), showing strong order-of-magnitude consistency with the microscopic structure of the potential. This document is a working note, not a final peer-reviewed publication. It reports the current state of an independent research program, shared publicly to establish priority, invite feedback, and facilitate collaboration. The work is not affiliated with any institution, and the author welcomes critical comments from the scientific community.