Specular Field Theory: Geometric Mapping of Quantum Scrambling and Predictive Modeling at the Informational Horizon

This working note presents a quantum circuit test of Specular Field Theory (SFT), a geometric framework in which the scrambling rate of a quantum system is linked to the curvature of a temporal manifold. Using a 5-qubit mirror circuit (Loschmidt echo) with a temporal modulation applied to the single-qubit rotation angles,we establish a quantitative mapping between the circuit modulation parameter (omega) and the temporal curvature (A).We analyze the entanglement contribution (E_ent) as an operational interpolation between the fidelity of recovery (F_rec0) in the bulk and the Shannon entropy (S_Shannon) at the boundary, governed by a sigmoidal transition. The study evaluates the model across three main aspects: Circuit Calibration: The mapping A(omega) is calibrated using baseline circuit constants (A_0 and alpha),establishing a consistent linear response in the bulk. Bulk-Boundary Transition: The transition between bulk and boundary regimes is analyzed using a sigmoidal weighting function, yielding a strong coefficient of determination (R^2 = 0.9974). Blind Prediction & Horizon Regularization: Parameters are calibrated exclusively on the linear bulk (omega <= 0.05), and the regularized formula predicts the scrambling rate in the critical regime (omega >= 0.08) with an average error of 7.93%, utilizing a boundary regularization scale (epsilon) that accounts for temporal volume saturation near the informational horizon (A = 1/2). The note also discusses finite-size scaling (showing the bulk intercept beta vanishing in the thermodynamic limit),directional circuit responses under time reversal (comparing forward and rewind evolutions), and the distinction between exact statevector simulations and future empirical validation on quantum hardware.