Directional Radiation–Matter Conversion and Modular–Weierstrass Closure: A Planck-Time Geometric-Impulse Model with a Near-Horizon Cutoff Branch

This paper develops a phenomenological model in which an expansion-like response arises from the coarse-grained radiation–matter conversion current of a local or regional luminous field rather than from an isolated photon. The framework combines a collective luminous-field closure, directional coupling time, a conditional near-horizon Planck-cutoff branch, and a near-square Modular–Weierstrass closure. A conversion event is postulated to excite a short geometric impulse of the effective spatial geometry, potentially approaching the Planck time. The resulting proper-separation rate is a metric response, not a particle or signal velocity. An effective conversion current is mapped through a normalized kernel to a congruence expansion scalar. Under an explicitly selected one-mode autonomous cubic closure, the normalized response admits a Weierstrass first integral. In a fixed normalized-curve gauge, the physical and modular invariant pairs coincide; more generally, isomorphic representatives share the same j-invariant. The branch-normalized modular coordinate is complex, with positive-real modulus κ that depends on branch and normalization. A 100-digit finite-sum benchmark gives κ ≈ 2.8121 × 10¹⁴; a converged Eisenstein-series evaluation differs by 0.073%. The framework is falsifiable through independently calibrated multi-region tests.