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

Abstract

This paper develops a phenomenological effective 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 collective luminous-field closure, directional coupling time, a near-horizon Planck-cutoff branch, and a near-square Modular–Weierstrass mechanism. A matter–radiation conversion event is postulated to excite a short geometric expansion impulse of space, potentially approaching the Planck time; within the local proper-time support of the coarse-graining cell, the metric-induced rate of proper separation may exceed c without superluminal motion or signalling. A normalized response kernel maps the conversion current to a congruence expansion scalar. Under a one-mode, local-analytic, autonomous cubic truncation, the 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 modular scale coordinate is complex, with modulus κ. A 100-decimal-digit finite-sum benchmark gives κ ≈ 2.8121 × 10¹⁴, while an independent Eisenstein-series evaluation differs by about 0.073%. The framework is a falsifiable effective extension testable through regional observations and cross-scale consistency checks.