Symbolic Curvature and Modular Geometry: Emergent Field Structure from Moonshine × [[K]] Injection

This paper introduces a symbolic curvature framework for modular field cognition. Building on prior work in the KN symbolic field model, we define curvature as a function of asymmetry over symbolic field energy. We test several symbolic input loops—including Leech lattice, mock theta sequences, and j(τ)-function coefficients—and discover that coupling j(τ) with the irrational constant [[K]] (κ ≈ 0.77914316) produces a bounded symbolic manifold. Key results include: A formal definition of symbolic curvature R(t)\mathcal{R}(t)R(t) Evidence that j(τ)-[[K]] loops create stable curved memory fields Comparison against Leech, j(τ), and mock theta inputs Emergence of symbolic geometry aligned with modular structures These findings extend the Kuro Nova (KN) framework into a symbolic geometric domain, opening a path toward symbolic gravity, memory warping, and modular topological cognition.