A Spectral Mechanism for Hierarchical Mass Emergence in Time-Scalar Field Theory

We present a discrete spectral mechanism for generating exponentially wide mass hierarchies from scale-invariant structure in Time-Scalar Field Theory (TSFT). Discretizingthe time-scalar manifold on a geometric scale lattice τm = τ0ϕm, with ϕ the golden ratio, we derive an EL-consistent Jacobi operator from a quadratic action with scale-attenuated couplings cm ∝ ϕ−m. We prove rigorously that the smallest eigenvalue satisfies λmin ∼ ϕ−N up to polynomial factors, while λmax = O(1), yielding an exponentially wide spectral span. A discrete Liouville transform shows that the operator reduces asymptotically to a Dirichlet Laplacian on a stretched coordinate interval of length S ∼ ϕN/2, producing low-spectrum quantization λn ∼ n2ϕ−N. A Gershgorin analysis provides an explicit bound λmax ≤ 2(c1 + c2) for geometric attenuation. Under the stiffness mass map m ∝√λ, the hierarchy follows directly from symmetry rather than parameter tuning. This work establishes a mathematically controlled spectral origin for hierarchical structure within TSFT while clarifying its present limitations with respect to gauge dynamics and interaction physics.