Structural Speed Limit.

What, exactly, limits a signal speed when the available data describe only diffusion, relaxation, and the first correction beyond the diffusive approximation? This preprint turns that question into a precise inverse problem and determines both what low-energy transport–spectral data can reveal about a causal cone and what they fundamentally leave undetermined. For finite-dimensional, linear, constant-coefficient passive symmetric-hyperbolic completions with one conserved mode, the paper derives the sharp minimum characteristic speed compatible with a directional diffusion coefficient, an active relaxation edge, and the quartic coefficient of the slow Bloch branch. The minimum is attained by a rigid canonical three-component realization. At the same time, connected cyclic realizations with the same low-energy data realize every larger characteristic speed. The result therefore identifies the exact lower boundary of causal realizability while proving that the actual propagation ceiling belongs to the full principal symbol, not to diffusion or the spectral gap alone. This distinction produces a layered theory of causal reconstruction. The excess above the minimum separates into three nonnegative operator defects; metric locking is characterized by an explicit divisibility criterion; a global Clifford minimizer is constructed when that criterion is satisfied; and finite causal depth provides an independent microscopic upper bound. When the measured speed of light is introduced as an external anchor, the admissible structures form a dimensionless causal simplex with a branch duality, a self-dual point, and a rigid universal barrier. Electromagnetic readout can transfer the principal cone to the photon sector only under explicit completeness conditions, and even complete readout does not determine the photon’s quartic dispersion. The paper also develops a spectral theory of the arrow of time. The instantaneous decay rate of a quadratic defect is shown to be a decreasing spectral mean whose derivative is exactly controlled by spectral variance. This yields visible-measure tomography, a Lyapunov law for the time-dependent causal budget, Hankel certificates of visible relaxation rank, and a separate Krylov hierarchy for the source-visible norm of the principal symbol. These results clarify why relaxation, causal orientation, and modular time are compatible but mathematically distinct structures. The work provides a self-contained interface between transport geometry, spectral stability, hyperbolic causality, and physical readout. Its central outcome is not a dimensional estimate of the speed of light, but an exact classification of the information required to reconstruct, select, and experimentally test a causal cone.