On the Spectral Conditions for the Existence of Matter

The microphysical problem in Unified Substrate Theory is whether a single substrate, tested by a fixed and falsifiable operational protocol, can force the existence of internally structured, gauge-coupled matter without discretionary assumptions. The present work resolves this problem at the operational level. The theory is defined by a closed triple consisting of an admissible substrate response operator, a complete multiscale spectral diagnostic, and declared null ensembles. A diagnostic completeness theorem establishes that exactly one logarithmic spectral generator exists within the admissible class. This excludes any spectral or geometric origin of multiplicity and forces all physical diversity into an internal sector. Diagnostic locality and stability then compel linear internal action, algebraic closure, compact gauge covariance, and anomaly consistency. The framework is falsifiable at the operator–diagnostic level; all microphysical structure follows conditionally and inevitably from its success