On the Structural Necessity of Logarithmic Periodicity in Unified Substrate Theory

Unified Substrate Theory (UST) identifies a fixed set of logarithmic frequencies that recur across gravitational, cosmological, and complex physical systems, appearing in effective response kernels in a universal log-periodic form. These frequencies are empirically stable, parameter-free, and recur across independent domains. In the published UST framework, they are interpreted as resonant eigenmodes of a substrate operator, while their microscopic origin is explicitly stated as an open theoretical problem.

This work demonstrates that the observed structure is not optional. It is shown that locality, causality, and finite energy density exclude continuous scale invariance in any microscopic theory admitting a spectral representation. If scale structure survives at all, it survives uniquely in discrete form. Discrete scale invariance implies a logarithmic lattice in momentum space, from which log-periodic oscillations follow necessarily. The quantity β is thereby fixed as a spectral property of the substrate operator already present in UST. The gravitational potential is defined as the long-wavelength linear response of the substrate.

The analysis shows that the β-spectrum of Unified Substrate Theory is a structural consequence of a local microscopic substrate rather than a phenomenological assumption. The remaining theoretical problem is reduced to the dynamical selection of stable spectral modes, in exact agreement with the open problems previously identified in UST.