Structural Compatibility of Exponential Stability: Spectral Gaps, Distinguishability Production, and Marginal Renormalization Flow

Not every dynamical law supports persistent physical structure. Parametrically localized, exponentially stable configurations require suppression of surrounding fluctuations together with controlled separation in configuration space. This interplay between contraction and distinguishability is structural and largely independent of microscopic realization.

We derive a geometric compatibility condition governing this balance. Consider a dynamical system admitting (i) a positive-definite metric of distinguishability and (ii) a contractive sector characterized by a spectral scale lambda. We show that, in scaling regimes controlled by a single dominant contraction scale, finite persistent structure requires quadratic compatibility between contraction and distinguishability production.

This condition is encoded in the invariant ratio
R = sigma / lambda^2,
where sigma denotes the invariant rate of distinguishability production. Finite persistent structure is possible only if R remains finite and nonzero throughout the stability regime.

As a concrete realization, we analyze renormalization-group flow in local quantum field theory. For marginal asymptotically free flows, the invariant ratio develops a finite interior balance scale associated with exponential infrared hierarchy. Within perturbatively renormalizable local relativistic non-Abelian gauge theories, marginality of the gauge coupling occurs uniquely in four spacetime dimensions.

Four-dimensional marginal gauge dynamics therefore provide a realized perturbative instance of a general structural principle: dynamically generated persistent structure requires quadratic compatibility between contraction and distinguishability production.