Entropy Under Coherence-Weighted Multiplicity: A Variational Extension of Boltzmann Statistics

We introduce a generalized formulation of statistical entropy based on coherence-weighted multiplicity, extending the classical Boltzmann counting framework by assigning state-dependent weights derived from an informational coherence field ΔC that modulates microstate accessibility. The resulting entropy functional reduces exactly to the classical Boltzmann form in the limit of uniform coherence and introduces a principled correction term that penalizes strong coherence gradients. By applying standard variational calculus under a normalization constraint, we derive a Helmholtz-type eigenmode equation as a stability condition for coherence-structured ensembles, establishing a conservative bridge between generalized entropy and spectral stability. We further define the Metamorphic Stability Index (MSI) as an operational diagnostic for the balance between structural organization and dynamic transition potential, and discuss testable implications for systems exhibiting hysteresis-like informational transport. The framework is fully compatible with equilibrium statistical mechanics, to which it reduces as a special case, and is grounded in the informational geometry of Viscous Time Theory (VTT).