Surface-Shell Geometric Arrest from Constrained Overlap Free Energies: A Conditional First-Principles Normal Form for Three-Dimensional Structural Glasses

This manuscript develops a conditional first-principles normal-form theory for geometric arrest in three-dimensional structural glasses. The theory begins from a microscopic Hamiltonian and a quenched reference configuration, lifts the description to an infinite-dimensional amorphous-state field, and then contracts this field through a reference-overlap projection to obtain an effective constrained-overlap free-energy functional. Under explicit quenched large-deviation, sector-stability, wall-law, shell-entropy, and no-cheap-channel hypotheses, the paper derives a surface-shell or space-time relay obstruction for high-to-low overlap relaxation. The main result is a geometric bottleneck theorem: in three-dimensional locally constrained overlap dynamics, a macroscopic decorrelation path must either expose a surface-order intermediate-overlap shell or pay an equivalent space-time relay action. Combined with reversible local dynamics and entropy-corrected conductance estimates, this obstruction yields spectral slowdown and an alpha-relaxation law expressed through independently measurable quantities such as configurational entropy, overlap-wall tension, mismatch cost, and cooperative length. The manuscript explicitly separates definitions, assumptions, conditional theorems, physical interpretations, and falsification criteria. It does not claim an unconditional solution of structural glass relaxation. Instead, it formulates a sharp normal-form mechanism and a reference-resolved route by which the mechanism can be tested, falsified, or realized in molecular, coarse-grained, kinetically constrained, or programmable synthetic glass systems.