Large deviations and optimal surface spectra of film interaction energies

A sample-dependent constraint couples every Fourier mode of a disordered surface to the interaction between confining walls. We determine the complete strong-disorder large-deviation crossover of this interaction in a three-dimensional mean-spherical film. After the film limit at fixed field strength, the rescaled radial powers obey an entropy large-deviation principle; contraction through two self-consistent masses gives the exact energy rate. Every positive energy has a unique, smooth optimal radial spectrum. A canonical Hessian estimate proves global identification and strict convexity of the rate, including the depleted lower branch. The Gaussian center connects to a depleted endpoint with cost e^(−2/3) and an enhanced tail with cost e^(4/3), with explicit constants. There is no finite-energy mode condensation. Conditioning on an atypical energy selects the optimal radial spectrum exponentially, and a simple rescaling collapses every fixed aspect ratio onto one curve. Averaging the constraint preserves the central Gaussian curvature but changes the support and the far-tail exponent. Acknowledgments. We acknowledge Gewu Intelligence Lab for providing the collaborative research environment in which this project was developed. W.J. acknowledges support from the Strategic Priority Research Program of the Chinese Academy of Sciences under Grant No. XDB1680102. This preprint has not been externally peer reviewed. The deposit includes the main article, complete supplementary proofs, and a curated source and reproducibility archive. The strong-disorder results concern sequential limits: the film limit at fixed strength is taken first.