Quantitative Biology

The Lagrangian Architecture of Probability: From Constraint Functionals to Semantic Manifolds

Authors: Stephen P. Smith
Comments: 12 Pages.

This essay develops a unified framework connecting classical Lagrangian mechanics, probability theory, and information geometry through the lens of semantic manifolds and variational principles. By treating probability distributions as dynamic semantic fields and constraints as variational operators, we demonstrate how Shannon entropy, Fisher information, and nested Markov blankets interact to structure inference across multiple scales. The resulting architecture formalizes semantic duality, linking local sensitivity to global resonance, and provides a rigorous model for recursive constraint propagation in complex systems. This framework suggests a metaphysical interpretation of inference, where meaning, structure, and dynamics emerge from the interplay of constraints, resonance, and symmetry.
Category: Quantitative Biology