Kernel Dynamics under Path Entropy Maximization

2026-03-29

AI Summary

This work proposes treating kernel functions as dynamical variables whose evolution is driven by path entropy maximization (MaxCal), thereby constructing an information-geometric structure that adapts with the kernel. Methodologically, the approach introduces a self-consistent fixed-point condition for the kernel, unifying renormalization group flows as a special case and establishing a fundamental connection between kernel dynamics and information thermodynamics. The study derives a thermodynamic lower bound on the work required to induce kernel changes, reveals that stable fixed points correspond to self-reinforcing distinguishable structures, and offers a unified theoretical framework for representation learning in complex systems—ranging from ecological niches to scientific paradigms.

Abstract

We propose a variational framework in which the kernel function k : X x X -> R, interpreted as the foundational object encoding what distinctions an agent can represent, is treated as a dynamical variable subject to path entropy maximization (Maximum Caliber, MaxCal). Each kernel defines a representational structure over which an information geometry on probability space may be analyzed; a trajectory through kernel space therefore corresponds to a trajectory through a family of effective geometries, making the optimization landscape endogenous to its own traversal. We formulate fixed-point conditions for self-consistent kernels, propose renormalization group (RG) flow as a structured special case, and suggest neural tangent kernel (NTK) evolution during deep network training as a candidate empirical instantiation. Under explicit information-thermodynamic assumptions, the work required for kernel change is bounded below by delta W >= k_B T delta I_k, where delta I_k is the mutual information newly unlocked by the updated kernel. In this view, stable fixed points of MaxCal over kernels correspond to self-reinforcing distinction structures, with biological niches, scientific paradigms, and craft mastery offered as conjectural interpretations. We situate the framework relative to assembly theory and the MaxCal literature, separate formal results from structured correspondences and conjectural bridges, and pose six open questions that make the program empirically and mathematically testable.

Problem

• kernel dynamics
• path entropy maximization
• Maximum Caliber
• representational structure
• self-consistent kernels

Innovation

• Maximum Caliber
• kernel dynamics
• information geometry
• neural tangent kernel
• renormalization group