Kernel Dynamics under Path Entropy Maximization

TL;DR

This paper introduces a variational framework treating kernel functions as dynamical variables under path entropy maximization, linking kernel evolution to information geometry and potential empirical neural network dynamics.

Contribution

It formulates a novel MaxCal-based approach to kernel dynamics, including fixed-point conditions, RG flow, and neural tangent kernel evolution, with thermodynamic bounds and interpretative insights.

Findings

Proposes a variational MaxCal framework for kernel evolution.

Identifies fixed points as self-reinforcing structures.

Suggests neural tangent kernel evolution as an empirical case.

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.