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Kernel Dynamics under Path Entropy Maximization

Jnaneshwar Das

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.

Kernel Dynamics under Path Entropy Maximization

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.

Paper Structure

This paper contains 25 sections, 1 theorem, 25 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Maximum Caliber
  4. RKHS Geometry and the Kernel Primitive
  5. Standing Assumptions
  6. The Space of Kernels
  7. The Endogenous Landscape
  8. MaxCal over Kernel Space
  9. A Minimal Two-Kernel Toy Model
  10. Constraints
  11. The MaxCal Kernel Distribution
  12. Thermodynamic Cost of Kernel Change
  13. Fixed Points and Their Stability
  14. Stability Criterion
  15. Special Cases
  16. ...and 10 more sections

Key Result

Proposition 1

RG flow can be represented as a special case of MaxCal over kernels in which: (i) the constraint is scale invariance of the partition function, and (ii) the reference measure $\mathcal{Q}$ is uniform over the renormalization group orbit.

Figures (2)

  • Figure 1: Illustration of a kernel trajectory $\gamma(t)$ in $\mathcal{K}$. Each kernel $k_t$ determines a representational substrate relative to which a metric $g_{k_t}$ on probability space $\mathcal{P}$ is analyzed, so moving through $\mathcal{K}$ simultaneously deforms the geometry on which inference proceeds.
  • Figure 2: Lake algal-bloom scenario for adaptive sample return. The bloom front advects from $b_t$ to $b_{t+\Delta t}$; adaptive-kernel planning shifts waypoints toward moving high-information boundaries while maintaining return-to-base feasibility. The same structure extends to an ASV--AUV team with coordinated dock/undock: surface waypoints place the stack for subsurface profiles, and rendezvous windows replace return-to-base alone as a binding feasibility constraint.

Theorems & Definitions (7)

  • Definition 1: Kernel space
  • Definition 2: Kernel path entropy
  • Definition 3: Self-consistent kernel
  • Conjecture 1
  • Proposition 1
  • Conjecture 2
  • Conjecture 3