Spectral Kernel Dynamics via Maximum Caliber: Fixed Points, Geodesics, and Phase Transitions

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.

The spectral kernel field equation R[k] = T[k] lacks a conservation-law analog. We prove (i) the fixed-point flow is strictly volume-expanding (tr DF > 0), precluding automatic conservation, and (ii) the conservation deficit per mode equals the Hessian stability margin exactly: D_m = -Delta'. Closing the deficit requires a scene-side compensating contribution, which we formalise as the distinction dynamics equation dc/dt = G[c, h_t], with MaxCal-optimal realisation G_opt. On fixed-topology 3D surface graphs we derive a conditional topology-preserving compression theorem: retaining k >= beta_0 + beta_1 modes (under a spectral-ordering assumption) preserves all Betti-number charges; we include a worked short-cycle counterexample (figure-eight) calibrating when the assumption fails. A triple necessary spectral diagnostic -- Fiedler-mode concentration, elevated curl energy, anomalous beta_1 -- is derived for planetary drainage networks at O(N) cost. Two internal real-data sequences serve as preliminary consistency checks; full benchmarks and adaptive-topology extensions are deferred.

High-capacity associative memories based on Kernel Logistic Regression (KLR) exhibit exceptional storage capabilities and robustness. Previous empirical studies identified a hyperparameter regime, the "Ridge of Optimization," where attractor stability is maximized. However, the geometric nature of this regime and the optimization dynamics required to reach it have remained unclear. In this paper, we investigate the static geometry of the parameter space and the learning trajectory of Gradient Descent (GD) in KLR-trained Hopfield networks. Using the eigenvalue spectrum of the Hessian, we reveal that the Ridge corresponds to a phase boundary located adjacent to a rank-1 spectral collapse, acting as a geometric singularity where the principal curvature is massively amplified. Furthermore, we demonstrate that the learning dynamics exhibit a transient self-stabilizing behavior driven by the Edge of Stability (EoS) phenomenon. Rather than seeking flat regions, the network parameters are driven toward a state where the local curvature dynamically equilibrates near the stability limit dictated by the learning rate, allowing the optimization to survive the initial instability. We provide analytical derivations for both the rank-1 asymptotic collapse and the dynamic feedback loop governing this equilibration. These findings suggest that optimal, high-capacity memory representations are not formed in flat minima, but are dynamically sculpted at the highly curved boundaries of geometric singularities.