Kernel Dynamics under Path Entropy Maximization

We derive a closed-form geometric functional for kernel dynamics on finite graphs by applying the Maximum Caliber (MaxCal) variational principle to the spectral transfer function h(lambda) of the graph Laplacian eigenbasis. The main result is that the MaxCal stationarity condition decouples into N one-dimensional problems with explicit solution: h*(lambda_l) = h_0(lambda_l) exp(-1 - T_l[h*]), yielding self-consistent (fixed-point) kernels via exponential tilting (Corollary 1), log-linear Fisher-Rao geodesics (Corollary 2), a diagonal Hessian stability criterion (Corollary 3), and an l^2_+ isometry for the spectral kernel space (Proposition 3). The spectral entropy H[h_t] provides a computable O(N) early-warning signal for network-structural phase transitions (Remark 7). All claims are numerically verified on the path graph P_8 with a Gaussian mutual-information source, using the open-source kernelcal library. The framework is grounded in a structural analogy with Einstein's field equations, used as a guiding template rather than an established equivalence; explicit limits are stated in Section 6.

The Free Energy Principle (FEP) is a leading framework for mathematically modeling self-organization and learning, while Integrated Information Theory (IIT) is a computational ontology of consciousness oriented around irreducible cause and effect. While conceptual unifications have been proposed and appear to be supported by empirical findings, the absence of a rigorous mathematical mapping places upper bounds on their precision and testability. This work proposes that information can be defined as the deviation $ψ$ of realized dynamics from a constrained maximum-caliber (MaxCal) path ensemble over a finite time horizon. Under this definition, each of the cause/effect repertoires central to IIT 3.0 emerge directly from MaxCal variational principles, allowing IIT's phenomenological calculus to be re-derived from constrained entropy-maximization (CMEP). This framework supplies a theoretical bridge to active inference, which is mathematically dual to CMEP under Langevin dynamics, and offers a principled route for extending IIT to new dynamical regimes. When the approach is applied under the Central Limit Theorem (CLT) for Markov chains and via large deviations theory (LDT) to Ising models, information $ψ$ is shown to be equivalent to prediction error under accompanying predictive coding models. This may hold relevance to the ``hill-shaped trajectory'' of $Φ$ observed in neuronal cultures adapting to sensory inputs. Together, these results provide a physically and mathematically grounded rationale for studying the convergence of FEP, IIT, and thermodynamic frameworks of cognition such as recent work grounding consciousness in violations of the Fluctuation-Dissipation Theorem (FDT).

The initialization of deep neural networks determines whether information and gradients can propagate across depth, yet a unified theory connecting these properties to learning dynamics remains elusive. Combining mean-field theory and random matrix theory, we establish a direct link between correlation propagation and the Neural Tangent Kernel (NTK) that governs learning in the sequential limit of infinitely wide, infinitely deep networks. Correlation propagation to infinite depth is possible only at a single, critical point in the weight-bias variance plane. At this point, we leverage the algebraic decay of the end-to-end Jacobian with depth to prove that the NTK becomes exactly proportional to the output correlation at infinite depth, tying together information propagation and learning dynamics. We further show that orthogonal initialization suppresses the leading finite-size corrections present under Gaussian initialization, clarifying the respective roles of the two initialization ensembles in this limit. These theoretical predictions are validated quantitatively on finite-width, finite-depth networks. Together, these results demonstrate that orthogonal initialization and criticality are required to control the asymptotic dynamics of deep learning.