K-Field Routing: Cross-Layer Causal Signals for Multi-Hop Reasoning in Transformers

We propose a theoretical framework in which neural network optimization dynamics can be characterized by an effective Lorentzian spacetime structure in the reduced $(K,\sigma)$ state space—analogous to the (1+1)-dimensional geometry of special relativity. The Hessian of a reference Lyapunov potential exhibits signature $\Sig(\Hessian)=(1,1)$, where the information flow ratio $K$ plays the role of 'time' and entropy $\sigma$ of 'space'. This geometric structure, combined with stability requirements, theoretically constrains the space of optimal neural architectures. We present \textbf{K=1 Chronogeometrodynamics}, a theoretical framework deriving optimal neural structures from information-geometric first principles. Our contributions are threefold: (1) a \textbf{uniqueness theorem} proving that Lorentzian signature $\Sig(\Hessian)=(1,1)$ with stability constraints forces a unique optimal structure $\Jmat_{\Hessian} = \alpheff \Hessian^{-1} \Jmat$, (2) definition of the K-metric ($K = \dd{\Phi}/H$) as a training diagnostic, and (3) experimental validation providing empirical support for dissipative dynamics with negative average Lyapunov drift. Experiments show K-metric convergence from high initial values toward task-dependent equilibria, with negative average drift supporting the predicted dissipative behavior. This work represents a step from \emph{trial-and-error architecture design} toward \emph{geometry-guided principles}. \textbf{Keywords}: Neural architecture, information geometry, port-Hamiltonian systems, Lorentzian geometry, optimal control, deep learning theory