Principia KAN: Decoupling Symbolic Routing from Coefficient Estimation for Stable Equation Recovery

Symbolic regression seeks both low prediction error and exact recovery of the generating structure. In differentiable recovery from a constrained dictionary, these objectives can diverge when candidate selection shares an amplitude parameter with coefficient allocation. The selector must then determine which basis is active while also distributing coefficient mass, a failure we formalize as the Shared-Amplitude-Driven Coupling of Symbolic Routing and Coefficient. We propose Principia KAN (PKAN), which retains the KAN principle of learnable univariate edge functions and constrains each edge in the recovery layer to a physics-informed dictionary. Its physical expectation layer uses a decoupled selector and gives every candidate a signed coefficient; their product is the effective contribution in physical units. The complete system combines temperature annealing in two stages and sparsity on effective contributions with constrained support projection and coefficient refitting. Across 100 tasks and six levels of observation noise, it reaches a 96.83% exact formula recovery rate (EFRR), defined here as exact recovery of the fixed dictionary term set. Under the same training and decoding protocol, strict direct readout before projection and refitting reaches 74.00% for PKAN and 59.17% for the shared amplitude control. The higher direct-readout recovery rate indicates that the decoupled parameterization makes the correct dictionary terms more identifiable inside the trained layer, a difference that the projection and refit decoder can conceal. We evaluate additive equations over three variables, using 36 unary bases per variable and benchmark tasks compatible with this candidate dictionary.