HodgePhaseNet: Discrete Hodge Decomposition of Semantic Flows in Transformer Attention

Kiyotaka Kasubuchi, Kazuo Fukiya

WavePhaseNet introduced frequency-resolved semantic representations, QuantumPhaseNet introduced gauge-covariant phase connections, and ModularPhaseNet made a finite cyclic phase channel and its cycle holonomy computable in an ordinary Transformer. These constructions are complementary, yet they do not by themselves separate an observed collection of pairwise relations into a globally consistent direction, circulation around local compositions, and circulation that persists around a larger unfilled cycle. HodgePhaseNet addresses this separation by representing semantic transitions as an antisymmetric real-valued 1-cochain on a finite oriented simplicial complex. With vertex-edge incidence matrix $B_1$ and edge-face incidence matrix $B_2$, every such field has a unique orthogonal decomposition $Ω= B_1^TΦ+ B_2Ψ+ H$, where $B_1H=0$ and $B_2^TH=0$. The three component fields are unique even when their potentials are not. We give explicit projectors, a sparse implementation, integrability and gauge results, a verified numerical example, and a redesigned attention interface. The attention bias is defined on allowed ordered query-key pairs, remains consistent under arbitrary choices of…