HodgePhaseNet: Discrete Hodge Decomposition of Semantic Flows in Transformer Attention

Kiyotaka Kasubuchi; Kazuo Fukiya

cs.LGOct 10, 2026

HodgePhaseNet: Discrete Hodge Decomposition of Semantic Flows in Transformer Attention

Authors: Kiyotaka Kasubuchi, Kazuo Fukiya

Abstract

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 edge orientation, and preserves a decoder's causal information boundary by constructing its complex from the visible prefix. Gauge invariance of the attention output is conditional on using invariant or appropriately compensated features. A simple difference of vertex embeddings has no coexact or harmonic component; a genuine relation encoder is therefore indispensable. Local circulation is not automatically contradiction, and harmonic flow is not automatically missing knowledge. These interpretations and any reduction in factual error require controlled empirical evaluation. This paper presents a mathematical and architectural proposal, not external natural-language performance results.

Figures & tables

Figure 1: Complementary roles of the earlier PhaseNet methods. Their features feed a separately learned real edge field; the Hodge layer separates that field into gradient, local coexact, and global harmonic components before returning structured features to attention.

Figure 2: One-head computation path. All decoder features and the complex are constructed from the visible prefix. Edge orientation is converted for each ordered query–key pair; the compensated gradient $D$ is used only when its gauge assumptions hold. The learned edge bias precedes the causal mask and softmax.

Figure 3: Five-vertex example of Section 7.1. The blue triangle $(1,2,3)$ is a filled face and carries the coexact field $C$ ; the orange triangle $(1,4,5)$ has no face and supports a harmonic cycle $H$ . Arrows follow the edge orientations in ( 24 ); the annotated values are component fields, not the full $\Omega$ .

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ModularPhaseNet: Finite-Cyclic Phase Geometry for Computable Semantic Hierarchy, Direction, and Context Consistency in Standard Transformers

Kiyotaka Kasubuchi, Kazuo Fukiya

Transformer Transformer Attention Hierarchical Representation Learning Equivariant Representation Learning

Open paper AskPDF arXiv

We propose ModularPhaseNet, a classical and integer-computable discretization of the continuous complex phase geometry introduced in QuantumPhaseNet. The real-valued hidden states of a standard Transformer are retained, while only an auxiliary phase channel is quantized into a cyclic subgroup G = <g> of order q | (p-1) in the multiplicative group of F_p. A continuous phase e^{i phi} is represented by z = g^a mod p; phase composition becomes group multiplication, relative phase becomes group division, conceptual hierarchy is induced by a filtration of cyclic quotients, semantic direction is represented by oriented relative group elements, and contextual consistency is measured by gauge-invariant cycle holonomy. The method introduces three components into an otherwise standard Transformer: a finite-phase encoder, a quotient-filtration hierarchy module, and a group-valued connection module. Their outputs enter self-attention as real-valued bias terms. Training uses distributions in the real group algebra or straight-through Gumbel-Softmax, whereas inference uses exact modular exponentiation and precomputed tables. No quantum hardware, complex-valued matrix multiplication, or discrete-logarithm computation is required. We prove quantization-distortion bounds, nesting of quotient-induced partitions, gauge invariance, a discrete integrability result for flat connections, and boundedness of the resulting attention output. The central empirical hypothesis is that these exact discrete invariants improve hierarchy recovery, discourse alignment, contradiction detection, and calibrated hallucination-risk prediction under a controlled compute budget. This paper reports the theory together with a pre-registered evaluation plan; the experiments described in Section 14 have not yet been carried out, and no empirical result is claimed here.

Abstract

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