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

• URL: http://arxiv.org/abs/2609.06000v1
• Date: Sat, 05 Sep 2026 10:02:23 GMT
• Title: ModularPhaseNet: Finite-Cyclic Phase Geometry for Computable Semantic Hierarchy, Direction, and Context Consistency in Standard Transformers
• Abstract summary: We propose a discretization of the continuous complex phase geometry introduced in QuantumPhaseNet.<n>No quantum hardware, complex-valued matrix multiplication, or discrete-logarithm computation is required.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

Related papers

• Quotient Dynamics, Effective Curvature, and Implicit Bias in Positive Quadratic Networks [0.0]
  Positive quadratic networks admit the lowrank representation f_U(x)=xtop UUtop x, where UinmathbbRdtimes r is identifiable only up to right multiplication.<n>We study how this quotient structure governs training dynamics, curvature, recovery, and bias.
  arXiv  Detail & Related papers  (2026-07-28T12:04:46Z)
• WavePhaseNet: A DFT-Based Method for Constructing Semantic Conceptual Hierarchy Structures (SCHS) [0.0]
  This paper reformulates Transformer/Attention mechanisms in Large Language Models.<n> Dimensionality Reduction GPT-4's 24,576-dimensional embedding space exhibits a 1/f spectral structure based on language self-similarity and Zipf's law.<n>Cohomological Consistency Control The reduced embedding space, constructed via cohomological regularization over overlapping local windows, allows defining a graph structure and cochain complex.
  arXiv  Detail & Related papers  (2026-02-16T03:07:41Z)
• Transformers learn factored representations [62.86679034549244]
  Transformers pretrained via next token prediction learn to factor their world into parts.<n>We formalize two representational hypotheses: (1) a representation in the product space of all factors, whose dimension grows exponentially with the number of parts.<n>We derive precise predictions about the geometric structure of activations for each, including the number of subspaces, their dimensionality, and the arrangement of context embeddings within them.<n>This provides a principled explanation for why transformers decompose the world into parts, and suggests that interpretable low dimensional structure may persist even in models trained on complex data.
  arXiv  Detail & Related papers  (2026-02-02T17:49:06Z)
• Latent Object Permanence: Topological Phase Transitions, Free-Energy Principles, and Renormalization Group Flows in Deep Transformer Manifolds [0.5729426778193398]
  We study the emergence of multi-step reasoning in deep Transformer language models through a geometric and statistical-physics lens.<n>We formalize the forward pass as a discrete coarse-graining map and relate the appearance of stable "concept basins" to fixed points of this renormalization-like dynamics.<n>The resulting low-entropy regime is characterized by a spectral tail collapse and by the formation of transient, reusable object-like structures in representation space.
  arXiv  Detail & Related papers  (2026-01-16T23:11:02Z)
• Transmutation based Quantum Simulation for Non-unitary Dynamics [35.35971148847751]
  We present a quantum algorithm for simulating dissipative diffusion dynamics generated by positive semidefinite operators of the form $A=Ldagger L$.<n>Our main tool is the Kannai transform, which represents the diffusion semigroup $e-TA$ as a Gaussian-weighted superposition of unitary wave propagators.
  arXiv  Detail & Related papers  (2026-01-07T05:47:22Z)
• Transformers Are Universally Consistent [14.904264782690639]
  We show that Transformers equipped with softmax-based nonlinear attention are uniformly consistent when tasked with executing Least Squares regression.<n>We derive upper bounds on the empirical error which, in the regime, decay at a provable rate of $mathcalO(t-1/2d)$, where $t$ denotes the number of input tokens and $d$ the embedding dimensionality.
  arXiv  Detail & Related papers  (2025-05-30T12:39:26Z)
• Exact renormalization group flow for matrix product density operators [0.0]
  We study a real-space renormalization group transformation of MPDOs represented by a circuit of local quantum channels. Unlike matrix product states (MPS), which always have a well-defined isometric renormalization transformation, we show that general MPDOs do not necessarily admit a converging exact renormalization group flow.
  arXiv  Detail & Related papers  (2024-10-30T05:09:51Z)
• Quantum Random Walks and Quantum Oscillator in an Infinite-Dimensional Phase Space [45.9982965995401]
  We consider quantum random walks in an infinite-dimensional phase space constructed using Weyl representation of the coordinate and momentum operators. We find conditions for their strong continuity and establish properties of their generators.
  arXiv  Detail & Related papers  (2024-06-15T17:39:32Z)
• DIFFormer: Scalable (Graph) Transformers Induced by Energy Constrained Diffusion [66.21290235237808]
  We introduce an energy constrained diffusion model which encodes a batch of instances from a dataset into evolutionary states. We provide rigorous theory that implies closed-form optimal estimates for the pairwise diffusion strength among arbitrary instance pairs. Experiments highlight the wide applicability of our model as a general-purpose encoder backbone with superior performance in various tasks.
  arXiv  Detail & Related papers  (2023-01-23T15:18:54Z)
• Unified Fourier-based Kernel and Nonlinearity Design for Equivariant Networks on Homogeneous Spaces [52.424621227687894]
  We introduce a unified framework for group equivariant networks on homogeneous spaces. We take advantage of the sparsity of Fourier coefficients of the lifted feature fields. We show that other methods treating features as the Fourier coefficients in the stabilizer subgroup are special cases of our activation.
  arXiv  Detail & Related papers  (2022-06-16T17:59:01Z)
• GroupifyVAE: from Group-based Definition to VAE-based Unsupervised Representation Disentanglement [91.9003001845855]
  VAE-based unsupervised disentanglement can not be achieved without introducing other inductive bias. We address VAE-based unsupervised disentanglement by leveraging the constraints derived from the Group Theory based definition as the non-probabilistic inductive bias. We train 1800 models covering the most prominent VAE-based models on five datasets to verify the effectiveness of our method.
  arXiv  Detail & Related papers  (2021-02-20T09:49:51Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.