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Complex amplitudes represent semantic states, while a covariant phase rate yields a semantic wavelength as a proxy for conceptual scale and low-frequency graph modes define document-level discourse direction. The theory proves local gauge invariance, quantum-block unitarity, boundedness and conditional stability of WavePhase Attention, and a calibratable hallucination-risk formulation, validated offline in a synthetic Validation Studio.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/quantumphasenet-gauge-covariant-geometric-and-quantum-spectral-semantic-concept-hierarchies-abstract/273430/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/quantumphasenet-gauge-covariant-geometric-and-quantum-spectral-semantic-concept-hierarchies-abstract/273430.png","ImageObject",300,407,{"name":92,"@type":93},"Finn","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-29","2026-09-15",true,{"@type":102,"interactionType":103,"userInteractionCount":52},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What is QuantumPhaseNet designed to model?","Question",{"text":112,"@type":113},"QuantumPhaseNet models context-dependent semantic states and their hierarchical organization. It uses complex amplitudes, semantic wavelength, and low-frequency graph modes to capture conceptual scale and document-level discourse direction.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How does the paper extend Transformer representations?",{"text":117,"@type":113},"It replaces a frame-dependent phase with a gauge-covariant phase, introduces a gauge connection and curvature on a complex semantic bundle, and adds spectral selection plus a calibratable hallucination-risk formulation.",{"name":119,"@type":110,"acceptedAnswer":120},"Is the “quantum advantage” claimed as a general result?",{"text":121,"@type":113},"No. The results provide conditional and synthetic evidence for classical quantum-inspired components, and the paper does not report unconditional quantum speedup in the evaluated setting.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},273430,1789471394,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":52,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":129,"read_time":144,"preview_page_count":4},34359740700684,"https://ap-avatar.wpscdn.com/avatar/1f400023980c374ae676?_k=1777273430885731487","arXiv :2608 . 15820v1 [ cs .CL] 16 Aug 2026  \nQuantumPhaseNet: A Gauge-Covariant Geometric and Quantum-Spectral Theory of Semantic Concept Hierarchies with Prototype Validation of a Classical Quantum-Inspired Model  \nKiyotaka Kasubuchi Kazuo Fukiya  \nJuly 30, 2026  \nAbstract  \nWe present QuantumPhaseNet, a gauge-covariant geometric and quantum-spectral extension of Transformer representations. Context-dependent semantic states are modeled as complex amplitudes; a covariant phase rate induces a semantic wavelength used as a proxy for conceptual scale; and low-frequency graph modes define a document-level discourse direction. The theoretical part establishes local gauge invariance, unitarity of the quantum block, boundedness and conditional stability of WavePhase Attention, and a calibratable hallucination-risk formulation. We also implemented a fully offline Validation Studio for the classical quantum-inspired pipeline in Section 14.1 and evaluated the five research questions in Section 16.1 on its built-in synthetic setting (n=240, observation noise 0.22, circuit noise 0.08, five seeds) . RQ1 yielded a wavelength-hierarchy Spearman correlation of 0.852 versus 0.707 for the baseline, 87.3% direction accuracy, and AUC 0.953 . RQ2 achieved discourse alignment 0.933 versus 0.589 and 41.2 versus  \n16.2 paragraphs before drift. RQ3 achieved AUROC 0.881 versus cosine 0.765 and phase-shuffle 0.536 .  \nRQ4 achieved error-detection AUROC 0.854 versus entropy 0.634, with Brier 0.150 and ECE 0.098 . RQ5 did not show quantum advantage: target probability and end-to-end cost efficiency were 25.5% and 0. 107, compared with 70 .7% and 0 .707 for the Chebyshev classical approximation. These results provide initial synthetic evidence for the classical quantum-inspired components, but not external validity or unconditional quantum speedup.  \nKeywords: Transformer; semantic manifold; gauge connection; covariant phase; semantic wavelength; concept hierarchy; graph Fourier transform; classical quantum-inspired computation; WavePhase Attention; hallucination; calibration; prototype validation  \n1 . Introduction  \n1.1 Background  \nTransformers compute dependencies within a sequence in parallel through self-attention. For an input representation 􀔍 ∈ ℝ􀖉×􀕿 , standard self-attention is  \n􀔆 = 􀔍􀔌􀕲 , 􀔀 = 􀔍􀔌􀕬 , 􀔋 = 􀔍􀔌􀕷 ,  \nAtt(􀔆, 􀔀, 􀔋 ) = softmax (  + 􀔂causal) 􀔋  \nas introduced by Vaswani et al. (2017) . Although this architecture is highly expressive, inner-product similarity alone does not explicitly reveal concept inclusion, levels of abstraction, parallel transport between contexts, document-level discourse direction, connection to evidence, or the geometry of a generation trajectory.  \nOrder Embeddings learn semantic hierarchies as explicit coordinate-wise orders (Vendrov et al. 2016), while Poincaré Embeddings exploit the correspondence between tree-like volume growth and negative curvature (Nickel and Kiela 2017) . Geometric deep learning incorporates symmetry and geometry into learning systems (Bronstein et al. 2021); graph Fourier analysis provides spectral tools on irregular domains (Shuman, Ricaud, and Vandergheynst 2013); and vector diffusion based on the connection Laplacian compares vectors that live in different tangent spaces (Singer and Wu 2012) . Quantum natural language processing likewise provides precedents for representing semantic composition with Hilbert spaces, category theory, tensor products, complex states, and quantum operators (Coecke, Sadrzadeh, and Clark 2010; Coecke et al. 2020) .  \nRather than merely juxtaposing these ideas, this paper integrates gauge-covariant phases of semantic states, a wavelength-based hierarchical scale, a low-frequency discourse direction, quantumspectral selection, attention, and hallucination risk into a single operator framework.  \n1.2 Terminology and Scope of Claims  \nWe call the overall model QuantumPhaseNet and its core attention mechanism WavePhase Attention. The present work extends our earlier","cbCaibg9QHUZSPJN","https://ap.wps.com/l/cbCaibg9QHUZSPJN","pdf",326853,31,"English","# Abstract\n## Terminology and Scope of Claims\n## Main Contributions\n## Introduction\n### Background\n### Definitions of Realizations and Claims","[{\"question\":\"What is QuantumPhaseNet designed to model?\",\"answer\":\"QuantumPhaseNet models context-dependent semantic states and their hierarchical organization. It uses complex amplitudes, semantic wavelength, and low-frequency graph modes to capture conceptual scale and document-level discourse direction.\"},{\"question\":\"How does the paper extend Transformer representations?\",\"answer\":\"It replaces a frame-dependent phase with a gauge-covariant phase, introduces a gauge connection and curvature on a complex semantic bundle, and adds spectral selection plus a calibratable hallucination-risk formulation.\"},{\"question\":\"Is the “quantum advantage” claimed as a general result?\",\"answer\":\"No. The results provide conditional and synthetic evidence for classical quantum-inspired components, and the paper does not report unconditional quantum speedup in the evaluated setting.\"}]","QuantumPhaseNet - Gauge-Covariant Geometric and Quantum-Spectral Semantic Concept Hierarchies - Abstract | PDF",78]