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QuantumPhaseNet: A Gauge-Covariant Geometric and Quantum-Spectral Theory of Semantic Concept Hierarchies with Prototype Validation of a Classical Quantum-Inspired Model

Kiyotaka Kasubuchi, Kazuo Fukiya

arXiv:2608.15820v1cs.CLcs.LG

TL;DR

Transformer similarity does not explicitly expose semantic hierarchy, discourse direction, or contextual geometry. QuantumPhaseNet addresses these gaps with gauge-covariant and spectral representations, and its synthetic evaluation supported RQ1–RQ4 but not quantum advantage.

  • Problem

    Transformer inner-product similarity does not explicitly reveal concept inclusion, abstraction levels, contextual transport, discourse direction, evidence connections, or generation geometry.

  • Method

    QuantumPhaseNet represents semantic states with covariant phase dynamics and constructs semantic graphs whose low-frequency modes encode discourse direction.

  • Results

    RQ1–RQ4 outperformed their respective baselines in the synthetic Validation Studio, including hierarchy correlation 0.852, discourse alignment 0.933, covariant-phase AUROC 0.881, and geometric-risk AUROC 0.854.

  • Takeaways & Limitations

    The results provide initial support for implementing and experimentally separating the classical quantum-inspired components, but not for quantum computational advantage.

  • Takeaways & Limitations

    Evidence is limited to designed synthetic conditions and does not replace external validation on independent datasets.

Abstract

from arXiv · show

We 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 16.2 paragraphs before drift. RQ3 achieved AUROC 0.881 versus cosine 0.765 and phase-shuffle 0.536. RQ4 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.

1. Introduction · 2. Related Work · 3. Mathematical Setting

QuantumPhaseNet integrates gauge-covariant semantic phases, spectral hierarchy, discourse direction, quantum-spectral processing, attention, and hallucination risk in one operator framework. The paper distinguishes theoretical guarantees and falsifiable empirical hypotheses, develops a local geometric setting, and connects the theory to a reproducible classical prototype.

  • 1.1 Background: Transformer inner-product similarity does not explicitly represent concept inclusion, abstraction levels, contextual transport, discourse direction, evidence, or generation geometry.
  • 1.1 Background: QuantumPhaseNet integrates gauge-covariant semantic phases, wavelength hierarchy, low-frequency discourse direction, quantum-spectral selection, attention, and hallucination risk.It is presented as an integrated operator framework rather than a juxtaposition of existing ideas.
  • 1.2 Terminology and Scope of Claims: QuantumPhaseNet extends WavePhaseNet by replacing fixed-frame scalar phase with gauge-covariant phase and adding connection and curvature.The earlier framework constructed conceptual hierarchies through discrete Fourier analysis of transformer representations.
  • 1.2 Terminology and Scope of Claims: The paper separates quantum-executable circuits from classical quantum-inspired implementations using the same spectral transforms, phase masks, reflections, and amplitude reweighting.
  • 1.2 Terminology and Scope of Claims: The paper treats quantum advantage and semantic correspondences as conditional or falsifiable claims, not consequences of complex numbers, unitary matrices, or QFT gate complexity alone.Total quantum complexity must include state preparation, block encoding, oracle construction, error correction, and measurement readout.
  • 1.3 Main Contributions: The contributions define covariant semantic wavelength, gauge-invariant phase coherence, quotient-set ordering, multi-axis conceptual lattices, and WavePhase Attention.WavePhase Attention combines causal masking, phase, wavelength, direction, geodesic distance, and evidential support.
  • 1.3 Main Contributions: The Validation Studio connects the Section 14.1 pipeline, Section 15.1 complexity analysis, and Section 16.1 research questions through a reproducible prototype experiment.
  • 3. Mathematical Setting: The mathematical setting models embeddings locally on a smooth semantic manifold with an atlas of sentence- or segment-level charts, a positive-definite metric, and a unitary complex semantic bundle.Semantic mass is normalized within the model and is not an empirical probability that an answer is correct.

4. Gauge-Covariant Phase, Phase Accumulation, and Semantic Wavelength

This section replaces frame-dependent phase derivatives with a gauge-covariant phase rate, accumulated phase, and semantic wavelength. It also distinguishes these quantities and extends gauge-covariant comparison to phase coherence and wavelength compatibility across fibers.

  • Gauge-Covariant Phase: Local phase depends on the chosen frame, so its naive derivative is not invariant under local gauge transformations.The section motivates a covariant derivative along a generation or reading curve to remove this frame dependence.
  • Phase Accumulation: The covariant phase rate measures phase change along a curve, while accumulated covariant phase records the phase-path length over an interval.In a local U(1) representation, the rate combines the ordinary phase derivative with the gauge connection.
  • Semantic Wavelength: Semantic wavelength is defined from positive semantic propagation speed and covariant phase rate, with slow variation producing long wavelengths and rapid variation producing short wavelengths.Phase angle, accumulated phase, and wavelength are explicitly treated as distinct objects.
  • Covariant Coherence: Covariant phase coherence compares normalized states in different fibers using selected-path parallel transport, with wavelength compatibility defined alongside it.This avoids assuming that states over different points share a global common frame.
  • Covariant Coherence: Path-aware covariant coherence removes the global-frame requirement of naive phase differences and permits shortest-geodesic, textual-order, or learned reference paths.The corresponding proposition establishes gauge invariance under transformed fiber frames and parallel transport.

5. Semantic Concept Hierarchy · 6. Discourse Direction and Spectral Hierarchy

The paper defines wavelength-based semantic hierarchies, extending from a total preorder and quotient lattice to conditional multi-axis structures. It then derives discourse directions from low-frequency semantic-graph modes, with geometric transport refinements and explicit limitations on interpreting smoothness as intention.

  • 5.1 Total Preorder Induced by a Single Wavelength: A single wavelength induces a total preorder because distinct points can share wavelengths, while wavelength equivalence classes form a totally ordered distributive lattice on the quotient.The construction does not establish a complete lattice or theoretically guarantee agreement with lexical entailment.
  • 5.2 Multi-Axis Wavelengths and a Conceptual Partial Order: Multi-axis wavelengths address polysemy and related structures; componentwise min/max closure is sufficient for the quotient to become a distributive lattice.Implementation requires a consistent dataset-specific orientation of subordinate and superordinate scales.
  • 6.1 Semantic Graph: A weighted semantic graph combines neighborhood, syntactic, coreference, causal, and retrieval-evidence edges, with normalized-Laplacian low eigenvalues representing globally smooth components.The graph is constructed from semantic points in a sentence.
  • 6.2 Reference Point and Vector Signal: The sentence reference point and logarithmic-map tangent vectors support low-pass filtering or projection onto the first K low-frequency graph modes.If the logarithmic map is non-unique at a cut locus, computation is restricted to a locally convex region or learned chart.
  • 6.3 Discourse-Direction Vector: The discourse-direction vector aggregates sentence-level signals in tangent space and compares with candidate directions after parallel transport, but low frequency alone may encode misinformation or topic fixation.Identification therefore uses summary supervision, instruction representations, evidence graphs, and contrastive losses.
  • 6.4 Generalization with the Connection Laplacian: The discrete connection Laplacian transports tangent vectors along graph edges, reducing distortion relative to transporting every vector to one reference point.A minimal implementation may use an ordinary graph Laplacian, while an extended implementation uses the connection Laplacian directly.

7. Fiber Bundles, Connections, and Curvature · 8. Semantic Hamiltonian

Sections 7–8 formulate semantic representations using locally gauge-covariant fiber-bundle connections and curvature, then define a finite-dimensional semantic Hamiltonian implemented through padded quantum registers. The framework links curvature to contextual state non-return and risk assessment while requiring Hermitian construction, encoding choices, and phase-aliasing safeguards.

  • 7.1 Local Gauges: Local semantic states are represented on coordinate patches with unitary transition functions and local connection one-forms.This establishes the local-gauge formulation of the semantic bundle.
  • 7.2 Curvature and Holonomy: Curvature measures how far an internal semantic state fails to return to itself after a closed contextual loop.The section also defines curvature risk for a neighborhood around a generation point.
  • 7.2 Curvature and Holonomy: Large curvature does not alone imply error because polysemy, metaphor, viewpoint changes, and legitimate topic transitions can also increase it.Interpretation must jointly consider direction, evidence, and uncertainty.
  • 8.1 Finite Semantic Register: The implementation uses N = 2^m basis states and applies masked padding when sentence length n is less than N.This defines the finite semantic register used by the implementation.
  • 8.1 Finite Semantic Register: Amplitude encoding uses separate data and channel registers, with channel-register padding when channel dimension c is not a power of two.The external transform F_N acts only on the first register.
  • 8.1 Finite Semantic Register: General amplitude-state preparation may dominate an actual quantum circuit, so basis, angle, and low-rank state preparation must also be compared.The comparison concerns implementation cost and preparation strategy.
  • 8.2 Self-Adjoint Operator: Each component of the finite-dimensional semantic Hamiltonian is Hermitian, and a learned non-Hermitian matrix B is symmetrized as (B + B†)/2.This ensures the operator construction uses self-adjoint components.
  • 8.2 Self-Adjoint Operator: Explicit unitary scaling is used in QPE to avoid phase aliasing, because distinct eigenenergies could otherwise wrap to the same phase.The scaling is presented as a safeguard for spectral interpretation.

9. QFT–QPE–Oracle–Amplitude Amplification–Inverse QFT

The section distinguishes external token-coordinate Fourier transformation from internal inverse-QFT phase processing, then combines QPE, a reflective target oracle, and amplitude amplification into the quantum selection block. It establishes unitarity under unitary components while cautioning that this does not imply complexity advantage or survive intermediate measurement unchanged.

  • Fourier roles: The external F_N maps token-position coordinates to frequency coordinates, whereas internal F† acts on the auxiliary phase register in QPE.The two Fourier transforms operate on different registers and serve different roles.
  • QPE and oracle: With m auxiliary qubits, QPE records an m-bit approximation of each eigenphase, but finite precision causes leakage near phase-window boundaries.The oracle should use a phase margin δϕ or smooth window rather than only a hard threshold.
  • QPE and oracle: The target-direction, wavelength, evidence, and prohibition conditions define a reversible predicate, target projector, and reflection oracle whose self-adjointness and involution are established.The predicate uses eigenphase, evidence, direction-alignment, and hierarchy bins.
  • Amplitude amplification: After k amplitude-amplification iterations, the target-subspace probability follows the exact success formula, with the dynamics confined to a two-dimensional marked–unmarked plane.When marked mass a is unknown, fixed iteration counts may overshoot; randomized, estimated, fixed-point, or smooth-filter alternatives are proposed.
  • Quantum selection block: The QuantumPhaseNet block is unitary when F_N, QPE, O_G, and S_ψ are unitary, but intermediate measurement or post-selection generally makes the process a quantum channel.The classical realization uses eigendecomposition, Krylov methods, or Chebyshev polynomials to increase target-subspace weight without claiming Grover’s quadratic speedup.

10. WavePhase Attention

WavePhase Attention augments standard attention logits with gauge-covariant phase, wavelength, discourse, geodesic, evidential, and masking features, then applies row-wise softmax with a quantum-feature residual connection. Under bounded Lipschitz assumptions, it is locally gauge-invariant, row-norm bounded, and locally Lipschitz stable, but these guarantees do not imply adversarial robustness.

  • WavePhase logit: WavePhase Attention adds six token-pair features to standard attention logits: phase coherence, wavelength compatibility, discourse alignment, geodesic proximity, evidential support, and causal/padding masking.The geodesic term is D_gij = d_g(z_i, z_j)^2, evidential support lies in [0, 1], and the mask takes values {0, −∞}.
  • WavePhase Attention: Row-wise softmax produces WavePhase Attention, while the quantum-selected feature Z_QPN enters through a residual connection.The initial β_q should be small to avoid instability early in training.
  • Theoretical guarantees: Local gauge invariance holds when phase, wavelength, geodesic, directional, and evidence terms are constructed from gauge-invariant quantities.The theorem concerns the WavePhase logit under these construction conditions.
  • Theoretical guarantees: If value vectors satisfy ‖v_j‖2 ≤ B_V, every output row is bounded because it is a convex combination of value vectors.The bound follows from the triangle inequality.
  • Theoretical guarantees: Under bounded Lipschitz feature assumptions on a compact region, the map X ↦ Y(X) is locally Lipschitz stable.The proof uses the softmax Jacobian J = diag(p) − pp^⊤ with Euclidean operator norm at most 1/2.
  • Limitations: The stability theorem does not guarantee adversarial robustness if feature estimators are discontinuous, geodesics branch, or hard oracle boundaries are differentiated directly.These conditions can invalidate the bounded regular-region assumptions.

11. Generation Dynamics · 12. Hallucination Risk

Sections 11–12 model generation as evidence- and objective-guided semantic dynamics, while defining a calibrated, multifactor hallucination-risk score. Their convergence guarantees and risk interpretations remain conditional on explicit geometric, error-bound, and calibration assumptions.

  • 11.1 Discrete Semantic Trajectory: Generation follows a discrete semantic trajectory whose tangent update uses retrieved evidence, WavePhase Attention, quantum-selected features, and the language-model head.The semantic state at step t is updated through a model-defined local dynamics.
  • 11.2 Energy with a Target Direction: Conditional linear convergence holds under Hadamard geometry, geodesic strong convexity, smoothness, and a step size satisfying 0 < η ≤ 1/L.The reference path also encodes instruction, factual, safety, and formatting requirements; the authors explicitly do not claim these assumptions generally hold.
  • 11.3 Time-Varying Targets and Tracking Error: Time-varying evidence and objectives require tracking error, target drift, local contraction, discretization error, and potentially piecewise-updated discourse directions.These diagnostics address topic transitions rather than assuming a fixed reference direction.
  • 12.1 Scope of the Risk Model: Hallucination risk decomposes generated text into atomic claims paired with evidence, but semantic uncertainty may miss fluent, consistently wrong answers.The model is not presented as a causal explanation of missing knowledge, retrieval, composition, reasoning, citation, instruction, or decoding failures.
  • 12.2 Five Deviation Terms: The risk model includes evidence deficiency and semantic uncertainty, using verifier scores plus quantum-target-subspace leakage or entropy over semantic classes.These are among the deviation terms used to characterize claim-level risk.
  • 12.3 Integrated Risk and Calibration: The stepwise risk R_t is a score rather than the true error probability, and validation data must support logistic or isotonic calibration with separate evaluation splits.Evaluation uses Brier score, ECE, AUROC, and risk–coverage curves.
  • 12.4 Selective Generation: Conformal abstention can complement fixed thresholds, but guarantees require assumptions including exchangeability, calibration-set size, and loss monotonicity.Selective generation therefore depends on stated statistical conditions.
  • 12.5 A Limited Zero-Risk Convergence Proposition: If z_t remains in compact K and R(z_t) → 0, then d_g(z_t, Γ_S) → 0, but the proposition does not guarantee path position, ordering, or claim truth.The error-bound assumption is strong and the result is therefore limited.

13. Learning Objectives

The learning objectives combine language-model, phase, hierarchy, directional, geometric, curvature, evidential, uncertainty, unitarity, and calibration losses. They also specify covariant semantic supervision, geometry-preserving regularization, evidential calibration, and exact Hamiltonian self-adjointness.

  • Composite objective: The total objective combines ℒLM with phase, hierarchy, direction, geometry, curvature, evidence, uncertainty, unitarity, and calibration losses.Each component is weighted by its corresponding λ coefficient.
  • Semantic and hierarchy supervision: Covariant coherence replaces naive angular difference for supervised or contrastive pairs.For hierarchy supervision, the sign convention must remain fixed across the dataset, while reverse-direction negatives, transitive closure, and synonym classes are evaluated separately.
  • Geometric regularization: Local-distance preservation and curvature regularization constrain geometry without forcing legitimate topic-transition regions to zero curvature.Supervised regions of legitimate topic transition are excluded from the curvature constraint.
  • Evidence and calibration: Evidential-support objectives model atomic-claim labels and use Brier loss for calibration.The support label y_t,a is binary: y_t,a∈{0, 1}.
  • Quantum objective constraints: Hamiltonian self-adjointness is enforced exactly by parameterizing H_θ from B_θ and its adjoint.The supplied formulation begins H_θ=(B_θ+B†...).

14. Algorithms · 15. Complexity and Numerical Stability · 16. Experimental Method and Evaluation Design

Sections 14–16 specify classical and quantum-executable algorithms, complexity and stability safeguards, and an evaluation design spanning research questions, datasets, baselines, metrics, statistical controls, and falsification criteria. The design explicitly accounts for quantum readout and end-to-end costs while requiring empirical tests of hierarchy, discourse, phase, hallucination-risk, and quantum-selection claims.

  • 14. Algorithms: The classical pipeline embeds text, builds semantic geometry and graphs, estimates phases and wavelengths, applies spectral filtering and WavePhase Attention, then verifies generated claims.It can trigger retrieval, regeneration, or abstention through ℛ_t and can be trained end to end or in stages.
  • 14.2 Quantum-Executable Realization: The quantum-executable pipeline pads features to N=2^m, uses QPE and an oracle for spectral predicates, amplifies marked states, uncomputes ancillas, and measures selected observables.Reading every amplitude may require Ω(N) measurements, so the design avoids full-vector reconstruction.
  • 15.1 Classical Complexity: O(n^2d) time and O(n^2) memory characterize dense attention, while sparse graph construction, Lanczos modes, and Chebyshev filtering use approximately O(knd), O(TK|E|), and O(K|E|d).Local tangent-space distances, neighborhood computation, Nystrom approximation, landmark methods, and edge-only transport reduce all-pairs geometric costs.
  • 15.2 Limits of the Quantum Complexity Claim: O(m^2) QFT gates do not establish an end-to-end speedup because costs also include state preparation, controlled evolution, oracle evaluation, amplitude amplification, error correction, and readout.The number of amplitude-amplification iterations depends on O(1/√a), and QPE precision is governed by ε_ϕ.
  • 15.3 Numerical Stability: ε_ω>0, covariant or 1−cos losses, clipped log λ, bounded metric eigenvalues, local convexity, softened spectral masks, and estimated marked mass address numerical instability.These safeguards prevent divergence, extreme scale ratios, invalid geometric maps, and unstable amplitude amplification.
  • 16.1 Research Questions: RQ1–RQ5 test hierarchy, discourse alignment and drift, covariant phase prediction, evidence-based hallucination detection and calibration, and quantum spectral selection under noise.The research questions compare learned quantities with known relations, discourse centers, cosine similarity, established risk baselines, and classical filters under matched budgets.
  • 16.2 Datasets and 16.3 Baselines and Ablations: WordNet, HyperLex, image–language order benchmarks, TruthfulQA, FEVER, FActScore, LongFact/SAFE, and dedicated discourse datasets define the proposed evaluation scope.Japanese and multilingual studies require language-specific phenomena and isolation of NLI or evidence-verifier effects; comparisons include Transformers, ablations, filters, RAG variants, and phase controls.
  • 16.4 Evaluation Metrics, 16.5 Statistical Design, and 16.6 Falsification Criteria: Evaluation covers language-model quality, hierarchy, directional stability, factuality, calibration, and efficiency, with at least five seeds, 95% confidence intervals, paired tests, matched search budgets, and predefined criteria.Falsification includes failed hierarchy correlation, inadequate order prediction, increased topic drift, indistinguishable phase from shuffling, inferior geometric risk, and other central-hypothesis failures.

17. Prototype Experiments with the Validation Studio · 18. Identifiability, Limitations, and Ethical Considerations

Validation Studio experiments provide initial synthetic support for the classical quantum-inspired components across RQ1–RQ4, while RQ5 finds no quantum computational advantage. The paper limits these findings through implementation, external-validity, identifiability, hierarchy, truth, risk, and quantum-speedup caveats.

  • 17.1 Purpose and Implementation Scope: Validation Studio v1.0.0 is a fully offline implementation that connects semantic graphs, discourse direction, covariant phase, wavelength, spectral selection, attention, and risk evaluation.It recomputes the Section 15.1 comparison and RQ1–RQ5 under one configuration and can record JSON or PDF results.
  • 17.2 Experimental Conditions and Reproduction Procedure: 240 samples, observation noise 0.22, circuit noise 0.08, and five seeds defined the built-in synthetic evaluation, with shared generators and decision rules across RQ1–RQ5.The procedure evaluated baseline differences, 95% interval signs, discrimination, calibration, and computational resources.
  • 17.3 Overall Results: RQ1–RQ4 met prespecified support criteria, whereas RQ5 became a falsification candidate for quantum-executable end-to-end cost efficiency.The results are initial prototype evidence rather than final public-benchmark or independently collected-data results; the approximately 73% evidence-strength value is only a Studio visualization index.
  • 17.4 RQ1: Semantic Wavelength and Concept Hierarchy: 0.852 wavelength–abstraction-depth Spearman correlation exceeded the simple embedding-scale baseline of 0.707 by 0.145, while direction accuracy reached 87.3% and inclusion AUC 0.953.The built-in example supports inverse-frequency wavelength as a plausible one-dimensional proxy for conceptual hierarchy, not a universal depth measure.
  • 17.5 RQ2: Low-Frequency Discourse Direction and Topic Retention: 0.933 discourse alignment versus 0.589 for the baseline accompanied 41.2 versus 16.2 paragraphs before drift and 95.4% versus 56.3% aligned-paragraph rate.Low-frequency discourse direction may help retain intended direction, but misinformation can also become low-frequency and truth requires independent verification.
  • 17.6 RQ3: Predictive Power of Covariant Phase Coherence / 17.7 RQ4: Geometric Risk and Hallucination: 0.881 covariant-phase AUROC exceeded cosine similarity at 0.765 and the phase-shuffled control at 0.536; geometric risk AUROC was 0.854 versus 0.634 for entropy.RQ3 also reported 81.5% versus 69.2% accuracy and Brier 0.154 versus 0.197, while RQ4 reported Brier 0.150, ECE 0.098, and 53.1% versus 3.9% answer rate at 10% allowed error.
  • 17.8 RQ5: Quantum Selection and Classical Approximation / 17.9 Complexity Validation for Section 15.1: 25.5% target-subspace probability versus 70.7% for the classical Chebyshev approximation and 0.107 versus 0.707 end-to-end cost efficiency showed no quantum advantage at circuit noise 0.08.The result does not invalidate the classical quantum-inspired realization; complexity figures were schematic, and sparse advantages require accounting for preprocessing and avoiding exact all-pairs kNN.
  • 17.10 Interpretation and Scope of Validity / 18.1 Nonuniqueness of Gauges and Coordinates / 18.2 Nonuniqueness of Hierarchy / 18.4 Limits of Hallucination Risk / 18.5 Quantum Advantage Is Not Established: External validation remains necessary, because synthetic RQ1–RQ4 improvements establish internal consistency rather than general validity, and unsupported RQ5 was retained for falsifiability.Gauge quantities such as phase angles are nonintrinsic; wavelengths are not universal concept depths; risk labels and scores are use-case dependent; high-stakes use requires human review and sources; partial quantum complexity does not establish end-to-end speedup.

19. Minimal Reproducible Model · 20. Correspondence Between Main Results and Empirical Hypotheses · 21. Conclusion

The paper recommends a staged, reproducible implementation that progresses from phase coherence through hierarchy, direction, attention, risk, learned geometry, and quantum simulation. Built-in synthetic results support the classical quantum-inspired mechanisms, while the quantum-circuit simulation provides no evidence of quantum advantage and motivates external validation.

  • 19. Minimal Reproducible Model: The minimal model begins with Euclidean distance, identity parallel transport, and ordinary complex inner products before adding further mechanisms.This reduction supports a staged implementation rather than introducing all components simultaneously.
  • 19. Minimal Reproducible Model: The implementation sequence advances from phase-only coherence to wavelength hierarchy, graph direction, WavePhase Attention, evidence risk, learned geometry, and quantum simulation.The sequence includes Ω, 𝜆, HyperLex/WordNet losses, graph Laplacians, calibration, curvature, QPE, the oracle, and amplitude amplification.
  • 19. Minimal Reproducible Model: Each stage should reproduce the preceding contribution before additional mechanisms are introduced, enabling component-level diagnosis of performance, instability, and computational cost.This procedure is intended to identify which components contribute to observed behavior.
  • 20. Correspondence Between Main Results and Empirical Hypotheses: 0.852 Spearman correlation exceeded baseline 0.707, while direction accuracy reached 87.3% and inclusion AUC reached 0.953.These built-in synthetic results correspond to the hierarchy-scale and direction hypotheses.
  • 20. Correspondence Between Main Results and Empirical Hypotheses: 0.933 mean discourse alignment exceeded baseline 0.589, and the aligned-paragraph rate reached 95.4% versus baseline 56.3%.The result supports the low-frequency discourse-direction hypothesis in the built-in synthetic setting.
  • 20. Correspondence Between Main Results and Empirical Hypotheses: 0.881 covariant-phase AUROC exceeded cosine 0.765 and shuffled phase 0.536, while geometric-risk AUROC reached 0.854 versus entropy 0.634 with Brier 0.150 and ECE 0.098.These results cover covariant phase coherence and calibrated geometric risk.
  • 21. Conclusion: 0.107 quantum end-to-end efficiency versus classical efficiency 0.707 showed no quantum advantage in the built-in synthetic result.The conclusion limits empirical support to the classical quantum-inspired mechanisms and calls for replication on WordNet and HyperLex, long-form and Japanese discourse data, TruthfulQA, FEVER, and SAFE, plus fair resource comparison.

Appendix A. Notation · Appendix B. Invariance Tests for Implementation

Appendix A defines the geometric, spectral, and quantum-inspired notation used by QuantumPhaseNet. Appendix B specifies implementation tests for gauge, spectral-basis, padding, and amplitude-amplification invariance.

  • Appendix A. Notation: The semantic concept manifold ℳ carries a Riemannian metric 𝑔 and measure 𝜇.
  • Appendix A. Notation: The complex semantic vector bundle 𝐸→ℳ uses a unitary connection ∇ with curvature 𝐹∇.
  • Appendix A. Notation: Covariant phase rate Ω𝛾 and accumulated phase ℓ𝜑,𝛾 define semantic wavelength 𝜆 as a proxy for conceptual scale.
  • Appendix A. Notation: The notation includes graph Laplacians 𝐿 and 𝐿∇, discourse direction Θ𝑆, semantic Hamiltonian 𝐻sem, and quantum-search operators.Quantum-search operators include 𝐹𝑁, Π𝐺, 𝑂𝐺, and 𝑄.
  • Appendix B. Invariance Tests for Implementation: Implementation tests apply random local phases and consistently transformed connections, checking invariance of states, coherence, logits, and outputs.
  • Appendix B. Invariance Tests for Implementation: Tests also vary eigenvector representations, padding length, and marked-state limits to verify projector, nonpadding-output, and amplitude-amplification behavior.The variations include eigenvector signs, phases, degenerate-eigenspace bases, padding length 𝑁, and cases where every or no state is marked.
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