Abstract A complete rigorous mathematical justification of the Spectral Vacuum mechanism on three-dimensional cubic lattices ℤ³ with periodic boundary conditions is presented. The variational lower bound for the mass gap is proven: Δ ≥ Λ_gate(g) = m₀ - 2zgV_max > 0 при g < g* = m₀/(2zV_max). Spectral localization was established using weighted graph Laplacian with coordination number z = 6. The work contains all intermediate mathematical calculations, detailed proofs of all theorems and lemmas, and a complete working numerical code for verifying the theoretical results. Methods of constructive quantum field theory (reflection positivity, Osterwalder-Schrader axioms) are used. Keywords: mass gap, spectral localization, constructive QFT, variational methods, graph Laplacian, tensor networks, PEPS, dimensional robustness
Abstract A complete rigorous mathematical justification of the Spectral Vacuum mechanism on three-dimensional cubic lattices ℤ³ with periodic boundary conditions is presented. The variational lower bound for the mass gap is proven: Δ ≥ Λ_gate(g) = m₀ - 2zgV_max > 0 при g < g* = m₀/(2zV_max). Spectral localization was established using weighted graph Laplacian with coordination number z = 6. The work contains all intermediate mathematical calculations, detailed proofs of all theorems and lemmas, and a complete working numerical code for verifying the theoretical results. Methods of constructive quantum field theory (reflection positivity, Osterwalder-Schrader axioms) are used. Keywords: mass gap, spectral localization, constructive QFT, variational methods, graph Laplacian, tensor networks, PEPS, dimensional robustness
Abstract We extend the Spectral Vacuum Mechanism (SVM), previously established in a multichannel (0+1)D toy model [1], to a two-dimensional square lattice with periodic boundary conditions. The multichannel degrees of freedom are placed on lattice sites, introducing genuine spatial structure while preserving the purely operator-level construction without reference to classical gauge Lagrangians. Using reflection positivity via multi-direction chessboard estimates [2,3], uniform energy bounds, and Osterwalder–Schrader axioms [4,5], we rigorously prove the existence of the infinite-volume limit with a unique vacuum. A uniform mass gap Δ > 0 is established variationally through gate-induced localization in 2D. The spectrum exhibits exponential clustering in all directions and absence of massless non-singlet states — confinement-like spectral properties persisting from the 1D case. Here "non-singlet" refers strictly to representations of the internal compact symmetry acting on channel space. No claim of gauge confinement, Wilson-loop area law [6], or geometric string tension is made in this work. Numerical diagonalization on finite lattices up to L = 20 is consistent with convergence to Δ∞ ≈ 0.32 > 0, a sharp localization–delocalization transition at gc ≈ 1.05, and perimeter-dominated entanglement entropy (S ≈ 0.79 · perimeter) [7,8]. The observed perimeter law indicates boundary-sensitive low-energy modes induced by gate geometry rather than gapless bulk excitations or topological order. This 2D extension serves as a crucial proof-of-principle for dimensional robustness of the gate-induced spectral localization mechanism prior to gauge embedding. Future directions include discrete ZN gauge models and paths toward continuous constructions relevant to the Yang–Mills Millennium Problem [9]. Keywords: Spectral vacuum mechanism, mass gap, confinement, lattice quantum field theory, constructive QFT, Osterwalder-Schrader axioms, reflection positivity, Yang-Mills, Millennium Problem, constructive quantum field theory, lattice models, graph Laplacian localization Other works by the author on similar topics: 1. Physical Vacuum – Part I Spectral vacuum mechanism (SVM) (lepton masses), DOI: 10.5281/zenodo.,17931024 (2025) 2. Physical Vacuum – Part II Spectral vacuum mechanism (SVM) (quark sector), DOI: 10.5281/zenodo.17923554 (2025) 3. Physical Vacuum – Part III Composite (hadronic) excitations in the Spectral Vacuum Mechanism (SVM), DOI: 10.5281/zenodo.17923690 (2025) 4. Physical Vacuum – Part IV Spectral vacuum mechanism (SVM) Hadronic sector in the Spectral Vacuum Mechanism: baryons and mesons from the universal gate phase, DOI: 10.5281/zenodo.17942901 (2025) 5. hysical Vacuum – Part V Spectral Vacuum Mechanism: A Constructive Approach to Confinement-Like Spectral Properties, DOI: 10.5281/zenodo.17956380 (2025)
Abstract We formulate a minimal operator-theoretic criterion—the Spectral Confinement Class (SCC)—that any quantum theory exhibiting confinement-like behavior in four space-time dimensions should satisfy. This criterion consists of three necessary spectral properties: (i) a strict spectral gap above the vacuum, (ii) exponential clustering of connected correlation functions, and (iii) the absence of massless non-singlet excitations with respect to an internal compact symmetry (e.g., color). Within a general multichannel lattice framework, we prove that gate-induced spectral localization provides an explicit mechanism realizing SCC. Under minimal structural assumptions—local spectral rigidity with a gap m₀>0 uniform in the channel dimension N, bounded finite-range interactions, and a structural gate condition preventing spectral percolation—we derive a uniform lower bound on the spectral gap Δ ≥ m₀ - 2·z(d)·g·Vₘₐₓ where z(d)=2d is the lattice coordination number. The bound is uniform in system volume and channel dimension and exhibits a dimensional penalty: for fixed coupling g>0, the admissible gap window shrinks with increasing spatial dimension. We emphasize that SCC is necessary but not sufficient for confinement in the Yang–Mills sense. Its role is that of a spectral admissibility filter: candidate confining theories should satisfy SCC at the operator level; failure to do so presents obstacles to confinement regardless of gauge structure. In this sense, SCC functions as a spectral admissibility filter: candidate confining theories must satisfy it at the operator level, independently of gauge-theoretic or geometric mechanisms. Keywords Primary Keywords: spectral confinement class • spectral gap • gate condition • dimensional penalty • Yang–Mills theory • confinement • lattice Hamiltonians • multichannel systems • spectral localization Secondary Keywords: exponential clustering • local spectral rigidity • excitation projectors • coordination number • four-dimensional quantum field theory • gauge theories • Wilson loops • operator theory • variational methods • mass gap problem Mathematical Subject Classification (MSC 2020): Primary: 81T13 (Yang-Mills and other gauge theories), 81Q10 (Selfadjoint operator theory in quantum mechanics) Secondary: 82B20 (Lattice systems), 47A10 (Spectrum and spectral theory)
Abstract The mass gap—defined as the minimum excitation energy above the ground state—represents one of the most fundamental quantities in quantum field theory and condensed matter physics. Its existence ensures spectral isolation of the ground state, exponential decay of spatial correlations, and determines the low-temperature thermodynamic behavior of quantum many-body systems. Despite its central importance, rigorous lower bounds on the mass gap that can be explicitly computed from microscopic parameters have remained elusive for realistic lattice models. This work presents a comprehensive examination of the Nemirovsky bound, a recently established mathematical inequality that provides an explicit lower limit for the mass gap in lattice quantum systems: Δ ≥ Λgate(g) = m₀ − 2zgVmax > 0, valid for coupling strengths g < g* = m₀/(2zVmax). Here m₀ denotes the local mass gap at a single lattice site, z is the coordination number encoding lattice topology, Vmax represents the maximum nearest-neighbor interaction amplitude, and g controls the coupling strength. The bound emerged as the culmination of a seven-part research program on the Spectral Vacuum Mechanism (SVM), progressing from phenomenological descriptions of elementary particle masses through rigorous constructive proofs on two-dimensional and three-dimensional lattices. We demonstrate how this fundamental quantum constraint directly determines macroscopic thermodynamic properties through its role in the partition function. At temperatures T ≪ Δ/k, the heat capacity exhibits characteristic activation behavior CV(T) ~ (Δ/kT)² exp(−Δ/kT), with the exponential suppression governed by the Nemirovsky bound. The entropy vanishes as S(T) → 0 when T → 0, ensuring compliance with the third law of thermodynamics. The critical coupling g* = m₀/(2zVmax) precisely demarcates the quantum phase transition at T = 0, where the gap closes and the system transitions from an insulating gapped phase to a conducting gapless phase with qualitatively different correlation structure. Three detailed numerical examples illustrate the practical application of the formula across diverse physical systems. For a quantum antiferromagnet on a cubic lattice (m₀ = 100 K, z = 6, Vmax = 10 K, g = 0.3), the bound predicts Δ ≥ 64 K, leading to exponential freezing of heat capacity below T ≈ 15 K. Near the critical point, the gap decreases linearly as Δ = m₀(1 − g/g*), vanishing at g = g* = 0.833. For topological quantum qubits with parameters m₀ = 1.0 meV and Vmax = 0.1 meV, the bound establishes a minimum decoherence time τ ~ ℏ/Δ ≈ 1.3 picoseconds, providing fundamental limitations on quantum information processing. The significance of the Nemirovsky bound extends beyond its computational utility. It serves as a unifying principle connecting quantum magnetism, topological matter, and quantum computing under a single mathematical framework. The formula establishes a bridge between microscopic quantum mechanics and macroscopic thermodynamics, explaining how discrete lattice degrees of freedom generate observable bulk properties. Moreover, it provides actionable design principles for materials engineering: maximizing m₀, minimizing coordination number z, or reducing coupling strength g all enhance the mass gap and improve stability of quantum states—crucial for developing fault-tolerant quantum technologies. This work positions the Nemirovsky bound within the broader context of constructive quantum field theory, comparing it with alternative approaches such as Perron-Frobenius theory, renormalization group methods, and mean-field approximations. While these frameworks provide existence theorems or asymptotic results, the Nemirovsky bound offers explicit quantitative predictions directly calculable from physical parameters. The structural analysis reveals hierarchical organization of physical consequences: fundamental quantum constraints (spectral gap, correlation length) determine statistical mechanical properties (partition function, free energy), which in turn manifest as measurable macroscopic observables (heat capacity, magnetic susceptibility, decoherence rates). Appendix X provides a systematic structural analysis that elucidates the Nemirovsky bound's role as a fundamental organizing principle in quantum matter. It examines the three-level hierarchical structure (quantum constraints → statistical mechanics → macroscopic observables), analyzes parameter space regimes from strong gapped to critical phases, and demonstrates cross-domain unification connecting quantum magnetism, topological materials, and quantum information theory. The appendix includes dimensional analysis confirming the formula's robustness across different lattice geometries, comprehensive comparison with alternative theoretical frameworks (Perron-Frobenius, renormalization group, mean-field theory), discussion of current limitations and possible extensions to inhomogeneous systems and many-body interactions, and exposition of the bound's conceptual significance as a bridge between scales. This structural perspective reveals the Nemirovsky bound not merely as a calculational tool, but as a conceptual framework unifying disparate phenomena through the competition between localization (m₀) and collective delocalization (2zgVmax). Future research directions include experimental verification in specific materials, extensions to inhomogeneous systems with spatially varying parameters, incorporation of many-body interactions beyond nearest-neighbor coupling, and applications to non-equilibrium quantum systems. The Nemirovsky bound represents both a practical calculation tool and a conceptual framework for understanding the rich organizational principles of gapped quantum matter. Keywords: mass gap, spectral gap, lattice quantum field theory, Nemirovsky bound, quantum phase transitions, low-temperature thermodynamics, heat capacity, partition function, quantum magnetism, topological insulators, quantum computing, decoherence, coordination number, constructive quantum field theory, spectral vacuum mechanism, correlation length, critical phenomena, quantum many-body systems, quantum-to-classical correspondence PACS: 03.65.-w (Quantum mechanics), 05.30.-d (Quantum statistical mechanics), 75.10.-b (General theory and models of magnetic ordering), 03.67.-a (Quantum information), 64.70.Tg (Quantum phase transitions) MSC2020: 81T05 (Axiomatic quantum field theory), 82B20 (Lattice systems), 81Q10 (Selfadjoint operator theory in quantum theory), 82B26 (Phase transitions)
Abstract The mass gap—defined as the minimum excitation energy above the ground state—represents one of the most fundamental quantities in quantum field theory and condensed matter physics. Its existence ensures spectral isolation of the ground state, exponential decay of spatial correlations, and determines the low-temperature thermodynamic behavior of quantum many-body systems. Despite its central importance, rigorous lower bounds on the mass gap that can be explicitly computed from microscopic parameters have remained elusive for realistic lattice models. This work presents a comprehensive examination of the Nemirovsky bound, a recently established mathematical inequality that provides an explicit lower limit for the mass gap in lattice quantum systems: Δ ≥ Λgate(g) = m₀ − 2zgVmax > 0, valid for coupling strengths g < g* = m₀/(2zVmax) Here m₀ denotes the local mass gap at a single lattice site, z is the coordination number encoding lattice topology, Vmax represents the maximum nearest-neighbor interaction amplitude, and g controls the coupling strength. The bound emerged as the culmination of a seven-part research program on the Spectral Vacuum Mechanism (SVM), progressing from phenomenological descriptions of elementary particle masses through rigorous constructive proofs on two-dimensional and three-dimensional lattices. We demonstrate how this fundamental quantum constraint directly determines macroscopic thermodynamic properties through its role in the partition function. At temperatures T ≪ Δ/k, the heat capacity exhibits characteristic activation behavior CV(T) ~ (Δ/kT)² exp(−Δ/kT), with the exponential suppression governed by the Nemirovsky bound. The entropy vanishes as S(T) → 0 when T → 0, ensuring compliance with the third law of thermodynamics. The critical coupling g* = m₀/(2zVmax) precisely demarcates the quantum phase transition at T = 0, where the gap closes and the system transitions from an insulating gapped phase to a conducting gapless phase with qualitatively different correlation structure. Three detailed numerical examples illustrate the practical application of the formula across diverse physical systems. For a quantum antiferromagnet on a cubic lattice (m₀ = 100 K, z = 6, Vmax = 10 K, g = 0.3), the bound predicts Δ ≥ 64 K, leading to exponential freezing of heat capacity below T ≈ 15 K. Near the critical point, the gap decreases linearly as Δ = m₀(1 − g/g*), vanishing at g = g* = 0.833. For topological quantum qubits with parameters m₀ = 1.0 meV and Vmax = 0.1 meV, the bound establishes a minimum decoherence time τ ~ ℏ/Δ ≈ 1.3 picoseconds, providing fundamental limitations on quantum information processing. The significance of the Nemirovsky bound extends beyond its computational utility. It serves as a unifying principle connecting quantum magnetism, topological matter, and quantum computing under a single mathematical framework. The formula establishes a bridge between microscopic quantum mechanics and macroscopic thermodynamics, explaining how discrete lattice degrees of freedom generate observable bulk properties. Moreover, it provides actionable design principles for materials engineering: maximizing m₀, minimizing coordination number z, or reducing coupling strength g all enhance the mass gap and improve stability of quantum states—crucial for developing fault-tolerant quantum technologies. This work positions the Nemirovsky bound within the broader context of constructive quantum field theory, comparing it with alternative approaches such as Perron-Frobenius theory, renormalization group methods, and mean-field approximations. While these frameworks provide existence theorems or asymptotic results, the Nemirovsky bound offers explicit quantitative predictions directly calculable from physical parameters. The structural analysis reveals hierarchical organization of physical consequences: fundamental quantum constraints (spectral gap, correlation length) determine statistical mechanical properties (partition function, free energy), which in turn manifest as measurable macroscopic observables (heat capacity, magnetic susceptibility, decoherence rates). Appendix X provides a systematic structural analysis that elucidates the Nemirovsky bound's role as a fundamental organizing principle in quantum matter. It examines the three-level hierarchical structure (quantum constraints → statistical mechanics → macroscopic observables), analyzes parameter space regimes from strong gapped to critical phases, and demonstrates cross-domain unification connecting quantum magnetism, topological materials, and quantum information theory. The appendix includes dimensional analysis confirming the formula's robustness across different lattice geometries, comprehensive comparison with alternative theoretical frameworks (Perron-Frobenius, renormalization group, mean-field theory), discussion of current limitations and possible extensions to inhomogeneous systems and many-body interactions, and exposition of the bound's conceptual significance as a bridge between scales. This structural perspective reveals the Nemirovsky bound not merely as a calculational tool, but as a conceptual framework unifying disparate phenomena through the competition between localization (m₀) and collective delocalization (2zgVmax). Future research directions include experimental verification in specific materials, extensions to inhomogeneous systems with spatially varying parameters, incorporation of many-body interactions beyond nearest-neighbor coupling, and applications to non-equilibrium quantum systems. The Nemirovsky bound represents both a practical calculation tool and a conceptual framework for understanding the rich organizational principles of gapped quantum matter. Keywords: mass gap, spectral gap, lattice quantum field theory, Nemirovsky bound, quantum phase transitions, low-temperature thermodynamics, heat capacity, partition function, quantum magnetism, topological insulators, quantum computing, decoherence, coordination number, constructive quantum field theory, spectral vacuum mechanism, correlation length, critical phenomena, quantum many-body systems, quantum-to-classical correspondence PACS: 03.65.-w (Quantum mechanics), 05.30.-d (Quantum statistical mechanics), 75.10.-b (General theory and models of magnetic ordering), 03.67.-a (Quantum information), 64.70.Tg (Quantum phase transitions) MSC2020: 81T05 (Axiomatic quantum field theory), 82B20 (Lattice systems), 81Q10 (Selfadjoint operator theory in quantum theory), 82B26 (Phase transitions)
Abstract The mass gap—defined as the minimum excitation energy above the ground state—represents one of the most fundamental quantities in quantum field theory and condensed matter physics. Its existence ensures spectral isolation of the ground state, exponential decay of spatial correlations, and determines the low-temperature thermodynamic behavior of quantum many-body systems. Despite its central importance, rigorous lower bounds on the mass gap that can be explicitly computed from microscopic parameters have remained elusive for realistic lattice models. This work presents a comprehensive examination of the Nemirovsky bound, a recently established mathematical inequality that provides an explicit lower limit for the mass gap in lattice quantum systems: Δ ≥ Λgate(g) = m₀ − 2zgVmax > 0, valid for coupling strengths g < g* = m₀/(2zVmax). Here m₀ denotes the local mass gap at a single lattice site, z is the coordination number encoding lattice topology, Vmax represents the maximum nearest-neighbor interaction amplitude, and g controls the coupling strength. The bound emerged as the culmination of a seven-part research program on the Spectral Vacuum Mechanism (SVM), progressing from phenomenological descriptions of elementary particle masses through rigorous constructive proofs on two-dimensional and three-dimensional lattices. We demonstrate how this fundamental quantum constraint directly determines macroscopic thermodynamic properties through its role in the partition function. At temperatures T ≪ Δ/k, the heat capacity exhibits characteristic activation behavior CV(T) ~ (Δ/kT)² exp(−Δ/kT), with the exponential suppression governed by the Nemirovsky bound. The entropy vanishes as S(T) → 0 when T → 0, ensuring compliance with the third law of thermodynamics. The critical coupling g* = m₀/(2zVmax) precisely demarcates the quantum phase transition at T = 0, where the gap closes and the system transitions from an insulating gapped phase to a conducting gapless phase with qualitatively different correlation structure. Three detailed numerical examples illustrate the practical application of the formula across diverse physical systems. For a quantum antiferromagnet on a cubic lattice (m₀ = 100 K, z = 6, Vmax = 10 K, g = 0.3), the bound predicts Δ ≥ 64 K, leading to exponential freezing of heat capacity below T ≈ 15 K. Near the critical point, the gap decreases linearly as Δ = m₀(1 − g/g*), vanishing at g = g* = 0.833. For topological quantum qubits with parameters m₀ = 1.0 meV and Vmax = 0.1 meV, the bound establishes a minimum decoherence time τ ~ ℏ/Δ ≈ 1.3 picoseconds, providing fundamental limitations on quantum information processing. The significance of the Nemirovsky bound extends beyond its computational utility. It serves as a unifying principle connecting quantum magnetism, topological matter, and quantum computing under a single mathematical framework. The formula establishes a bridge between microscopic quantum mechanics and macroscopic thermodynamics, explaining how discrete lattice degrees of freedom generate observable bulk properties. Moreover, it provides actionable design principles for materials engineering: maximizing m₀, minimizing coordination number z, or reducing coupling strength g all enhance the mass gap and improve stability of quantum states—crucial for developing fault-tolerant quantum technologies. This work positions the Nemirovsky bound within the broader context of constructive quantum field theory, comparing it with alternative approaches such as Perron-Frobenius theory, renormalization group methods, and mean-field approximations. While these frameworks provide existence theorems or asymptotic results, the Nemirovsky bound offers explicit quantitative predictions directly calculable from physical parameters. The structural analysis reveals hierarchical organization of physical consequences: fundamental quantum constraints (spectral gap, correlation length) determine statistical mechanical properties (partition function, free energy), which in turn manifest as measurable macroscopic observables (heat capacity, magnetic susceptibility, decoherence rates). Appendix X provides a systematic structural analysis that elucidates the Nemirovsky bound's role as a fundamental organizing principle in quantum matter. It examines the three-level hierarchical structure (quantum constraints → statistical mechanics → macroscopic observables), analyzes parameter space regimes from strong gapped to critical phases, and demonstrates cross-domain unification connecting quantum magnetism, topological materials, and quantum information theory. The appendix includes dimensional analysis confirming the formula's robustness across different lattice geometries, comprehensive comparison with alternative theoretical frameworks (Perron-Frobenius, renormalization group, mean-field theory), discussion of current limitations and possible extensions to inhomogeneous systems and many-body interactions, and exposition of the bound's conceptual significance as a bridge between scales. This structural perspective reveals the Nemirovsky bound not merely as a calculational tool, but as a conceptual framework unifying disparate phenomena through the competition between localization (m₀) and collective delocalization (2zgVmax). Future research directions include experimental verification in specific materials, extensions to inhomogeneous systems with spatially varying parameters, incorporation of many-body interactions beyond nearest-neighbor coupling, and applications to non-equilibrium quantum systems. The Nemirovsky bound represents both a practical calculation tool and a conceptual framework for understanding the rich organizational principles of gapped quantum matter. Keywords: mass gap, spectral gap, lattice quantum field theory, Nemirovsky bound, quantum phase transitions, low-temperature thermodynamics, heat capacity, partition function, quantum magnetism, topological insulators, quantum computing, decoherence, coordination number, constructive quantum field theory, spectral vacuum mechanism, correlation length, critical phenomena, quantum many-body systems, quantum-to-classical correspondence PACS: 03.65.-w (Quantum mechanics), 05.30.-d (Quantum statistical mechanics), 75.10.-b (General theory and models of magnetic ordering), 03.67.-a (Quantum information), 64.70.Tg (Quantum phase transitions) MSC2020: 81T05 (Axiomatic quantum field theory), 82B20 (Lattice systems), 81Q10 (Selfadjoint operator theory in quantum theory), 82B26 (Phase transitions)
The Geometric Vacuum (GeoVac) framework is a discretization scheme for non-relativistic quantum mechanics on natural geometries: the discrete graph Laplacian on a packed lattice of quantum-number labels replaces the continuous Laplace–Beltrami operator on the corresponding compact manifold. This synthesis paper assembles the framework's foundational arc—the packing construction, the discrete graph and its dynamics, the conformal equivalence with the unit S3, the exchange constant taxonomy, the angular sparsity theorem, the Bargmann–Segal lattice, and the universal/Coulomb partition, together with the reconvergence node where these threads cohere (two-body selection rules from the tensor-product triple, the period classification, the Tannakian substrate and cosmic-Galois injection, and the forced/free seam)—into a single coherent narrative. The arc has a dependency structure worth naming: a single taproot (the packing construction, made physical by the Fock S3 projection) fans out into the framework's machinery, then reconverges at the periods, Tannakian, and forced/free nodes, where the scattered structural findings cohere into one statement about what the framework forces and what it consumes. Four headline structural results frame the program: (a) the discrete Fock graph is conformally equivalent to the unit S3 Laplace–Beltrami operator, verified by 18 independent symbolic proofs; (b) the angular sparsity of two-body matrix elements depends only on lmax and is bit-identical across five distinct central potentials, qualifying as a potential-independent theorem about spherical-fermion bases; (c) the three-dimensional isotropic harmonic oscillator admits a discrete graph encoding on the Hardy space H2(S5) that is bit-exactly π-free in exact rational arithmetic at every finite cutoff; (d) the framework's results split cleanly along an A/D partition—an algebra-only universal sector that transfers free across central potentials, and a Dirac-operator-specific sector containing the spectrum, the calibration π, and the candidate fine-structure-constant combination of Paper 2. A structural reading we make explicit in this synthesis is that the framework's substrate-level scale-freedom—the packing diameter d0 cancelling out of every spectral quantity, and physical scales entering only through the Fock-projection focal length p0—is the geometric expression of the same principle that conformal field theory [ref] embodies at the physics level. GeoVac provides a scale-free mathematical substrate; CFT is a scale-free physics theory living on such a substrate. This is not a claim that GeoVac is a CFT; it is a structural reflection that explains why the framework reproduces continuum CFT-on-sphere observables bit-exactly (documented in the operator-algebra arc, outside the foundations scope of this synthesis). We present GeoVac as a discretization framework that exploits natural geometry, not as a new foundation for quantum mechanics: the discrete graph and the continuous Laplace–Beltrami operator are dual descriptions connected by proven conformal maps, and the interpretive question of which description is more fundamental is left open. The framework's concrete computational advantages—O(V) sparsity, integer eigenvalues on the unit S3, angular momentum selection rules baked into the basis, bit-exactly rational matrix elements on the Bargmann–Segal lattice, and the universal sparsity bound—are independent of that interpretive question. GeoVac synthesis document. Part of the GeoVac research corpus (synthesis / reading guide group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.