Spectral Vacuum Mechanism — PART XXVII Metric, Curvature, Topology and Dimensionality from Spectral Overlaps in the SVM Framework

ABSTRACT We present Part XXVII of the Spectral Vacuum Mechanism (SVM) program: a constructive, resource-light route from vacuum spectral data to emergent geometry. Starting from localized vacuum eigenmodes {psi_i} and their pairwise overlaps, we define a correlation-decay distance d_spec(i,j) = -(1/alpha) * log | | with alpha the inverse localization length (estimated as the slope of -log| | versus gate distance in the strong-localization window, thus fixing the overall length scale), and develop a distance-first framework in which metric structure, curvature proxies, and effective dimensionality are reconstructed without assuming a background manifold. The core localization control parameter is lambda = g/m0. We take lambda < 0.1 as the energy-domain counterpart of the Part XVIII mixing-stability criterion epsilon_mix < 0.1, and use it as the operational localization gate for geometric observables in Part XXVII. In the strong localization regime (lambda < 0.1), overlap-factorization errors are perturbative and d_spec behaves as a controlled near-metric: symmetry and positivity hold directly, while triangle-inequality violations are bounded and scale as O(lambda) with an ensemble-dependent constant (benchmarked numerically in Section 10). On this basis we introduce a practical reconstruction pipeline: (i) local metric tensors are obtained from distance geometry (local quadratic fits / embedding stability) rather than assumed coordinates; (ii) curvature-like response is diagnosed by U(1) phase transport derived from overlap phases, defining loop holonomy Hol(C) and Berry phase gamma(C) on minimal cycles; (iii) topology is probed by a robust Z2 holonomy diagnostic in regimes where gamma(C3) is sharply peaked near pi (mod 2pi) on minimal triangles, yielding Hol(C3) ≈ -1 within tolerance and providing a concrete sign/holonomy protection mechanism compatible with Spin^c-type interpretations (as a compatibility statement, not a completed theorem); and (iv) dimension is defined operationally and shown to be scale-dependent via two estimators—ball-growth N(R) and diffusion/heat-kernel return probability P(t)—with explicit finite-size and refinement-stability requirements. A critical correction is enforced throughout: finite gate motifs (e.g., K4) cannot define an asymptotic (R→infinity) dimension; only finite-scale effective slopes are meaningful, and continuum claims require convergence under refinement. The resulting output is an audit-ready validation protocol based solely on overlaps and phases—triangle-defect statistics, cycle-holonomy distributions, and effective-dimension curves—connecting emergent geometry to the same spectral mechanism that underlies mass gaps, confinement-friendly localization, and robust sign/holonomy structures in SVM. MAIN RESULTS (1) Spectral distance from overlaps: d_spec(i,j) = -(1/alpha) * log | | provides a universal correlation-decay notion of distance derived directly from vacuum eigenmodes. (2) Controlled near-metric scaling: for lambda = g/m0 < 0.1, triangle-inequality violations are bounded and scale as O(lambda), with an ensemble-dependent constant; benchmark families yield small defects (Section 10.1). (3) Distance-first metric reconstruction: local metric tensors g_mu nu are reconstructed from distance geometry (local quadratic fits/embedding stability) without assuming a background manifold (Section 4). (4) Curvature proxies from holonomy: overlap phases define U(1) transport U_ij, loop holonomy Hol(C), and Berry phase gamma(C), giving a constructive curvature-like diagnostic already on minimal cycles (Section 5; Appendix A). (5) Robust Z2 holonomy diagnostic: in frustrated localization regimes, minimal triangles show gamma(C3) sharply peaked near pi (mod 2pi), implying Hol(C3) ≈ -1 within tolerance and yielding a stable Z2 sign/holonomy invariant compatible with a Spin^c-type interpretation (compatibility claim, not a completed theorem). (6) Effective (scale-dependent) dimension: dimension is defined operationally via ball-growth d_eff(R) and diffusion d_eff(t); dimensional flow is measurable and must be reported over stable windows (Section 6). (7) Critical finite-size correction: finite motifs (including K4) do not define an asymptotic dimension; any quoted exponent is a finite-window effective slope and must be treated as estimator-dependent (Appendix B). (8) Audit-ready numerical validation: a resource-light validation suite is specified: triangle defect vs lambda, holonomy distributions on small cycles, and effective-dimension curves with refinement-stability tests (Section 10). KEYWORDS Spectral Vacuum Mechanism; emergent geometry; spectral distance; localization; near-metric; triangle defect; Berry phase; holonomy; curvature proxy; Spin^c interpretation; effective dimension; diffusion/heat kernel; graph Laplacian; distance geometry; refinement stability; audit-ready validation. Other works by the author on this topic: Spectral Vacuum Mechanism — Part XIV: Spectral Confinement as a Necessary Condition for Quantum Field Theory. Confinement Gate‑Induced Spectral Localization and Dimensional Constraints, Zenodo. DOI: 10.5281/zenodo.18140235 (2026). Spectral Vacuum Mechanism — Part XV: Unification of the Mass Formula in SVM Particles of the Standard Model, Zenodo. DOI: 10.5281/zenodo.18207487 (2026). Spectral Vacuum Mechanism — Part XVI: Spectral Confinement under Truncated SU(2) Gauge Embedding: Preservation of the Spectral Confinement Class, Zenodo. DOI: 10.5281/zenodo.18225421 (2026). Spectral Vacuum Mechanism — Part XVII: Spectral Confinement under Truncated SU(3) Gauge Embedding: Toward a Constructive QCD‑like Framework, Zenodo. DOI: 10.5281/zenodo.18280887 (2026). Spectral Vacuum Mechanism — Part XVIII: Continuum Trajectory and Low‑Energy Self‑Consistency under SU(3) Truncation, Zenodo. DOI: 10.5281/zenodo.18415826 (2026). Spectral Vacuum Mechanism — Part XIX: Gauss‑Law Certificates and Audit Artifacts under SU(3) Truncation, Zenodo. DOI: 10.5281/zenodo.18422292 (2026). Spectral Vacuum Mechanism — Part XX: SU(3) Truncation Removal: Controlled j_max → ∞ at Fixed (a, V) in the Physical Sector, Zenodo. DOI: 10.5281/zenodo.18434530 (2026). Spectral Vacuum Mechanism — Part XXI: Thermodynamic Limit (V → ∞) at Fixed Lattice Spacing in the Gauss‑Law Sector, Zenodo. DOI: 10.5281/zenodo.18444149 (2026). Spectral Vacuum Mechanism — Part XXII: Ultraviolet Stability and the Continuum Limit, Zenodo. DOI: 10.5281/zenodo.18448953 (2026). Spectral Vacuum Mechanism — Part XXIII: At Finite Density: Hamiltonian Deformation and Phase Transitions, Zenodo. DOI: 10.5281/zenodo.18450115 (2026). Spectral Vacuum Mechanism — Part XXIV: Validation of the Continuum Trajectory: Kinetic Scaling, Gauss-Law Purity, Solver Robustness, and Failure Map, Published February 2, 2026 | Version v1, Zenodo. DOI: 10.5281/zenodo.18459836 (2026). Spectral Vacuum Mechanism — Part XXV: Spectral Observables in the Continuum SU(3) Hamiltonian: Correlators, Gauss-filter Operators, Susceptibilities, and Observable-Level Audit, Published February 5, 2026 | Version v1, Zenodo. DOI: 10.5281/zenodo.18498737 (2026). Spectral Vacuum Mechanism — Part XXVI:Confinement Without Area Law: Spectral Diagnostics in the Hamiltonian SU(3) Framework, Zenodo. DOI: 10.5281/zenodo.18519299 (2026)