SPECTRAL VACUUM MECHANISM — PART XXXVIII The Spectral Foundation of Fundamental Constants:Rigorous Derivation and Natural Calibration

Abstract We establish a structural reduction of gravitational and quantum scales within the vacuum Hessian spectral action S[H] = −α Tr[H²] + β Tr[H⁴]. Rather than treating Newton’s constant G and Planck’s constant ℏ as independent postulates, we show that both emerge as effective infrared parameters governed by the same spectral rigidity couplings α (geometric stiffness) and β (nonlinear stabilization), up to a single unavoidable dimensional reconstruction scale required by general dimensional analysis. Using Seeley–DeWitt heat-kernel asymptotics, the quadratic spectral sector generates the Einstein–Hilbert term with G_eff ∝ 1/α, demonstrating that gravitational strength is controlled by spectral geometric rigidity. The equilibrium spectrum of the vacuum Hessian exhibits a gap Δ = √(α/(2β)), which sets the intrinsic rigidity threshold of the spectral substrate. From the geometric quantization mechanism developed in Part XXXVII, Planck’s constant arises as the minimal symplectic area of a reconstructed phase-space cell, ℏ_eff = C_phase · Δ ∝ √(α/β), where C_phase is an order-unity geometric factor determined by phase-correlation statistics of the reconstructed vacuum ensemble. Imposing the experimentally measured normalization of G and ℏ fixes the dimensionless couplings to α̃ = 6π ≈ 18.9, β̃ ≈ 1.9, both naturally of order unity in Planck units, without fine-tuning. The associated spectral gap lies at the Planck scale, indicating a single ultraviolet rigidity threshold underlying both gravity and quantization. The framework therefore reduces gravitational and quantum constants to emergent parameters of a unified spectral vacuum structure with only one fundamental dimensional scale, while preserving full consistency with the geometric quantization mechanism established in Part XXXVII. Keywords: Fundamental constants, Newton constant, Planck constant, spectral action, heat-kernel expansion, Seeley-DeWitt coefficients, dimensional analysis, natural calibration, emergent gravity, emergent quantization, stochastic spectral flow, Planck scale, vacuum rigidity Other works by the author on a related topic:: Spectral Vacuum Mechanism (SVM) Series [1] M. Nemirovsky, Spectral Vacuum Mechanism — Part XXX: Foundations and Geodesic Slack Theorem, Zenodo. DOI: 10.5281/zenodo.18636847 (2026) [2] M. Nemirovsky, Spectral Vacuum Mechanism — Part XXXI: Emergence of Geometry from Spectral Data, Zenodo. DOI: 10.5281/zenodo.18647374 (2026) [3] M. Nemirovsky, Spectral Vacuum Mechanism — Part XXXII: Gauge Structure and Yang-Mills Theory, Zenodo. DOI: 10.5281/zenodo.18657365 (2026) [4] M. Nemirovsky, Spectral Vacuum Mechanism — Part XXXIII: Classical Spectral Flow and Ricci Flow Projection, Zenodo. DOI: 10.5281/zenodo.18695432 (2026) [5] M. Nemirovsky, Spectral Vacuum Mechanism — Part XXXIV: Quantum Spectral Flow and Stochastic Quantization, Zenodo. DOI: 10.5281/zenodo.18704453 (2026) [6] M. Nemirovsky, Spectral Vacuum Mechanism — Part XXXV: Spectral Strings and Worldsheet Dynamics, Zenodo. DOI: 10.5281/zenodo.18726006 (2026) [7] M. Nemirovsky, Spectral Vacuum Mechanism — Part XXXVI: Spectral Origin of String Tension and Confinement, Zenodo. DOI: 10.5281/zenodo.18749131 (2026) [8] M. Nemirovsky, Spectral Vacuum Mechanism — Part XXXVII: Phase Geometry as the Structural Origin of Quantization and Infrared Curvature, Zenodo. DOI: 10.5281/zenodo.18769976 (2026)