SPECTRAL VACUUM MECHANISM — PART XXXII Spectral Field Equations:Discrete Einstein–Yang–Mills Stationarity and Spectral Dynamics

Abstract This work extends the Spectral Vacuum Mechanism (SVM) from static spectral functionals (Parts XXIX–XXXI) to dynamical field equations. We derive the complete system of discrete Einstein–Yang–Mills equations from the stationarity principle applied to the unified spectral action S_SVM = α a₂(H) + β a₄(H). Main achievements: • Discrete Einstein equations from metric variation: ℰ_ij(R,δ) = β T^spec_ij • Discrete Yang–Mills equations from gauge variation: 𝒥_C(R,A) = 0 • Spectral stress-energy tensor T^spec_ij encoding gauge backreaction on geometry • Singularity resolution mechanism: SCC-2 constraints prevent R → ∞, replacing singularities with stable spectral cores Spectral flow dynamics: gradient descent on unified action combines Ricci flow and Yang–Mills flow The key conceptual advance: geometry (a₂) and gauge (a₄) are not independent sectors but coupled through variational principle. Gauge field stress acts as source in Einstein equations; geometric curvature modifies Yang–Mills propagation. This realizes Einstein's dream of 'unified field theory' at the spectral level. We prove consistency with SCC-2 criteria (Part XXXI), establish uniqueness of stationary solutions under regularity conditions, and demonstrate that classical singularities (black holes, Big Bang) are replaced by finite-density spectral structures. Complete numerical implementation validates the framework on test geometries. Part XXXII lays the foundation for the study of spectral cosmology, black hole interiors, and quantum field dynamics in subsequent papers and is the first in the series where the spectral vacuum is no longer a static object, but a dynamical system governed by coupled Einstein-Yang-Mills equations derived from a single operator. Keywords: spectral field equations, discrete Einstein equations, Yang-Mills gauge theory, unified action principle, singularity resolution, spectral flow, stress-energy tensor, variational methods 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) Spectral Vacuum Mechanism — Part XXVII:Metric, Curvature, Topology and Dimensionality from Spectral Overlaps in the SVM Framework, Zenodo. DOI: 10.5281/zenodo.18560958 (2026) Spectral Vacuum Mechanism — Part XXVIII:Emergent Gauge Structure from Spectral Overlaps:From Local Phase Freedom to U(1)/SU(n) Connections and Berry-like Holonomies, Zenodo. DOI: 10.5281/zenodo.18599116 (2026) Spectral Vacuum Mechanism — Part XIX:Spectral Yang-Mills Dynamics:Field Equations, Running Coupling, and Confinement from Vacuum Geometry, Zenodo. DOI: 10.5281/zenodo.18624234 (2026) Spectral Vacuum Mechanism — Part XXX: Emergent Gravity from Spectral Geometry:Einstein-Hilbert Action from Vacuum Hessian Heat Kernel, Zenodo. DOI: 10.5281/zenodo.18636847 (2026) Spectral Vacuum Mechanism — Part XXXI: Spectral Continuum Limit:Curvature Defects, Coefficient a₂(H), and Operator Criteria for Emergence of Einstein-Hilbert Action, Zenodo. DOI: 10.5281/zenodo.18647374 (2026)