SPECTRAL VACUUM MECHANISM — PART XLI Variational–RG Characterization of the Ultraviolet Dimension

Abstract

The dimensionality of spacetime is usually postulated rather than derived. In this work we show that within the spectral reconstruction protocol the ultraviolet spacetime dimension is selected dynamically as a compatibility optimum of the spectral data.

The analysis is based on the ultraviolet heat-trace expansion and the Seeley–DeWitt reconstruction basis span{1, t, t²}. We introduce a compatibility functional

Ψ(d) = sup_{W_uv} ε_rel(d, W_uv)

which measures the maximal reconstruction residual across admissible ultraviolet windows. We demonstrate that this functional possesses a unique minimum at dimension d = 4.

The effective spectral dimension defined from the heat trace,

d_eff(E) = −2 d ln P(E^{−1}) / d ln E

generates an associated spectral beta-function

β_spec(d_eff) = d d_eff / d ln E

We prove that the compatibility minimum coincides with the ultraviolet fixed point of this flow. Furthermore, we establish a spectral monotonicity principle showing that the compatibility functional acts as a Lyapunov quantity, ensuring irreversible approach to the ultraviolet dimension.

The result establishes a variational–RG characterization of the ultraviolet dimension: within the admissible spectral reconstruction class the ultraviolet spacetime dimension is uniquely selected as the point where spectral incompatibility is minimized, the spectral beta-function vanishes, and the compatibility flow reaches its stable endpoint.

Keywords: spectral geometry, spectral dimension, heat-trace expansion, Seeley–DeWitt coefficients, spectral reconstruction, compatibility functional, ultraviolet fixed point, renormalization group, variational principle, quantum gravity.

Other works by the author on this topic::

[1] M. Nemirovsky, Spectral Vacuum Mechanism – Part XXVII: Metric from Spectral Overlaps, Zenodo. DOI: 10.5281/zenodo.18560958 (2026)

[2] M. Nemirovsky, Spectral Vacuum Mechanism – Part XXVIII: Emergent Gauge Structure, Zenodo. DOI: 10.5281/zenodo.18599116 (2026)

[3] M. Nemirovsky, Spectral Vacuum Mechanism – Part XXX: Emergent Gravity, Zenodo. DOI: 10.5281/zenodo.18636847 (2026)

[4] M. Nemirovsky, Spectral Vacuum Mechanism – Part XXXVII: Phase Geometry and Quantization, Zenodo. DOI: 10.5281/zenodo.18769976 (2026)

[5] M. Nemirovsky, Spectral Vacuum Mechanism – Part XXXIX: UV–4D Emergence as a Stable Spectral Property of the SVM Vacuum Hessian, DOI: 10.5281/zenodo.18861827 (2026)

[6] M. Nemirovsky, Spectral Vacuum Mechanism – PART XL Spectral Compatibility and Dimensional Necessity: Internal Consistency of the UV–4D Framework, DOI: 10.5281/zenodo.18889765 (2026)