Overview
This article studies a covariant projective entropy functional defined by a minimal elliptic Laplace-type operator on a four-dimensional Riemannian manifold. The goal is to identify the leading local geometric content of the renormalized metric variation in a controlled infrared regime, while keeping higher-derivative, non-local, and anomaly terms explicit.
Defining the projective entropy as $S_\Pi[g]=\tfrac12\log\det' A_g$, with $A_g=-\nabla_g^2$, the functional is regularized through the spectral zeta function and renormalized by subtraction of heat-kernel poles. The resulting renormalized spectral stress tensor admits a derivative-order decomposition whose leading local two-derivative term is proportional to the Einstein tensor.
Core contributions
- Covariant spectral functional: $S_\Pi[g]=\tfrac12\log\det'(-\nabla_g^2)$ with ellipticity and spectral well-posedness.
- Zeta regularization and pole structure: heat-kernel coefficients determine local counterterms, including the Einstein–Hilbert sector.
- Renormalized metric variation: the variation decomposes into an Einstein term, cosmological term, quadratic-curvature sector, non-local form factors, and anomaly contribution.
- Infrared dominance: for $R\,\ell_{\mathrm{sp}}^2\ll 1$, the local two-derivative Einstein response dominates over four-derivative and non-local corrections.
- Induced Newton coupling: the effective gravitational coupling scales as $G_N\sim 16\pi^2\ell_{\mathrm{sp}}^2$ and follows the standard induced-gravity logic.
- Born–Infeld ultraviolet completion: the UV completion is uniquely determined (under coherence extensivity) to be of determinantal Born–Infeld form, providing the complete effective geometric action.
Renormalized spectral stress tensor
The renormalized metric variation defines a covariant spectral stress tensor $\mathcal{T}^{\Pi,\mathrm{ren}}_{\mu\nu}=-\tfrac{2}{\sqrt{g}}\delta S_\Pi^{\mathrm{ren}}/\delta g^{\mu\nu}$. In four dimensions, its structure can be organized by derivative order: a local Einstein tensor term, local quadratic-curvature terms, non-local form factors, and a conformal anomaly contribution.
The Einstein tensor arises from the counterterm associated with the $a_2$ heat-kernel pole, while the quadratic sector originates from $a_4$ and yields Bach-like four-derivative structures upon variation.
Infrared hierarchy
In the weak-curvature regime, where dimensionless curvature invariants satisfy $R\,\ell_{\mathrm{sp}}^2\ll 1$, the renormalized effective response admits a covariant derivative expansion. Higher-derivative local terms and non-local form factors are suppressed by additional powers of $\ell_{\mathrm{sp}}^2$, making the local two-derivative Einstein response the dominant infrared contribution.
Born–Infeld ultraviolet completion
The paper establishes that the ultraviolet completion of the infrared Einstein sector is uniquely determined under a coherence extensivity hypothesis. The complete effective action takes the determinantal Born–Infeld form
$\sqrt{-\det(g_{\mu\nu}+\ell_{\mathrm{sp}}^2 R_{\mu\nu})}-\sqrt{-g}$,
an Eddington-inspired structure where Einstein gravity appears as the universal two-derivative infrared sector and the Born–Infeld extension provides the unique admissible spectral ultraviolet completion. This result does not rely on a specific microscopic model but follows from general properties of the renormalized spectral functional.
The induced Newton coupling scales parametrically as $G_N\sim 16\pi^2\ell_{\mathrm{sp}}^2$, up to scheme-dependent numerical factors.
Newtonian limit
The resolvent-based mechanism also recovers the familiar long-range Newtonian profile in the appropriate limit. In three spatial dimensions, the Green kernel of a Laplace-type elliptic operator exhibits a universal $1/r$ decay. Consequently, the weak-perturbation variation $\delta S_\Pi=\tfrac12\mathrm{Tr}(A^{-1}\delta A)$ inherits a Newtonian spatial profile from resolvent structure.
Within the covariant setting, this behavior appears as the static, weak-field manifestation of the same spectral framework whose renormalized metric variation yields the infrared-dominant Einstein response and Born–Infeld ultraviolet completion.
Relation to the Cosmochrony program
This article is formulated as a self-contained spectral-geometry analysis. It is compatible with broader Cosmochrony motivations, but it is intended to be readable and evaluable on its own, without requiring the full pre-geometric framework.
References
Jérôme Beau. Infrared Einstein Response from a Renormalized Spectral Entropy Functional. 10.5281/zenodo.18818721