Conditions for an Infrared Einstein Sector from Spectral Geometry

Overview

This article asks under which conditions a minimal elliptic Laplace-type operator on a four-dimensional Riemannian manifold can support a local infrared Einstein sector — and, just as importantly, which parts of gravity such an operator does not determine.

Defining the projective entropy as $S_\Pi[g]=\tfrac12\log\det' A_g$, with $A_g=-\nabla_g^2$, the analysis separates two regularization statements that must not be conflated. A physical proper-time cutoff $\Lambda=\ell_{\mathrm{sp}}^{-1}$ carries the power-sensitive local terms $a_0\Lambda^4$ and $a_2\Lambda^2$; zeta regularization carries the finite determinant and the logarithmic scale dependence, governed in four dimensions by $a_4=\zeta_A(0)$.

What is not claimed. One minimal scalar contributes $\Delta c_{\mathrm{EH}}^{\Lambda}=-\Lambda^2/[12(4\pi)^2]$ to the Einstein–Hilbert coefficient — a negative value. The finite renormalized coefficient, its sign, and hence a positive Newton constant are matching data fixed by a renormalization condition and the complete operator content. They are not predicted by this determinant alone.

Scope statement. This page provides a structured summary. The authoritative technical reference is the preprint linked above.

Core contributions

Covariant spectral functional: $S_\Pi[g]=\tfrac12\log\det'(-\nabla_g^2)$ with ellipticity and spectral well-posedness.
Separation of regularization schemes: a proper-time cutoff carries $a_0\Lambda^4$ and $a_2\Lambda^2$; zeta regularization carries the finite and logarithmic dependence, governed by $a_4=\zeta_A(0)$. The two answer different questions and must not be merged.
Renormalized metric variation: once the coefficients are supplied, the variation decomposes into Einstein, cosmological, higher-derivative, and non-local sectors.
Infrared dominance: the local two-derivative Einstein term dominates when $R\,L_4^2\ll 1$, with $L_4^2=|\beta^{\mathrm{ren}}/c_{\mathrm{EH}}^{\mathrm{ren}}|$ set by the ratio of renormalized coefficients — not by $\ell_{\mathrm{sp}}$ alone.
The Newton coupling is a matching datum: the cutoff fixes the magnitude of the one-operator contribution, which is negative. The sign and finite value of $c_{\mathrm{EH}}^{\mathrm{ren}}$ depend on the complete operator content and a renormalization condition; the observed $G_N$ is not derived here.
Born–Infeld ultraviolet completion: under the separate coherence-extensivity hypothesis, the tensorial completion of a supplied Einstein infrared term remains of determinantal Born–Infeld form.

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, a cosmological term, local quadratic-curvature terms, and non-local form factors.

The Einstein tensor arises from the $a_2$ sector, while the quadratic sector originates from $a_4$ and yields Bach-like four-derivative structures upon variation. For the minimal operator these are the same terms: the logarithmic variation of $a_4$ already belongs to the local four-derivative sector, so no separate anomaly tensor is counted alongside it.

Infrared hierarchy

In the weak-curvature regime the renormalized effective response admits a covariant derivative expansion. The hierarchy is controlled by the ratio of renormalized coefficients rather than by $\ell_{\mathrm{sp}}$ on its own: writing $L_4^2=|\beta^{\mathrm{ren}}/c_{\mathrm{EH}}^{\mathrm{ren}}|$, the local two-derivative Einstein response dominates over four-derivative and non-local corrections when $R\,L_4^2\ll 1$.

Born–Infeld ultraviolet completion

Under a separate coherence-extensivity hypothesis, the tensorial completion of a supplied Einstein infrared term remains of determinantal Born–Infeld form

$\sqrt{-\det(g_{\mu\nu}+\ell_{\mathrm{sp}}^2 R_{\mu\nu})}-\sqrt{-g}$,

an Eddington-inspired structure in which Einstein gravity appears as the two-derivative infrared sector. The statement is conditional on the hypothesis and on the supplied Einstein term; it does not rely on a specific microscopic model.

If the complete matched coefficient happens to be dominated by cutoff contributions and is positive, then dimensional scaling gives $G_N=O(\ell_{\mathrm{sp}}^2)$ with a field-content-dependent loop factor. That is a conditional matching regime, not a derivation: neither the sign nor the numerical coefficient follows from the single minimal scalar determinant.

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. Conditions for an Infrared Einstein Sector from Spectral Geometry. 10.5281/zenodo.18818721