The Spectral Gravity Sub-Programme

Overview

The emergent geometry sub-programme (Presentation Note 2) reconstructs the effective Lorentzian metric $g^{\mu\nu} = 2\eta^{\mu\nu}$ as a forced output of admissibility. What dynamics does this metric obey, and why does it take the Einstein form?

The spectral gravity sub-programme answers this in four stages built on a single functional, the projective spectral entropy $\mathcal{S}_\Pi[g] = \tfrac{1}{2}\log\det' A_g$ associated with the minimal admissible Laplace-type operator $A_g = -\nabla_g^2$. The Einstein tensor arises as the infrared $a_2$ pole of the renormalized metric variation; the field equations $G_{\mu\nu} = 8\pi G_N T_{\mu\nu}^{(\Pi)}$ are recovered as the Euler–Lagrange condition of a spectral equilibrium principle; the construction is then causally completed in Lorentzian signature and ultraviolet-completed by an Eddington-inspired Born–Infeld action. This note covers the gravitational sector in the absence of matter; the joint gauge–gravity stratification belongs to Presentation Note 8.

Four stages, one functional. $\mathcal{S}_\Pi[g]$ is the only input. The $a_2$ pole gives the Einstein tensor with induced Newton constant $G_N \sim 16\pi^2 \ell_{\mathrm{sp}}^2$. The local spectral multiplier $\beta(x)^{-1} = \beta_0^{-1} - \tfrac{1}{6}R(x) + O(\nabla^2 R)$ reformulates the field equations as a spectral equilibrium condition (no horizon, no Rindler wedge, no Raychaudhuri theorem required). The Lorentzian completion propagates two transverse-traceless helicity-$\pm 2$ modes with a universal $k^4$ correction $\gamma = 1/30$ in the natural spectral scheme. The unique UV completion is the Eddington-inspired Born–Infeld action, conditional on Hypothesis [H-ext] (Gravity Theorem 1).

The structural chain

$\mathcal{S}_\Pi[g] = \tfrac{1}{2}\log\det' A_g \;\Longrightarrow\; \delta_g\mathcal{S}_\Pi \ni c_{\mathrm{EH}}\, G_{\mu\nu} \;\Longrightarrow\; G_{\mu\nu} = 8\pi G_N T_{\mu\nu}^{(\Pi)} \;\Longrightarrow\; \omega^2 = c^2 k^2 - \gamma\,\ell_{\mathrm{sp}}^2 k^4 \;\Longrightarrow\; \sqrt{-\det(g + \ell_{\mathrm{sp}}^2 R)}.$

Four stages arise from distinct constituent papers: the IR hierarchy and induced Newton constant from the heat-kernel $a_2$ coefficient of $\mathcal{S}_\Pi$ (Gravity 3.0); the local spectral multiplier field and the Einstein equation as the Euler–Lagrange condition of a spectral equilibrium principle (Thermodynamics); the Schwinger–Keldysh Lorentzian completion with two helicity-$\pm 2$ TT graviton modes and the $\gamma = 1/30$ dispersion correction (Lorentz/CausalPropagation); and the Eddington-inspired Born–Infeld action as the unique admissible tensorial UV completion under conditions (C1)–(C5) and Hypothesis [H-ext] (Gravity Theorem 1, BornInfeld).

Papers of the sub-programme

IR Einstein response and induced Newton constant.

Gravity 3.0 — establishes $\mathcal{S}_\Pi[g] = \tfrac{1}{2}\log\det' A_g$ on a four-dimensional Riemannian manifold and derives the renormalized metric variation. Results: IR hierarchy with the $a_2$ Einstein term infrared-dominant in $R\ell_{\mathrm{sp}}^2 \ll 1$ (structural); induced Newton constant $G_N \sim 16\pi^2 \ell_{\mathrm{sp}}^2$ (structural); BI tensorial uniqueness theorem under (C1)–(C5) and [H-ext] (conditional); structural necessity of non-injective projection for $\mathcal{S}_\Pi > 0$.

Spectral equilibrium and the Einstein equation.

Thermodynamics — develops the local thermodynamic interpretation of $\mathcal{S}_\Pi[g]$. Results: local spectral multiplier $\beta(x)^{-1} = \beta_0^{-1} - \tfrac{1}{6}R(x) + O(\nabla^2 R)$ from the diagonal heat kernel; spectral first law $\delta\mathcal{S}_\Pi = \int \beta^{-1}\delta E_\Pi$ as a consequence of the Seeley–DeWitt hierarchy; Einstein equation $G_{\mu\nu} = 8\pi G_N T_{\mu\nu}^{(\Pi)}$ as the Euler–Lagrange condition of a spectral equilibrium principle, with no horizon, Rindler wedge, or Raychaudhuri theorem invoked.

Causal completion and gravitational waves.

Lorentz / CausalPropagation — constructs the Lorentzian completion via a Schwinger–Keldysh prescription. Results: real retarded quadratic kernel; two transverse-traceless helicity-$\pm 2$ graviton modes in the $R\ell_{\mathrm{sp}}^2 \ll 1$, $\omega \ll \ell_{\mathrm{sp}}^{-1}$ regime (scalar and vector sectors non-dynamical); dispersion relation $\omega^2 = c^2 k^2 - \gamma \ell_{\mathrm{sp}}^2 k^4$ with $\gamma = 1/(180\zeta) = 1/30$ in the natural spectral scheme; bound on $\ell_{\mathrm{sp}}$ from gravitational-wave catalogues.

Born–Infeld UV completion.

BornInfeld — establishes the scalar BI uniqueness: the saturation bound $|\partial_t\chi| \leq c_{\mathrm{BI}}$ selects a square-root action density in the one-dimensional sector (proved). Combined with Gravity Theorem 1, the tensorial extension yields the Eddington-inspired BI action $\mathcal{S}^{\mathrm{BI}} \propto \int[\sqrt{-\det(g + \ell_{\mathrm{sp}}^2 R_{\mu\nu})} - \sqrt{-g}]$. Establishes the parity involution $\chi \mapsto -\chi$, forcing the minimal fibre condition $\Pi^{-1}(y) \supseteq \{\chi, -\chi\}$ feeding into O18.

Inputs and outputs

Upstream inputs. Non-injectivity as a structural necessity for $\mathcal{S}_\Pi > 0$ (ENI); the effective four-dimensional Lorentzian manifold $(M, g^{\mu\nu} = 2\eta^{\mu\nu})$ with minimal admissible Laplace-type operator $A_g = -\nabla_g^2$ from the emergent geometry sub-programme (Presentation Note 2, Q5b–Q11); the spectral length scale $\ell_{\mathrm{sp}}$ from Branch I and the Born–Infeld saturation constant $c_{\mathrm{BI}}$ of the projection dynamics; the BI parity involution and minimal fibre condition from BornInfeld and O18.

Outputs. The Einstein equations $G_{\mu\nu} = 8\pi G_N T_{\mu\nu}^{(\Pi)}$ in the matter-free IR; induced Newton constant $G_N \sim 16\pi^2 \ell_{\mathrm{sp}}^2$; the Eddington-inspired Born–Infeld action conditional on [H-ext]; two helicity-$\pm 2$ transverse-traceless graviton modes; the $\omega^2 = c^2 k^2 - \gamma \ell_{\mathrm{sp}}^2 k^4$ dispersion with $\gamma = 1/30$; bound on $\ell_{\mathrm{sp}}$ from GW data; the $a_2 \to$ gravity, $a_4 \to$ gauge spectral stratification — consumed by Q12, Q13, the gauge–gravity stratification sub-programme (Note 8), and cosmological/phenomenological applications.

Status

Gravity in the matter-free infrared is closed structurally: the $a_2$ Einstein term dominates under $R\ell_{\mathrm{sp}}^2 \ll 1$ (Gravity 3.0 §4.5); the Einstein equation is recovered as a spectral equilibrium condition (Thermodynamics Theorem 4.3); two TT helicity-$\pm 2$ modes with $\gamma = 1/30$ dispersion are derived in Lorentzian signature (Lorentz/CausalPropagation §3). Scalar BI uniqueness and the parity involution are proved unconditionally (BornInfeld), as are the spectral carrier identification and the scalar BI reduction (Gravity Lemmas 2–3). The tensorial BI uniqueness theorem (Gravity Theorem 1) is conditional on Hypothesis [H-ext] (admissible coherence extensivity, Lemma 1 only). Three items remain open: the analytical proof of [H-ext]; the full Lorentzian Einstein equilibrium beyond linearisation; and the coupled $G_{\mu\nu} = 8\pi G_N T_{\mu\nu}$ with explicit matter, pending the Fermionic Matter Sub-Programme (Note 6). This note does not address Yang–Mills dynamics — that belongs to the gauge–gravity stratification sub-programme (Presentation Note 8).