This repository contains the source of the Spectral Gravity Presentation Note Cosmochrony paper The Spectral Gravity Sub-Programme — Presentation Note 4

This work is a structured entry point to the spectral gravity sub-programme of the Cosmochrony corpus, not a summary of results. It maps the constituent papers, identifies the internal phases, records the status of every result as proved, structural, numerical, or open, and states the remaining open deliverables.

Central Question

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 the admissibility structure of the non-injective projection $\Pi$.

Why does the projective spectral entropy functional $\mathcal{S}_\Pi[g] = \tfrac{1}{2}\log\det' A_g$ produce the Einstein equations as its metric variation, and how is this dynamics completed causally and ultraviolet-wise?

The sub-programme establishes that the Einstein equations $G_{\mu\nu} = 8\pi G_N T_{\mu\nu}^{(\Pi)}$ emerge as the infrared-dominant $a_2$ response of the renormalized spectral entropy functional, not as a postulated dynamics. This note concerns the derivation of gravitational dynamics in the matter-free IR; the gauge--gravity stratification belongs to Presentation Note 8.

Logical 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 conceptually distinct stages:

1. IR Einstein response — the $a_2$ pole of the renormalized metric variation is infrared-dominant in $R\ell_{\mathrm{sp}}^2 \ll 1$ (Gravity 3.0).
2. Spectral equilibrium — the Einstein field equations as Euler--Lagrange of $\mathcal{S}_\Pi$ at fixed projected matter content, with no horizon or Rindler structure invoked (Thermodynamics).
3. Causal completion — Lorentzian Schwinger--Keldysh prescription, two transverse-traceless helicity-$\pm 2$ graviton modes, dispersion $\omega^2 = c^2 k^2 - \gamma \ell_{\mathrm{sp}}^2 k^4$ with $\gamma = 1/30$ (Lorentz/CausalPropagation).
4. UV completion — the Eddington-inspired Born--Infeld action as the unique tensorial completion of the Einstein--Hilbert sector under conditions (C1)--(C5) and Hypothesis [H-ext] (BornInfeld + Gravity Theorem 1).

Constituent Papers

#	Paper	Stage	Local path
1	Gravity 3.0 (Beau2026h) — Infrared Einstein Response from a Renormalized Spectral Entropy Functional	IR Einstein response, induced $G_N \sim 16\pi^2 \ell_{\mathrm{sp}}^2$, BI tensorial uniqueness	../gravity/
2	Thermodynamics (Beau2026Thermodynamics) — local spectral first law, Einstein equation as spectral equilibrium	Spectral equilibrium	../thermodynamics/
3	Lorentz / CausalPropagation (Beau2026i) — Lorentzian completion, gravitons, $k^4$ dispersion	Causal completion	../lorentz-paper/
4	BornInfeld (Beau2026c) — scalar BI uniqueness, parity involution $\chi \mapsto -\chi$	UV completion	../born-infeld-paper/

Status of Results

Proved (unconditional):

• Scalar BI uniqueness and parity involution (BornInfeld).
• Spectral carrier identification: $X = \ell_{\mathrm{sp}}^2 g^{-1}R$ (Gravity Lemma 3).
• Scalar BI reduction $h(\lambda) = \tfrac{1}{2}\log(1 + \lambda)$ (Gravity Lemma 2).

Structural:

• IR hierarchy and IR dominance of $G_{\mu\nu}$ (Gravity §4.5).
• Induced Newton constant $G_N \sim 16\pi^2 \ell_{\mathrm{sp}}^2$ (Gravity §5).
• Spectral multiplier field and first law (Thermodynamics §2--3).
• Einstein equation as spectral equilibrium (Thermodynamics Theorem 4.3).
• Lorentzian Schwinger--Keldysh kernel, two helicity-$\pm 2$ modes, $\gamma = 1/30$ dispersion (Lorentz/CausalPropagation §2--3).

Conditional on [H-ext]:

• Tensorial BI uniqueness (Gravity Theorem 1, Lemma 1 only).

Open Deliverables

1. Analytical proof of [H-ext] (admissible coherence extensivity) — promotes Gravity Theorem 1 to unconditional.
2. Full Lorentzian Einstein equilibrium — extend Thermodynamics Theorem 4.3 beyond Riemannian signature, integrating the Schwinger--Keldysh prescription.
3. Coupled $G_{\mu\nu} = 8\pi G_N T_{\mu\nu}$ with matter — pending the Fermionic Matter Sub-Programme (Note 6).

Compilation

bash compile.sh

Produces out/SpectralGravityNote.pdf.

Citation

To be assigned upon first Zenodo publication.