The Spectral Gravity Sub-Programme

The spectral gravity sub-programme of the Cosmochrony corpus addresses a single central question: under which conditions does the projective spectral entropy functional ${\mathcal{S}_\Pi}[g] = \tfrac{1}{2}\log\det' A_g$ support an infrared Einstein sector for the effective metric? The sub-programme separates the physical proper-time cutoff (power-sensitive terms $a_0\Lambda^4$, $a_2\Lambda^2$) from zeta regularization (finite and logarithmic terms governed by $a_4$), decomposes the renormalized metric variation into Einstein, cosmological, higher-derivative, and non-local sectors, and states the exact conditions under which the Einstein term dominates. The renormalized couplings — including the sign and value of Newton's constant — are matching data, not predictions. Conditionally on supplied coefficients, stationarity of the difference functional $\Gamma_\Pi = {\mathcal{S}_\Pi}^{\mathrm{ren}} - W_\Pi$ yields the Einstein–matter coupling $G_{\mu\nu} + \Lambda_{\mathrm{ren}} g_{\mu\nu} = 8\pi{G_{\mathrm{N}}} T_{\mu\nu}$. On the posited local four-derivative Lorentzian truncation, the massless graviton branch is exactly luminal, and the ultraviolet completion of determinantal Born–Infeld type is an admissible member of an explicitly parametrised family, not a uniquely selected one. The sub-programme is constituted by four papers: Gravity, Thermodynamics, Lorentz/CausalPropagation, and BornInfeld. Interpretive status. The sub-programme establishes which local structures spectral geometry allows and where every remaining gap lies — the matching principle for the couplings, the integrability bridge to any thermodynamic reading, the in-in construction of a Lorentzian spectral action, and the selection principle for the ultraviolet completion. Gravity emerging as a spectral response remains a structural possibility, not an established result.