This repository contains the source of the Cosmochrony Gravity paper.
The paper separates two distinct regularization statements:
- a physical proper-time cutoff produces power-sensitive local terms;
- zeta regularization controls the finite determinant and logarithmic scale dependence.
For a four-dimensional minimal scalar Laplacian,
[ \operatorname{Tr}(e^{-tA_g}) \sim \frac{1}{(4\pi t)^2} \int_M \mathrm d^4x,\sqrt g, \left(a_0+t a_2+t^2a_4+\cdots\right), ]
with
[ a_0=1, \qquad a_2=\frac{R}{6}, ]
and
[ a_4=\frac{1}{360} \left( 12\nabla^2R+5R^2-2R_{\mu\nu}R^{\mu\nu} +2R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} \right). ]
The factor ((4\pi t)^{-2}) is outside the coefficient series and is not repeated inside (a_4).
With the proper-time cutoff (\Lambda=\ell_{\mathrm{sp}}^{-1}),
[ S_{\Pi,\mathrm{loc}}^\Lambda =-\frac{1}{2(4\pi)^2} \int_M\mathrm d^4x,\sqrt g \left[ \frac{\Lambda^4}{2}a_0 +\Lambda^2a_2 +\log!\left(\frac{\Lambda^2}{\mu^2}\right)a_4 \right]. ]
For (S_\Pi=+\tfrac12\log\det' A_g), one minimal scalar therefore contributes
[ \Delta c_{\mathrm{EH}}^\Lambda =-\frac{\Lambda^2}{12(4\pi)^2}. ]
Zeta regularization gives instead
[ \frac{\mathrm d S_\Pi^\zeta}{\mathrm d\log\mu} =-\zeta_A(0), ]
which is governed in four dimensions by the integrated (a_4) coefficient. It does not produce an (a_2\mu^2R) term.
The renormalized metric variation may contain
[ \delta S_\Pi^{\mathrm{ren}} =\int_M\mathrm d^4x,\sqrt g \left[ c_{\mathrm{EH}}^{\mathrm{ren}}G_{\mu\nu} +c_\Lambda^{\mathrm{ren}}g_{\mu\nu} +\beta^{\mathrm{ren}}B_{\mu\nu} +\cdots \right]\delta g^{\mu\nu}. ]
The observed coupling is defined by the matching condition
[ c_{\mathrm{EH}}^{\mathrm{ren}}=\frac{1}{16\pi G_N}. ]
The heat-kernel expansion determines the cutoff-sensitive contribution but not the finite value or sign of this coefficient. Additional operator content and a renormalization condition are required to obtain the observed positive Newton constant.
The local Einstein term dominates the four-derivative sector when
[ RL_4^2\ll1, \qquad L_4^2=\left|\frac{\beta^{\mathrm{ren}}}{c_{\mathrm{EH}}^{\mathrm{ren}}}\right|. ]
Under cutoff-dominated matching without cancellations, (L_4=O(\ell_{\mathrm{sp}})) and (G_N=O(\ell_{\mathrm{sp}}^2)). This is conditional scaling, not a numerical prediction.
Under the separate coherence-extensivity hypothesis, the determinantal Born--Infeld density
[ \sqrt{-\det(g_{\mu\nu}+\ell_{\mathrm{sp}}^2R_{\mu\nu})}-\sqrt{-g} ]
is an admissible tensorial completion of a supplied Einstein infrared term, but the structural conditions do not select it uniquely: the admissible completers form the family (\Phi(X)=\exp\sum_i h(\lambda_i)) with (h) constrained only at the origin and at the saturation boundary, containing in particular (\Phi_a(X)=\sqrt{\det(I+X)},e^{a,\mathrm{tr}(X^2)}) for every real (a). This classification does not determine the finite Einstein coefficient.
Spectral geometry fixes the available local tensor structures and their cutoff sensitivities. It does not yet derive the positive finite gravitational coupling. The remaining problem is to construct an independent matching principle from the complete projected operator content.
J. Beau, Conditions for an Infrared Einstein Sector from Spectral Geometry, Zenodo, 2026.
Portions of the development benefited from iterative interactions with large language models used as analytical assistants. All claims, interpretations, and final formulations remain the author's responsibility.