Infrared Scaling of the Cosmological Constant and Statistical Realization of Vacuum Stress

We develop a three-window framework for interpreting the smallness of the observed cosmological constant.

{Window I} derives an infrared (IR) scaling law for the geometric cosmological term from an RG-improved Einstein--Hilbert action with running couplings $G_k$ and $\Lambda_k$ and the cosmological scale identification $k=\zeta H$.

Varying an RG-improved action (rather than inserting running couplings ad hoc) yields modified Friedmann equations that remain consistent with the Bianchi identity.

If the dimensionless couplings approach an IR fixed point, the dimensionful cosmological term obeys $\Lambda_k \propto k^2 \propto H^2$ and admits a self-consistent de Sitter regime.

 

{Window II} introduces a separate statistical ``realization window'': gravity is assumed to couple effectively to a coarse-grained vacuum-stress invariant whose realized contribution is exponentially suppressed by large-deviation statistics.

This provides a phenomenological mechanism for why only a small fraction of microscopic vacuum stress contributes to macroscopic curvature, without altering the tensorial structure of Einstein's equations.

 

{Window III} supplies a correlator-based derivation of the coarse-grained variance $\sigma_\chi^2(k)$ from an explicit model of the connected two-point function of the stress invariant.

This yields a controlled scaling law for the large-deviation rate $I(k)$, clarifies when the infrared suppression is automatic, and identifies the parametric conditions needed to match the observed hierarchy.

The result is not presented as a complete microscopic solution of the cosmological constant problem, but as a structurally minimal reformulation in which the hierarchy arises from IR geometric scaling and statistical realization in a coarse-grained gravitational description.