It from Bit as Curvature: Einstein--Hilbert Action from 3D Shannon Geometry

Once one accepts information theory as primitive, the three-dimensional Shannon measurement geometry developed in "A Geometric Invariant for 3D Shannon--Nyquist Measurement Geometry'' (henceforth Paper I) admits a natural geometric uplift. In Paper I a Hodge-theoretic measurement geometry on $0$-forms in three dimensions leads to a dimensionless 3D Shannon invariant $\alpha(L_{\mathrm{meas}})$, which plays the role of an effective signal-to-noise ratio scale for multi-mode Gaussian channels.

In this paper we show that the same structures, phrased from the outset on a Hodge $3^*/4^*$ footing, single out a unique leading curvature penalty. On each three-dimensional slice the only local, quadratic, gauge-invariant aliasing functional compatible with the measurement axioms has geometric content proportional to the curvature $3$-form $R^{(3)} *_3 1$. Integrating this slice-wise penalty over a foliation defined by a unit flow yields a four-dimensional curvature $4$-form of extrinsic-curvature type that is equivalent in the bulk to the Einstein--Hilbert action for an effective metric $g^{\mathrm{eff}}_{AB}$.

We work at a meta level: no new physical interpretation is introduced. The sole role of the 3D Shannon invariant $\alpha(L_{\mathrm{meas}})$ is to fix the overall geometric prefactor of the resulting four-dimensional action. Subsequent work can therefore treat our results as imported theorems about curvature penalties and effective actions, with a single pointer back to Paper~I for the definition of $\alpha(L_{\mathrm{meas}})$.