Notes on a Gaussian-Based Distribution Algebra for the Non-linear Wave Equation of the Shift Vector in Quantum Foam

We develop a non-linear distributional renormalisation algebra for Gaussian Quantum Foam, built from sequences of scaled Gaussians on spacelike hypersurfaces of homotopic, globally hyperbolic spacetimes and their distributional limits.

The algebra is closed under multiplication and second-order differentiation, with all non-linear operations defined on smooth representatives before taking the limit. Applied to the non-linear scalar-field wave equation for the shift vector, the wave operator converges to a linear combination of $\delta$ and $\delta''$, encoding a sharply localised curvature impulse that displaces the vacuum; in the correspondence limit, the equation reduces to the massless Klein–Gordon equation.

 Classical singularities are replaced by a well-defined distributional structure: the scalar Ricci projection is non-negative on the singular support and converges to a positive $\delta$–$\delta''$ combination, while away from the support, in the emerging classical spacetime, the strong energy condition is violated on open sets. The trace of the extrinsic curvature, the mean curvature, and the null expansions vanish on the support (no trapped surfaces). For finite values of the sequence index, there exist open neighbourhoods in which both the inward and outward null expansions are strictly negative; thus, locally and in a classical context, trapped surfaces can occur in those regions.

The level sets of the global time function, together with their normal, become asymptotically null, yielding a limiting unstable characteristic hypersurface that fixes evolution by null data and forbids any extension
into chronology-violating regions.

Finally, it is argued that, within this framework, a gravity-induced spontaneous state reduction restores the Equivalence Principle in the emerging classical spacetimes.