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

After having received a favourable referee report, I'm now pushing to get it ready for publication—assuming the board aligns with the referee. Working in near-total isolation is brutal; rooting out typos and basic mistakes is a painstaking slog. A few more refactor sessions remain before the final version 100; While I wait for the transition from autumn to winter, it's important to clarify the role of test functions as probes and the properties they must satisfy. We can't select test functions arbitrarily if we want to probe a physical quantity. Any measurement occupies a spatial volume element and therefore involves averaging; in distribution language, that is a linear functional acting on the space of admissible test functions; Back home from the mountains and on my way to work, I simplified my example about the breach of strong causality. End of the break—but an example was needed for the time-machine problem: I formulated it but cut short the refactoring of the formalities—old-school C++ coder that I am. The example was needed, at least for me. If future spacetime engineers try to tamper with and warp spacetime locally in an initially globally hyperbolic region into a time machine—don't. A geon consists of self-gravitating massless particles; it's an unstable setup, and you don't want to blow it up. A geon forms a Cauchy horizon, and the KRW theorem applies inside the reliability horizon. Thus Hawking's Chronology Protection Conjecture likely holds in quantum foam, and I'm back to work on the time-machine problem; It is autumn in Sweden—another late session, but worth it: I showed that general relativity in Quantum Foam carries its own quantum theory. Since spring and early summer 2025, I've made real progress after realising that Gaussians can bypass the long-standing problem of products of distributions. The field equation still eluded me until I sat down after work—having just dealt with securities amid the turmoil following the Rose Garden announcement—and defined a nonlinear, renormalised distributional algebra; that was the turning point: I could finally construct the field equation of Quantum Foam. It felt like a spark—the beginning of time (and the end of singularities), as Wheeler put it: "For time is not a primordial and precise concept; it must be secondary, derivative, and approximate." My work aims at a beginning of time in a frozen state, with the bare mass. In that sense, the project is ended and complete—though I know such a statement is always false and that, with time, more will come. Still, the chapter on singularities—and the fact that none of Penrose's conditions for avoiding singularities are required once one works in distributional geometry—marks a natural end to a long quest. It has been a journey spanning more than thirty years: from early work on unstable photon rings near rotating black holes, through the time-machine problem, and then a long absence from research until 2022. Even then I was thinking, sketching, searching for a resolution. The multiple versions of this work are intentional, allowing precise tracking of theoretical refinements and making it easier to compare, reuse, and reference key developments across the Quantum Foam project;