Euclidean Condensate Theory (ECT): Emergence of Spacetime, Quantum Mechanics, and Gravity from Spontaneous O(4) Symmetry Breaking

Abstract

Euclidean Condensate Theory (ECT) is a research programme asking whether physical time, Lorentzian propagation, quantum structure, and gravitational response can arise as effective sectors of a medium defined initially on a four-dimensional Euclidean carrier. Its minimal concrete model contains a real scalar field and adopts, as a physically motivated structural input, an ordered state with a nonzero scalar gradient; the dynamics that select, stabilise, and form this state remain open. The gradient direction has an exact (O(3)) stabiliser. For independently supplied coefficients, the scalar principal tensor has Lorentzian signature exactly when (\beta(\beta-\alpha)<0), equivalently (\alpha>\beta>0) on the retained (\beta>0) branch. This is an exact result within the declared effective model, but it does not by itself derive physical time, a universal observable metric, or a common causal cone for matter, photons, and tensor gravity. The preprint develops sector-specific completion programmes for gravity and cosmology, galactic response, quantum and open-system structure, and gauge and matter sectors. Throughout, it distinguishes exact mathematics in explicitly stated models, conditional structural inputs and closures, phenomenological fits and imported benchmarks, and unresolved physical ownership. The controlled outputs include proofs of class, reproducible diagnostics, and scoped negative results. The galactic-response closure, for example, yields exact internal algebra and a conditional slope-four baryonic Tully–Fisher relation only after a separately supplied gravity bridge; the bridge, the origin of the acceleration scale, and a predictive population likelihood remain open. The quantum programme does not yet derive physical Born probabilities, unique outcomes, or an ECT-specific Bell forward law. The physical tensor action, Newton matching, common causal structure, and physical gauge and matter sectors likewise remain open. ECT is therefore not presented as a completed ultraviolet theory, a derivation of all known physics from one scalar, or a source of fully derived ECT-specific numerical predictions. It is an auditable common-origin architecture with explicit assumptions, conditional results, open problems, and route-specific acceptance and rejection criteria. Release-scoped calculation, validation, and figure-generation code, together with data contracts and provenance, is publicly available at https://github.com/chufelo/ECT-preprint-code. Narrative companion: https://doi.org/10.5281/zenodo.19430795