Geometric Constraint Evolution and Large-Scale Cosmological Correlations: A Mathematical Research Programme

We present a mathematical research programme investigating whether the large-scale statistical correlations observed in the cosmic web admit a natural description through the evolution of geometric constraints, boundary conditions, and time-dependent multipole structures within the framework of General Relativity. Three principal contributions are made. First, we introduce a systematic conceptual reframing in which physically imprecise language — such as “energy transfer between galaxies” or “information propagation” — is replaced by mathematically disciplined counterparts grounded in the ADM formalism [3], the Weyl curvature decomposition [4,42], and post-Newtonian multipole theory [5,6,41]. Second, we introduce the notion of a locally stable geometric domain (Definition 3.1) — a causally convex open subset of spacetime characterised by bounded multipole evolution, boundary regularity, and a quantified geometric isolation condition expressed via the Bel-Robinson tensor [27,28] — as a mathematically rigorous replacement for the informal concept of an “information cocoon,” together with a controlled approximation theorem (Lemma 3.1) relating the reduced geometric state to the full Cauchy data. Third, we identify nine open problems (OP-1 through OP-9), each stated as a precise, falsifiable mathematical question with explicit verification criteria, spanning the Weyl curvature power spectrum, controlled approximation bounds [11,12], multipole evolution with external tidal coupling [35], tidal tensor correlation functions [33,34], BMS framework generalisation to de Sitter backgrounds [25,26], boundary constraint equations, topological characterisation of the cosmic web via persistent homology [15,16,17], Weyl curvature and filamentary structure, and holonomy signatures in cosmological observables [30]. No new physical forces, hidden transport mechanisms, or modifications to General Relativity are proposed. All speculative statements are explicitly classified using a consistent eight-level epistemic labelling system. The programme is designed to be falsifiable at each stage and to connect with established mathematical physics, including post-Newtonian theory [18,41], numerical relativity [31,32], cosmological perturbation theory [1,13], and topological data analysis [15].