Constructive Solution Theory on Frozen Organizational Geometry: From Covariant Field Equations to Admissible Generation, Branches, and Effective Limits

Abstract

This paper develops a constructive solution theory for the organizational field system established in Paper V. The configuration manifold, base metric, affine connection, and curvature are retained as frozen inherited geometry, while the organizational field, an independent field connection when present, and constraint multipliers constitute the solution variables. The analysis distinguishes positive-definite Riemannian sectors, where the minimal problem is elliptic, from Lorentzian sectors, where an evolution direction, hyperbolicity, admissible initial data, and constraint propagation are additionally required. Candidate spaces, residual maps, boundary data, solution-consistent generators, regularity classes, weak limits, perturbative hierarchies, local solution manifolds, bifurcations, and effective projections are defined relative to a declared analytical sector. The nonlinear continuous realization introduced in Paper V is then used to derive explicit curvature-dependent equilibrium branches and a locally convergent generation map. The resulting framework establishes relative constructive closure: every solution-theoretic object remains traceable to Papers 0–V together with clearly stated functional-analytic assumptions, without modifying the frozen geometry or claiming universal existence, uniqueness, or physical adequacy.