Mathematical Foundations of Organizational Field Theory

This article establishes the foundational mathematical architecture of Organizational Field Theory (OFT) through a hierarchically constructed configuration-space framework. The objective is deliberately limited: the present work does not claim a complete analytical or physical theory, but defines the logical language, admissibility criteria, construction axioms, primitive mathematical objects, projected variational structure, emergent geometry, stability framework, and spectral perspective required for the systematic development of the research program. The formalism is governed by a constructive principle: no mathematical object is introduced unless its role is required by the preceding logical structure. An admissible state is represented by the ordered collection S = (Ω, κ, Φ^μ, Ψ^ν) and belongs to a constrained configuration space Q_OFT. Arbitrary variations are projected onto the admissible tangent structure by Π_OFT, while a real-valued functional A_OFT: Q_OFT → ℝ provides the global variational object from which subsequent governing equations are to be derived. The configuration metric, connection, curvature, Hessian, and spectral problem are introduced at the foundational level without prematurely fixing their detailed analytical realizations. The scope of this manuscript is defined by a deliberate separation between foundational construction and analytical realization. The latter is assigned to a sequence of subsequent papers, each built upon the common mathematical core established here. Paper 0 therefore functions as the reference architecture of the OFT research program: logically closed at the level of its construction rules, yet analytically open to rigorous extension.