The Formalism of Domain-Bound Dissolution: Path Integrals and the Emergence of the Cosmological Constant Λ from the P1 Postulate

This work is the first part of a three‑paper series that develops the physical and mathematical consequences of Postulate P1, extending the conceptual architecture established in the six‑paper sequence The Articulation of Postulate P1. Whereas the earlier sequence clarified the ontological and representational foundations of fundamental undefinedness, the present paper initiates the formal continuation of that framework. The central contribution of this paper is a reinterpretation of the variational principle under P1. Because no representational domain can achieve closure, the classical condition δS = 0 cannot hold globally. The Euler–Lagrange equation is therefore reformulated as a boundary formalism in which the extremum condition is replaced by a structural remainder term χ. This term expresses the inherent incompleteness of representation and marks the transition between classical behavior (χ → 0) and quantum behavior (χ ≠ 0). The path‑integral formalism is retained but reinterpreted: its integration domain cannot be ontological under P1, and χ becomes the structural signature of domain‑bound openness. The paper introduces two key structural quantities: the minimal representable coherence Cmin⁡, which defines the threshold of manifestation, and the meta‑structural remainder χ, which expresses the impossibility of representational closure. Their distinction clarifies the emergence of residual constants such as Λ without invoking vacuum energy or physical substrates. The cosmological constant appears as a representational residue of dissolution rather than a dynamical field. This paper establishes the formal and conceptual foundations for the trilogy. Paper 2 will develop the dynamical consequences for quantum systems, and Paper 3 will synthesize the full physical picture, showing how established theories arise as limiting cases of maximal representational coherence.