Reformulated Solution of O1 in the EPPQ Framework: Context, Interface, and Continuum Reconstruction

We reformulate the open problem O1 in the EPPQ framework as a contextual interface reconstruction problem. The original EPPQ program starts from a primordial distinction α and the Principle of Absolute Historicity (PAH), with a local correction operator Θ selecting admissible states by minimizing an inconsistency functional I. The present work incorporates the advances obtained in the project: (i) a decomposition of each state into an updated visible frame and a contextual residual layer, (ii) an interface functional whose effective visible cost is obtained by marginalization over contextual fibers, (iii) a rigorous reduction of O1 to the geometric-analytic hypotheses LEP4 and LEP5, and (iv) a corrected conditional route from discrete relational dynamics to Mosco convergence and continuum geometry. This first part develops the model in a mathematically explicit way. We define the state space, the contextual fibers, the interface functional, and the effective inconsistency functional. We then prove the first uniform geometric consequence: bounded degree of minimizers. The subsequent sections set up the compactness and finite-range hypotheses needed for the later steps (doubling, Poincaré, LEP4, LEP5, and Mosco convergence), which are reserved for Part II.