The Modal Triplet Theory Program A0: A Structural Theory of Reduced Description

We present a clean, non-circular formulation of Modal Triplet Theory (MTT), a conditional framework governing when stable reduced descriptions exist, how they relate, and why no global reduced description is possible. The theory is formulated abstractly in terms of projection, admissibility, overlap consistency, and contractive induced dynamics, without assuming spacetime, geometry, fields, or dynamics. We introduce admissible domains, coherence capacity, admissibility barriers, and an encoding atlas organizing all reduced descriptions as local charts. We prove core structural results including universality under contractive projection, irreversibility of reduced dynamics, conditional emergence of probability, and the impossibility of global reconstruction. A categorical formulation of admissible encodings is developed, leading to a global chart-of-charts structure that replaces any single global reduced description. This paper establishes the foundational vocabulary, axioms, and interpretive rules for the entire Modal Triplet Theory corpus.