Probability as Structural Underdetermination

This working paper develops the Probability branch of Phase Compatibility Theory by asking a foundational question that standard probability formalisms do not, by themselves, uniquely answer: what physical structure determines the alternatives to which probability is assigned? The proposed ordering begins with recursive whole formation. Presently instantiated participants enter lawful relations that may close into a new physical whole, and that whole may subsequently participate in higher-order organisation. Each formed whole constrains the relations physically accessible from its present organisation. Available energy, conservation laws, coupling conditions and selection constraints further determine which channels are open. Compatibility then restricts those accessible relations to the continuations capable of lawfully forming, persisting or reorganising. Probability enters only after this structure has been established, as the numerical weighting of unresolved admissible alternatives. A realised relation may form a new whole and thereby alter the accessible continuation set on which the next probability distribution is defined. Probability is therefore treated not merely as repeated selection from a permanently fixed sample space, but as part of a recursive sequence in which physical organisation changes the structure of subsequent possibility. “Compatibility streams” describe lawful sequences of recursively formed wholes and their branch points; they do not imply that every mathematically expressible branch exists as a separately realised world. The paper maintains a strict distinction between source formation, physical accessibility, compatibility, probability weighting, observer access, registration and marginalisation. Observer-resolved probabilities may differ because observers possess different physical access relations or resolutions. Improved ensemble precision, coherence, averaging or repeated measurement normally refines the observer map around an unchanged source transition; it does not create a new source probability structure unless a distinct whole-level state, mode or transition is physically identified. The manuscript also records a candidate duality–orthogonality representation in which complementary inversion may correspond to a relative phase of 180° and orthogonal resolvability to 90°. This remains a proposed representational bridge rather than an established physical law. Version 0.4 is an unaudited working governing-theory draft. It does not derive the Born rule, establish a universal probability-weighting exponent, identify energy with Phase, equate negative Phase with antimatter, or report an empirical Probability result. The Phase-native formation, accessibility and weighting laws remain open to independent derivation and prospective testing.