Foundations of Infratier Physics: Closure Fibers and the Dynamics Below Structural Equivalence

Orientation. Closure identifies structural equivalence at a chosen resolution. Infratier analysis studies the distinctions that remain below that equivalence. Dynamics determines which of those distinctions must be restored for prediction. The term infratier is used here as an organizational term for structured distinctions lying below a specified equivalence or quotient resolution. It is not an additional closure axiom or primitive mathematical object. In the present framework, the primary infratier object is a closure fiber whose members share one closure identity while potentially differing in saturation, defect, robustness, pathway, observability, or future closure behavior. Closure operators are classical mathematical objects: extensive, monotone, idempotent maps that assign presentations to closed realizations. Their associated equivalence relation partitions a domain into closure fibers containing potentially many distinct presentations of the same closed identity. Classical closure theory naturally emphasizes the closed sector and the quotient induced by this equivalence. The present work asks what dynamically relevant structure can remain inside those equivalence classes. The canonical predictive state is distinguished from realization-dependent fiber coordinates such as saturation defect, robustness, latency, pathway state, and hysteresis state. A chosen coordinate may be finer than the predictive quotient, coarser than it, or represent it exactly. Predictive minimality is therefore target-relative and must not be conflated with presentation reconstruction, saturation recognition, or computational minimality. The resulting structure is proposed as a mathematical foundation for Infratier Closure Physics: the study of mathematically or physically significant organization that persists below a coarse closure-equivalence resolution. The work does not claim novelty for closure operators, quotient constructions, future-output equivalence, factor maps, or generic state minimization. Its distinctive contribution is the closure-specific organization of these ideas around closure fibers and the exact question of which distinctions closure coarse-graining forgets, which dynamics requires for prediction, and which stronger targets require additional coordinates. Keywords: closure operators; closure fibers; predictive equivalence; quotient dynamics; coarse-graining; hidden state; state reduction; dynamic subfibers; predictive depth; reclosure; closure transition; hysteresis; mathematical physics; Infratier Closure Physics