Unified Substrate Closure under Admissible Regime Determination

This work completes a constructive determination of physical mechanics under an explicit admissibility hypothesis: a regime qualifies as physical only if it admits a governing equation, a numerical calibration, and a directly testable observable. Regimes that fail this equation–number–observable criterion do not define predictive physics within the present framework.

Under this hypothesis, the admissible expressions of a single physical substrate are sharply constrained. Exactly five regimes satisfy the criterion: Propagation (light), Ordering (time and causality), Localization (matter), Coupling (gravity), and Global Drift (cosmology). Each regime is derived independently, calibrated arithmetically, and associated with an observable channel. No regime introduces additional primitive structures or unfixed constants; all numerical content is fixed internally by calibration.

The central result is that, under the same admissibility hypothesis, these five regimes are not independent constructions. They arise as controlled limits of a single mechanical law. The unifying object is the principal phase of a substrate excitation, evolved with respect to an induced ordering parameter rather than a primitive time. From this phase, canonical energy–momentum are defined, and admissible dynamics reduce to a single ordering-covariant Hamilton–Jacobi master constraint involving three induced scalars only: an ordering-distortion potential, a localization gap, and a global ordering drift.

Within this admissible class, the master constraint constitutes the minimal closure compatible with all five calibrated regimes. In the gapless limit it enforces null propagation with a single invariant speed. With nonzero gap it yields gapped dispersion and the inertial mass relation. With ordering distortion it reproduces gravitational time dilation and weak-field coupling. With drift of the ordering background it yields redshift as an ordering ratio and enforces the observed one-mode cosmological structure. Any extension of the constraint by additional terms or degrees of freedom violates at least one regime calibration and is therefore excluded under the stated hypothesis.

The resulting framework is a closed mechanics conditional on the admissibility criterion: one substrate, one ordering observable, one invariant propagation speed, one localization gap–mass identity, one infrared coupling closure, and one global drift law, unified by a single master constraint whose form and coefficients are fixed by calibration rather than postulated. Under these conditions, the five regimes exhaust the admissible ways in which the substrate can manifest in physical observables.