Invariant Closure and Representation Preservation: A Geometric Foundation for Space, Momentum, Energy, Action, and Time

Modern physics relates space, momentum, energy, action, and time through experimentally established equations, yet these quantities are ordinarily introduced as independent primitives whose mathematical kinship is discovered only after the fact. This paper develops a representation-based geometric framework that reverses that order. Two principles are postulated: the existence of an invariant closure — a completed geometric cycle possessing a fixed closure period — and a Representation Preservation Principle, according to which every observable quantity is one member of a complementary pair whose combined magnitude reproduces the invariant closure. From these principles a representation geometry is developed in which the closure phase, which governs the evolution of a single closure, and the representation angle, which governs the partition of the closure between complementary observables, are shown to be independent geometric degrees of freedom. A Representation Preservation Theorem then proves that every global dimensional normalization of this geometry preserves both the Pythagorean relation among the complementary representations and the representation angle itself. Successive application of the theorem generates a hierarchy — geometry, velocity, momentum, energy, action, and time — in which each level follows from the previous one through a single normalization constant rather than an independent physical postulate. Within this hierarchy the Planck relation, the de Broglie relation, the Compton wavelength, the propagation relation, and the relativistic energy–momentum relation are recovered as specific normalizations of one invariant structure. The framework leaves these established relations unchanged; its contribution is the identification of the common geometric origin responsible for their shared mathematical form.