Dynamic Gate Numbers: The Mathematical Structure of Irreversible Constraints on Future Admissibility

Dynamic Gate Numbers: Trajectory-Valued Quantities with Admissible Transitions This paper presents an axiomatic theory of Dynamic Gate Numbers, a class of quantitative objects designed to replace static scalar values in systems where cost, difficulty, or value depends on state and history. Classical arithmetic treats numbers as observer-independent constants; this work formalizes the conditions under which such a treatment necessarily fails. A Dynamic Gate Number is defined as an ordered pair (Nx,Γ)(N_x, \Gamma)(Nx,Γ), where NxN_xNx is a state-dependent cost trajectory and Γ\GammaΓ is a gate operator selecting admissible state transitions. Quantities are evaluated only along gate-admissible trajectories, making constraint and quantity inseparable components of a single mathematical object. From five axioms governing state spaces, cost functions, transition structures, gate operators, and nontriviality, the paper proves three fundamental results: Degeneracy of static numbers — classical static numbers arise only as degenerate cases under transition invariance or enforced amnesia. Impossibility of universal scalar representation — no single scalar function can faithfully represent the preorder induced by trajectory-valued, gate-constrained quantities. Necessity of nontrivial gates — non-degenerate dynamical quantities require gates satisfying ∅⊊Γ(s)⊊Σ\varnothing \subsetneq \Gamma(s) \subsetneq \Sigma∅⊊Γ(s)⊊Σ; fully open or fully closed gates inevitably produce degeneracy. The framework provides a common mathematical substrate underlying previously developed resonance theory and Yang Jihoon Dynamics, which are shown to instantiate the same axiomatic structure. These connections are presented as formal mappings rather than empirical validations. This work is purely mathematical. It contains no simulations, numerical experiments, algorithmic implementations, or empirical claims. All results follow from structural necessity within the axiomatic system. Interpretation is intentionally left open.