NSk--MathFoundation The Mathematical Foundation of the NSk/ψ Program: Distinguishability, Refinement, Averaging, and Transition Calculus

NSk--MathFoundation establishes the foundational mathematical apparatus of the NSk/ψ Program. The module starts from minimal distinguishability and constructs, from within the system, abstraction classes, partitions, refinement, stabilization, construction cost, mathematical objects, arithmetic, transition operators, averaging, variance, coarse-graining entropy, and compatible finite readouts. Its central principle is that mathematics is not assumed as a pre-existing structure. Instead, mathematical apparatus becomes available only when the corresponding constructive conditions are satisfied. To formalize this principle, the module introduces structural gates, the gate activation principle, the downstream realization record, and the FIN--LIMIT gate. These tools specify when a fragment of mathematical structure is legally activated, when a downstream module may import it, and how a limiting object can be recognized through compatible finite readouts without postulating completed infinity at the outset. The module also defines interfaces to later components of the NSk/ψ Program, including REAL, Lattice, Geo, Algebra, Cost, Entropy, Physicality, Micro, and other downstream modules. All theorem-level results are proved internally from earlier definitions and results of the module. Classical terminology, when referenced, is used only for orientation and comparison, not as an external source of proof.