NSk--REAL--PURE: Realizational Mathematics without Actual Infinity -- The REAL Layer and the REAL/FORM Separation

This preprint introduces the NSk--REAL--PURE module, a self-contained PURE-layer foundation of Realizational Mathematics within the Nowak–Stachowiak NSk/ψ (NSk/phi) programme. The central thesis is a strict separation between REAL (finite, realizational, non-ideal) and FORM (idealized, continuous, or limit-based mathematics used only as an external descriptive extension). In particular, the module avoids actual infinity, limiting procedures, and σ-additivity as internal principles. The theory starts from a single primitive notion of windows together with finitary refinement operations (partitions, parent maps, and scale steps). From these primitives and axioms, the module derives: (i) a finitary additive calculus on realizational values, (ii) a refinement calculus (tower/total-variance identities in finite form), and (iii) a PURE entropy defined only on finite window distributions (formally distinct from Shannon’s informational setting). Functions such as log are used solely in the external scalar metalanguage to evaluate finite sums, never to construct REAL objects or justify limit transitions. A key outcome is the A4 engine: a finitary free-energy functional and a telescoping descent certificate along admissible refinement chains, yielding an explicit AnchorWitness record (the “anchor exists” result at the PURE level). Physical interpretation is intentionally deferred: the module exports contracts that higher CORE/EXEC modules may instantiate via realization-specific gates, without adding new axioms here. This module is intended as a PURE-level module for the NSk/ψ programme.