Status. This record is a formal CORE module of the NSk/ψ programme in the FIN layer. It provides a purely finite (discrete) “calculus on refinement”: weighted L² structures, projections, and step/chain decompositions on window partitions linked by refinement maps. The module is designed as a reusable engine for later analytic blocks (in particular Poincaré/Log–Sobolev style arguments built on conditional expectations), while keeping the assumptions minimal and explicit. Scope. We work on a refinement step u₂ ≼ u₁ between two window partitions, with parent masses p (u₁), child masses p (u₂), and the associated conditional distributions q·|₉. The objects are finite tag functions a ∈ ℝ^Jᵤ (no geometry, no PDEs, no functional analysis on manifolds). All identities are proved by finite summation and the refinement interface. Core construction. The module introduces: (1) the weighted inner product and norm on each level u, ⟨a, b⟩ᵤ: = Σ₉∈₉㶃 pⱼ (u) aⱼ bⱼ, ‖a‖ᵤ²: = Σ₉∈₉㶃 pⱼ (u) aⱼ², (2) the refinement operators on tags: a lift L₁₂ (parent-to-child replication) and the conditional expectation / averaging operator Avg₂₁ (child-to-parent coarse graining), (3) the step increment Δ a: = a − L₁₂ Avg₂₁ a and the associated projection viewpoint. Main results. The central outputs are the orthogonal decompositions that make the refinement calculus usable as a drop-in analytic engine: • Step Pythagoras / energy decomposition: ‖a‖ₔ䃒² = ‖Avg₂₁ a‖ₔ䃑² + ‖Δ a‖ₔ䃒², implying that Avg₂₁ is a contraction in weighted L². • Chain (multi-step) decomposition: along a refinement chain uN ≼ … ≼ u₀, the total energy splits into orthogonal increments, yielding canonical telescoping relations and variance/energy bookkeeping that is stable under composition. • Deterministic steps (ΔS = 0): under the standard uncertainty contracts (E0–E2) imported from NSk–Axioms–CORE, a zero-entropy-increment step forces degeneracy of the conditional distributions and leads to an isometry behaviour on the appropriate subspace (no “new information” is introduced by the refinement). Interface and reuse. The module is written as a CORE interface: later blocks can cite it as the canonical finite L²/projection calculus over refinement trees, without re-proving weighted orthogonality and step/chain decompositions. It is compatible with the global CORE interface NSk–MAIN–CORE and uses the refinement axioms from NSk–Axioms–CORE. For the global CORE entry points and the PURE axiomatic base, see: NSk–MAIN–CORE (global CORE interface): DOI: 10. 5281/zenodo. 17897274NSk–Axioms–CORE (PURE axiomatic base): DOI: 10. 5281/zenodo. 17989950NSk--Ontology--MANIFEST (programme map / ontology): DOI: 10. 5281/zenodo. 18064782 Versioning. This FIN–Calculus module is intended to remain stable as a reusable engine. New analytic applications (PLS inequalities, spectral estimates, geometry coupling) should appear in separate records that cite the present module, rather than extending it with application-specific assumptions.
Nowak et al. (Wed,) studied this question.