NSk--A4 Critical Thresholds for Affine Descent Families

When does the functional F_α = Ẽ − αS decrease along every admissible step of a discrete dynamical system? This module gives a sharp, fully elementary answer for one-parameter affine families: the set of parameters for which F_θ = F₀ − θB (with B ≥ 0) is a descent functional is exactly the half-line [θcrit, ∞), where the critical threshold θcrit is the infimum of all admissible parameters. For the canonical energy-entropic instance F_α = Ẽ − αS this threshold equals the supremum of the step-wise ratios ΔẼ/ΔS over all steps with positive entropy increment — a formula computable directly from the local data of the system. The module is organised in four layers. PRE-PURE fixes the minimal combinatorial vocabulary: admissible systems, chains, increments, and step costs, with no geometry, probability, or physical assumptions. PURE develops the abstract theory of descent certificates for affine families: the half-line structure of the admissible parameter set, margin estimates (the cost per step is at least η·B whenever α exceeds the threshold by η), Err+ certificates for approximately satisfied conditions, global telescoping inequalities, cost summability on infinite chains, and variable-parameter schedules. CORE instantiates the scheme for F_α = Ẽ − αS — a discrete analogue of statistical free energy — proving that αcrit = supΔẼ/ΔS: ΔS > 0 is the exact admissibility boundary, and deriving all certificate variants in closed form. EXEC translates the results into operational language, clarifying the boundary cases (αcrit = ±∞) and the behaviour under approximate satisfaction and varying schedules. An appendix records the module's role in the NSk/ψ programme and its optional import contracts with the NSk–MathFoundation and NSk–Entropy modules. All theorems and proofs are finite and combinatorial. No convexity, differentiability, or metric structure is assumed anywhere.