NSk--A4 Critical Thresholds for Affine Descent Families (archival conceptual version)

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 [θ_crit, ∞), where the critical threshold θ_crit 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 α_crit = sup{ΔẼ/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 (α_crit = ±∞) 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.This record preserves the public development of the NSk–A4 line, first published on 9 December 2025 and subsequently revised in Versions 2 and 3. It is retained as an archival formulation of the energy–entropy descent mechanism and the associated critical-threshold construction. This apparatus was subsequently reorganized, strengthened, and developed in the standalone paper CriticalDescent, which now provides the current formulation of the theory.