The Hierarchical Non-Classical Measure (HNCM): A Capacity-Induced Neutrosophic Framework for Quantifying Indeterminacy

We introduce the Hierarchical Non-Classical Measure (HNCM), a framework that quantifies indeterminacy by deriving—rather than assigning—a neutrosophic triple (T, I, F) from a single monotone normalized capacity and its conjugate dual. The normalization T + I + F = 1 then becomes a theorem: truth is the capacity (belief), falsity its complement (disbelief), and indeterminacy the exact belief–plausibility gap of Dempster–Shafer theory and imprecise probability. For Sugeno λ-measures we prove the closed form I(A) = λ T(A) F(A), which makes the construction well posed if and only if λ ≥ 0 and continuously recovers additive probability as λ → 0. A general 2-additive construction encodes criterion-specific redundancy and complementarity through the cut-identity I(A) = Σ_cut m_ij, and the channel is lifted to arbitrary (possibly uncountable) measure spaces, where it equals the credal-interval width of a coherent lower probability and, for distorted probabilities ν = h∘P, has the closed form I(A) = 1 − h(P(A)) − h(1 − P(A)). Every reported quantity is regenerated by a single seed-fixed program, with the structural identities verified to machine precision and the Choquet integral cross-validated by independent estimators; the framework is characterized by variance-based Sobol indices, Monte-Carlo uncertainty quantification, robustness sweeps, and a reproducible multi-criteria decision study with a λ-sensitivity analysis of the ranking. A dedicated scientific risk assessment and a falsifiability roadmap delimit the domain of validity. The contribution is not the resolution of open scientific problems but a principled, internally consistent, and independently reproducible reconstruction that removes the arbitrariness of prior neutrosophic constructions.