Projection, Measure, and Idempotent Relations: Independent Axioms and a Fixed-Point Coupling Law

TL;DR

This paper develops a minimal axiom system within ZFC for pre-structural data involving measures, relations, and idempotent maps, establishing independence, fixed-point solutions, and classification results.

Contribution

It introduces a novel minimal axiom system for pre-structural data, proves independence of axioms, and characterizes solutions via fixed-point reformulation and classification theorems.

Findings

Axioms I-III are mutually independent, with explicit separating models.

The coupling law admits a unique fixed-point solution expressed in closed form.

Under certain conditions, the admissibility problem reduces to the identity-retraction case, leading to classification of measure classes.

Abstract

We introduce a minimal ZFC-internal axiom system for pre-structural data (X, A, mu, mu^{otimes 2}, R, I, Pi_R, G, E_0, eta), where Pi_R : X -> R is a designated map and G subset X x X is a measurable relation; admissible structural models are those pre-structural data satisfying Axioms I-III, which couple a finitely additive measure, an idempotent retraction, and an idempotent symmetric relation through a single coupling law (Axiom III). The axiom system is satisfiable in ZFC via explicit finite and countable models, including finite families with eta neq 0. The three axioms, and the three subclauses of Axiom III, are mutually independent, witnessed by explicit separating models. The coupling law admits a fixed-point reformulation: it is the unique bounded finitely additive solution of a Banach-contraction equation f = T_eta f determined by (mu, Pi_R, eta), with closed form f_*(B) = mu(B) + (eta/(1-eta)) mu(Pi_R^{-1}(B)) and a Neumann-series expansion. Admissible structural models with a common eta form a category Struct_eta in which Pi_R and G appear as idempotents analogous to the two sides of a monad-comonad pair. Under fiber measurability together with either a finiteness hypothesis (R-fin) or countability plus sigma-additivity (R-ctbl), a quotient-factorization theorem reduces the general admissibility problem to the identity-retraction case Pi_R = id_X; in that case, under the hypotheses of Theorem 5.6 (pi-id-classification), each G-equivalence class C_k satisfies mu(C_k) in {0, (1-eta)^{-1}}.