Projection, Measure, and Idempotent Relations: Collapse, Rigidity, and a Fixed-Point Coupling Law

Abstract

We introduce a minimal ZFC-internal axiom system for pre-structural data (X, A, mu, mu^{otimes2}, R, I, Pi_R, G, E_0, eta), coupling a finitely additive measure mu, an idempotent retraction Pi_R : X -> R, and an idempotent symmetric relation G through a single coupling law (Axiom III). Our central result is a collapse theorem: every admissible model is concentrated on the representative sector R, namely mu(X\R)=0, with no full-partition hypothesis. As immediate consequences, eta<1 holds automatically and the two-point load is rigidly determined, mu^{otimes2}((B x X) cap G) = mu(B)/(1-eta), so it is not an independent datum once (mu, eta) are fixed. A further consequence is component quantization: every measurable G-equivalence class C has mass mu(C) in {0, (1-eta)^{-1}}; as an arithmetic corollary, when finitely many positive-mass classes exhaust the measure their count equals (1-eta)E_0, a positive integer, tying the scale E_0 and the rate eta together. We establish consistency in ZFC by explicit finite, countable, and continuous (Lebesgue) models with eta neq 0, and prove mutual independence of the three axioms and of the three subclauses of Axiom III: collapse is driven by invariance III(b) alone, eta<1 and load rigidity add the coupling law III(c) and the retraction property (Axiom I), and Axiom II enters only at quantization. Finally we give a fixed-point reformulation of the coupling law as the unique bounded finitely additive solution of a Banach contraction f = T_eta f, and a null-extension factorization exhibiting every admissible model as its identity-retraction core extended by mu- and mu^{otimes2}-null data.