Measurability as Retentive Emergence: A Structural Criterion for Measurement Beyond Observation

This paper introduces a minimal operational criterion for the emergence of measurability in systems dominated by noise, dissipation, or observational incompleteness. Within the ψ-Architecture framework, measurability is defined not by the presence of an observer, instrument, or semantic interpretation, but by the system’s intrinsic capacity to retain structural difference (Δψ). We show that a system becomes measurable if and only if Δψ remains invariant under bounded degradation of observation, including coarse-graining, noise increase, or partial loss of resolution. This transition coincides with crossing the retentive Ξ-threshold, beyond which distinctions become system-intrinsic and reproducible. The resulting ψ-measurability threshold reframes measurement as an emergent consequence of retention rather than an a priori epistemic act. The criterion is domain-independent and applies uniformly across physical, chemical, and informational systems. This work is accompanied by a short companion paper that clarifies observability as a derived consequence of retentive measurability within the same framework.