A number is usually treated as a value that can be carried from one context to another without changing what it is. This assumption is extraordinarily useful, but it is not always ontologically sufficient. In many physical, geometric, biological, and computational systems, the value of a quantity cannot be separated from the direction in which it is evaluated, the scale at which it is resolved, or the invariant structure it preserves under transformation. This paper introduces Noetherian Finsler Numbers as closure-coordinate objects for such systems. A Noetherian Finsler Number, or NFN, is not proposed as a replacement for real numbers, complex numbers, vectors, tensors, spinors, operators, Finsler metrics, or Noetherian invariants. Rather, it is proposed as a higher-order representational object for cases in which value is closure-conditioned. In its minimal form, an NFN is written 𝑁𝑓 = (𝑀, 𝐝, 𝑆, 𝒬), where 𝑀 denotes magnitude, 𝐝 denotes directionality or phase-flow, 𝑆 denotes scale or closure regime, and 𝒬 denotes the conserved invariant or closure charge that preserves identity under admissible transformation. The framework joins three principles. From Finsler geometry, it inherits the idea that metric structure may depend on direction. From Noetherian symmetry, it inherits the idea that invariance under transformation generates conserved structure. From closure ontology, it inherits the idea that identity is not static sameness, but persistence under admissible transformation. The central claim is that NFNs apply where the conditions of value are part of the value. They provide a formal language for anisotropic systems, multiscale dynamics, closure-preserving transport, quantum coherence, turbulence, biological boundaries, topological protection, and invariant-aware artificial intelligence. In this sense, NFNs may be understood as numbers with closure memory: number-like objects that record not only value, but the conditions under which value remains itself. Keywords Noetherian Finsler Numbers; closure coordinates; Unified Coherence Closure Framework; closure ontology; Finsler geometry; Noether’s theorem; anisotropic geometry; scale-dependent systems; conserved invariants; closure memory; closure identity; multiscale dynamics; quantum coherence; turbulence; biological boundaries; invariant-aware AI.
Philip Lilien (Fri,) studied this question.