The Unreachability Principle: From Metric Invariants to Operational Selmer Geometry

We introduce a framework where mathematical objects are treated as operationally unobtainable attractors approached through sequences of finite approximations. For classical constants and algebraic integers, this yields the Unreachability Principle, where attractors are organized by a metric threading depth. Empirical testing validates this principle for Mahler measures (n = 2256, perfect Ω_p detection). However, large-scale analysis of elliptic curves (n = 15,112) reveals a fundamental breakdown of metric organization: rank and canonical height show negligible correlation with metric invariants (R2 ≤ 0.09), falsifying initial scaling predictions. We prove this failure is a structural necessity: rank is a coherence dimension, not a metric invariant. Consequently, we propose Operational Selmer Geometry, where Selmer groups represent coherent operational threads at finite depth. Within this framework, the Birch–Swinnerton-Dyer conjecture is reframed as a stability theorem: analytic instability equals algebraic coherence. This shift positions Selmer theory not as an auxiliary tool, but as the fundamental language of unobtainable arithmetic objects.