The Rhys Canon: Five Structural Limits on Internal Selection

This deposit records The Rhys Canon, a collection of five independent, conservative impossibility theorems that identify precise structural limits on internal selection in formal systems. Across mathematics, complexity theory, spectral analysis, asymptotic frameworks, and dynamical systems, many foundational problems implicitly require more than existence or consistency. They require selection: a distinguished scale, a generic instance, an integer invariant, a finite value, or a determination of which solutions are admissible or physically relevant. The Rhys Canon isolates five settings in which such selection is provably impossible without extending the underlying ontology. The five canon laws are: Rhys I — No Canonical MetricPurely arithmetic frameworks cannot internally define a representation-independent scale, normalization, or metric capable of selecting a distinguished analytic value. Rhys II — Shannon–TypicalityWorst-case complexity theory admits no intrinsic notion of typical, generic, or prevalent instances without importing external probability geometry. Rhys III — Spectral–IndexSmooth spectral or analytic data alone cannot yield nontrivial integer-valued indices; discreteness requires singular or topological structure. Rhys IV — Asymptotic–RenormalizationAsymptotic or divergent frameworks cannot canonically select finite values without a renormalization scheme or reference scale. Rhys V — Observer / AdmissibilityDynamical systems whose language encodes only equations of motion cannot internally determine which solutions are stable, physically relevant, or admissible; such selection requires external witness or admissibility structure. Each theorem is ontology-relative, mutually independent, and conservative: crossing any boundary is possible, but only by explicitly extending the framework. No claims are made regarding undecidability, inconsistency, or the truth of specific conjectures. The Canon provides a unifying diagnostic for why certain long-standing problems resist solution—not due to technical difficulty, but because they presuppose structure that the native ontology cannot supply. A classification table mapping major problems (including RH, P vs NP, BSD, Navier–Stokes, and Yang–Mills) to their corresponding boundaries is included. This document serves as the law-level foundation for related boundary atlases and program-level results deposited separately.