Observer Equivariance as an Interpretation of Quantum Theory: Symmetry, Born weights, and Fourier uncertainty from a single invariance principle

Observer Equivariance (OE) is proposed as an interpretation of quantum theory based on a single structural principle: objectivity is invariance under changes of observer perspective. The framework is formulated using a projection p:\mathcal O \to \mathcal S, where \mathcal O is a groupoid of observer perspectives (descriptions) and \mathcal Srepresents structural content intended to be objective. Within this setting, the paper links three familiar pillars of quantum theory as consequences of one invariance requirement: Symmetry. Physical symmetries are characterized as automorphisms of \mathcal S that are liftable to \mathcal O in a way compatible with the relevant gauge freedom. This recovers Wigner/Uhlhorn-type constraints from a general “liftability” viewpoint. Born weights. Event weights are constrained by invariance under a minimal internal “mixing” operation (unitary change of basis within a fixed structural event), together with coarse graining and phase invariance. A proof sketch shows how this yields a functional equation that forces quadratic (Born-type) weights. Uncertainty. Translational perspective changes motivate a Fourier/Pontryagin-duality analysis in which Plancherel/Parseval norm preservation selects an L^2 structure and yields standard Fourier uncertainty relations as representation-theoretic constraints. The article positions OE relative to QBism and Relational Quantum Mechanics: it shares the view that “facts” and state assignments are perspective-indexed, while differing in that symmetry constraints and the probability calculus are treated as structural necessities derived from invariance, rather than as purely epistemic or decision-theoretic postulates. DOI (this version): 10.5281/zenodo.18221625 Concept DOI (all versions): 10.5281/zenodo.18221624 AI use: AI systems (ChatGPT, Gemini, Grok, and Claude) were used for drafting assistance, language editing, and mathematical polishing. The author reviewed the manuscript and takes full responsibility for all claims and any errors.