Symmetry-Driven Renormalization in the Observer-Equivariance Framework

This working paper develops a symmetry-driven renormalization scheme within the Observer-Equivariance (OE) framework, where observer perspectives are taken as ontologically primary and physical law is formulated as equivalence of descriptions under change of perspective. The key idea is that renormalization is not just a technical tool to tame UV divergences, but a structural constraint on how observer perspectives can be coarse-grained while preserving symmetry, anomaly freedom, and a well-defined quantum measure. Mathematically, the paper introduces a G–equivariant prequantum line bundle over the path space of a globally hyperbolic spacetime and imposes the vanishing of an equivariant first Chern class c_1^G(L) as an anomaly-freedom condition. Coarse-graining is implemented by G–equivariant spectral projectors defined via the Laplace–Beltrami operator in a background field setting. A central lemma shows that such coarse-graining preserves the equivalence class of the observer-equivariant measure up to a global phase, implying that the renormalization flow is confined to an anomaly-free, observer-equivariant submanifold of theory space. The scheme is then tested in pure gravity. At one loop, the Einstein gravity divergences can be absorbed into a finite G–invariant parameter section, in line with standard results. At two loops, the Goroff–Sagnotti cubic Weyl term is treated as a direction in coupling space which, according to a precise conjecture, points out of the observer-equivariant, anomaly-free submanifold. In this sense, an OE-compatible fixed point would require the Goroff–Sagnotti coupling to vanish. The paper also sketches a modular version of coarse-graining based on Tomita–Takesaki theory for local von Neumann algebras, and introduces a Landau–Ginzburg–type order parameter for “perspective condensation”, modelling the emergence of semiclassical spacetime as a phase transition in the space of observer perspectives. This article is part of an ongoing series on the Observer-Equivariance framework and is intended as a working, conceptually oriented contribution rather than a fully rigorous proof of all technical claims.