Gauge‑Reduced Discrete Noether Invariants in the Sigma Calculus

DescriptionThis work introduces the Holonomy–Tension Charge, a new discrete conserved quantity for systems exhibiting log‑periodic renormalization‑group (RG) behaviour. The charge is defined as the overlap Q = ∑ₑ∈E σₑʰᵃʳᵐ · (σₑʰᵃʳᵐ − h₁σₑʰᵃʳᵐ)between a harmonic tension field and its holonomic image. Under the conditions of weak periodicity (WLP) and Hodge‑isometry (HI) the quantity remains invariant along the entire RG orbit, providing a discrete analogue of Noether’s theorem for cyclic RG flows. It serves as a robust diagnostic for numerical RG solvers, a one‑number compression handle for log‑periodic data, and a prospective classifier of universality sectors in lattice and network models.Potential applications range from critical phenomena over multiscale tensor methods to adaptive regularisation schemes in deep‑learning architectures. This construction is model-agnostic and can be applied to any discrete field theory that admits a quadratic gauge-invariant action and a finite Hodge-isometric symmetry.Typical application domains include lattice electromagnetism, spring networks, and topological spin chains, where the invariant serves as a validation scalar or structural diagnostic. The formalism requires no reference to continuum field theory and is suitable for both theoretical analysis and algorithmic implementation. Version note.This version corrects several formal inconsistencies, clarifies the assumptions and notation, and improves the presentation of the general and categorical parts. The main result and overall scope are unchanged. This revision adds a categorical reformulation of the reduced discrete symmetry underlying the conserved holonomy charge (Appendix B). A category of reduced sigma–holonomy data is introduced, with Hodge-isometric intertwiners as morphisms. The conserved scalar is thereby recast as a canonical invariant of reduced holonomy objects and is shown to be preserved both functorially and along discrete holonomy orbits. The main reduced-sector conservation result is unchanged; the revision sharpens its structural and mathematical formulation.Added remark 1.10 (Physical interpretation of the isometry assumption)