What question did this study set out to answer?

This work seeks to derive gauge invariance from concepts rather than impose it as an axiom, revealing connections between color charge and gravitational effects.

June 8, 2026Open Access

Paper LIX: Gauge Invariance Without Gauge Axioms: Local SU(3)C as Fano Automorphism In- variance, Topological Protection of Colour Quantum Numbers, and the Decoupling of Colour from Gravity at Sub-Planck Energies

Key Points

This work seeks to derive gauge invariance from concepts rather than impose it as an axiom, revealing connections between color charge and gravitational effects.
Utilized the brane-bulk octonionic framework to derive local SU(3)C invariance.
Analyzed Fano automorphisms and topological properties of the Fano lattice.
Calculated the ratios of energy scales to demonstrate the decoupling of colour from gravity.
Colour quantum numbers are conserved under gravitational effects below the Planck scale.
Gravitational curvature effects are negated at nuclear scales by over 51 orders of magnitude.
The total covariant derivative factorizes, allowing independent analysis of spacetime and colour connections.

Abstract

Standard gauge theory imposes local SU (3) C invariance as an axiom, then derives gluons as the required compensation elds. We show that in the brane-bulk octonionic framework, the entire gauge structure is derived rather than imposed. The argument proceeds in four steps. (1) Gauge transformations are Fano automor- phisms. Local SU (3) C colour rotations are precisely the local G2 automorphisms of the Fano lattice that preserve the e7 horizon direction. These transformations cost exactly zero energy because the innite shear modulus of the auxetic ν = −1 brane fabric (Pa- per LIV) penalises only symmetry-breaking deformations, never symmetry-preserving ones. Gauge invariance is therefore not imposed it is the statement that zero-energy lattice transformations are undetectable. (2) Colour is a topological winding num- ber. Colour charge is an integer winding of the Fano lattice nodes in the e4, e5, e6 sector. Topological integers cannot change under any smooth deformation. Gravita- tional curvature is a smooth deformation. Therefore colour quantum numbers are ex- actly conserved under any gravitational eect below the topology-changing threshold of Paper XLV (Planck-scale singularity formation). (3) The ratio Ecurv/EFano ∼10−51 at nuclear scales. The Fano lattice stiness (∼ΛQCD ∼440 MeV) exceeds gravitational curvature eects at quark scales by 51 orders of magnitude. Minimal coupling of colour to gravity is not merely negligible it is topologically forbidden for any continuous curvature. Gravity and colour decouple exactly. (4) The total covariant deriva- tive factorises. The spacetime connection and the colour connection act on indepen- dent tensor factors and commute. This factorisation holds exactly at all energies below E⋆∼Λ2 QCD/mPl ∼10−16 GeV. The eight gluons emerge as the propagating phonons of the Fano colour sector (eigenvalue-7 modes of the Fano Laplacian, Paper XLII), with the colour-singlet e7 mode conned to the horizon and absent from the physical spectrum reproducing the absence of a ninth massless gluon without additional input. Two open computations remain: the running of αs from the Fano phonon census, and the formal derivation of θQCD ≲δM/M from the Z3 tilt. Four new predictions follow (Predictions 106109). Part of the One-Octonion Brane-Bulk Framework series. Anchor DOI: 10. 5281/zenodo. 19120873. Community: one-octonion-brane-bulk. Author: Bharathi Dasan Jagadeesan, M. D. , University of Minnesota. ORCID: 0000-0002-1143-941X.

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