Yang-Mills as the rank-1 self-sourced cell of a structural classification of fundamental interactions

Maxwell theory is linear because the electromagnetic field carries stress-energy but not electric charge: the field does not source itself. General relativity is nonlinear because the gravitational field carries the universal source, stress-energy, and therefore sources itself. This paper asks for the rank-1 analogue of the second situation. What structure is forced when a massless rank-1 gauge potential's own field carries the coupling label that sources it, restricted to local two-derivative classical Lagrangians? The argument runs three structural demands together: frame-independence (the equation holds in every Lorentz frame), index matching (both sides of a tensor equation transform the same way), and self-sourcing (a potential whose own field carries the source that drives its equation appears on its own source side). Applied to a massless rank-1 potential of this kind, the three force a fixed-point equation in which the potential enters both as field and as source. Self-sourcing then requires a non-trivial cubic vertex coupling three copies of the potential to a bilinear operation on the coupling labels. The explicit form of that vertex (Lorentz-antisymmetric structure, internal-antisymmetric structure constants fabc = −facb) is supplied by the BRST-cohomological deformation theorem for free massless spin-1 fields (Barnich and Henneaux 1993; Henneaux 1998; Barnich, Brandt, and Henneaux 2000) under the local two-derivative classical Lagrangian assumption. The Jacobi identity is adopted as the standard sufficient closure that makes the bracket a Lie bracket and the labels a Lie algebra; whether self-sourcing alone uniquely forces Jacobi without invoking the deformation programme is left open. Given the Jacobi-closing bracket, the rank-1 analogue of the Feynman-Deser-Ogievetsky-Polubarinov consistency argument used at rank 2 for general relativity, in the cohomological-deformation form, identifies Yang-Mills as the unique nonlinear completion modulo field redefinitions and gauge-invariant spectator vertices. Maxwell is the fabc = 0 abelian limit; general relativity sits in the rank-2 self-sourced cell of the same 2×2 classification by rank and self-sourcing. The result is the classical, massless structural core. The specific Lie algebra each gauge sector carries, mediator masses, chiral matter assignments, generations and mixings, confinement, and the quantum theory all remain outside scope.