Gauge–Gravity Spectral Synthesis: A Conditional Einstein–Yang–Mills System from Projective Spectral Entropy

The companion Gravity and Q12 papers derived, respectively, the Einstein tensor as the horizontal a₂ response and the local Yang–Mills equations as the vertical a₄ response of the projective spectral entropy functional S_Π[g,A] = 1/2log' Ag,A. The present paper joins the two variations within a single renormalized local action and establishes a conditional Einstein–Yang–Mills system. Varying the matched action with respect to the connection at fixed metric gives D_μ Faμν = 0; varying the same tr_ρ(F²) term with respect to the metric gives its energy–momentum tensor, through (1/√g)\,δ\!∫\!√g\,tr_ρ(F²)/δ gμν = 2τμν with τμν = tr_ρ(FμρF_ν^ρ - 14 gμνF²). Since tr_ρ(F²) = I_ρ\,FᵃμνFa\,μν, canonical matching fixes the products cEH = 1/(16π GN) and cF I_ρ = 1/(4gYM²), where I_ρ is the Dynkin index of the fibre representation; the Einstein equation with Yang–Mills source then holds by definition of TYM, with 8π GN = 1/(2cEH) depending on the Einstein coefficient alone. Nothing here predicts a coupling: GN, gYM and the cosmological coefficient Λᵣₑₙ, which does contribute to the metric equation, are independent renormalized matching data. The derivation is Euclidean, and identifying the result with the Lorentzian Einstein–Yang–Mills equations requires a continuation not performed here. For the a₆ sector we give only a representative spanning list of the invariants that can occur, modulo integrations by parts and Bianchi identities: in four dimensions a₆ produces no ultraviolet divergence, and the renormalized dimension-six coefficients are themselves matching data. Statements about Λ⁴, Λ² and logΛ behaviour refer specifically to a proper-time cutoff and are not scheme-independent. The gauge group G_Π is taken as input from the spectral admissibility sub-programme. Interpretive status. The reading of gravity and gauge dynamics as stratified by Seeley–DeWitt order rather than unified by an enlarged symmetry group is offered as an interpretive outlook, not a theorem.