A4DAction-BasedDerivationoftheEinstein–MaxwellEquations Coupling,Symmetries,andLimit

Wepresentaprecisionextensionofthecoupledfour-dimensionalEinstein–Maxwell frameworkthat consistentlyincorporatesaboundaryaction,mixed(Robin)electromagneticboundaryconditions,anda topologicalθ-term[3].Withinthiscovariantvariationalsetupwederiveaclosed-formmodifiedfirst law fordyonicReissner–Nordström,δM= κ, 8πG δAH+ΦeδQphys, e +ΦmδQm+Wλ,whereQphys, e =Qe+θ, 2π Qm (Wittenshift)andWλ=λH √σAaδAaencodestheboundaryworkunderRobindata[2].Thestatement isobtainedusingtheIyer–Waldcovariantphase-spaceformalismandisconsistentwiththestandard first lawintheDirichlet limit (Wλ=0)andwithθ=0 [1]. Wealsoclarifyconventions, horizon potentials, ensemble interpretations, andSmarrscaling, andprovideappendicesonintegrabilityand duality.Ourresultssharpenthelow-energythermodynamicsofgravity–electromagnetismasaboundaryandtopology-refinedcoupledtheory.