Part IV: Fermionic and Gauge Extensions of the Regulated Araki–BKM Framework , Mixed Matter Systems, One-Loop Einstein–Hilbert Response, and Separation of LIE from Gravitational Matching

This part extends the finite-cutoff Araki–Bogoliubov–Kubo–Mori (BKM) framework tomixed bosonic–fermionic systems and to gauge fields. The operator-algebraic construction isformulated on a Z2-graded field algebra with an even faithful normal reference state; physicalobservables are taken in the even subnet. The modular flow preserves the grading, and thepositivity of the BKM Hessian for bounded modular-analytic even perturbations follows fromAraki perturbation theory and non-negativity of relative entropy.A bounded functional-calculus regulator for a smeared stress tensor and a Gaussianmodular smoothing are constructed. The existence of the linearized regulated BKM sourceis kept as an explicit finite-cutoff assumption; it is not inferred merely from boundednessof fM(x) = M tanh(x/M) when the stress tensor is an unbounded affiliated operator. Thesubsequent Euclidean calculation is used as a one-loop effective-action comparison. Itslocal Einstein–Hilbert coefficient is not identified here with the independently defined staticKMS/LIE area coefficient.The one-loop matter calculation is performed explicitly for Dirac, Majorana, Weyl,Maxwell, Yang–Mills, and non-minimally coupled scalar sectors. For the explicitly assumedminimal Standard Model bookkeeping spectrum–three generations, no right-handed neutrinos,one Higgs doublet, and gauge group SU(3) × SU(2) × U(1)–the leading matter-inducedEinstein–Hilbert coefficient in the adopted common proper-time scheme is1 / ℏGSM EH,ind = ((1/12)− 2ξ) Λ2 / π.This is a minimal-SM one-loop effective-action coefficient, not a spectrum-completenesstheorem and not a derivation of the observed Newton constant.The Einstein–Hilbert part of the regulated replica effective action gives the correspondinggeometric replica/Wald surface term AΣ/(4ℏGEH). For non-minimally coupled scalars thisdoes not by itself equal the complete von Neumann matter entropy. For Maxwell andYang–Mills fields the bulk gauge-fixed determinant must be distinguished from the physicalgauge edge/contact sector. A complete multi-field physical entropy theorem is therefore notclaimed.We reserve G(a)LIE for the information coupling defined by the independently constructedstatic-KMS finite-cutoff LIE theorem, GSM EH,ind for the minimal-SM induced Einstein–Hilbertcoefficient calculated here, and GN for the measured gravitational coupling. Their equality requires additional common-regulator, renormalization, spectrum-completeness, and renormalization-group/ threshold input. Only under the deliberately strong leading-ordersaturation assumption does one obtain the diagnostic relation Λ ≃ MPl root (π / (1/12 − 2ξ)).This relation is not claimed to be regulator independent or predictive near the cancellationpoint ξ = 1/24, where neglected terms need not remain subleading.