Part IV: Fermionic and Gauge Extensions of the Regulated Araki–BKM Framework

This part extends the finite-cutoff Araki–Bogoliubov–Kubo–Mori (BKM) framework tomixed bosonic–fermionic systems and to gauge fields. The operator-algebraic construction is formulated on a Z2-graded von Neumann algebra with an even faithful normal reference state. The modular flow is required to preserve the local grading. The positivity of the BKM Hessian for bounded modular-analytic even perturbations is established from Araki perturbation theory and the non-negativity of relative entropy. A bounded functional-calculus regulator for a smeared stress tensor and a Gaussian modular smoothing are constructed. The existence of the linearized regulated BKM source is kept as an explicit finite-cutoff assumption; it is not inferred merely from the boundedness of the function fM(x) = M tanh(x/M) when the underlying stress tensor is an unbounded affiliated operator. The one-loop matter calculation is then performed explicitly for Dirac, Majorana, Weyl, Maxwell, Yang–Mills, and non-minimally coupled scalar sectors. The bosonic and fermionic determinant signs are kept distinct from the raw Seeley–DeWitt coefficient. For the Standard Model field content consisting of three generations, no right-handed neutrinos, one Higgs doublet, and the gauge group SU(3) × SU(2) × U(1), the leading matter-induced Einstein–Hilbert coefficient in the adopted proper-time cutoff scheme is 1 / ℏGSMind = ((1/ 12)− 2ξ) Λ2 / π.The replica calculation is deliberately stated at the level actually controlled by the preceding construction. The Einstein–Hilbert part of the regulated replica effective actiongives the geometric surface term AΣ/(4ℏG). For non-minimally coupled scalars this is an effective-action/Wald-type contribution and is not, without further contact-term analysis, identified with the complete von Neumann matter entropy. For Maxwell and Yang–Mills fields, the bulk determinant must also be distinguished from the gauge edge/contact sector. A complete physical gauge entropy is therefore not claimed here. Finally, the identification of the full informational coefficient with Newton’s constant and the saturation of that coefficient by the leading matter-induced term are stated as separate physical matching assumptions. Finite local R terms, bare terms, and omitted sectors are not fixed by the leading a1Λ2 calculation. Under leading-order saturation,Λ ≃ MPl root( π / (1/12 − 2ξ)), so that on the branch 0 ≤ ξ < 1/24, at the same matching accuracy, Λ ≳√12π MPl, ℓUV ≲ ℓPl√12π ≃ 0.163 ℓPl.This scale relation is conditional on the cutoff/renormalization scheme and on interpreting Λ as a physical microscopic scale.