Finite-Cutoff Local Information Equilibrium and Araki–BKM Stability in Local AQFT

We prove a finite-cutoff, fixed-bifurcation form of Local Information Equilibrium (LIE) forthe shift-invariant sector of the four-dimensional massless minimally coupled scalar on a fixedde Sitter reference geometry. The proof uses a single symmetry-preserving Gaussian regulatorwith continuous static time, a shape-regular finite-range spatial lattice, symplectic removal of therigid shift mode, and the exact static KMS state. The same microscopic Hamiltonian determinesthe reference state, modular Hamiltonian, leading area response, and shell susceptibility. Theleading entropy response is obtained from the tangent/Rindler limit, where the self-similar stencilyields a finite positive regulator coefficient and excludes an independent leading shear term by itssquare-lattice dihedral symmetry. We prove the previously open dyadic modular-susceptibilityestimate using the exact hyperbolic optical geometry of the de Sitter static patch, Gaussiangeneralized-eigenvalue variation, local energy-form control, radial–angular channel decomposition,and discrete Agmon decay. This establishes the required cutoff-uniform shell bound without a globalspectral-gap assumption or auxiliary scalar mass. Combining the shell theorem with the controlledFermi expansion gives the finite-cutoff fixed-bifurcation LIE balance, including a logarithmicallyenhanced curvature remainder. The remaining open extensions are the current-algebra entropyidentification, small-ball/local-horizon universality, identification of the information coupling withNewton’s constant, and removal of the physical cutoff in the type-III limit. The Euclidean replicaconstruction is used only as an independent consistency check.