Finite-Dimensional Modular Theory, Non-Commutative Relative-Entropy Geometry, and Resilient Quantum Mirror Descent for Holographic State Reconstruction (Version 3)

Abstract We establish an exact mathematical framework connecting finite-dimensional Tomita–Takesaki modular theory, Kubo–Mori–Bogoliubov (KMB) Riemannian geometry, and entropic optimization for quantum states. First, we prove that the second directional derivative (Hessian) of the Umegaki quantum relative entropy at a KMS thermal state is identically the KMB inner product defined by logarithmic-mean divided differences. Second, we establish an analytical, strictly positive lower bound for the modified logarithmic Sobolev inequality (mLSI) constant α1 ≥ 2λgap(L) / [ln(1 / λmin(ρKMS)) + 2] for Lindblad generators satisfying paired Alicki detailed balance. Third, we formulate relative-entropy quantum mirror descent with faithful-state regularization for holographic Petz map state inversion. Key Benchmarks & Verification (Dilaton Studio Engine) • Modular Stationarity Residuals: ||L(ρKMS)||F = 4.46 × 10^-16 (Exact Machine Limit)• Daleckii–Krein Logarithmic Derivative Accuracy: Within 2.55 × 10^-7• Relative-Entropy Hessian Agreement: Within 1.22 × 10^-7• Holographic Petz Horizon Reconstruction: 94.95% quantum fidelity recovery of infalling quantum information across evaporating horizons Version 3 Release Notes • Refined mathematical proofs for the mLSI lower bound and regularized quantum mirror descent convergence.• Updated complete multi-page document layout and numerical benchmark tables.• Corrected internal document cross-references and version metadata.