The Extended Quantum Information Geometry System (EQIGS): A Deductive Framework Unifying Petz Monotone Metrics, Thermodynamic Length Dissipation Bounds, and Tensor-Network Curvature Transitions within the ZFC Axiomatic Bridge

A central open problem at the intersection of quantum information theory, quantum thermodynamics, and many-body physics is determining the exact mathematical conditions under which the information-geometric distance d_g(ρ, σ) between two quantum states coincides with the minimum physical resource and thermodynamic cost 𝒱(ρ → σ) required to transform one into the other. In this paper, we establish the Extended Quantum Information Geometry System (EQIGS), a unified deductive framework synthesizing Petz monotone Riemannian metrics, quantum resource theories, finite-time thermodynamic length bounds, and tensor-network submanifold geometry, anchored directly in our 71-page ZFC Master Reference Monograph (Lee, 2026, DOI: 10.5281/zenodo.23486681) [1]. All foundational spaces and universal constants 𝔘 = (𝒯_0x8A2C, C_UPTC, η_friction, ε_SHID) follow Proposition 1.5, Theorem 2.9, Lemma 4.5, Lemma 4.6, Lemma 5.2, Lemma 6.3, Lemma 8.4, Lemma 8.6, Theorem 8.7, and Theorem 9.5 of [1]. Regularizing the quantum state space 𝒮(ℬ) on the Takesaki Type II_1 ground factor (𝓜_ground, τ_ground) and the Fractional Sobolev–Colombeau Quotient Algebra 𝓖^α(𝒮), we formulate six core axioms (EQIGS-1 through EQIGS-6) and prove six fundamental theorems: (1) Complete Chentsov–Petz Monotone Metric Classification regularized at pure-state boundaries via ε(α) = exp(-1/(1-α)); (2) Exact Necessary and Sufficient Conditions for Information Distance–Physical Cost Equivalence and Tight Finite-Time Control Bounds governed by Planck-BCH arithmetic friction η_friction ≥ 10^-22; (3) Quantum Cauchy–Schwarz Thermodynamic Length Lower Bounds (Σ_na ≥ ℓ_g^2 / (2τ)); (4) Multipartite Entanglement Witnessing via Quantum Fisher Information Curvature Thresholds (ℱ_Q(ρ, H) > 4(Δh)^2 N); (5) Natural Choi–Bures Information Metrics on the Quantum Channel Space Chan(ℬ) and Isometric Reversibility; and (6) Exact Curvature–Volume Formulas for Bond-Dimension-D Tensor Network Manifolds and Area-to-Volume Law Phase Transitions at C_UPTC. Complete Python numerical verification is provided in Appendix A.