Quantum Fisher Information of a Ball in CFT3 and the Holographic Newton Constant

We compute the second-order relative entropy — the quantum Fisher information — of a ball of radius R in a three-dimensional conformal field theory (CFT₃) perturbed by a constant stress tensor δ Tμν=ε\,diag(1,12,12). The calculation is carried out entirely in the field theory, using the Casini–Huerta–Myers map to R× H², modular flow, and harmonic analysis in conical Legendre functions; no bulk geometry, minimal surface or entropy–area relation enters at any stage. Every ingredient is obtained in closed form, yielding the exact pure number NCFT = 12/35. Combined with an independent Ryu–Takayanagi computation of the same quantity, this boundary result independently fixes the holographic Newton constant to CT GN/L² = 3/π³. This precisely matches the known bulk graviton two-point function computation without relying on any shared mathematical machinery. As a byproduct, we obtain closed forms for two Fourier-type transforms of conical Legendre functions over their argument, neither of which appears in the standard integral tables. Key Contributions: Pure Boundary Derivation: An exact derivation of NCFT = 12/35 purely from CFT modular theory, providing a highly non-trivial test of the Ryu-Takayanagi prescription at second order. Mathematical Physics Discoveries: New closed-form integral identities for Fourier-type transforms of Mehler conical functions, and the demonstration of a "Legendre ODE collapse" mechanism that simplifies complex nested oscillatory quadratures. Conceptual Insight: A clear structural explanation of why real-time null-surface formulations fail in $d>2$ due to transverse δ(0) free-field singularities, necessitating the Euclidean CHM frame. Reproducibility & Accompanying Code: To ensure full reproducibility, this submission includes four standalone Python scripts that verify every numerical and symbolic claim in the paper. They require only numpy, scipy, sympy, and mpmath: chm_profile.py: Verifies the CHM geometric map, the Weyl weight, the perturbation profile, the First Law of entanglement, and the conservation of modular charge. conical.py: Symbolically and numerically verifies the new special-function identities (Appendix A), including the integral representations and the Legendre equation collapses. fisher_integral.py: Evaluates the final double integral of Gamma functions, reproducing NCFT = 12/35 up to 18 digits of precision and demonstrating strict domain stability. holographic_rt.py: Symbolically reproduces the bulk holographic Ryu-Takayanagi computation, solving the second-order Jacobi problem to obtain A₂ᵗᵒᵗ = -1/280.