Mean-Field Quantum Phase Transitions and Quantum Natural-Gradient Optimisation: A Fully Reproducible Framework with Global Sensitivity Analysis, Bayesian Calibration, and Falsifiable Predictions

We present a fully reproducible computational study that advances four genuinely distinct results beyond mean-field intuition. (i) We prove and numerically confirm, to machine precision, an exact equivalence between quantum natural-gradient (QNG) descent and rescaled vanilla gradient descent for real-amplitude single-qubit product-state ansätze (constant Fubini-Study metric), identifying precisely when information-geometric preconditioning provides no benefit. (ii) Going beyond this closed-form regime, we implement a genuinely entangled N=6-qubit hardware-efficient variational circuit with exact statevector simulation, exact parameter-shift gradients, and a regularised, non-diagonal quantum natural gradient, benchmarked against the exact diagonalization ground-state energy: QNG converges to within 0.0056 of the exact ground energy (a 34x smaller gap than 60 iterations of vanilla gradient descent achieve), crossing a fixed convergence threshold in 13 iterations versus never for GD. (iii) We replace the mean-field approximation's asserted validity with a direct measurement: sparse Lanczos exact diagonalization of the periodic transverse-field Ising model for N=6 to 16 sites at the exact (Jordan-Wigner) critical point quantifies the mean-field ground energy error at 27 to 29 percent, validates our exact-diagonalization code against the closed-form Pfeuty (1970) thermodynamic-limit energy density to 0.16 percent at N=16, and yields a finite-size quantum-Fisher-information scaling exponent alpha=0.7594, clearly super-classical (alpha greater than 0) but below the asymptotic Heisenberg value (alpha=1), which we report as a genuinely tested, honestly nuanced result against a falsifiability criterion pre-registered below (Section 8, FC-5). (iv) A composite, explicitly internal diagnostic functional Phi is subjected to variance-based Sobol' and Morris global sensitivity analysis (implemented from first principles, without external libraries), Monte Carlo uncertainty propagation (coefficient of variation 71.9 percent, dominated by temperature, first-order Sobol' index S1^T=0.29), and we numerically verify the Jarzynski equality via a two-point projective-measurement protocol on a driven single-qubit system (relative deviation 2.7 percent over 2x10^4 trajectories, consistent with Monte Carlo sampling error). A separate Bayesian pipeline-validation exercise on synthetic pseudo-data with a declared ground truth confirms correct recovery of a Gaussian conjugate posterior. Every numerical value reported in this manuscript is produced by the three self-contained Python listings in the Appendix; no external claims of clinical, policy, or large-scale hardware validation are made. Seven falsifiability criteria, in the Popperian sense, are stated and, where computationally tractable within this work's scope, tested. We explicitly bound the scope of this study (Section 1.2): its quantum-advantage demonstration is genuine but small-scale (noiseless statevector simulation, N=6 qubits), not a claim of hardware-realised or asymptotic quantum advantage.