Certified AI Physics Discovery and Autonomous Experimentation (CAIPD): Unifying Curvature-Filtered Law Identification, Conservation-Guaranteed Simulation, and Active Counterexample Falsification within the ZFC Axiomatic Bridge

While machine learning and symbolic regression increasingly propose candidate equations and surrogate simulators from physical data, a fundamental epistemological gap remains: statistical data fitting does not equate to the discovery of a physical law, nor does adding a soft physics penalty to a loss function guarantee simulation correctness, stability, or out-of-distribution (OOD) validity. In this paper, we establish Certified AI Physics Discovery and Autonomous Experimentation (CAIPD), a deductive methodology and mathematical framework anchored in our 71-page ZFC Master Reference Monograph (Lee, 2026, DOI: 10.5281/zenodo.23486681) [1] and vertically integrating the Causal Operator Learning System (COLS, DOI: 10.5281/zenodo.23556758) [2], Self-Verifying Knowledge Evolution Systems (SVKES, DOI: 10.5281/zenodo.23557138) [3], the High-Dimensional Geometric Probability System (HDGPS, DOI: 10.5281/zenodo.23557360) [4], the Dynamic Topological Critical Transition System (DTCTS, DOI: 10.5281/zenodo.23557507) [5], and the Extended Quantum Information Geometry System (EQIGS, DOI: 10.5281/zenodo.23557698) [6]. Governed by the universal quadruple 𝔘 = (𝒯_0x8A2C, 𝐂_UPTC, η_friction, ϵ_SHID), we formulate six core axioms (CAIPD-1 through CAIPD-6) and prove six fundamental theorems: (1) Curvature-Filtered Rejection of Spurious Statistical Correlations (d_stat ≠ d_geo) and Sample-Complexity Acceleration (∝ 1/κ_eff); (2) Necessary and Sufficient Conditions for Verifiable Physics Simulation combining exact UROA conservation (ℬ_ϵ), functional do-calculus, and rigorous convergence-error bounds; (3) Information-Optimal Autonomous Experiment Selection via Pareto Counterexample Falsification (O(ln(m/δ)/κ_eff) budget scaling); (4) Lifetime Statistical Safety (≤ α_global) and Tarski–Löb Stratified Immunity against self-referential corruption of physical principles; (5) Chaotic Trajectory vs. Topological Invariant Predictability Horizon Separation; and (6) Counterfactual Inverse Design of Metamaterials and Multi-Experiment Precision Constant Verification. Complete Python numerical verification is provided in Appendix A.