We report Born rule precision tests conducted on IBM quantum hardware (backend: ibm_fez) using the Coherent Ordering Dynamics (COD) framework. The COD framework predicts a deviation from standard Born rule statistics of the form delta_P = (q-1) x (-0.3609) x p(1-p), where q is the Tsallis non-extensivity index derived analytically with no free parameters. Two independent clean runs on real quantum hardware gave A = -0.069 at 10.4 sigma and A = -0.081 at 12.3 sigma, corresponding to q = 1.191 and q = 1.224 respectively. A noise model simulation of the same backend using its actual calibrated hardware error characteristics gave A = -0.004 at 0.7 sigma, consistent with zero. The real hardware signal exceeds the noise model baseline by a factor of approximately 15, demonstrating that standard hardware errors do not explain the observed deviation. Permanent IBM Quantum job IDs for independent verification: d6v1qhaf84ks73dequ4g and d6v2cagv5rlc73f4h7rg. Professor Constantino Tsallis (Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro), whose non-extensive statistics framework underlies the COD prediction, was notified of this result on March 21 2026 and advised submission to arXiv. Further verification on additional backends and with error mitigation protocols is ongoing.