Statistical Validation of a Phase-Gradient Relational Clock in Cobalt–Corrin Active-Space Models: A Comparison Across Placeholder, Literature, and Ab Initio Molecular Geometries

We present a systematic numerical investigation of a phase-gradient observable, tau_pre, as a candidate relational clock of Page-Wootters type in truncated complete-active-space configuration-interaction (CASCI) models of cobalt-corrin systems. Four molecular geometries of increasing chemical grounding are examined: a minimal placeholder CoN4 structure, the simplified cob(I)alamin core of Kornobis et al., the ground- and excited-state cyanocobalamin coordinates of Ghodrati et al., and a 26-atom corrin-inclusive fragment derived from Brunold-group coordinates. For each geometry the mean Schrödinger residual of conditional evolution under tau_pre is compared with the residual obtained under the external laboratory parameter s, at active-space sizes corresponding to 10, 12, and 14 qubits. A statistical control based on 200 randomized monotonic time reparameterizations is employed throughout. In every geometry and at every qubit count tested, the residual ratio of tau_pre lies outside the full null distribution (z ≈ 3.3–8.6σ). Literature-derived geometries exhibit substantially smaller degradation of statistical significance with increasing Hilbert-space dimension (Kornobis, Ghodrati S0) or none at all (Brunold S0, which strengthens), in contrast to the placeholder model, where significance weakens as the active space grows. Explicit per-step ab initio Hamiltonian reconstruction confirms that linear interpolation between endpoint geometries introduces negligible bias relative to genuine trajectory recomputation. While these results establish the numerical robustness of tau_pre as an operational observable, its status as a genuine generator of time translations remains an open theoretical question. Keywords: Page-Wootters formalism, relational time, phase-gradient clock, CASCI, cobalt-corrin, vitamin B12, quantum foundations, residual diagnostic