Chronometric Closure Paper VII: Operational Derivation and Stability of the Distinguishability Factor

Chronometric Closure Paper VII addresses the foundational burden of operationally deriving and stabilizing the distinguishability factor D within the CIFT-CMM-CUP programme. The paper constructs the transition-local quantity Dk as a calibrated finite-transition distinguishability functional over admissible finite-rank record sectors, with primary form Dk = Nl dB^2(rho_k, rho_k+1), where dB is the Bures distance and Nl = 1/FQ,ref(l) is the default calibration inherited from CIFT. The central result is a conditional theorem showing that, under finite-rank admissibility, positivity and normalization, operational comparability, non-background parameterization, data-processing monotonicity, coarse-graining stability, and bounded calibration, Dk is non-negative, finite, operationally interpretable, monotone under admissible information-loss channels, compatible with Cl, and suitable for insertion into the Paper VI integrand. A further conditional refinement shows that, where an admissible local record-state assignment and transition localization are available, Dk may be read as the calibrated pullback/restriction of the local distinguishability structure along a record-transition edge. The paper does not claim a universal distinguishability-field theorem for all physical systems.