Chronometric Closure Paper IX: Record-Algebraic Coarse-Graining, Weak-Loss Universality, and Conditional Stability of the Positive-χ Branch

The CIFT–CMM–CUP programme defines the local chronometric field as a record-supported, coarse-grained chronometric scalar built from distinguishability, record stability, and admissible coarse-graining. Earlier closure papers strengthened the source sector, modular-record projection, benchmark empirical control, source-to-clock mapping, finite chronometric accumulation, non-circular history measures, distinguishability, and low-residual recovery architecture. A central residual burden remained: the coarse-graining map was specified axiomatically rather than realized from microscopic record-state data, and the positive manifestation branch depended on compatibility between coarse-grained record data and the dynamical maintenance source. This paper develops a conditional structural closure for that burden. First, it constructs the coarse-graining map from record-algebraic data using either exact conditional expectations or conditional-expectation-type completely positive projections with controlled error. Second, it defines a protocol record-density representative for admissible local record-distinguishability densities and proves that the induced map satisfies the CIFT admissibility conditions under explicit hypotheses. Third, it proves leading-order common-mode robustness: local-moment equivalent coarse-grainings produce the same leading coherence-selective clock residual up to controlled and algebraic-projection errors. Fourth, it narrows the record-factor result to calibrated single-sector weak-loss universality, thereby preserving the leading signature without claiming global uniqueness of relative entropy. Fifth, it formulates chronometric maintenance as a Lyapunov dissipativity assumption and shows that this assumption is nonempty in a linearized Gaussian–Ohmic/Markovian sector. Finally, using a logarithmic positive-branch variable, it proves conditional small-data future stability of the positive-χ branch in a controlled hyperbolic model sector. The result is not a completed theory of quantum gravity. It is a conditional closure theorem: if the record-algebraic, weak-loss, common-mode, and dissipative maintenance assumptions hold, then the local chronometric field is no longer an arbitrary coarse-grained scalar, the leading experimental residual is not scheme-dependent in the controlled regime, and the positive manifestation branch remains stable under the stated assumptions.