Operator A10-R: reconstruction of the neutrino mass scale after the A10 reproducibility audit

This publication presents the corrected and extended version of Operator A10, denoted as A10-R1, developed after a full reproducibility audit of the previous form of the operator. The objective of the work is to organize the formalism for reconstructing the neutrino mass scale in such a way that every numerical quantity follows from explicitly defined parameters, correct dimensional relations, reproducible calculations, and falsifiability criteria. The starting point is the demonstration that the original form of Operator A10 does not independently reproduce the declared value (0.0472,\mathrm{eV}). Literal substitution of the published components leads instead to (0.948995,\mathrm{eV}), indicating a missing normalization, an implicit offset, or an interpretive ambiguity between the atmospheric scale and the full mass eigenvalue (m_3). Therefore, the previous A10 is assigned in this work the status of a formally incomplete reconstruction hypothesis rather than an independently verified numerical prediction. The new A10-R operator is not a cosmetic correction of the previous formula. It is formulated as an apparatus for reconstructing a physically admissible neutrino mass spectrum, based on explicit dimensional conditions, a physicality gate, separation of mass conventions, and a complete transition from oscillation data to absolute observables. A central role is played by the E–A rank–2 branch with the condition (\det M_\nu=0), which leads to reproducible values:[\Sigma m_\nu \simeq 58.710229,\mathrm{meV},][m_\beta \simeq 8.875953,\mathrm{meV},][m_{\beta\beta}\in[1.496649,3.738888],\mathrm{meV}.] The publication includes full central calculations, an arithmetic audit of Operator A10, rank–2 reconstruction, uncertainty propagation using a Monte Carlo simulation with (300000) trials, and a separate Majorana-phase simulation with (100000) trials. It also introduces PASS/FAIL tables for the operator variants, a parameter registry, equation descriptions, reproducibility criteria, and a replication artifact enabling independent verification of the results. The significance of the work lies in transforming the problematic A10 operator into the controlled A10-R1 formalism, in which numerical predictions are not treated as declarations but as results passing explicit gates of physicality, dimensional consistency, reproducibility, and falsifiability. The article therefore provides a technical and scientific closure of the A10 audit and forms a basis for further research on the neutrino sector within the PJM–GTSFC–USC–GTCW framework.