Physics-Constrained Reconstructions of Sunspot Number from Millennial-Scale Annual Heliospheric Modulation Potential.

UNLABELLED: Long-term solar magnetic variability governs the space climate conditions. These variations also modulate the galactic cosmic ray influx reaching Earth and subsequently regulate the production of cosmogenic isotopes in Earth's atmosphere. By harnessing the multi-millennial records of the abundances of these radio-isotopes stored in natural terrestrial reservoirs, such as tree wood and ice sheets, it is possible to reconstruct past solar magnetic activity predating the telescopic era. However, existing regression-based reconstruction methods using isotope data from time series extracted from the reservoirs are often applied to conditions outside the valid parameter range. This can result in a number of problems, including negative sunspot numbers during low solar activity phases, which correspond to statistically permissible extrapolation artefacts but have no physical meaning. To address this, we developed a Monte-Carlo inversion framework consisting of a sequence of physics-based semi-empirical forward models, supplemented by an Approximate Bayesian Computation-based posterior selection, to infer the most plausible temporal evolution of solar magnetic parameters. Our approach produces physically consistent, uncertainty-quantified reconstructions of annually resolved parameters, including sunspot numbers. We apply the method to two modulation potential datasets, namely geomagnetic record from 1845 - 2020, and an annual 14C estimate over the period 971 - 1932. Our new reconstruction method is designed to yield strictly non-negative sunspot numbers and shows close agreement with direct observations within uncertainty during the telescopic era. These results offer valuable insights into long-term solar variability and provide improved constraints for long-term solar dynamo modelling. The annual-scale reconstructions are also directly usable for solar irradiance estimation, terrestrial climate modelling, and studies of long-term Sun-Earth coupling. SUPPLEMENTARY INFORMATION: The online version contains supplementary material available at https://doi.org/10.1007/s11207-026-02731-0.