  Our direct knowledge of solar eruptive events is limited to several decades
and does not include extreme events, which can only be studied by the indirect
proxy method over millennia, or by a large number of sun-like stars. There is a
gap, spanning 1--2 orders of magnitude, in the strength of events between
directly observed and reconstructed ones. Here, we study the proxy-method
sensitivity to identify extreme solar particle events (SPEs). First, the
strongest directly observed SPE (23-Feb-1956), used as a reference for
proxy-based reconstructions, was revisited using the newly developed method.
Next, the sensitivity of the cosmogenic-isotope method to detect a reference
SPE was assessed against the precision and number of individual isotopic
records, showing that it is too weak by a factor $\approx$30 to be reliably
identified in a single record. Uncertainties of 10Be and 14C data are shown to
be dominated by local/regional patterns and measurement errors, respectively.
By combining several proxy records, an SPE 4--5 times stronger than the
reference one can be potentially detected, increasing the present-day
sensitivity by an order of magnitude. This will allow filling the observational
gap in SPE strength distribution, thus enriching statistics of extreme events
from 3--4 presently known ones to several tens. This will provide a solid basis
for research in the field of extreme events, both for fundamental science, viz.
solar and stellar physics, and practical applications, such as the risk
assessments of severe space-based hazards for the modern technological society.
