  The immune response to an acute primary infection is a coupled process of
antigen proliferation, molecular recognition by naive B cells, and their
subsequent proliferation and antibody shedding. This process contains a
fundamental problem: the recognition of an exponentially time-dependent antigen
signal. Here we show that B cells can efficiently recognise new antigens by a
tuned kinetic proofreading mechanism, where the molecular recognition machinery
is adapted to the complexity of the immune repertoire. This process produces
potent, specific and fast recognition of antigens, maintaining a spectrum of
genetically distinct B cell lineages as input for affinity maturation. We show
that the proliferation-recognition dynamics of a primary infection is a
generalised Luria-Delbr\"uck process, akin to the dynamics of the classic
fluctuation experiment. This map establishes a link between signal recognition
dynamics and evolution. We derive the resulting statistics of the activated
immune repertoire: antigen binding affinity, expected size, and frequency of
active B cell clones are related by power laws, which define the class of
generalised Luria-Delbr\"uck processes. Their exponents depend on the antigen
and B cell proliferation rate, the number of proofreading steps, and the
lineage density of the naive repertoire. We extend the model to include
spatio-temporal processes, including the diffusion-recognition dynamics of a
vaccination. Empirical data of activated mouse immune repertoires are found to
be consistent with activation involving about three proofreading steps. The
model predicts key clinical characteristics of acute infections and
vaccinations, including the emergence of elite neutralisers and the effects of
immune ageing. More broadly, our results establish infections and vaccinations
as a new probe into the global architecture and functional principles of immune
repertoires.
