A well-posed hybrid minisuperspace model for conformal cyclic cosmology

Conformal cyclic cosmology (CCC) proposes that the exponentially expanded future of one aeon is conformally identified with the initial hypersurface of the next. At the level of a reduced Friedmann–Lemaitre–Robertson–Walker (FLRW) model, this suggests a dynamical structure that is neither a single smooth flow nor a merely prescribed matching rule: within each aeon the variables satisfy differential equations, while the conformal crossover acts as a transition between boundary data and new initial data. We formulate this structure as a deterministic hybrid dynamical system on a compactified FLRW minisuperspace. The main contribution is a mathematical formulation of the CCC crossover as a regular hybrid transition. The compactified scale variable identifies a late-aeon conformal section with a closed guard, while the matter and geometric variables are restricted to a compact constraint manifold representing admissible conformal boundary data. A conformal reset operator maps this boundary data to an initial section for the next aeon. We introduce explicit conformal admissibility assumptions–regularity of the compactified flow, compatibility with the Friedmann constraint, transversality of the crossover section, and reset invariance of the physical constraint set–under which the reduced CCC system satisfies the hybrid basic conditions, has maximal solutions from every admissible state, is forward invariant on the physical state space, and has no crossover deadlock. A further lower-bound condition on the post-crossover section excludes Zeno accumulation of aeon transitions. The formulation also yields an induced aeon-to-aeon return map on the crossover section. We prove a local spectral-radius criterion for asymptotic stability of cyclic crossover data: if the linearization of the return map at a fixed crossover state has spectral radius smaller than one, then nearby boundary data generate infinite aeon sequences converging to the same cyclic pattern. The theory is illustrated by a compactified scalar-field FLRW model with a positive cosmological constant and an exponential potential. In this example the scale compactification, the Friedmann constraint manifold, the de Sitter-type future boundary, and the reset placement can be verified explicitly. The paper does not claim to prove the full CCC proposal; rather, it supplies a well-posed mathematical reduction in which conformal crossover and local aeon-to-aeon stability can be studied while retaining their interpretation as transitions between conformal boundary data and post-crossover initial data.