The Redshift Catastrophe—Structural Incompatibility Within FLRW Cosmology: A Formal Constraint on Early–Late Mapping

This work establishes a functional-analytic obstruction within the standard cosmological framework. Concretely, we model the set of all possible FLRW expansion histories H(z; p) as a finite-dimensional manifold immersed in the space of smooth functions. The process of extracting cosmological observables - the co- moving distance DA and the Alcock-Paczynski parameter FAP - defines a smooth map Φ from this parameter manifold into an infinite-dimensional Banach space of constraints C = R × C([a, b]) × RN . Our central theorem shows that the image of Φ is a finite-dimensional submanifold of C, and thus has zero measure in the infinite-dimensional setting. Physically, this means that a generic set of combined early and late-universe observations - represented by a point in C - has zero probability of lying exactly on the FLRW manifold. The resulting inevitable incoherence is not due to noise or systematics, but to the inherent geometric rigidity of the FLRW metric ansatz. This result formalises why joint early and late-time cosmological constraints generically produce irreconcilable tensions within the standard framework.