Finite-Horizon Structures IX: Critical Loci and Singular Geometry of the Projective Y-Structure

Finite-Horizon Structures IX develops the intrinsic singular layer of the Finite-Horizon Structures programme for a chosen smooth positive representative of an underlying projective Y-structure. While the preceding articles establish the regular differential, measure-theoretic, dynamical, propagative, and rigidity layers on the noncritical locus, the present work studies the complementary critical locus, where the first-order geometry carried by the coherence one-form no longer determines the local structure. The article introduces the basic singular objects attached to the chosen representative, including critical points, critical values, critical levels, second-order jets, and Hessian-type data. It then develops the second-order singular geometry needed to classify critical behaviour into three main regimes: nondegenerate Morse singularities, Morse-Bott critical submanifolds, and genuinely degenerate singularities requiring higher-order analysis. A central theme of the paper is the effect of these singularities on the superlevel filtration defined by the chosen representative. In particular, the article shows how critical levels act as the natural singular thresholds of the filtration, where local geometric or topological transitions may occur. Conversely, away from the critical spectrum and under appropriate properness assumptions, the filtration is locally trivial. The paper also studies covariance under structural equivalence, distinguishing strictly projective singular objects from representative-dependent but projectively covariant data. The result is a local and structural theory of critical loci and singular superlevel geometry that completes the regular framework developed in the earlier parts of the series and prepares the next stage of the programme: singular propagation and intrinsic transition geometry.This article forms part of the Ranesis framework developed by Alexandre Ramakers.