Sheaf-of-Contexts Identity: Global Coherence, H1 Obstructions, QFI-Lifted Invariants, and Murphy-Scheduled Evidence

Identity is a glued object. For any agent distributed across audiences, stances, times, media, and idioms, we formalize identity as the existence (or least-obstruction approximation) of a global section that glues local context sections across a cover of contexts. The SCI framework supplies: (1) a site-of-contexts Ctx with covers and restrictions; (2) a (pre)sheaf Fof identity-bearing sections and Cech cochains; (3) an obstruction metric Ob derived from the Cech 1-coboundary (δs)ij; (4) a QFI-lift ensuring identity invariants survive coarse-graining and idiom translation; (5) Murphy horizons Lc that schedule probes on overlaps where seams most likely surface. We develop the mathematics (sheaves on sites, Cech-derived comparisons, stability under base change, and non-abelian extensions to stacks), operational algorithms (restriction learning, obstruction solvers, overlap graphs and Laplacian bounds), and experiments (human/model) with falsifiable predictions. SCI is orthogonal to First-in-Cache Identity (FCI): SCI captures cross-context coherence; FCI captures within-stream temporal continuity. Together they form a two-axis invariant theory of identity. Executive Summary. Identity is not a single store of truth but a consistency claim across contexts. SCI quantifies that claim with cohomology (gluing exists ⇔H1 =0), exports it across idioms with information-geometric invariants (QFI), and tells you where/when to test (Murphy). The result is an operational physics of selfhood: seams become measurable, portable, schedulable.