Manifold Rarity and Quantitative Telos: A Sheaf-Geometric Utility with QFI-Stable Instrumentation

We develop a portable, testable calculus for coherent novelty. The core quantity is a manifold rarity score MR(x) that prices jointly rare, internally coherent composites in a representation manifold, and a quantitative telos field τ(x) that points along directions where rare coherence actually pays off. Technically, MR combines (i) near-mass rarity, (ii) local dimension regularization, and (iii) a coherence guard; telos combines (i) information-geometric curvature and (ii) spectral-tilt sensitivities into a direction of value. We add a sheaf-theoretic gluing residual Ob that penalizes moves which fail to globalize across contexts. The main results: (1) invariance and stability under quotient flows; (2) rate-function scheduling (Murphy horizons) for discovering high-MR targets at bounded risk; (3) an action–betweenness correspondence showing where to probe; (4) falsifiable predictions linking rarity–telos to adoption speed and discovery latency. Algorithms (rarity, gluing, telos estimation) and PGFPlots figures are provided.