Geometric–Structural Layer Closure (Pasev) II: Operators and Relational Geometry

This work extends the foundational framework by introducing operators acting on sections of the fibered structure, enabling relational geometry across the base space. A connection operator is defined: ∇ : Γ(E) → Γ(T*O ⊗ E) Curvature is introduced as: R(v, w)s = ∇v∇ws − ∇w∇vs A fiberwise metric structure is defined to provide measurement: gα : Fα × Fα → ℝ All operators are defined in a strictly algebraic, non-differential setting, without assuming a smooth manifold structure. Variation is interpreted structurally rather than through infinitesimal calculus. Compatibility conditions ensure: • Fiber preservation • Projection invariance • Structural consistency across layers This layer establishes a complete relational geometric system while remaining within the constraints of the foundational structure.