Boundary–Complete-Loop Theory: LBR Topology and Boundary-Adapted Analysis

We develop Boundary–Complete-Loop Theory as a self-contained constructive framework for topology and analysis. Its primitive objects are normalized states consisting of typed boundaries, directional skeletons, graded raw-loop systems, attaching relations, decorations, and finite presentations or compatible exhaustions. Locality is defined through stable probes and reconstructive covers rather than point sets or prescribed open subsets. Continuous morphisms are characterized by the pullback stability of local probes. The theory internally constructs homotopies, chain complexes, homology, cohomology, cup and cap operations, duality certificates, fibers, lifting structures, manifold-recognition procedures, and certified topology-changing operations. Raw complete loops, closed cycles, fillable cycles, and homology classes are kept strictly distinct: \[ Λ_kʳᵃʷ(X)≠ H_kLBR(X). \] Homotopy is defined by certified elementary deformations and interval-state cylinders before homological invariance is established. Fibers arise as pullback states and retain their induced boundaries, vertical loops, transport data, and decorations. For finite presentations, the framework supplies terminating algorithms for normalization, continuity verification, homotopy checking, homology and cohomology computation, fiber construction, obstruction detection, local manifold recognition, and surgery updates. Results requiring additional realization, confluence, or conservativity hypotheses are explicitly separated from internally proved statements. **Keywords** Boundary–Complete-Loop Theory; Boundary–Loop Rigidity; constructive topology; normalized states; raw loops; LBR homotopy; LBR homology; LBR cohomology; fiber states; obstruction theory; certified computation; topology change.