Boundary–Complete-Loop Theory Volume III: LBR Abstract Topological Algebra Subtitle Integral-Loop Equations, Root Extensions, Group Actions, and Algebraic Topological Families

We develop LBR abstract topological algebra from the integral-loop equation \[ 0Λ Λ⊗Z R T_Λ0. \] A finite LBR state is algebraized through the loop group algebra \(R[Λ]\), while boundary, attaching, neutral, decoration, and filling relations are retained as separate typed ideals. Ordered words and realization certificates are preserved because the resulting quotient algebra alone does not determine the complete LBR state. Using effective equation probes, we construct integral-loop equation states and contravariant coordinate operations. Finite-index lattice inclusions are converted, through certified Smith normal forms, into root groups and monomial root equations. Coefficient roots and loop roots are treated separately. Character actions on root variables lead to computable invariant algebras, quotient states, and criteria identifying root-character groups with full deck-transformation groups. We further develop typed base change, conservative finite descent, effective integral closure, and parameterized algebraic LBR families. Their fibers retain boundary, raw-loop, attaching, filling, decoration, branch, and normalization data. Algebraic discriminants provide candidate change loci, while genuine topological change is certified only by comparison of complete topological normal forms. This distinguishes coefficient extension, loop-lattice extension, contraction, splitting, merging, degeneration, and topological phase change. For finite effective classes with terminating confluent normalization and complete coverage certificates, the constructions yield decision procedures with explicit success or obstruction certificates. Necessary and sufficient finite criteria are established for seven principal completeness problems: root realization, deck identification, invariant generation, integral descent, topology-discriminant completeness, effective integral closure, and compiler coverage. Keywords Boundary–Complete-Loop theory; LBR topology; integral-loop equations; loop group algebras; Laurent algebras; root-lattice extensions; Smith normal form; group actions; invariant algebras; deck transformations; effective descent; integral closure; algebraic topological families; topology discriminants; certified computation.