Boundary–Loop Rigidity: Symbolic Classification, Local Dissection, and Boundary-Adapted Solution Theory

Abstract

We introduce Boundary–Loop Rigidity (LBR), a formalism that separates boundary data, complete-loop data, discrete symbols, and continuous realization parameters. Its algebraic model is a split exact sequence whose loop quotient is identified with the boundary space. When the loop group is infinite cyclic, every element has a unique normal form consisting of a boundary representative and an integral multiple of a fixed loop. For regular plane curves, the integer coordinate is the relative winding of a lifted tangent angle and agrees, in the closed case, with the degree of the unit tangent map. Local objects are encoded by boundary ports, directional symbols, loop generators, and attaching relations. Verified interiors may be replaced by condensed interface states, with composition governed by a gluing cocycle. A well-founded complexity gives termination for LBR-finite presentations, while compatible finite exhaustions provide finite-stage states and stabilized local invariants for selected infinite objects. Under local-model, descent, conservativity, and confluence hypotheses, the resulting standard-piece presentation determines the original object in the declared category. For signed strings in dimension \(N\), coordinate-permutation orbits are indexed by multiplicities \((p,q,r)\). An additional mixed-chirality axiom gives the conditional count \(D(N)=N^2+N+1\). In the metric-rigid coupling model, the generic normalized parameter dimension is \(\max\{0,\min(p,q)-1\}\), and the number of family-bearing chiral branches is \(K(N)=(N-2)(N-3)\) for \(N\ge4\). Almost-abelian Lie algebras provide decorated realizations of every formal triple. Typed replacement rules describe surgery on normalized presentations. For finite rule libraries with decidable admissibility predicates, one-step surgery reduces to finite certificate search and algebraic state update. The analytic counterpart is the boundary lifting \(X=E(G)\oplus\ker\mathcal T\), combined with Schur-complement condensation and homogeneous spectral expansions for ordinary and partial differential equations. **Keywords** Boundary–Loop Rigidity; symbolic classification; complete-loop dissection; boundary condensation; signed skeletons; gluing cocycles; algebraic surgery; spectral methods.