Conditional Reconstruction Closure A: Typed Closure and Composable Loss Budget for Local Constraint Networks

The typed foundation of local constraint networks converts three common failure modes—object-type confusion, non-composability of loss budgets, and non-commutativity of structuraloperators—into separately statable, machine-checkable theorem obligations. The entire papertakes a constraint network with type labels as its substrate: the res-track carries nonnegativeweights, probability normalization, and budget bookkeeping; the alg-track carries signed weights,Abelian group operations, and cochain structure; the two tracks are separated at the definitionallevel, and cross-track mixed computation is forbidden. On this substrate, the closure operatoris instantiated as a monotone propagation operator on the lattice of constraint sets; the leastfixed-point theorem on complete lattices thereby acquires a concrete network semantics, andthe fixed-point set forms a complete lattice under the induced order; the minimal informationloss budget is defined on channels induced by deterministic restricted morphisms, composablesubadditivity holds along composition in the category of restricted morphisms, and conservativeaddition of feasible budgets yields a cross-scale bookkeeping rule with known direction; thenoncommutative residual of projective structural maps is witnessed by explicit rational matrices,and the edge-by-edge vanishing of commutators along a path is a sufficient condition for multistep path independence; the closure process is divided into five types of outcomes accordingto the lattice layer, the pro-system layer, and the presheaf-gluing layer, and their respectivevalidity conditions and mutual non-implication relations are registered item by item; the finitedimensional prototype of hard-constraint condensation under the block-diagonal assumptionyields exact identities for the nullity jump, the Moore–Penrose pseudoinverse limit, and thepenalty-rescaling degeneration. All finite examples and parametric enumerations are verified byexact rational arithmetic or 60-digit arbitrary-precision comparison; definitional assertions andstructural arguments are marked truthfully, and machine-check labels do not overflow the scopestated in their entry text. This paper does not derive physical field equations, does not claimthat all closures terminate finitely, and does not extrapolate finite-dimensional bookkeeping intoinfinite-dimensional theorems.