Controlled Partial Reconstruction Width (CPR-WIDTH-V2.1)

Abstract

Controlled Partial Reconstruction Width (CPR-WIDTH) is introduced as a reconstruction-oriented structural framework for finite combinatorial problems. Instead of measuring the complete candidate space, CPR-WIDTH measures the maximum amount of task-relevant processed information that must remain simultaneously active during a sound and complete reconstruction process. For a finite problem instance, an admissible reconstruction ordering induces reachable prefix states, task-relative future signatures, minimum task-sufficient active interfaces, and corresponding raw, reachable, and semantic state spaces. This yields a structural dependency chain from reconstruction width to semantic-state cardinality and complete computational cost. The framework distinguishes the operational, unnormalized interface width from its normalized reporting form and separates structural compression from runtime conclusions. A complete operation-count model is developed with six disjoint components: construction, reduction, semantic transition, reconstruction, verification, and output. Four boundary conditions are formulated for explicit uniform representation, polynomial constructibility, task-relative soundness and completeness, and polynomial semantic-state and transition-structure control. Polynomial-time tractability is obtained only when these conditions are combined with polynomial bounds for all six runtime components. Representative realizations are given for Boolean satisfiability, constraint satisfaction, graph coloring, tensor representations, and digital arithmetic. For graph-based realizations, classical separator-style boundaries are treated as candidate interfaces whose exact CPR status requires task sufficiency and, for equality claims, cardinality minimality. For the ripple-adder family, a uniform all-input carry-transition schema is established with exact operational interface width one for all n\ge2, constant carry-state alphabet, linear semantic-state and transition structure, and complete operation count \Theta(n) for the declared schema task. The fixed-target counting experiment is treated separately under its own task. The framework does not establish universal tractability, a general polynomial-time algorithm for NP-complete problems, or P=NP. Its contribution is a formal method for separating global candidate-space size, active reconstruction width, semantic-state structure, and complete computational cost within a task-relative reconstruction theory.

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