Controlled Partial Reconstruction Width (CPR-Width-V2.0)

Abstract

Controlled Partial Reconstruction Width (CPR-Width) is a reconstruction-oriented structural parameter for finite combinatorial problems. Instead of measuring the complete candidate space, CPR-Width measures the maximum number of task-relevant components that must remain simultaneously active during a sound and complete reconstruction process. Let P be a finite problem instance and let π be an admissible reconstruction ordering. At each reconstruction stage, an active interface records the processed components whose information can still influence an admissible future continuation. The unnormalized CPR-Width is defined from the maximum cardinality of this interface over the complete reconstruction process and is minimized over admissible orderings; a fixed offset convention then yields the normalized CPR-Width ω_rec used in the structural chain below. The framework distinguishes raw interface states, reachable states, and task-relative semantic states obtained through reconstruction-safe signature equivalence, yielding the structural chain ω_rec(P,π,τ;M) → S_CPR(P,π,τ;M) → N_CPR(P,π,τ;M) (with τ and M suppressed when fixed) connecting reconstruction width, semantic-state growth, and complete computational cost. A full runtime model separately accounts for construction, reduction, semantic transition, reconstruction, verification, and output costs. Four boundary conditions are then stated for explicit representation, polynomial constructibility, sound and complete reconstruction, and polynomial semantic-state control. Representative realizations are given for Boolean satisfiability, constraint satisfaction, graph coloring, tensor representations, and digital arithmetic, including a dedicated theory of controlled and constant CPR-Width subclasses. Exact finite verification is provided for the full-adder model, while a uniform family-level result is established for the ripple-adder family. This work establishes a formal framework for analyzing controlled partial reconstruction. It does not establish universal tractability, a general polynomial-time algorithm for NP-complete problems, or a proof of P = NP.

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