Controlled Partial Reconstruction Width (CPR-WIDTH-V1.1)

Abstract

Controlled Partial Reconstruction Width (CPR-Width) is introduced as 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. For a finite problem instance P and an admissible reconstruction ordering π, an active reconstruction interface records exactly those processed components whose information can still influence an admissible future continuation. CPR-Width is defined as the maximum cardinality of this interface during reconstruction and is minimized over all admissible orderings. The framework distinguishes raw interface states, reachable states, and task-relative semantic states obtained through future-extension equivalence. This yields the structural chain ω_rec(P, π) → S_CPR(P, π) → N_CPR(P, π), (0.1) which connects reconstruction width, semantic-state growth, and computational cost. A complete runtime model separates construction, reduction, semantic transition, reconstruction, verification, and output costs. Four boundary conditions characterize explicit representation, polynomial constructibility, sound and complete reconstruction, and polynomial semantic-state control. Representative realizations are presented for Boolean satisfiability, constraint satisfaction, graph coloring, tensor representations, and digital arithmetic. Exact finite verification is provided for the full adder and ripple-adder models. The framework establishes a formal basis 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. Keywords: Controlled Partial Reconstruction Width; CPR-Width; active reconstruction interface; semantic-state compression; future-extension equivalence; structural width; finite combinatorial reconstruction; runtime analysis; constraint satisfaction; graph coloring. Scope and Non-Claims. The results apply only to the explicitly defined reconstruction models and to problem families satisfying all stated boundary conditions and runtime assumptions. A small reconstruction width alone does not imply polynomial runtime, and structural compression alone does not imply a wall-clock advantage. No NP-complete language is proved in this work to satisfy the complete CPR tractability conditions.

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