Controlled Partial Reconstruction Width (CPR-Width-V2)

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 quantifies the maximum amount of task-relevant information that must remain simultaneously active during a sound and complete reconstruction process. For a finite problem instance P and an admissible reconstruction ordering \pi, the framework introduces active reconstruction interfaces together with semantic-state compression, yielding the structural chain \omega_{\mathrm{rec}}(P,\pi) \rightarrow S_{\mathrm{CPR}}(P,\pi) \rightarrow N_{\mathrm{CPR}}(P,\pi), which links reconstruction width, semantic-state growth, and computational cost. A complete runtime model distinguishes construction, reduction, semantic transition, reconstruction, verification, and output costs. Four explicit boundary conditions characterize when reconstruction remains polynomially controllable. The framework is illustrated by representative realizations for Boolean satisfiability, constraint satisfaction, graph coloring, tensor representations, and digital arithmetic, including exact finite analyses of full-adder and ripple-adder circuits. CPR-Width establishes a formal framework for analyzing controlled partial reconstruction and task-relative structural complexity. It does not claim universal tractability, a general polynomial-time algorithm for NP-complete problems, or a proof of P=NP. Instead, it provides a rigorous mathematical foundation for reconstruction-oriented structural analysis of finite combinatorial systems.

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