Controlled Partial Reconstruction Width (CPR-WIDTH-V1)

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. 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. CPR-Width is defined from the maximum cardinality of this interface over the complete reconstruction process and is minimized over 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) connecting reconstruction width, semantic-state growth, and complete computational cost. A full runtime model is developed that 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. Exact finite verification is provided for the full adder and ripple-adder models.

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