Dimensional Reduction via Controlled Combinatorics (DRCC-V2-REW2)

Abstract

Dimensional Reduction via Controlled Combinatorics (DRCC) is presented as a finite mathematical framework for analyzing controlled structural reduction and reconstruction in combinatorial search spaces. For a finite problem instance P with candidate space X(P), DRCC defines a controlled map to a reduced structural space Z(P), a reduced state, an associated reconstruction space, and the fragmentation and reconstruction dimensions. Tensor representations serve as structural carriers: tensor restrictions, tensor projections, tensor fragments, tensor-equivalence classes, and tensor-output classes encode local relations while preserving the information required for admissible reconstruction. Computational effort is decomposed into collapse and reconstruction costs and compared with an explicitly stated baseline. A count-level gain is accepted only when the complete DRCC count is smaller than the baseline count. Wall-clock and asymptotic claims require separate implementations, measurements, and complexity bounds. The framework is examined through finite case studies involving the Full Adder, the Dadda multiplier and digital systems, graph coloring, Boolean satisfiability, constraint satisfaction, travelling-salesman-type routing, and housing selection. These cases illustrate structural-class reduction, exact or bounded reconstruction spaces, reproducible cost accounting, and evidence levels that distinguish structural, count-level, and empirical results. DRCC is positioned relative to enumeration, backtracking, constraint propagation, kernelization, symmetry reduction, structural-width methods, numerical tensor methods, and established digital-arithmetic architectures. The work does not claim P = NP, a universal speedup, or polynomial solvability from tensorization alone. Its contribution is a precise language and verification protocol for determining when controlled structural reduction preserves admissible solutions and may yield a finite computational advantage under an explicit cost model.

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