Dimensional Reduction via Controlled Combinatorics (DRCC-V2)

Abstract

This condensed manuscript presents a runtime-oriented formulation of Dimensional Reduction via Controlled Combinatorics (DRCC-V2). DRCC is a finite, discrete framework for studying reconstruction problems through controlled structural reduction. The guiding principle is to preserve the information required for admissible reconstruction and to collapse distinctions that are irrelevant to the reconstruction task. The manuscript defines the classical candidate space X(P), the fragmentation dimension d_frag(P) = |X(P)|, the controlled reduction map C_P: X(P) -> Z(P), the reconstruction space Ω_P(ζ) = {x in X_adm(P): C_P(x) = ζ}, and the reconstruction dimension d_rec(P,ζ) = |Ω_P(ζ)|. The operational stability condition is 0 < d_rec(P,ζ) ≤ d_frag(P) < ∞ The runtime model compares classical enumeration with a DRCC decomposition into collapse and reconstruction: T_DRCC = T_Collapse + T_Reconstruction. The runtime gain is defined as G(P,ζ) = T_Classic(P) / T_DRCC(P,ζ). The transition point W = (n_w, R_w[s]) marks the discrete size at which the classical and DRCC runtime models coincide. The manuscript applies this framework to finite case studies including housing selection, travelling-salesman-type routing, constraint satisfaction problems, satisfiability, graph 3-coloring, and the full adder. These examples are used to illustrate how controlled collapse, reconstruction dimension, runtime gain, and transition behavior can be computed or bounded in concrete settings. The work does not claim a proof of P versus NP. It does not claim a universal speedup for graph coloring, satisfiability, constraint satisfaction, or routing problems. Its contribution is a compact mathematical language and runtime accounting framework for identifying when controlled structural reduction may become advantageous relative to classical enumeration.

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