Theory and Measure of Decomposability

This paper develops a general theory of decomposability for structured systems, problems, and phenomena. The guiding intuition is familiar across mathematics, computation, and science: a large global object is often manageable only when it can be divided into local parts that admit separate analysis, while the dependence between parts is carried by a comparatively small interface. What is usually missing is a common mathematical language for saying when this is truly the case, how strongly it holds, and how it should be measured.

We define decomposability relative to a partition of the primitive objects of a system. A partition is useful only if three things happen simultaneously: local solvability is preserved inside each block, the interface required for global consistency remains bounded, and recomposition of local solutions is stable. This leads naturally to a decomposability profile with five components: local difficulty, interface complexity, coupling strength, reconstruction error, and recomposition stability. Exact decomposability appears as the zero-error case; approximate decomposability allows controlled distortion.

The paper's main conceptual claim is that decomposability is not reducible to problem size reduction alone. A split is valuable only when the interface between blocks is information-economical and the recomposition map is well-conditioned. This reframes many familiar distinctions: line-count versus modularity in proof systems, variable splitting versus true separability in optimization, and descriptive partitioning versus causal modularity in scientific models.

We then outline a constructive theorem program. Under exact interface sufficiency, global solutions factor through local feasible sets and a finite interface state. Under compressed-interface conditions, a low-dimensional interface summary is enough for approximate reconstruction. Under bounded coupling, iterative recomposition converges to the monolithic solution with explicit stability control. Spectral methods, graph Laplacians, and Schur-complement operators provide one important special realization of this program, but they are treated here as consequences of the more general theory rather than as its defining language.

The result is intended as a foundational object language for decomposition-aware reasoning across theorem proving, verification, optimization, planning, and scientific modeling. The longer-term goal is to turn decomposition from an art of good guesses into a measurable and eventually optimizable property.