Traditional organizational theory treats centralization and decentralization as structuraldesign choices, yet provides no unified mathematical framework for explaining how these choicesshape communication, coordination, and adaptation. This paper develops a geometric theory ofdecentralization by representing organizational information flow as a differentiable Riemannianmanifold (M,gκ) parameterized by a centralization scalar κ ∈ 0,1. Three canonical choicesfor the communication metric gκ are specified — resistance distance, diffusion, and Fisherinformation metrics — and the induced curvature tensor Rκ = Riem(gκ) is derived analyticallyfor each. Decentralization is formalized through the canonical functionalD(M,gκ,Rκ) = ∥Rκ(x)∥dVgκ (x),Mwhich is shown to be monotone-decreasing in κ under mild regularity conditions. A curvaturecoordination operator maps geometric invariants to coordination cost and adaptive capacity,yielding a curvature-driven performance frontier. The critical transition κc separating mechanistic,organic, and distributed self-management (DSMS) regimes is characterized as the unique κ ∈ (0,1)at which the spectral gap of the Ricci operator Ric♯gκ vanishes to leading order. A complete measurement framework connects each geometric construct to observable organizational variables: κ to span-of-control and decision- rights indices, K(κ) to betweenness-centrality statistics,and Jcoord to communication-overhead and decision-latency proxies. The resulting theoryyields falsifiable predictions, a principled design optimization, and a geometric interpretation ofcontingency theory, providing a mathematically rigorous and empirically tractable foundationfor analyzing and designing complex organizations.
Usman Zafar (Mon,) studied this question.