Specification Impossibility in Organizational Design: A High-Dimensional Geometric Analysis

Organizational design frameworks prescribe configurations across multiple interdependent levels yet no existing theory quantifies the fundamental limits of exhaustive specification. This paper applies high-dimensional geometry to prove that comprehensive organizational specification is geometrically impossible. OrgSchema Theory’s (OST) 8x6 activation matrix is formalized as a 48-dimensional specification space \([0,1]^{48}\); three core results follow. First, the Coverage Impossibility Theorem: at resolution \(\varepsilon = 0.1\) per dimension, there are \(10^{48}\) distinguishable specifications, each covering a ball of volume \(V_{48}(0.1) \approx 1.38 \times 10^{-60}\); even \(10^{20}\) specifications cover only \(10^{-40}\) of the space. Second, the Effective Dimensionality Theorem: OST’s cascade model reduces effective dimensionality from 48 to \(d_{\text{eff}} = 15.75\) at \(\gamma = 0.5\), a 67% reduction. Third, the Forkability Theorem: the fork model decomposes the specification space into shared and private subspaces, formalizing franchises and open-source ecosystems. An information-theoretic analysis shows that full specification requires 159.4 bits, far exceeding human working-memory capacity, establishing that cascade and fork compression is cognitively necessary. These geometric results are strictly complementary to Simon’s bounded-rationality argument: cognitive limits make optimization difficult; geometric impossibility precludes exhaustive coverage regardless of cognitive capacity. The paper distinguishes specification impossibility from NK landscape search complexity, providing the first formal volumetric justification for OST’s specialization requirement within the organizational design literature. Includes zharnikov-2026h-r5-specification-impossibility.yaml (Paper Spec v0.1.0) – a machine-readable specification of the paper's claims, assumptions, and dependencies. The paper's full machine-first bundle (the SPINE claim/dependency graph and the ONTOLOGY term module) lives in the public repository; see https://github.com/spectralbranding/paper-spec for the standard. This PDF is generated programmatically from that machine-first source under a research-as-repository model.