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

Organizational design frameworks prescribe configurations across interdependent levels, yet none quantifies the limits of exhaustive specification. This paper applies high-dimensional geometry to OrgSchema Theory’s (OST) 8x6 activation matrix, formalized as a 48-dimensional specification space [0, 1]48. Three results follow. The Coverage Impossibility Theorem: at resolution ε = 0.1 there are 1048 distinguishable specifications, each covering a ball of volume V48(0.1) ≈ 1.38 × 10−60, so even 1020 specifications cover 10−40 of the space. The Effective Dimensionality Theorem: the cascade model reduces effective dimensionality to deff = 15.75 at an illustrative γ = 0.5, a 67% reduction. The Forkability Theorem: the fork model decomposes the space into shared and private subspaces, formalizing franchises and open-source ecosystems. An information-theoretic analysis sets 159.4 bits of full specification against a working-memory capacity near 23 bits, a ratio that reduces to the dimension count over the chunk count and is resolution-free. These geometric results are complementary to Simon’s bounded-rationality argument: cognitive limits make optimization difficult, whereas geometric impossibility precludes exhaustive coverage regardless of cognitive capacity. The paper distinguishes specification impossibility from NK landscape search complexity, supplying the first formal volumetric justification for OST’s specialization requirement. Includes zharnikov-2026h-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 github.com/spectralbranding/paper-spec for the standard. This PDF is generated programmatically from that machine-first source under a research-as-repository model.