Operator Algebra of Institutional Alignment: A Mathematical Framework for Diagnosing Institutional Failure

Roshan Ghadamian
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Roshan Ghadamian: Institute for Regenerative Systems Architecture

IRSA Working Papers from Institute for Regenerative Systems Architecture

Abstract: Institutions fail not for want of resources or expertise but because capital cycles are misaligned with the temporal structure of institutional missions. Ghadamian formalises that misalignment through two operators: a decoupling operator Δ removing dependence on fragility cycles, and an alignment operator Λ synchronising capital with mission cycles. Their definitions are given; their algebraic behaviour — how they compose, commute, interfere and fail — is not. This paper supplies it. Working in the space of square-integrable cycle functions, we show that Δ and Λ are orthogonal projections onto closed subspaces, that their composition A = ΛΔ is the alignment transform, and that the positive operator A*A has spectrum in [0,1] whose eigenvalues are the squared cosines of the principal angles between the fragility-invariant and mission-cycle subspaces. Those angles are the exact sense in which alignment admits degrees. A norm-based alignment index follows with range [0,1] by construction, and cross-domain interference appears as the non-vanishing of commutators between the alignment operators of different mission domains.

Keywords: operator algebra; institutional alignment; orthogonal projection; principal angles; spectral decomposition; commutators (search for similar items in EconPapers)
JEL-codes: C02 D02 L20 O43 (search for similar items in EconPapers)
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