The Geometry of Business: A First-Principles Derivation of the Structural Constant of the Single-Product Operating Cycle

This paper identifies the single-product operating cycle — one product, one supplier, one customer — as the minimal closed financial system and derives its structural constant from first principles. Using only two inputs, the discount rate rr r and the growth rate gg g, a single derivation produces the dimensionless structural multiplier Λ = 1/(r−g). For typical developed-market parameters (r = 9%, g = 3%), this yields Λ ≈ 16.7, consistent with observed enterprise value multiples across major asset classes without parameter fitting. The annual planning cycle is established as the natural unit of time by six independent structural arguments. The viability condition r > g defines the stable operating range; its breach defines the geometry of failure. The asymmetric failure principle identifies soft boundaries as the direction of endogenous failure, readable from the governing valuation relation before failure occurs. Eight explicit limitations are stated, including the absence of a firm-level empirical test and the restriction of the framework to the minimal single-product unit. Two falsifiable predictions are specified for empirical testing. This paper is part of the DEP research series (ORCID: 0009-0001-4493-3473). It is consistent with the descriptive language of the Dynamic Equilibrium Principle but does not rely on it for its validity.