Description This work presents a purely structural proof that odd perfect numbers do not exist. Building on Euler’s classical representationN = p^{4k+1} m², p ≡ 1 (mod 4),we analyze the defining equation σ(N) = 2N through p-adic valuation transfer, multiplicative order constraints, and exponent parity conditions. The key observation is that all p-adic valuation required by σ(N) = 2N must be supplied by σ(m²), while each prime power divisor of m contributes to this valuation only in discrete units determined by order-theoretic constraints. These discrete valuation contributions are shown to be incompatible with the exponent structure imposed by the Eulerian form. The argument is entirely non-computational and does not rely on finite enumeration, size bounds, or numerical verification. The contradiction arises purely from valuation structure, multiplicative orders, and exponent constraints. As a consequence, no configuration within the Eulerian framework can satisfy all necessary conditions simultaneously, leading to a structural nonexistence result for odd perfect numbers. This Zenodo version presents the full logical conclusion explicitly. A more conservative formulation, avoiding an explicit nonexistence claim, has been prepared separately for journal submission. Keywords odd perfect numbersnumber theoryEulerian formdivisor sum functionp-adic valuationmultiplicative orderexponent constraintsstructural eliminationnon-computational proofvaluation theory Optional Metadata Related identifiersIs supplement to:Structural Elimination of a Touchard Branch in Odd Perfect Numbers (RP14) CommunitiesMathematicsNumber Theory LicenseCC BY 4.0 Version noteThis version explicitly states the nonexistence conclusion.A journal-safe variant formulates the result as a structural collapse of the Eulerian framework.