Parity Interpolation Theorem - The interpolation exponent of a two-channel constitutive law is fixed by time-reversal parity.

A medium responding through a reactive channel and a relaxational channel inseries admits a one-parameter family of constitutive interpolations,K = K₀X/(1 + Xⁿ)^(1/n), where X is the ratio of the two channels' compliances.The asymptotic requirements K → K₀ as X → ∞ and K → K₀X as X → 0 hold for everyn > 0, so they fix the family but not the member; in applications the exponent ischosen empirically, with n = 1 and n = 2 both in common use. This note shows that the exponent is not free. For a linear response whoseequation of motion contains exactly one odd-order time derivative, time-reversalparity places storage and loss on orthogonal axes of a single complex response,leaving no mixing angle available. Compliances carrying a common flux add asvectors in that plane, so the interpolation exponent is the norm exponent of theresponse plane; parity makes that plane Euclidean, and n = 2 follows uniquely,giving K(X) = K₀X/√(1 + X²). The case n = 1 is not merely disfavoured butstructurally inconsistent: reaching it requires the dissipative channel to beeven-order in time, which removes the relaxation the second channel wasintroduced to carry. At the crossover X = 1 the two candidate laws differ byexactly √2 — a parameter-free discriminant, since no property of the mediumenters the comparison. The result is conditional on its stated hypotheses throughout, and is a statementabout a class of constitutive laws rather than about any particular medium. Itarose as an internal step in a wider single-medium framework, and is stated hereindependently because the proof uses only the parity of the response operator andthe geometry of compliance addition.