The Shadow Principle: A Single Involution from Scale to Sky

The shadow framework is organized by one transported involution. It appears first as inversion on the multiplicative line, preserving Haar measure on (ℝ⁺,×). Under the Mellin transform, in the half-density normalization appropriate to the classical variable s, it becomes the reflection s←→1 - s. In analytic number theory this is the symmetry of the completed zeta function and, more generally, of the adelic zeta integral of Tate's thesis. In celestial holography it becomes the shadow transform Δ → 2 - Δ, whose unitary locus is the principal series Re(Δ) = 1. On the compact Grassmannian model Gr(2,4) it is the orthogonal-complement involution on two-planes, represented in Plücker coordinates by the Hodge-complement on decomposable two-vectors. At null infinity it appears as the antiunitary boundary operation T symmetry, which sends 1 + iλ to 1 - iλ by complex conjugation. This paper separates what is proved here from what is imported from the larger construction. The formal core is an abstract conjugacy theorem for transported involutions. The direct results prove: Haar inversion on (ℝ⁺,×); the Mellin half-density reflection; the unitary Mellin axis Re(s) = ½; the arithmetic shadow principle identifying the functional equation ξ(s) = ξ(1-s) as the Mellin image of Haar inversion after Poisson summation; the conjugacy of s ←→ 1 - s with Δ ←→ 2 - Δ under Δ = 2s; the spectral action of boundary T on principal-series weights; and measure preservation of the Grassmannian complement. Together these results give a commutative shadow diagram linking scale, arithmetic, celestial representation theory, Grassmannian geometry, and null-infinity physics.