The Observer as Hypotenuse: Closure of the 2π Gauge Residue via Real–Adjoint Sheet Orthogonality at the C7 Junction

In the Iterated Klein Bottle Lattice (IKBL) framework, the derivation of the fine-structure constant yields the primary result α⁻¹ = 320π/7 − 2π = 306π/7 ≈ 137.332. The subtraction of 2π in this expression has been argued on topological grounds in prior work but not derived from first principles. This supplement provides that derivation. We identify that the Real sheet ℛ and the Adjoint sheet 𝒜 of the IKBL manifold are orthogonal at the junction hypersurface Σ, forming a unit right triangle in the space of sheet orientations. The observer, functioning as an SU(2) filter frame, occupies the hypotenuse of this triangle — equidistant from both sheets — with sampling efficiency S_obs(π) = |cos(π/4)| = 1/√2 at the…