The Interior Observer Cosmological Framework - Paper 15: The Rosetta Boundary Observable Theorem — Structural Derivation of Δ = x⁴(1+γ²) from Quaternionic Half-Density Carriers

Paper 15 of the Interior Observer (IO) cosmological framework derives the Rosetta identity Δ = x⁴(1+γ²) conditionally within the IO framework through three structural theorems. The Quaternionic Norm Theorem establishes that the unique admissible SU(2)-invariant quadratic observable on the horizon boundary carrier is the quaternionic reduced norm, forcing Q(q_γ) = 1+γ². The Half-Density Carrier Theorem shows that background-free same-space pairing forces density weight w = 1/2, and the CMP conformal transgression delivers the scalar coefficient t(η) = x(η)². A Boundary No-Go Theorem establishes that the compactness ratio x = r_s/R_U cannot be produced locally on the horizon and must be delivered from the bulk. These compose into the Rosetta Boundary Observable Theorem (Theorem 11): under assumptions A0–A7, with the Conformal Modular Principle as the one irreducible geometric/modular semiclassical principle and the Semiclassical Separability Ansatz as the carrier-factorization axiom, the observer-epoch boundary observable is O(η) = x(η)⁴(1+γ²)|vol_{S²}|. The identity is epoch-evaluated, not a dynamical law along the full Oppenheimer-Snyder cycloid. Five revision cycles with two independent adversarial computational reviewers verified the complete proof chain. v1.1 - Paper 19 correction annotations. Abstract updated with two founding premises. Paper 15's own results (Rosetta Boundary Observable Theorem, Quaternionic Norm Theorem, Half-Density Carrier Theorem, Boundary No-Go Theorem) are pure boundary mathematics and entirely unaffected. Appendix A catalog entries for background-dependent results (projection map, N_eff = Δ, Ω_k = −0.006, H₀ composite) annotated as superseded by Paper 19.