Paper 16 of the Interior Observer (IO) Cosmological Framework resolves the last open seam in the dark energy derivation chain: why the gauge projection factor enters as √(1+γ²) (degree 1) rather than (1+γ²) (degree 2). Paper 9 labeled this the "phenomenological one-leg ansatz." This paper provides conditional closure through seven new theorems (transfer homogeneity, form-degree response, carrier representation, slice-family collapse, axiom elimination, two-fiber transfer degree, and a no-universal-1-form selector), a composite uniqueness theorem combining marginal trapping locality with bipartite transfer structure, and a universal transport no-go derived from framework self-consistency (GTTP violated at 20.15σ). Under the explicit premise package P1–P6, √(1+γ²) is the numerically unique correction factor for a local boundary-to-bulk transfer observable. The result is DERIVED/THEOREM conditional on the premise package — the same epistemic structure as Paper 15's Rosetta Boundary Observable Theorem. Twenty-seven failed derivation approaches are documented. Supplement S1 contains the self-consistent transport no-go computation. https://dfife.github.io/index.html v1.5 (April 2026): Appendix A replaced with Paper 15 v1.4 clean foundation + Paper 16 A.1.15 new results. K_Λ corrected from 49.85 to 48.50 (one-leg ansatz removed, Paper 9 v1.5). "Energy projection" → "matter projection." Status labels homogenized. Author block added. v1.4 (April 2026): Schur branch correction. All mixed-branch derivation chains (H₀ = 67.58, H₀,modified = 68.51) marked SUPERSEDED with Schur definitive reference (H₀ = 68.91, Paper 29). θ* updated to IO-native 0.599° (Papers 20–21). Appendix Steps 29, 58 and comparison lines updated. The gauge projection degree analysis (§§3–5) is structural and branch-independent; only the numerical starting values change. Title page standardized. v1.3 (March 2026): Cycloid parameterization correction applied to Appendix A foundational package. All contracting-convention values updated to expanding convention. v1.2 correction notice removed from abstract. Composite uniqueness theorem and all main body results are convention-independent. Appendix steps renumbered sequentially (1–81). Title page reformatted. See Paper 21 v1.1 for the full audit. v1.2 - Paper 19 correction annotations. Abstract updated with two founding premises. Core results entirely unaffected: composite uniqueness theorem, P1-P6, degree-1/degree-2 transfer machinery, marginal trapping locality, 27 dead approaches. P6 universal transport no-go (GTTP at 20.15σ) revalidated unchanged on Paper 19 background. Paper 19 uses Paper 16 as link 5 of 6 in the RT chain deriving the IO Friedmann equation. Appendix entries for background-dependent quantities (projection map, N_eff = Δ, Ω_k) superseded.