We construct a coercive realization of the Yang-Mills mass gap using Hilbert—de Sitter Spectral Geometry (HdSSG): the operator-theoretic framework built on SO(1,4) principal series representations, de Sitter causal-diamond exhaustion, and Mosco convergence of quadratic forms.The central result is a canonical equivariant unitary embedding (H⊥, L) → Π_{3/2}(SO(1,4)) via a Poisson-kernel intertwiner W_ξ, where L = Π⊥(-Δ_dS + 2gρ_s)Π⊥ is the HdSSG mass-gap operator. Positivity L > 0 and compact resolvent follow from elliptic theory and Kato-Rellich; the spectral gap Δ_HdSSG > 0 is the Casimir eigenvalue of Π_{3/2}.A GP condensate Y-junction serves as the coordinate realization, supplying the curvature constants C_A = 0.5060, C_F = 0.7200, C_K = 0.3643 from the normalized vortex ODE analytically. The curvature lock C_K < 1 gives cu2081 = 0.6357 > 0.The mass gap is established through: local surjectivity via ker(P_R*) = {0} (Fredholm adjoint coercion), global injectivity via Mosco limit and spectral positivity, density from spectral truncation, and Weyl-sequence exclusion by the same coercive estimate.Result: Δ_YM = inf σ(H_YM|_{Ω⊥}) ≥ cu2081 · Δ_HdSSG > 0, cu2081 = 0.6357. No internal hypotheses remain. Two axioms, no experimentally tuned constants.Full paper: https://doi.org/10.5281/zenodo.20349116