The classical Budyko framework provides a foundational description of long-term hydroclimatic partitioning but assumes hydrological closure, making it structurally inadequate for basins influenced by glacier melt, groundwater depletion, reservoir regulation, or inter-basin water transfers. This study presents the ShakyaQX (SQX) manifold — a generalized three-dimensional extension of the Budyko hypothesis that represents hydroclimatic partitioning as a surface in the state space (x, γ, φ), where x = PET/P is the climatic aridity index, γ is a landscape response parameter, and φ = 1 + ΔS/P is an orthogonal openness coordinate derived from the generalized water balance. By introducing φ as a true independent geometric axis rather than embedding it in the climatic coordinate, SQX eliminates the hyperbolic singularity that afflicts all input-rescaling (IR) approaches ([[EQUATION]] → ∞ as P + ΔS → 0) and avoids the piecewise bimodal formulations required by Gao et al. (2020) and He et al. (2025). The SQX formulation is proven to satisfy strict monotonicity and global concavity, to maintain bounded sensitivity [[EQUATION]]), and to reduce exactly to the Mezentsev-Choudhury-Yang curve when φ = 1.For the special case γ = 1, SQX reduces to the harmonic sum of x and φ, directly linking Budyko theory to the dual-resistance (Penman-Monteith) framework at the basin scale. Application to the Brahmaputra Basin at Golpara (2002-2021) yields [[EQUATION]] at annual scale.At monthly scale, 63 of 242 observations satisfy the SQX admissibility conditions (φ > 0, ET ≥ 0, [[EQUATION]]), yielding [[EQUATION]] with NSE = 0.650 and KGE = 0.651. The three admissibility violation years (2010, 2013, 2017) are diagnosed as probable GRACE overestimation events, demonstrating that the SQX admissibility bound functions as a sensitive detector of satellite storage uncertainty in cryosphere-influenced basins. SQX outperforms IR on four of five performance metrics and provides a structurally robust, physically interpretable foundation for hydroclimatic analysis under non-stationary and anthropogenically influenced conditions.