Decomposition of shared-parameter contributions in multi-probe cosmological inference: implications for Ωₘ, Ω_Λ, and cross-probe consistency

Joint cosmological inference combines Type Ia supernovae, BAO, and CMB measurements whose individual pipelines honestly carry their uncertainty pair-structure (expressed + bound) end-to-end, yet the joint fit silently collapses that pair-structure at the shared-parameter layer through scalar-only prior specification and shared-latent reuse. This paper decomposes the Δχ² attributable to shared-parameter contributions in a standard SN + BAO + CMB pipeline and quantifies the void — the gap between a full-pair representation and its scalar-collapsed counterpart — at Δχ² = 279 across the joint-probe recombination step. The decomposition is performed under the eb-binary carrier set (Martin 2026a, characterization theorem RSOS-260797), which forces pair-structure through all six axioms A1–A6; scalar collapse at any step produces a loss function measurable in this form. We locate the structural leak at the joint-inference boundary rather than inside any individual pipeline. A drop-one-probe sensitivity ensemble confirms the decomposition is not driven by any single dataset. Implications for Ωₘ, Ω_Λ, and cross-probe consistency in light of DESI DR1 are discussed. The Δχ² = 279 figure is the first concrete operational measurement of the framework's void construct and is diagnostic rather than prescriptive: it quantifies how much the current recombination architecture costs, without specifying which replacement is preferable. Manuscript: MN-26-1117-P (MNRAS, submitted).