Abstract
<title>Abstract</title> <p>We present two general statistical results for combining and diagnosing published cosmological measurements, and demonstrate both on real data from two independent tensions. First, we derive a closed-form expression for the variance inflation that results from combining measurements that are not fully independent — as occurs, in practice, whenever an aggregate multi-route estimate is combined with its own constituent routes. We prove that this reduces, in the equal-variance uniform-correlation case, to the classical survey-statistics design effect DEFF = 1 + (N − 1)ρ (Kish 1965), give the general unequal-variance form as a sandwich-variance identity, and discuss its relationship to generalized least squares and inverse-variance meta-analysis. Applied to the Hubble tension, published route-level H₀ values (SH0ES Cepheid–SN Ia, TRGB, Tully–Fisher) combined naively alongside a published seven-route covariance summary (Chen & Wang 2026) imply an effective overlap ρ_eff ≈ 0.84, which inflates the apparent tension from the correct 5.32σ to ≈ 6.5σ purely as a statistical artifact. Second, we formalize the falsification-region diagnostic as a general structural statement: when candidate systematics act as additive linear shifts on affected measurements, contours of constant tension significance are hyperplanes in systematic-parameter space, giving a reusable, low-dimensional visualization for any two-parameter systematic-sensitivity question. We demonstrate this on the H₀ tension (SH0ES/TRGB/Tully–Fisher vs. Planck), where the map shows the two most-cited residual calibration systematics cannot bring the tension below 3σ anywhere in their literature-plausible range, and on the S₈ tension (DES Y6 vs. Planck), where the identical framework, applied to baryonic-feedback and intrinsic-alignment amplitude, shows the tension can plausibly fall below 1σ within literature-motivated systematic ranges. The same mathematical structure thus correctly distinguishes a tension that resists known systematics from one that is plausibly explained by them — and, being generic to correlated-measurement combination, applies equally outside cosmology. As joint-probe cosmological analyses grow more complex, checking for exactly this kind of overlap is a low-cost statistical-hygiene step that keeps evidence that has already been counted once from being counted again as independent confirmation.</p>