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Cosmological tensions, most prominently the H0 and S8 discrepancies, are now understood as global consistency conditions on the underlying cosmological model rather than as isolated anomalies. We formalise this insight as a cosmological consistency triangle with three vertices that any viable cosmological model must populate simultaneously. The first two vertices, background expansion (H0) and present-day growth amplitude (S8), are universal across deviation classes. The third vertex is theory-specific: we give a formal definition that fixes it, up to reparametrisation, by the irreducible function F(z) introduced by the deviation, and demonstrate via counter-examples that the conditions have non-trivial content. We apply the framework to two contrasting deviations from ΛCDM. For f(Q) gravity, the third vertex is the growth shape, summarised by the growth index γ, against which the recent literature is surveyed. For interacting dark energy, it is the perturbation-level coupling diagnostic: we develop an original linear-perturbation analysis of the compartmentalisation model and present numerical predictions for fσ8(z) showing that the linear and non-linear coupling families are degenerate in redshift-shape at matched dimensionless coupling strength, with third-vertex discrimination acting amplitudinally in the strict quasi-static limit. From the third-vertex requirement we derive a five-point diagnostic checklist for IDE-class analyses, with brief applications to early and dynamical dark energy. The framework clarifies why background-only fits constrain at most two vertices of the triangle; DESI DR3, Euclid, the Simons Observatory, CMB-S4 and Einstein-Telescope-class sirens will decisively test whether any current proposal satisfies all three vertices simultaneously.