Abstract: The IPCC, NASA, NOAA, and EPA state that essentially all of the observed rise in atmospheric CO₂ since the pre-industrial era originates from fossil-fuel combustion. This paper shows that claim is mathematically inconsistent with the isotopic record those same agencies rely on. The Keeling plot of δ¹³C against 1/C has an intercept δ¹³C_I equal to the flux-weighted isotopic signature of the net carbon addition to the atmosphere. Applied by Koutsoyiannis (2024) to instrumental δ¹³C records from four Scripps CO₂ Program stations (Barrow, La Jolla, Mauna Loa, South Pole) spanning about 1978 to present, this construction gives δ¹³C_I = −13.2 ± 0.2‰. Caribbean sclerosponge δ¹³C reconstructions of surface-water dissolved inorganic carbon by Böhm et al. (2002) extend this value back to approximately 1500 AD with no significant drift. Independent Keeling regression of the Law Dome ice-core record by Köhler et al. (2006) yields the same intercept with r² = 0.96. The two-endmember mixture equation δ¹³C_I = f·δ¹³C_F + (1 − f)·δ¹³C_N, using the standard fossil signature δ¹³C_F ≈ −28‰ and the physically admissible range −13.2‰ ≤ δ¹³C_N ≤ −8‰, forces the fossil fraction f of the net addition into the interval [0, 0.26]. Even under the counterfactual relaxation δ¹³C_N = 0‰, well outside any physically realizable natural-reservoir signature, f is bounded above by 0.471. The five-century invariance of δ¹³C_I across a roughly 10⁴-fold increase in fossil emissions further collapses the admissible band to f ≈ 0. The claim of fossil dominance is therefore not merely empirically weak; it is inconsistent with the arithmetic of the isotopic constraint applied to the standard measurement record. The rejection of the conclusion requires the rejection of one of three inputs: elementary mass-balance algebra, the standard isotopic endmember values used throughout the carbon-cycle literature, or the primary Scripps and sclerosponge datasets themselves.