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 δ13CI equal to the flux-weighted isotopic signature of the net carbon addition to the atmosphere. Applied by Koutsoyiannis (2024a) [16] 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 δ13CI = −13.2 ± 0.2‰. Caribbean sclerosponge δ13C reconstructions of surface-water dissolved inorganic carbon by Böhm et al. (2002) [2] extend this value back to approximately 1500 AD with no significant drift. Independent Keeling regression of the Law Dome ice-core record by Kohler et al. (2006) [14] yields the same intercept with r2 = 0.96. The two-endmember mixture equation δ13CI = ƒ δ13CF + (1− ƒ ) δ13CN , using the standard fossil signature δ13CF ≈ −28‰ and the physically admissible range −13.2‰ ≤ δ13CN = 0‰, well outside any physically realizable natural- reservoir signature, ƒ is bounded above by 0.471. The five-century invariance of δ13CI across a roughly 104 -fold increase in fossil emissions further collapses the admissible band to ƒ ≈ 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.
