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 𝛿 13C 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 = 𝑓 𝛿13C𝐹 + (1− 𝑓 ) 𝛿13C𝑁 , using the standard fossil signature 𝛿13C𝐹 ≈ −28‰ and the physically admissible range −13.2‰ ≤ 𝛿13C𝑁 ≤ −8‰, forces the fossil fraction 𝑓 of the net addition into the interval [0, 0.26]. Even under the counterfactual relaxation 𝛿13C𝑁 = 0‰, well outside any physically realizable naturalreservoir signature, 𝑓 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 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.