Some argue that using CO2 from direct air capture (DAC) in enhanced oil recovery (CO2-EOR) can produce carbon-neutral oil by permanently storing more CO2 than is emitted when using the extracted fossil fuels. However, existing analyses often provide case-specific insights based on short-term operations without considering the full life cycle of reservoir exploitation, including primary, secondary, and tertiary (EOR) recovery phases. Here, we present a general, top-down approach based on mass and volume conservation to assess the potential carbon footprint of oil production, applicable to different temporal perspectives of reservoir exploitation. Supported by field data from 16 EOR projects, our analysis shows that 30% of projects appear carbon-neutral when EOR is considered in isolation, but they all become significantly carbon-positive when the full reservoir lifetime is considered. The volume of emitted CO2 exceeds the pore space freed for storage by at least a factor of three, making carbon-neutral oil physically unattainable in conventional reservoirs. The favorable conditions for low-carbon oil production during CO2-EOR exist solely because of extensive prior oil extraction and water injection, and only residual oil zones may truly offer potential for carbon-neutral oil due to their low oil saturation and lack of legacy emissions. While omitting legacy emissions from previously depleted fields may be justifiable and may enable claims of carbon neutrality during the EOR phase, newly developed fields, i.e., developed now or in the future, should be held accountable for the full life-cycle emissions they generate. This necessitates clear and transparent accounting policy frameworks. Although CO2-EOR may reduce oil's carbon footprint, promoting it as a pathway to carbon-neutrality risks legitimizing continued fossil fuel production, ultimately undermining global climate targets.