Marine radiocarbon integrates air--sea gas exchange, mixed-layer ventilation, and upper-ocean transport, but global observations are sparse and isotope-enabled Earth system simulations remain costly. We develop CarnoNet (Marine Carbon Neural Operator Network) for model-based reconstruction of CESM2-LENS marine surface abiotic radiocarbon from upper-ocean state variables and atmospheric boundary forcing. The target is the air--sea radiocarbon disequilibrium, \(Y=\Delta^{14}C_{\mathrm{atm}}-\Delta^{14}C_{\mathrm{ocean}}\), evaluated with CESM2 Large Ensemble historical data (1850--2015) and SSP3-7.0 data (2015--2035). The primary experiments use ensemble-mean fields; reported transfer is therefore interpreted as forced-response reconstruction rather than member-level internal-variability prediction. CarnoNet combines causal time-stacked predictors, a U-Net backbone, an optional Tucker-decomposed Fourier Neural Operator (TFNO) mixing layer, and a bounded boundary-condition gate. The model is used as a diagnostic mapping from ocean and boundary-condition drivers to surface radiocarbon disequilibrium. An atmospheric-forcing climatological baseline separates the prescribed atmospheric signal from the residual ocean disequilibrium. The selected calibrated configuration achieves an MAE of 7.28 and RMSE of 10.27 on the untouched SSP test remainder and reduces residual RMSE by 64\% relative to the atmospheric-forcing climatology. Targeted three-seed repeats show that time-stacked memory is the most reproducible component; boundary-condition gating improves temporal correlation but gives seed-dependent pointwise-error and spatial-correlation gains; and the TFNO mixing layer is not uniformly superior to the identity/U-Net baseline. A four-member SSP370 transfer assessment further shows that member-specific \(C_0\) calibration improves several RMSE and spatial-correlation metrics relative to uncalibrated member transfer but degrades temporal correlation and does not resolve member-level temporal variability, particularly in the North Atlantic deep-convection proxy region. These results support CarnoNet as a compact reconstruction approach for the forced component of surface radiocarbon, with clearly bounded use for individual-member variability.