Interpreting the $^{14}N$ abundance in Galactic open clusters Trumpler 20, NGC 4815, and NGC 6705: constraints from the extended CNOF cycle

Nitrogen abundances observed in evolved stars provide a sensitive probe of proton-capture nucleosynthesis and internal mixing processes operating under stellar interior conditions. In this work, we investigate the origin of nitrogen enrichment in the Galactic open clusters Trumpler~20, NGC~4815, and NGC~6705 using a theoretical framework based on the extended CNOF nuclear reaction cycle. We solve a nuclear reaction network representative of advanced hydrogen-burning environments to characterise the composition of nuclear-processed material, rather than modelling full stellar evolution or in-situ surface burning. The calculations are performed over a range of temperatures ($T_9 = 0.10-0.40$ K) and hydrogen mass fractions ($X_{\mathrm{H}} = 0.70-0.60$), corresponding to progressive hydrogen consumption. The resulting abundance contours define a well-constrained theoretical nitrogen enrichment space, with $^{14}\mathrm{N}$ emerging as the dominant bottleneck nucleus of the cycle. We interpret the computed abundances as effective surface enrichments, assuming partial transport and dilution of processed material within stellar envelopes. The predicted nitrogen abundances are compared with high-resolution spectroscopic measurements of cluster member stars from the literature. A statistical analysis yields strong positive correlations between theoretical predictions and observed $[\mathrm{N/H}]$ values, with correlation coefficients of $r = 0.96$, $0.97$, and $0.95$ for Trumpler~20, NGC~4815, and NGC~6705, respectively. The corresponding mean absolute deviations are $0.049$, $0.052$, and $0.050$ dex, which are comparable to typical observational uncertainties. The agreement suggests that the dominant nucleosynthetic pathways captured by the extended CNOF cycle can account for the observed nitrogen enrichment in these clusters, with possible secondary contributions from evolutionary mixing processes. These results support a primarily primordial nucleosynthetic origin for nitrogen, modulated by later stellar evolution.