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Aims. The light curves of the microlensing events MOA-2022-BLG-091 and KMT-2024-BLG-1209 exhibit anomalies with very similar features. These anomalies appear near the peaks of the light curves, where the magnifications are moderately high, and are distinguished by weak caustic-crossing features with minimal distortion while the source remains inside the caustic. To achieve a deeper understanding of these anomalies, we conducted a comprehensive analysis of the lensing events. Methods. We carried out binary-lens modeling with a thorough exploration of the parameter space. This analysis revealed that the anomalies in both events are of planetary origin, although their exact interpretation is complicated by different types of degeneracy. In the case of MOA-2022-BLG-091, the main difficulty in the interpretation of the anomaly arises from a newly identified degeneracy related to the uncertain angle at which the source trajectory intersects the planet–host axis. For KMT-2024-BLG-1209, the interpretation is affected by the previously known inner-outer degeneracy, which leads to ambiguity between solutions in which the source passes through either the inner or outer caustic region relative to the planet host. Results. Bayesian analysis indicates that the planets in both lens systems are giant planets with masses about two to four times that of Jupiter, orbiting early K-type main-sequence stars. Both systems are likely located in the Galactic disk at a distance of around 4 kiloparsecs. The degeneracy in KMT-2024-BLG-1209 is challenging to resolve because it stems from intrinsic similarities in the caustic structures of the degenerate solutions. In contrast, the degeneracy in MOA-2022-BLG-091, which occurs by chance rather than from inherent characteristics, is expected to be resolved by the future space based Roman RGES microlensing survey. 

Abstract The current studies of microlensing planets are limited by small number statistics. Follow-up observations of high-magnification microlensing events can efficiently form a statistical planetary sample. Since 2020, the Korea Microlensing Telescope Network (KMTNet) and the Las Cumbres Observatory (LCO) global network have been conducting a follow-up program for high-magnification KMTNet events. Here, we report the detection and analysis of a microlensing planetary event, KMT-2023-BLG-1431, for which the subtle (0.05 mag) and short-lived (5 hr) planetary signature was characterized by the follow-up from KMTNet and LCO. A binary-lens single-source (2L1S) analysis reveals a planet/host mass ratio ofq= (0.72 ± 0.07) × 10⁻⁴, and the single-lens binary-source (1L2S) model is excluded by Δχ²= 80. A Bayesian analysis using a Galactic model yields estimates of the host star mass of $M_{\mathrm{host}}={0.57}_{-0.29}^{+0.33}{M}_{\odot }$, the planetary mass of $M_{\mathrm{planet}}={13.5}_{-6.8}^{+8.1}{M}_{\oplus }$, and the lens distance of $D_{\mathrm{L}}={6.9}_{-1.7}^{+0.8}$kpc. The projected planet-host separation of $a_{\perp }={2.3}_{-0.5}^{+0.5}$au or $a_{\perp }={3.2}_{-0.8}^{+0.7}$au, subject to the close/wide degeneracy. We also find that without the follow-up data, the survey-only data cannot break the degeneracy of central/resonant caustics and the degeneracy of 2L1S/1L2S models, showing the importance of follow-up observations for current microlensing surveys.