## 1 Introduction  
Thermal errors in machine tools (MTs) have been a subject of considerable research since their first identification by [PERSON] in the early 1960s [1]. These errors are widely acknowledged as a dominant factor that influences MT accuracy, accounting for between 60% and 75% of overall geometric deviations, as noted by [PERSON] et al. [2]. Due to their significant impact on precision manufacturing, investigations into thermal effects remain an active and evolving research domain [3], with numerous studies being published annually that explore innovative modelling, control, and compensation strategies. Early studies, such as those by [PERSON] et al. [4], introduced fundamental concepts in thermal error compensation, while [PERSON] et al. [5] provided a classification framework distinguishing direct and indirect measurement methods for geometric errors. Subsequent work by [PERSON] et al. [6] concentrated on volumetric accuracy assessments using indirect techniques. More recent contributions, such as those by [PERSON] et al. [7] and [PERSON] et al. [3], have expanded the field by addressing thermal error modelling and machine tool calibration. Additionally, the integration of machine learning into compensation methodologies, as demonstrated by [PERSON] et al. [8], has introduced novel adaptive approaches for mitigating thermal effects in machining environments. Thermal distortions that affect the tool centre point (TCP) can stem from two principal sources: internal heat generation within the machine structure and external environmental influences. Internal factors include thermal loads from motors, bearings, ballscrews, and frictional interactions during machining, which lead to localised temperature variations within the system. Conversely, external factors arise from ambient temperature fluctuations, and induce thermal expansion through convective heat transfer mechanisms. These external influences, often dictated by daily and seasonal cycles, have become increasingly significant as the manufacturing of larger components continues to demand tighter tolerances and higher precision [9]. The challenge of mitigating ambient thermal effects has been well documented in the literature [10; 11; 12], particularly given the high capital investment and operational costs associated with maintaining controlled workshop environments [13]. While historical research has predominantly focused on the thermal effects originating within machine components, there has been a recent shift towards analysing the impact of environmental thermal variations. For instance, [PERSON] et al. [13] conducted extensive investigations in controlled conditions, placing an MT inside a climatic chamber to evaluate its response to stable ambient temperatures. More recent studies, such as those by [PERSON] et al. [14; 15], have examined seasonal temperature variations and their influence on MT spindles, proposing compensation models to mitigate these deviations. Similarly, [PERSON] et al. [16] developed a long-term monitoring method for tracking thermal errors in three-axis machine tools using an INVAR-based artefact that was pre-characterised under stable conditions. Additionally, [PERSON] et al. [12] implemented a dynamic R-test, based on [PERSON]'s methodology [17], to quantify five-axis machine tool deformations under fluctuating ambient conditions. Advancements in digital twin technology have also contributed to the improved characterisation of thermal deviations. [PERSON] et al. [18; 19] developed a digital twin framework for predicting volumetric thermal distortions by leveraging rapid thermal characterisation techniques. Beyond these methodologies, multilateration-based measurements employing tracking interferometers (TIs) have been explored as an alternative non-contact calibration method [20]. This approach estimates positional deviations by measuring distances between a retro-reflector at the TCP and fixed interferometers at the machine base.