## 1 Introduction
The initial phase angle, \(\delta\), of the disk is located between the fixed point, A, of the rotating system and the disk notch. The initial phase angle, \(\varphi\), of the magnetic field, \(\mathbf{B}\), is located between the (fixed) vertical sensor direction and the \(\mathbf{B_{horz}}\) direction, which depends on the fragment placement on the disk. M*, is oriented perpendicularly to the sensor along the direction of \(\mathbf{B}_{\text{Earth}}\). Using the PSD method, constant phase differences and weak magnetic fields, with constant directionality and values of a few tens of nT can be found. After the detection of the fragments magnetic field (Figure 2), the directionality of the base and body in co-belonging fragments is compared. The sensor remains in the same horizontal position, perpendicular to the vertical component and the horizontal component of the geomagnetic field. During the vase formation at the pottery wheel, circular grooves remain on the fragment surface. Parallel straight lines at an arbitrary direction (\(\sigma\)) are marked on the base fragments (Figure 3a), which suggests their joining direction. Regarding the fragments from the vase bodies (Figure 3b), their direction (\(\sigma\)) is selected along the surface grooves of the pottery wheel. The fragments are positioned on a leveled angle-measuring apparatus, with the sensor axis aligned along the direction (\(\sigma\)), at a position where \(\mathbf{B}_{\text{horz}}\) diverges at an arbitrary angle, \(\varphi\), from the sensor direction, while the disk is rotated manually until its reading is zeroed. At this position, where \(\mathbf{B}_{\text{horz}}\) is vertical to the sensor direction, its direction is marked. Neighboring fragments are oriented in a joining manner because their magnetic field, acquired before their fracturing, is due to the thermoremanent magnetization along the direction geomagnetic field during their cooling in the kiln. While finding the joining orientation of the base excavation fragments is not of essential practical value, because it is more easily performed empirically, it can be concluded from the above that neither the magnetic anisotropy of the material nor the irregular fragment shape prevent their joining from the orientation of their magnetic field; in fact this happens with remarkable precision. This is not the case for the distant body fragments, which come in a variety of shapes and are collected in large numbers. This is to be expected, because the thermomagnetic directionality varies depending on the orientation of the vase regions with respect to the geomagnetic field, and it is formed by the rotational symmetry of the ceramic vases. Figure 2: Illustration of the horizontal component detection apparatus from vase fragments. Fragments from 6 ceramic vases (Figure 4a) are placed on the horizontal angle-measuring apparatus disk, with the marked reference direction (\(\sigma\)) on their surface matching the fixed sensor direction. The component parallel to the fragment surfaces (\(\textbf{B}^{//}\)) and the angle of divergence (\(\varphi\)) of the reference direction (\(\sigma\)) with the fixed sensor direction (Figure 5a) are measured at the rotational position where the reading displays its maximal positive value. Representative measurements of the angle \(\varphi\) in body (**a**) and base fragments (**b**) of vase 1 are presented in Figure 6.  
Figure 4: (**a**). For the systematic study of the magnetic field directionality, single-sensor measurements from base and body fragments from 6 ceramic vases of the previous two centuries, fired in a traditional kiln, were taken. (**b**). The body fragments measurements were taken at the vertices of a lattice formed by the grooves, which provide the reference directions (\(\sigma\)), and vertical lines to them. (**c**). The base fragments measurements are taken on parallel lines, along the arbitrarily chosen joining direction (\(\sigma\)) of the fragments.