## 1 Introduction
The aggregate results and conclusions of calculating magnetization from magnetic field measurements on samples and fragments of the base of vases 1-6 in Section 6, were applied for the calculation of magnetization (see Supplementary Files) in irregular fragments of the body vase 1-6, using the methodology described in Section 7. In the 8 th section, the mathematical relationships of the calculated magnetization changes in the fragments' reference systems in vases with cylindrical (Section 8.1) and arbitrary rotational (Section 8.2) symmetry are described. By using rotational transformations of magnetization components in the reference systems where field measurements were taken (Section 8.3), the location of fragments was identified in the bodies of the six vases. Finally, Section 9 contains some proposals for future research. Detection of the Weak Magnetic Field of Ceramic Fragments and Its Direction Using a Fluxgate Sensor  
The most important problem in detecting the weak field of fragments, on the order of a few tens of nT, is its overlay with the Earth magnetic field, which is about 1000 times stronger, as well as the lab magnetic background. For the detection of the weak magnetic signal [34; 35; 36] of fragments within the geomagnetic field, a fluxgate sensor (model MAG 03 IE three independent axes fluxgate magnetometer, Bartington, Witney, Oxon, UK) [37] at a fixed horizontal position (Figure 1) is oriented perpendicularly to the vertical line (\(\mathbf{B_{vert}}\)) and to the horizontal component (\(\mathbf{B_{horz}}\)) of the Earth's magnetic field (\(\mathbf{B_{earth}}\)), such that its reading is zero before the fragment placement. When the sensor distance from the fragment surface is greater than 1.5-2 cm, its reading becomes zero. Accuracy of measurements at a practical level depends: a) on the ability of aligning the x-sensor with respect to the vessel body grooves, which can not be performed with an error less than 1 nT. For this reason, measurements on the fragments of the vase body are obtained with a 1 nT error (See Supplementary Files), while the maximum sensor accuracy is 0.1-0.2 nT. Therefore, sensors of greater accuracy (eg SQUIDsensors), would not give better results regarding the computation of \(\kappa\omega\) angles, which determine the position of the fragments in the vase bodies. b) on he contact of the fluxgate sensors (of length 3 cm) with the shell surface (see Section 9: Conclusions and Proposals for Future Research), which decreases as their curvature increases.. Thus the precision of the method could be improved or extended to vase fragments of smaller size, using shorter sensors. For the weak magnetic field extraction from the unwanted lab magnetic background noise the Phase Sensitive Detection (PSD) method [38] was used. The electric signal from the disk rotation, at a certain frequency, f, is used as a reference signal to the phase detector [39]. The direct electric current (1 mV/7 nT) from each fragment, detected by the magnetometer sensor, is converted during the disk rotation into alternating current and used as an input to the phase detector. The electric signal from the horizontal rotating component, \(\mathbf{B_{horz}}\), of each fragment is differentiated from the magnetic interferences because it is detected by the phase detector exclusively at the disk rotation frequency, \(f\). If the magnetic field of each fragment displays constant directionality, then its magnitude and initial angular phase difference (\(\delta\)-\(\varphi\)), are recorded, after the processing of the sinusoidal signals at the phase detector. During the disk rotation, the ceramic material induced magnetization,  
Figure 1: Illustration of the Earth magnetic field (\(\mathbf{B_{earth}}\)) and of the detected projection of the horizontal component (\(\mathbf{B_{horz}}\)) of the fragment field onto the sensor direction during the initial placement of the fragment on the rotating disk.