## 3 Results  
Starting from the special case, the synthesis parameters are selected as \(p=q=0\), in other words, \(t_{1}=t_{3}\), and \(t_{2}=t_{4}\). As shown in Figure 1b, this case simply doubles the size of the unit cell and the Brillouin zone is folded. Such band folding results in a linear crossover of the first (third) and second (fourth) along the wave vector direction at \(k=\pm\pi/a\). Away from the point (\(p=0\) and \(q=0\)), band gaps are formed since the degeneration resulting from the band folding is removed, as exemplified by the SC with (\(p=0.76\) and \(q=0\)) (blue dotted lines). It can be inferred from Figure 1a that the increase and decrease of \(p\) (or \(q\)) from \(p=0\) (or \(q=0\)) is not symmetric in the one-dimensional structure. Therefore, the dispersion relation \(\omega\) (\(p\), \(q\), \(k\)) is expanded at \(p=q=0\), the first-order terms \(\Delta p\) and \(\Delta q\) will play the dominant role and give linear dispersion in the \(p\) and \(q\) directions. Based on the above analysis, the intersections at (0, 0, \(\pm\pi/a\)) are WPs in this synthetic dimension. The intersection between two adjacent bands can also form at the wave number \(k=0\) or \(k=\pm\pi/a\) when \(p=0\) or \(q=0\), as guarded by space-reflection symmetry of the 1D PC plate. Let me give an example, an accidental linear degeneracy can also form at \(k=0\) with \(p=0.76\) and \(q=0\) as shown in Figure 1b. These degenerate points are also WPs in this synthetic dimension, which are further verified by the conic intersections in the bulk dispersions depicted in \(p\)-\(q\) space, as shown in Figure 1c. It is noted that the mirror symmetry of our framework ensures that when a WP exists at (\(p\), \(q\), \(k\)), there must be corresponding WPs with the same frequency and the same topological charge at (\(\pm p\), \(\pm q\), \(\pm k\)). The main difference between our system and the real Weyl crystals is that the total charge does not vanish because the synthesis parameters are not periodic. To further confirm the synthesized WPs, the frequency-dependent TS through 1D PC plates with different geometrical parameters are investigated. Here we pay attention to the frequency range around the WP formed by the first and second bands. Figure 3a-c shows 95 independent phononic crystal models (including 19 different p and five different q) and each model consists of 10 periods. The simulation results shows that when \(q\) gradually  
Figure 2: The simulation model for TS. approaches 0, the bandwidth of the transmission gap decreases and disappears. It shows obvious linear crossover near the expected frequency and structural parameters. It is also worth noting that the band structures with negative \(q\) are the same as the band structures with the positive \"\(q\)\", which leads to the same TS. The RP of a PC plate, which is surrounded by a perfectly matched layer and a rigid boundary, is also studied. Regarding the effective Hamiltonian, the non-trivial topology of the synthetic WP results in the formation of an eddy current-like RP structure in the parameter space, where the number of quantized phase windings is identical with the corresponding Weyl charge [2]. The distributions of RP are shown Figure 3a, where the frequency 0.306 MHz of the Weyl point between band 1 and band 2 is used. The phase distribution in the \(p\)-\(q\) space clearly shows a vortex. The reflection simulation proved the non-trivial phase distribution. Robust RP singularity is a new characteristic of topologically stable PC plates. At the same time, the vortex phase structure ensures the presence of interface states, and each WP is associated with an interface state. In Figure 4b, the energy density spectrum of the phononic crystal plate with seventeen periods at \((p,q)=(0.35, 0.45)\), normalized by the flat panel is calculated. There is a significant peak value appearing at 0.305 MHz, which is in line with the interface transition frequency in Figure 4a, which verifies the topologically non-trivial interface state.