Pauli webs spun by transversal $\mathrm{\mid }Y⟩$ state initialisation

Originally motivated by the (fold-)transversal related initialisation of logical surface code $\mathrm{\mid }Y⟩$ states from [arXiv:1603.02286, arXiv:2302.07395, arXiv:2302.12292], which was then explicitly extended to the fold-transversal $S$ gate implementation in [arXiv:2412.01391] for the rotated surface code, we employ ZX-calculus and Pauli web to understand the $\mathrm{\mid }Y⟩=S\mathrm{\mid }+⟩$ state transversal initialisation scheme.