= Jacobi abstruse identity
{c}
{title2=$\vartheta_3^4-\vartheta_4^4-\vartheta_2^4=0$}
The identity between <Jacobi theta functions> has the oscillator-product form $\prod_{r\in\mathbb N-1/2}(1+q^r)^8-\prod_{r\in\mathbb N-1/2}(1-q^r)^8=16q^{1/2}\prod_{n\ge1}(1+q^n)^8$ for $|q|<1$. In a critical <RNS string>, dividing this identity by the eight-boson oscillator product proves equality of <GSO projection>-retained <Neveu–Schwarz sector> and <Ramond sector> state multiplicities at every mass level.
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