Regularized formula for positive integer surgery

Let the $+p$-surgery formula of Gukov and Manolescu be given, and let the regularization be defined using the identity

$$\frac{1}{\prod_{j=1}^{n}(x+x^{-1}-q^j-q^{-j})}\bigg\vert_{x^u\mapsto \delta_{b,u(\operatorname{mod} p)}q^{-\frac{u^2}{p}}} = \frac{(-1)^n q^{\frac{n(n+1)}{2}}}{(q^{n+1})_n}q^{\frac{b(p-b)}{p}}P_{n}^{p,b}(q^{-1}).$$

Regularized $+p$-surgery formula. When the $+p$-surgery formula of Gukov and Manolescu converges, it should be used; when it does not converge, the displayed identity should regularize it, provided that the regularization converges. This proposes a way to extend the positive-surgery formula beyond its original domain of convergence, with the convergence of the regularized expression remaining the essential condition.