Almost sure local well-posedness for the nonlinear Schr\"odinger equations on $\Bbb T^d$ with non-algebraic nonlinearity

We study the Cauchy problem for the nonlinear Schr\"odinger equation on $\mathbb T^d$ with random initial data and a general non-algebraic power-type nonlinearity. We establish almost sure local well-posedness in every spatial dimension and for the whole mass-supercritical range allowed by the natural condition $0<s_{\mathrm c}<1+a$. The main new ingredient is a frequency-gaining probabilistic refinement of the Galilean bilinear estimates recently developed by Kwak and Kwon \cite{KwakKwon}. In the random setting, the gauge decomposition gives rise to three new types of terms: a mean-free coefficient, an opposite-phase interaction, and a scalar remainder. We control them by new resonance counting and large deviation arguments, and close the local theory through a phase-adapted two-component contraction. In the energy-critical case, our result extends the low-dimensional algebraic theories of Nahmod--Staffilani \cite{NahmodStaffilani15} and Yue \cite{Yue21} to every dimension $d\geq3$, including the higher-dimensional non-algebraic models.