Finite-difference zeta function regularisation and spectral weighting in effective actions

Standard zeta function regularisation enforces a scale-independent prescription for spectral aggregation, effectively fixing the relative weight of spectral contributions. We relax this constraint by replacing the derivative at $s=0$ with a finite-difference construction based on ${\zeta }_A(0)$ and ${\zeta }_A(q-1)$. In finite systems, it gives rise in the macroscopic limit to Tsallis-type quantities and a $q$-controlled information-geometric structure. In infinite dimensions, it yields an effective action whose variation $\delta {\mathrm{\Gamma }}_q=\mathrm{T}\mathrm{r}(A^{-q}\delta A)$ realises scale-dependent spectral weighting. Within this framework, zeta function regularisation, effective action, nonextensive scaling, and information geometry emerge as manifestations of a common principle of finite-difference spectral aggregation.