Silting reduction, relative AGK's construction and Higgs construction

We introduce the notion of a Calabi--Yau quadruple as a generalization of Iyama--Yang's Calabi--Yau triple. For each $(d+1)$-Calabi--Yau quadruple, we show that the associated Higgs category is a $d$-Calabi--Yau Frobenius extriangulated category, which moreover admits a canonical $d$-cluster-tilting subcategory. Concrete examples arise from the construction of relative cluster categories and Higgs categories in the setting of ice quivers with potentials, as well as from the singularity category of an isolated singularity. As an application, we prove that both the relative Amiot--Guo--Keller's construction and the Higgs construction of a $(d+1)$-Calabi--Yau quadruple take silting reduction to Calabi--Yau reduction.