The associated graded of the K-adic filtration

Replacing \(T\) by \(T_{/X}\) reduces us to \(X=*\). Put \(T':=T/K^{n+1}\), and let \(J\) be the congruence in \(T'\) generated by the image of \(K^n\). Then \(T/K^n\iso T'/J\). Since the quotient functor preserves acyclic products and \(2n\geq n+1\), we have in \(T'\). The layer \((K^n/K^{n+1})_*\) therefore identifies with \((J/J^2)_*\), the full subcategory of \(T'\) spanned by objects \(E\) for which \(E\to *\) belongs to \(J\).This subcategory is closed under pullbacks and pushouts in \(T'\). For pullbacks, use base-change stability of \(J\) and then compose with the terminal map of one of the factors; for pushouts, use closure of \(J\) under colimits in the arrow category. Consider a commutative square in the subcategory. Every morphism in the square belongs to \(J\) by the three-for-two property. If the square is cocartesian, the generalized Blakers–Massey theorem for the modality \((J,J^\perp)\) shows that its cartesian gap map belongs to \(J^2=\Iso\), so the square is cartesian. The dual Blakers–Massey theorem proves the converse. Thus the layer is pre-stable. The terminal object of \(T'\) belongs to it, and adjoining a point makes it pointed; a pointed pre-stable category is stable.