The Categorical Imperative Is the Unique j-Closed Fixed Point

Overview

VII.T35 proves that the Categorical Imperative (CI) is the unique j-closed fixed point of the τ-digestion operator j_dig: j_dig(CI) = CI. The Stage CI theorem (VII.P21) shows that the minimal j-closed fixed point is unique by the Knaster-Tarski fixed-point theorem. The CI is not a cultural convention or an arbitrary ethical choice but a structural necessity forced by the E₃ enrichment level.

Detail

Kant’s Categorical Imperative (‘Act only according to that maxim whereby you can at the same time will that it should become a universal law’) is traditionally presented as a foundational moral principle derived from practical reason. Book VII proves it is the unique fixed point of the τ-digestion operator. The τ-digestion operator j_dig acts on E₃ objects (self-describing systems that include human moral agents). A fixed point of j_dig is a principle that, when universalised (applied to itself), returns itself — a self-consistent maxim. VII.T35 proves that the space of candidate maxims has exactly one minimal fixed point, and that this fixed point is precisely the universal maxim ‘act only on principles you can universalise.’ The proof uses the Knaster-Tarski fixed-point theorem: j_dig is monotone on the lattice of normative principles ordered by generality, and the minimal fixed point is unique. Stage CI (VII.P21) establishes that this fixed point is reached in finite stages. The result is that ethics is derived from the structure of E₃ — the enrichment level at which self-describing systems (humans, AIs, other moral agents) first appear — rather than postulated from practical reason or moral intuition.

Result Statement

VII.T35: The CI is a j-closed fixed point: j_dig(CI) = CI. Stage CI (VII.P21): The minimal j-closed fixed point is unique by Knaster-Tarski. The CI is not a cultural convention but a structural necessity at E₃.