WHY SELF-REFERENTIAL MAINTENANCE NECESSARILY PRODUCES THRESHOLD DYNAMICS: A Categorical Derivation of IGT Dynamics from the Theory of Polynomial Functor Coalgebras (IGCDT)

Structured Abstract Background The IRM Impossibility Theorem (companion Article 1) proves that finite-noise self-maintenance produces strictly positive expected drift. The IGT Principle establishes empirically that SRSM systems exhibit threshold dynamics with internal precursors. Neither result answers the deepest structural question: why does SRSM architecture produce threshold dynamics at all, independently of any noise model? Gap No categorical derivation has established that threshold dynamics and internal precursors are structurally necessary for SRSM systems, as opposed to being probabilistic consequences of finite noise. The relationship between the Gödel–Turing–Lawvere family and the dynamical behaviour of SRSM systems has not been formalised at the categorical level. Approach We show that SRSM systems are precisely the terminal coalgebras of polynomial functors on appropriate categories of state spaces, and that the structure of such coalgebras — specifically, the absence of a natural stable retraction — is the categorical expression of IGT dynamics. The central result (IGCDT) is stated, with the core argument developed in two lemmas. Results The IGT Categorical Derivation Theorem (IGCDT) states that for any polynomial functor F: C → C whose coefficients include the state object A, the terminal coalgebra has no natural transformation stable under the functor's self-modification. Three corollaries: (1) IGCDT subsumes the IRM theorem; (2) the Gödel–Turing–Lawvere–IRM family is unified; (3) the generative/degenerative character of IGT transitions is categorically determined. Proof sketches for the core lemmas are provided; complete proofs require additional categorical machinery specified in §8. Implications SRSM systems are a natural kind defined by the polynomial functor coalgebra structure. IGT dynamics is not a probabilistic regularity but a categorical necessity. All five empirical internal precursor measures are domain-specific instantiations of GAMMA morphism variance increase in terminal coalgebra sequences — providing the first formally derived characterisation of what the precursor measures are actually measuring. Keywords: coalgebra, polynomial functor, terminal coalgebra, Lambek's lemma, self-referential systems, IGT dynamics, IRM theorem, autopoiesis, categorical dynamics, Gödel–Turing–Lawvere family, ALGUILAS-AI