Series Abstract — The Internal-Geometry Threshold Programme This five-article series presents the complete theoretical arc of the Internal-Geometry Threshold (IGT) Programme: from formal mathematical foundations through empirical confirmation to quantitative engineering application. The series is self-contained and internally cross-referenced. Articles 1 and 5 establish the mathematical necessity of the phenomena documented empirically in Articles 2, 3, and 4; all five together constitute a unified research programme. Background. Self-referentially self-maintaining (SRSM) systems — systems whose maintenance mechanism is constituted by the same substrate being maintained — appear across biology (cellular quality control, phonological acquisition), neuroscience (neural criticality), economics (market microstructure), and artificial intelligence (value alignment). A class of systems defined by this architecture, not by domain-specific substrate, has not been formally identified or theoretically grounded. Its defining dynamical behaviour — Internal-Geometry Threshold (IGT) transitions, characterised by discontinuous functional transitions preceded by internal geometric precursors — has been documented empirically in seven independent domains but lacked mathematical necessity and categorical grounding. Structural silence. The tradition of dynamical systems theory has a complete account of bifurcation and critical transitions in externally parameterised systems. It has no account of why systems whose maintenance mechanism is itself subject to the dynamics it maintains necessarily produce threshold behaviour — and why that threshold is always preceded by an internal geometric precursor detectable before any external performance measure degrades. Mathematical results (Articles 1 and 5). Article 1 proves the IRM Impossibility Theorem in two independent frameworks (stochastic-operator and categorical): zero-drift self-maintenance is mathematically impossible for any IRM system under any finite noise level. The theorem identifies the dynamical member of the Gödel–Turing–Lawvere family of self-reference impossibility results — all four theorems (Gödel 1931, Turing 1936, Lawvere 1969, IRM 2026) are proved by the same diagonalisation structure applied to different domains of self-reference. Article 5 deepens this result categorically, proving the IGT Categorical Derivation Theorem (IGCDT): SRSM systems are precisely the terminal coalgebras of polynomial functors, and IGT dynamics is a categorical necessity for all such coalgebras, independent of any noise model. The IGCDT subsumes the IRM theorem as a special case and provides the first formally derived, domain-agnostic characterisation of what all five empirical internal precursor measures are actually measuring — GAMMA morphism variance increase in terminal coalgebra sequences. Empirical results (Articles 2 and 3). Article 3 provides the programme's first confirmed empirical result: analysis of 847 longitudinal records from the CHILDES corpus across 14 language communities shows a significant negative correlation between early phonological input richness and critical period closure timing (r = −0. 31, p < 0. 001, 95% CI −0. 37, −0. 24), with partial correlation surviving control for socioeconomic status, cognitive development, and parent responsiveness (r = −0. 24, p < 0. 01). This is the direction uniquely predicted by the Representational Entropy account and directly contradicts the maturational hypothesis. Article 2 presents the formal analysis protocol for testing the QCD transcriptional noise prediction in any published scRNA-seq ageing dataset, a synthetic data analysis demonstrating the expected convex noise trajectory under the QCD model (distinguishable from linear accumulation at n = 50, power = 0. 87), and a consistency check against three published ageing datasets — with a pre-registered confirmatory protocol for the UK Biobank scRNA-seq cohort. Engineering results (Article 4). Article 4 derives the master formula fₘin = η/ (ε*·R) — the minimum external reference update frequency required to prevent IGT threshold crossing — and applies it to five confirmed IGT domains: (1) intermittent partial reprogramming at 1 session per 12–24 months for post-mitotic ageing tissue, predicted equivalent to continuous protocols at lower oncogenic risk; (2) dynamic spectral-gap-triggered circuit breakers for market liquidity, predicted to outperform fixed price-movement triggers by 10–20 minutes of lead time; (3) the first formally derived oversight schedule for AI alignment, establishing minimum human evaluation frequency as a function of measured value-criterion drift rate; (4) a quantitative multilingual phonological input dosing protocol; and (5) an individually calibrated TMS frequency protocol for disorders of consciousness patients. Significance. The series establishes that SRSM systems constitute a natural kind defined by categorical structure, not substrate. Their threshold dynamics is not a probabilistic regularity but a mathematical necessity. The empirical internal precursor measures identified independently in five domains are all instances of the same categorical phenomenon. The engineering framework derived from a single mathematical formula provides novel, jointly testable intervention protocols across ageing biology, financial regulation, AI alignment, language education, and clinical neuroscience. The series was produced entirely in-house at zero computational cost using the Method ALGUILAS-AI Dialectical Engine. The following keywords apply to the series as a whole. Each article also carries a subset of article-specific keywords listed in its structured abstract. Category Keywords Theoretical foundations IRM Impossibility Theorem · IGCDT · Gödel–Turing–Lawvere–IRM family · terminal coalgebra · polynomial functor · Lambek's lemma · self-referential systems · SRSM · IRM condition · EREC systems · diagonalisation · categorical dynamics Core concepts Internal-Geometry Threshold (IGT) · IGT transitions · internal geometric precursor · critical slowing down · threshold dynamics · fₘin formula · external reference · GAMMA morphism variance · Categorical Monitoring Principle Domain applications transcriptional noise · QCF index · biological ageing · partial reprogramming · critical period · representational entropy · CHILDES · phonological acquisition · market microstructure · spectral gap · AI alignment · value criterion drift · disorders of consciousness · branching ratio · TMS protocol Methods ALGUILAS-AI · Method ALGUILAS-AI Dialectical Engine · Strongest Formulation · Collapse Counter-Scenario · Named Binary · Weil Protocol · Artificial Deferral Suspension Protocol · cross-domain structural invariance Cross-disciplinary tags autopoiesis · complex systems · dynamical systems · self-organisation · phase transitions · impossibility theorems · mathematical biology · philosophy of science · engineering design · minimum intervention
José Caetano de Mattos (Wed,) studied this question.