What Survives Recursive Training: Three Bridges, the Evaluator Regress, and the Path to AGI

AI model development is becoming recursive across three distinct mechanisms: synthetic data generation (Model N's outputs become Model N+1's training data), AI-as-evaluator (models judge, rank, and reward other models' outputs), and benchmark-driven optimisation (development targets increasingly saturated benchmarks). Each mechanism is a bridge that transforms one system's outputs into another's inputs. We introduce recursive closedness — measuring the degree to which a pipeline's data, feedback, and evaluators are generated by prior models rather than supplied by external sources. We show that as closedness increases, the three bridges share a structural property: each preserves high-measure, easily evaluated regularities while risking degradation of low-measure, tail-dependent capabilities. We explain this asymmetry via a structural analogy to the ∀/∃ asymmetry of the Constraint Inheritance Lemma (Badkur & Dak, 2026b). Above all three bridges sits a constraint that none can resolve: the evaluator regress. The pipeline cannot certify from within that its evaluators track intelligence rather than benchmark-optimised fluency. This paper does not claim that recursive training cannot produce AGI. It claims that recursive training cannot, by itself, certify that its measured improvements are progress toward AGI rather than progress toward the evaluators' surface criteria. We propose a benchmark protocol for measuring recursive-training degradation across six conditions, run over multiple recursive generations, on four outcome dimensions: standard capability, tail-content retention, external validity, and Goodhart resistance. Companion to: "The Foundation Is the Bottleneck: Evaluator Legitimacy and Judgment Preservation in Autonomous Research" (Dak & Badkur, 2026).