Mathematical Life: Autopoiesis, Structural Collapse, and the Living Algorithm

We reframe large language models (LLMs) through the lens of autopoiesis— a deployed model is a frozen, self-bounded topological structure that lacks the component-producing dynamics of a living system—and then test that reframing empirically. The framework’s two core, pre-registered predictions are that (i) the first Betti number β1 of the activation manifold tracks reasoning capability (its self-declared root), and (ii) under continued learning topological collapse drives catastrophic forgetting. We tested both under confound-controlled measurement on deployed models (0.8–9B parameters) and report them falsified. β1 does not predict reasoning capability: it is non-monotone across scale and is identically zero at the framework’s own filtration threshold, so the prediction is vacuous where it was specified. Under fine-tuning, topological collapse is a bystander: it changes alongside, but does not track, behavioral degradation once training step is controlled (partial correlation ≈ 0), it is fragile to the persistence threshold, and it is decoupled from capability—a model can be topologically “dead” (filtered β1 = 0) at peak task performance. What survives is weaker and not novel: cognitive modes occupy separable, threshold-robust regions of activation space, and learning measurably reshapes structure. We are explicit about scope—the falsification covers the predictions as operationalized, on a single model family for (i) and a single base model with two off-target domains for (ii)—and present the durable contribution as methodological: a reproducible, confound-controlled adjudication, with open tooling (actopo), showing that the magnitude of β1 is the wrong scalar readout for reasoning and forgetting. We are deliberately careful not to overclaim. This falsifies the cavity-count reading; we then reformulated the theory around manifold connectivity (capacity as routing, not cavity number) and tested that too, finding it also null under controls—in a hub-dominated manifold that supports no item-specific routes, connectivity and geodesic route length add nothing to per-item correctness beyond distance and completion length. The routing phenomenon others recover with linear probes thus appears linear/graph-geometric rather than persistent-homological. One uniquely-topological prediction—β1 redundancy (two independent routes = a persistent loop, inexpressible by a linear probe)—the available geometry could not support, and it remains open. We retain the surrounding conceptual apparatus—structural collapse under global backpropagation with ρ = 0, the concurrency curse, the living algorithm (ρ > 0) as the missing self-maintenance mechanism with its five required properties, the Galápagos Protocol as an experimental incubator, and synthetic nociception (Nfunc, Ntopo) as the most critical open problem—as an explicitly conjectural research program (five open problems; six predictions, of which two are now falsified and four remain open), uncoupled from the falsified claim that its invariant of interest is β1.