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22.1.2 Can infinitesimal learning jump?

The title of this book’s final chapter creates an apparent tension with the claim that scientific creativity may require a jump. If LINCS is infinitesimal, is it confined to small improvements inside a fixed conceptual neighborhood?

The answer depends on what kind of jump is at issue. An infinitesimal method can describe a finite change whenever local directions integrate along a path. At a coarse temporal or observational resolution, the endpoint can look abrupt even though the trajectory is locally resolved. Long-run evolutionary change sometimes has this character: accumulated variation, inheritance, selection, drift, and ecological interaction can produce a large phenotypic or population-level separation. This example is suggestive, not a universal reduction. Mutations and lineage events may themselves be discrete, and evolutionary dynamics can cross thresholds or reorganize developmental and ecological regimes.

Other jumps are critical. A small change moves a system through a bifurcation, phase boundary, or loss of stability, after which its qualitative organization differs. Infinitesimal analysis can reveal the approaching instability, null mode, curvature, or failure of closure. It does not make the two regimes identical, and a first-order probe at a singular point may be insufficient.

The strongest jumps are declarational. A new generator, ontology, mechanism, or dual description may inhabit no smooth path inside the current model category. Quantization is not merely a small parameter update to a classical declaration; a biological mechanism transported from population theory is not already a tangent vector in the old biological vocabulary; and a bulk–boundary duality relates two theories rather than moving infinitesimally inside either one. Such proposals belong to the outer category of declarations and correspondences developed below.

Kind of apparent jump	Role of infinitesimal structure	Additional requirement
Integrated change	Supplies local directions and transports obstruction information along a path	Existence and admission of a finite integrated repair
Critical transition	Detects instability, interaction, or loss of closure near a regime boundary	Higher-order probes and validation on both sides of the transition
Declaration change	Localizes persistent pressure that the present language cannot absorb	A discrete proposal, transport of prior evidence, and independent admission

Table 22.1 Infinitesimal diagnosis supports several kinds of change without reducing every jump to a smooth trajectory.

This yields the appropriate claim for LINCS: infinitesimal diagnosis is valuable without being generatively complete. It provides typed local evidence, separates meaningful from null directions, and constrains where a larger repair must act. A proposal mechanism may then integrate those directions, cross a detected boundary, or make a discrete meta-repair. The admission mechanism must determine whether the resulting finite or frame-changing proposal actually explains the obstruction.

Design principle

Use infinitesimal structure as a microscope and a transport calculus, not as an ontology asserting that all novelty is continuous. A jump that leaves the current model space must remain visible as a change of declaration.