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4.10 From tangent diagnosis to learning
When numerical observations are available, a computational realization may use
The categorical framework permits \(\lambda =0\). A nonzero tangent term is an empirical and semantic admission decision, not an axiom.
Before a tangent signal influences repair, the experiment must establish the semantics of its probes, quotient null directions, evaluate it on held-out support, and keep base, tangent, task, and safety outcomes separately visible.
The next chapter specializes tangent directions to smooth intervention protocols. This turns the abstract question “where does the factorization fail under variation?” into a causal one: do the declared intervention directions close under sequential local perturbation? A normal bracket component then marks a specific explanatory gap—a behavior generated by visible interventions but not expressible in their declared span. Lie-bracket closure receives this causal interpretation only under additional statistical and structural assumptions.