Singularity Vanishing Theorem for Neurovarieties and Causality

We analyze the identifiability problem in general machine learning and establish the singularity-vanishing theorem. We first extend the classic Alexander theorem to a foundational double-point theorem for generic polynomial networks by characterizing the singular locus of their associated ideal sheaves. Building upon these algebro-geometric insights, we revisit and reformulate causal representation learning (CRL) to provide the first formal proof of identifiability under strictly non-injective transformations. We introduce algebraic extension modules paired with models or a hypothetical functional space, and show that valid interventions yield a sufficient number of independent conditions to obstruct the double-point scheme, thereby establishing a general singularity vanishing theorem and resolving a long-standing conjecture in identifiable representation learning. Furthermore, by anchoring representation learning in rigid algebraic geometry, our work implicitly echoes the recent call by Fluri et al. for the 'Hilbert Problems of AI' from geometry. We envision that our results provide a novel, grounded theoretical perspective on causality and world models—the ultimate objectives of sciences such as biology and physics—while potentially opening new avenues for theoretical guarantees across broader machine learning domains.