Fast and Faithful: Scalable Neuro-Symbolic Learning and Reasoning with Differentiable Fuzzy 𝓔𝓛++

The unification of neural learning and symbolic reasoning remains a foundational challenge in AI, forcing a persistent trade-off between logical rigor and computational scale. The field has largely diverged into two paths: expressive frameworks rooted in first-order logic that are formally sound but computationally intractable, and scalable methods, such as geometric embeddings, that sacrifice formal logical guarantees for efficiency. In this paper, we introduce DF-ɛℒ++, an end-to-end differentiable framework that transcends this trade-off by unifying PTIME-complete reasoning with neural learning. Our primary contribution is a complete, theoretically-grounded methodology centered on a unified semantic-loss framework: we unite the tractable structure of the Description Logic ɛℒ++ with a Product-based fuzzy semantics, deriving our learning objective directly from the corresponding Goguen implication to ensure high logical fidelity. This principled semantic core is made robust and practical by two supporting innovations: a normalization strategy that re-architects complex axioms for stable optimization, and a novel domain construction technique that prevents model collapse to ensure non-trivial reasoning. Validated on massive, real-world ontologies like SNOMED CT (377K concepts), DF-ɛℒ++ demonstrates a unique synergy of scale and performance: it remains computationally efficient where expressive systems fail, while decisively outperforming dominant scalable baselines in a range of knowledge base completion tasks with up to a 42% relative improvement in Hits@1. This work establishes a new, provably sound, and scalable pathway for a new generation of neuro-symbolic systems that are both empirically powerful and logically reliable.