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The integration of symbolic reasoning into gradient-based learning remains the central challengeof Artificial Intelligence, often characterised as the “System 1 versus System 2” dichotomy. CurrentNeuro-Symbolic approaches fail to resolve this tension because the interface between discrete logicalpredicates and continuous neural manifolds is fundamentally non-differentiable. Existing hybridarchitectures rely on heuristic glue or fragile approximation methods that lack rigorous theoreticalguarantees. We propose Differentiable Topological Synthesis (DTS), a unified theoretical framework thatredefines logical predicates not as discrete integers, but as Jacobian invariants within a continuousdynamical system. By formalising the “Neuro-Symbolic Gap” as a Geodesic topological distanceon a Riemannian manifold, we demonstrate that logical rules can be analytically synthesised viaadjoint sensitivity methods on the logic manifold, allowing for standard gradient descent withoutthe information loss inherent in discretisation. We argue for Proof-Gradient Steering, a mechanismwhere the gradient of a formal proof directly modulates the neural vector field, enforcing causalconsistency and logical entailment as geometric constraints on the optimisation landscape. Finally, we propose Topological Regularisation, a framework that treats logical constraints asgeometric barriers. While absolute safety is undecidable in general Turing complete systems, weargue that mapping logical error to the Riemannian metric creates a strong inductive bias that significantlyreduces the probability of logical hallucination. This offers a rigorous scientific perspectiveon bridging the gap between statistical learning and formal reasoning.