Topological Regularisation: Improving Neuro-Symbolic Robustness via Differentiable Logic Manifolds

The integration of symbolic reasoning into gradient-based learning remains the central challenge
of Artificial Intelligence, often characterised as the “System 1 versus System 2” dichotomy. Current
Neuro-Symbolic approaches fail to resolve this tension because the interface between discrete logical
predicates and continuous neural manifolds is fundamentally non-differentiable. Existing hybrid
architectures rely on heuristic glue or fragile approximation methods that lack rigorous theoretical
guarantees.

We propose Differentiable Topological Synthesis (DTS), a unified theoretical framework that
redefines logical predicates not as discrete integers, but as Jacobian invariants within a continuous
dynamical system. By formalising the “Neuro-Symbolic Gap” as a Geodesic topological distance
on a Riemannian manifold, we demonstrate that logical rules can be analytically synthesised via
adjoint sensitivity methods on the logic manifold, allowing for standard gradient descent without
the information loss inherent in discretisation. We argue for Proof-Gradient Steering, a mechanism
where the gradient of a formal proof directly modulates the neural vector field, enforcing causal
consistency and logical entailment as geometric constraints on the optimisation landscape.

Finally, we propose Topological Regularisation, a framework that treats logical constraints as
geometric barriers. While absolute safety is undecidable in general Turing complete systems, we
argue that mapping logical error to the Riemannian metric creates a strong inductive bias that significantly
reduces the probability of logical hallucination. This offers a rigorous scientific perspective
on bridging the gap between statistical learning and formal reasoning.