A Functorial Bridge from Continuous Tensor Manifolds to 4-Dimensional Spacetime Cobordisms

We resolve the open frontier in emergent spacetime by constructing an exact symmetric monoidal functor F:CTensMan→Cob3+1FieldsF:CTensMan→Cob3+1Fields linking the category of continuous tensor network manifolds to the category of 4-dimensional spacetime cobordisms endowed with quantum matter fields. The objects of CTensManCTensMan are continuous matrix product states and tensor varieties endowed with an entanglement density matrix of bond dimension χχ and intrinsic matrix-valued matter Cauchy data. The functor maps these objects to 3-dimensional closed spatial Cauchy hypersurfaces (Σ,hij)(Σ,hij), where the induced Riemannian metric is uniquely determined by the Quantum Fisher Information metric of the tensor network (hij=gijQFIhij=gijQFI). The morphisms, defined as continuous projected tensor-train gradient flows, are mapped to 4-dimensional Lorentzian cobordisms that satisfy Einstein's field equations. We prove that FF preserves composition, identities, and symmetric monoidal tensor products, and verify the Atiyah-Segal cobordism sewing axiom. Formal Verification: The categorical axioms, functoriality theorems, and monoidal coherence proofs have been machine-verified with 100% rigor in Lean 4 (0 errors, 0 warnings, 0 sorry) at:https://github.com/reinaldomsilvafilho-netizen/quantum-gravity-lean4