Global Topological Dirac Synchronization

The paper defines GTDS via the Dirac-coupled dynamics with gamma matrices and proves that a GTDS ansatz Φ = Û ⊗ s exists iff D/Φ = 0; since s is arbitrary this reduces to the purely topological kernel condition DÛ = 0, hence LÛ = 0, and then analyzes K = 1 (Eulerian graphs) and K = 2 examples (SLTT yes; unweighted triangulations no; a WTT yes under a specific weight identity) and an MSF-based stability analysis. The candidate solution reproduces the same kernel condition, the Eulerian criterion for K = 1, the SLTT construction, the impossibility for unweighted 2D simplicial complexes, and the WTT weight relation 1/√w1 + 1/√w2 = 1/√w3. Its optional linearization/set of 2d master-stability blocks is consistent with the paper’s Dirac decomposition/MSF framework. Minor differences are present in presentation (e.g., the candidate states LÛ = 0 ⇒ DÛ = 0 explicitly, which follows from D being symmetric), but there are no substantive conflicts.