Topological ℤ₂ index of the microtubule: the seam as a parity obstruction and structural verification of seamless lattices

Microtubules are nanotubes of αβ-tubulin heterodimers whose architecture is de- termined by two integers: the protofilament number N and the start number S. In the dominant B lattice, lateral contacts between protofilaments are homotypic (α–α, β–β) except at the seam, a line of heterotypic α–β contacts present in the canonical 13-3 microtubule. The seam is usually described as a contingent feature of assembly. We prove here that it is an exact topological obstruction: the α/β label on the cylindrical lattice (N,S) is globally consistent if and only if S is even, and in general the number of seams satisfies n≡S (mod 2). We verify the index against all published structural configurations we could locate —nine, including taxol-induced multi-seam architectures and transitions with lattice holes— without a single parity counterexample. Theassociatedprediction—thateven-Sarchitectures,inparticular 15-4 and 16-4, admit fully helical (seamless) lattices— is confirmed by independent cryo-EM reconstructions: the 15-4 of Xenopus laevis is resolved at 3.1 Å as fully helical (EMD-50945), while in the same dataset the 14-3 and 15-3 (odd S) do have a seam, an ideal internal control. We further show that the index is a property of the chemical label and not a dynamical defect: it implies neither mechanical weak- ness (consistent with rigidity measurements) nor localized modes. We close with an experimental proposal, with a pre-registered bidirectional prediction, to decide whether the seam has a functional readout on catastrophe dynamics.