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 determined by two integers: the protofilament number N and the start number S. In the dominant B lattice, lateral contacts 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 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). The index is verified against all locatable published structural configurations — nine single-(N,S) architectures, including taxol-induced multi-seam 13-3 tubes (3 or 5 seams, both odd as required) and transitions with lattice holes — with zero parity counterexamples. The associated prediction, formulated before examining the reconstructions, is confirmed by independent cryo-EM: the 15-4 of Xenopus laevis is fully helical (seamless) at 3.1 Å (EMD-50945), while the 14-3 and 15-3 in the same dataset (odd S) do have a seam — an ideal internal control. The index is a property of the chemical label, not a dynamical defect: it implies neither mechanical weakness nor localized modes. An experimental proposal with a pre-registered bidirectional prediction (catastrophe frequency of fully helical vs pseudo-helical tubes at fixed N) is included, with an explicit power calculation. Version 2 — correction notice. Table 1 printed "2–5 seams (taxol)" against odd S = 3: as tabulated, the row was a parity counterexample to the paper's own theorem. Restated as 3 or 5 seams (the observed odd values), with tubes whose seam number varies along their length explicitly excluded from the test rather than counted as agreeing. The proposed experiment's n ≥ 50 per arm is now backed by a power calculation: it reaches 80% power only for CV ≤ 0.35; at a realistic CV ≈ 0.6 the requirement is n ≈ 150 per arm. The Z2 index theorem itself was verified and is unchanged. Two references corrected against the literature (Sandblad et al. 2006 is Cell 127, 1415–1424; the Mal3 structural-plasticity paper is von Loeffelholz et al., Nat Commun 8, 2110).