You cannot comb a hairy ball flat without a cowlick — somewhere the hair must stand up (that’s why there’s always a calm point in a cyclone field on Earth). But you CAN comb a hairy doughnut perfectly flat, everywhere, no cowlick. The difference is one number: the Euler characteristic. Slide between the sphere (χ=2, always a cowlick) and the torus (χ=0, combable).
The Poincaré–Hopf theorem: for a continuous tangent vector field on a closed surface, the sum of the indices of its zeros equals the surface’s Euler characteristic χ. The sphere has χ=2, so a tangent field MUST have zeros whose indices sum to 2 — the Hairy Ball Theorem, at least one cowlick. The torus has χ=0, so a nowhere-zero field is allowed and exists: the constant ‘around the tube’ field ∂/∂φ never vanishes — the doughnut combs flat. A fail-loud self-check throws unless χ(torus)=0 (combable) and χ(sphere)=2 (not). ◆ real topology, node-verified.
The instrument shows the constant poloidal field on the torus (never zero) and a dipole-like field on the sphere (two zeros summing to 2); a clean illustration of the index sum = χ, not a claim about any specific physical field.