A Null-Energy-Condition Constraint for a Stationary Cylindrical Metric with Closed Timelike Circles

We consider the Lorentzian metric ds^2 = -(dt - W(t,r) dphi)^2 + dr^2 + r^2 dphi^2 + dz^2, where phi is periodic and W is a sufficiently regular function. Circles with constant t, r, and z are timelike when |W(t,r)| > r. In an orthonormal frame, a pair of null vectors yields the necessary null-energy-condition (NEC) inequality |W_r - r W_rr| <= W_r^2. For a stationary profile W = W(r), this inequality can be written as |q'(r)| <= q(r)^2 with q = W'/r. We prove that, if this inequality holds for every r > 0, the profile extends regularly to the axis with W(0) = 0, and W(r)/r tends to zero as r tends to infinity, then W vanishes identically. The result is restricted to the stated metric ansatz and boundary assumptions; it is not a general no-go theorem for closed timelike curves. A SymPy script accompanies the manuscript to recalculate the displayed Einstein-tensor components. Hello, my name is Anakin, I’m 20 years old, and I’m an independent author interested in theoretical physics, particularly general relativity and the nature of spacetime. This manuscript is an attempt to explore a specific mathematical question involving closed timelike curves and the null energy condition. I’m sharing this preliminary work to make the calculations available for examination, discussion, and independent verification. I welcome constructive feedback, corrections, and suggestions for further investigation.