Curvature-induced smectic-C order of tangentially anchored hard spherocylinders on a sphere with a rigidly locked director field

We study the strict locked-orientation limit of hard spherocylinders on a sphere, in which the rod axes are rigidly locked to a prescribed tangential director field. The bulk hard-rod phase diagram contains no smectic-C (Sm-C) phase, so any coherent tilt found here is curvature-induced. On the sphere, the smectic layers can fail to close in only two ways. Assigning each of these two channels a cost, through a minimal ratio-symmetric mismatch function, yields a hierarchy of geometric statements: the lower edge of the smectic-area window at 45° is a channel-swap symmetry point; the upper edge at 58.3° is a testable channel-saturation hypothesis; the smectic-A to Sm-C boundary is given by an explicit analytic expression; and, independently of the cost function, local geometry sets the rod tilt angle through the rod-to-radius ratio, modulated by a chirality envelope peaking near 24°. Locked-orientation Monte Carlo simulations across fifteen geometries support these predictions with no fitted elastic constants; the envelope exponent is a stated golden-ratio input. The smectic-area fraction peaks at 55°, and a coherent Sm-C window is detected.