Nelson (2026a) established that fingerprint ridge singularities can be represented as topological charges of a U (1) line bundle, with first Chern numbers computed on local planar charts enclosing each defect. That result was local. The present work extends the framework to the curved fingertip surface as a whole. The volar surface of the distal phalanx is a topological disk with Euler characteristic chi = 1. The Poincare-Hopf theorem for manifolds with boundary, combined with the observed anatomical boundary condition, requires that total nematic charge vanishes on the complete anatomical volar surface. Merrett and Witte's (2025) finding that 95. 5% of fingerprints exhibit charge neutrality on cropped images is reinterpreted as confirmation of the interior half of this identity; their boundary winding deficit (WdM = 0. 71 +/- 0. 12 vs. the predicted WdM = 1) is most naturally interpreted as the signature of image-frame truncation rather than topological violation. The fiber bundle over the punctured disk is constructed explicitly: n + 1 charts, U (1) transition functions whose winding numbers are the individual Chern numbers, and a connection 1-form omega whose curvature 2-form is KdA, with K the Gaussian curvature. This connection enters the Napoli and Vergori (2012) elastic energy decomposition, coupling the ridge director field to both intrinsic and extrinsic surface curvature. Three results emerge that are invisible on a flat patch. First, the Gaussian curvature is estimated to contribute a geometric phase correction on the order of 10-15% of defect winding (intrinsic + extrinsic). Second, the Vitelli and Turner (2004) geometric potential positions defects according to the curvature distribution: cores are attracted to the apex, deltas are repelled to the flanks. This provides a geometric explanation for whorl placement. Third, these curvature effects are strongest where Gaussian curvature amplitude is maximal, which occurs during ridge formation, when the volar pad curvature is estimated at 9-16 times adult values. The ridge pattern preserves the imprint of the curvature potential present when it formed. Three falsifiable predictions are stated with named datasets and explicit kill conditions. The first, absolute charge neutrality on full-surface scans, is testable immediately against NIST Special Database 302. The framework remains abelian and U (1) throughout; the upgrade from S1 to S2 fiber and from U (1) to SO (3) structure group is deferred to subsequent work.
Yvonne Nelson (Fri,) studied this question.