Safe Cuts for Two-Rim Convex Bands: A geometric proof and Lean formalization

Let 𝐴 and 𝐵 be convex hulls of finite planar point sets, each with nonempty interior, and let ℎ > 0. We prove that the lateral surface of 𝐾 = conv((𝐵 × {0}) ∪ (𝐴 × {ℎ})) admits a static planar development after cutting one original lateral hinge. The development is continuous on an explicit cut-open face quotient, is an affine Euclidean isometry on each original facet, and has pairwise disjoint developed face interiors. Boundary contact is permitted. Neither nesting nor a prescribed sign pattern of the intrinsic turns is assumed, and original triangular lateral facets are included. The proof combines a folded-turn projection inequality derived from rim closure, radial support about the fixed point of the full affine development circuit, and an extremal-seam argument that separates distinct height traces. Degenerate rim edges are treated by selecting one seam and one family of full face maps before restoring the original faces from positive trims. Zero rotational defect is handled separately by a Cartesian monotonicity argument. Four public Lean declarations express the result without an external band-unfolding premise. We specify the formal surface, the nonoverlap convention, the verification evidence, and the remaining distinctions from arbitrary-slab formulations, cap attachment, and continuous unfolding motion.