This study is grounded in prior work on program induction framework with a structured latent program space, called Program Lattice Auto Encoder(PLAE). It preserves compositional structure by training an encoder where programs and their compositions correspond to integer linear combinations of program bases, forming a discrete program lattice that captures the geometric structure of compositional reasoning. Based on it, this paper proposes a novel extension of the PLAE aimed at improving generalization and efficiency by choosing a cylindrical lattice latent space instead of plane, which can represent invariant programs. The core hypothesis is that only isometric transformations conserve compositional properties of lattice structure and therefore developable surfaces such as a cylinder or cone are permissible as embedding space. Moreover, through demonstrating a contradiction of lattice on conical manifolds, it conclude that only cylinder is a possible embedding manifold for lattice structure.