Unknotted Double Helix-Like Projection via Phase-Shifted Two-Lap Radial Modulation

This record presents a minimal derived construction of a two-strand configuration whose planar projection exhibits double helix–like visual intertwining while remaining topologically trivial in three-dimensional space. The construction is obtained by introducing a phase-shifted copy of a previously established unknotted closed curve with trefoil-like projection (Yorisuna, 2026, v1.1). No modification is made to the base curve itself; the second strand is defined solely by a rigid phase shift of the original parametrization. Both strands are traversed over two full laps (t ∈ [0, 4π]) and share an identical, fixed depth-ordering rule: the positive phase (first lap) lies above the negative phase (second lap) at all apparent overlaps in planar projection. As a result, visually complex helical intertwining and apparent crossings arise entirely from phase-shifted indexing and projection-induced overlap, rather than from topological linking or physical intersection. In three dimensions, the configuration consists of two disjoint unknotted components, with all apparent crossings resolved by depth separation inherited from the base construction. No cumulative noise, perturbation, or stochastic deformation is introduced; the extension relies exclusively on deterministic phase shifting and inherited ordering constraints. This record is intended as a clean, minimal example demonstrating that double-helix–like visual complexity can emerge without topological entanglement. It is provided as a separate, derived work to preserve the canonical integrity of the original single-strand construction. Related Identifier:The base construction (single-strand unknotted trefoil-like projection, v1.1) is referenced as the source from which this two-strand configuration is derived.