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Theoretical analysis reveals a closed resonant field configuration with odd framing parity in continuous textures, indicating that parity changes require physical defects.
Field textures can carry discrete topological information while their energy changes continuously. This paperproposes the Klein Spiral: a closed resonant configuration whose transverse director field around a loophas odd framing parity, represented by the first Stiefel-Whitney class w1 = 1. Changing that parity requires a defect, an amplitude zero, a boundary interaction, or another transition that interrupts the director field.
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