Cyclicity of interaction-frame transformations

We study the structure of rotation sequences composed of piecewise displacements by an angle <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>β</a:mi> </a:math> about arbitrary axes in three dimensions. We find that when <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mi>β</b:mi> <b:mo>=</b:mo> <b:mn>2</b:mn> <b:mi>π</b:mi> <b:mo>/</b:mo> <b:mi>m</b:mi> </b:mrow> </b:math> , repeated toggling-frame transformations return the sequence to its original form after <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mi>m</c:mi> </c:math> applications. We develop this cyclicity property to unify several families of error-tuned control sequences used across quantum technologies, including NMR, MRI, atomic clocks, and atom-scale computing. For the <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mi>m</d:mi> <d:mo>=</d:mo> <d:mn>2</d:mn> </d:mrow> </d:math> case, we show a duality between narrowband and broadband <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mi>π</e:mi> </e:math> -element sequences. For <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mrow> <f:mi>m</f:mi> <f:mo>></f:mo> <f:mn>2</f:mn> </f:mrow> </f:math> , we relate the cyclic structure to polyhedral symmetry, leading to original sequence designs based on <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mi>m</g:mi> </g:math> -fold rotational axes. The result may guide the construction of robust control protocols grounded in geometric and group-theoretic principles.