Twisted Rings and Twisted Signals: The Sector Calculus of Rotational Twists
Year: 2026
Type: preprint
Abstract: A twist can live in a signal or in a ring. A twisted signal returns after one period multiplied by a fixed phase. A twisted ring is a solid ring whose transported frame returns turned by an angle. This article shows that the two are one structure. On the flat model of the ring, the part of a field with a given rotation charge is a twisted signal along the ring, and its phase is the charge times the turning angle. The ring spectrum is therefore a union of twisted-signal spectra, and the transform of the MKT canon (the Möbius–Klein Transform series of articles) for one fixed twist diagonalizes each charge sector along the ring. The real form of a twist is a rotation of a real plane, and the half turn is the only nontrivial twist that a single real component can carry; this places the canon's mirror twist inside the circle of rotational twists. The Möbius strip has two thickenings: it is swept out by a diameter of the half-turned solid torus, which is orientable, and it is a slice of the solid Klein bottle, which is not. Tensor components carry charges that add, and a rotation adds no pseudotensor sign; the curl switches sectors only under a reflection. Operators that commute with rotations keep the charge, and the circular derivatives raise or lower it by one. A geometric twist preserves time-reversal symmetry, so each level has a partner of opposite charge and opposite wave number; an external flux breaks the pairs except at half a flux quantum; and a full turn of the twist moves the ladder of each charge by that charge. Read as a function of the twist, the family of transforms is the Zak transform. Integration over the twist inverts it, the twist is Fourier dual to the winding number, and sampling the twist at the roots of unity keeps the windings only modulo their number. The fractional parts of mode numbers then measure the turning angle and an external holonomy, with the charge-zero sector as a reference and an aliasing in the charge. On rings of Hill type, with one channel and nearest-neighbour coupling, the real twists are the band edges; on rings with longer or multichannel couplings they need not be. Analyticity in the twist gives exponential localization; position and twist together force a zero of the Zak transform, and two twist angles carry a Chern number. As a worked example, a ring of two orbitals whose frame turns by a quarter turn per circuit has two quarter-flux sectors that are time-reversal partners; when its bonds rotate the frame and its sites treat the two orbitals alike, its levels come in exact pairs, and a symmetric orbital mixing or an on-site splitting removes this guarantee. Most ingredients are known in separate fields. They are derived here from the canon, collected in one dictionary and checked by a deterministic verification script.
Author: Mark, Laszlo
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221936023