quarterTurnSector

Recognition works on an eight-tick discrete register (the T7 octave). Modes are indices in $\mathrm{Fin},8$. A structured sector packages a kept-mode finset together with two constraints: every odd mode must be kept, and the DC mode $0$ must be dropped. That is the local meaning of the structure used here.

The canonical odd-mode set is ${1,3,5,7}$. Membership in that set is exactly odd parity of the index (proved by exhaustive case split on $\mathrm{Fin},8$). The cyclic shift $T$ on eight-tick signals, $(Tf)(k)=f(k+1\bmod 8)$, is the discrete time generator in the surrounding complex-structure forcing development; sectors select which Fourier modes survive under projector-then-shift updates built from that generator.

Definitional instance of the structured-sector record. The kept-mode field is set to the canonical odd set ${1,3,5,7}$. Odd-mode inclusion is a one-line rewrite: for odd $k$, the parity membership lemma converts oddness into finset membership. DC exclusion is a direct simplification against the concrete finset (zero is not among $1,3,5,7$).

This is the paper's minimal structured sector: keep exactly the odd modes. It sits in the Recognition Operator module as the default mode filter for projector-followed-by-shift updates on the eight-tick register (T7 octave). Downstream use sites are not yet wired in this graph snapshot, but the object is the named sector any later operator or Hamiltonian-emergence argument would instantiate when it needs the quarter-turn core without extra even modes.

In framework terms it is the smallest sector compatible with the structured-sector axioms: full odd content for the quarter-turn complex structure, DC removed so the mean mode does not pollute the evolution. It does not itself force $D=3$ or the J-cost; it only fixes the mode support those later steps act on.