cyclic_shift

The module builds the 8-point DFT as the canonical unitary basis for the Recognition Science eight-tick cycle. That period $\tau_0=2^D$ with $D=3$ is forced by the foundation chain (T7/T8). The DFT-8 matrix is built from the primitive root $\omega=e^{-2\pi i/8}$, and its columns are the spectral modes.

The cyclic shift is the discrete time-translation generator on those 8-vectors. Upstream, the same idea appears as the shift $T$ on Signal8: $(Tf)(k)=f(k+1\bmod 8)$, called the fundamental discrete time-evolution generator. Parallel forms exist in the eight-tick Weyl and macroscopic-ledger layers; this definition is the spectral-module native version on $\mathrm{Fin},8\to\mathbb{C}$.

Together with the DFT modes, $S$ is what the spectral theory diagonalizes: eigenvectors are the DFT columns, eigenvalues powers of $\omega$.

Pure definition, not a proved theorem. The map sends $v$ to the function $t\mapsto v(\langle(t.val+1)\bmod 8\rangle)$, with the modular bound discharged by Nat.mod_lt and norm_num. No lemmas are applied; the term is the pointwise shift.

This is the time-translation operator that DFT-8 is built to diagonalize. Downstream, the discrete difference diff8 is $(S-I)v$, and diffEnergy8_mode identifies the mode energy with $|\omega^k-1|^2$, which is the $\sin^2(\pi k/8)$ factor in gap-weight projections.

In the recognition-operator layer it seeds iterated shift, linearity of iterates, and the eigenvector law $S^n(\mathrm{mode}_k)=(\omega^k)^n\cdot\mathrm{mode}_k$. Gravity and ledger modules re-export it as the one-tick recognition update on Signal8.

Framework landmark: T7 eight-tick octave. Without a clean $S$, the claims that DFT diagonalizes shift, separates DC from neutral modes $k=1..7$, and supplies the $\varphi$-lattice spectral basis do not type-check.