DFTCliffordBridge
plain-language theorem explainer
Packages a group homomorphism from 8-point DFT mode indices to the Z/8Z Clifford grading of Cl₈, together with the period-eight condition on shift eigenvalues. Cited by anyone linking the RS eight-tick cycle to Bott periodicity and spinor representations. Pure structure interface; the canonical instance is supplied separately by the mode-to-grading map.
Claim. A DFT--Clifford bridge is a map $m:\mathbb{Z}/8\mathbb{Z}\to\mathbb{Z}/8\mathbb{Z}$ sending DFT mode indices to Clifford grades such that $m(k+k')=m(k)+m(k')$ (addition mod 8) and $m(k)+8=m(k)$ for every mode $k$.
background
The module connects Recognition Science's eight-tick structure to Clifford algebras via Bott periodicity: $\mathrm{Cl}_{n+8}\cong\mathrm{Cl}_n\otimes\mathrm{Cl}_8$, with the 8-tick DFT identified with the grading of $\mathrm{Cl}_8$, and $\mathrm{Spin}(3)\cong\mathrm{SU}(2)$ supplying the $D=3$ spinor side.
The grading group is $\mathbb{Z}/8\mathbb{Z}$. On the spectral side, the 8-point DFT uses the primitive root $\omega=e^{-2\pi i/8}$; the cyclic-shift eigenvalue at mode $k$ is $\zeta^k$. The bridge asserts that this mode index is the Clifford grade.
Physically this is why the eight-tick octave (forcing chain T7) is forced: real spinor representations, $KO$-theory, and the division-algebra tower all repeat with period 8.
proof idea
No proof body: this is a structure declaration. It bundles three fields: a map from $\mathrm{Fin},8$ into the grading group $\mathbb{Z}/8\mathbb{Z}$, an additivity axiom (homomorphism for addition mod 8), and the period-eight identity on grades. Inhabitants are built later by supplying a concrete map (the mode-to-grading map) and discharging the two axioms.
why it matters
This is the interface that makes the DFT--Clifford identification a first-class object. Downstream, the canonical bridge fills the fields with the mode-to-grading map and the grading-additivity lemma; the complete 8-tick Clifford bridge then packages Bott periodicity, this DFT bridge, and the $\mathrm{Cl}_3\cong M_2(\mathbb{C})$ spinor isomorphism in one record.
In the forcing chain it anchors T7 (eight-tick octave): the same period-8 that appears in Bott periodicity and real spinors is exactly the DFT cycle length. Without this bridge, the claim that "8 ticks because Clifford period 8" remains informal.
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