canonicalBridge
plain-language theorem explainer
Packages the 8-point DFT mode index into a Clifford Z/8Z grade as a single bridge instance: mode-to-grade map, additive compatibility mod 8, and period-eight eigenvalue identity. Cited by anyone assembling the full 8-tick ↔ Bott-periodicity link. Construction is a structure literal wiring two prior lemmas and a one-line ZMod identity.
Claim. The canonical DFT–Clifford bridge is the structure whose mode-to-grade map sends each 8-point DFT mode $k\in\{0,\ldots,7\}$ to the corresponding grade in $\mathbb{Z}/8\mathbb{Z}$, preserves addition modulo 8, and satisfies $\mathrm{grade}(k)+8=\mathrm{grade}(k)$ for every mode $k$.
background
The module links Recognition Science's eight-tick cycle to Clifford algebra via Bott periodicity: $\mathrm{Cl}_{n+8}\cong\mathrm{Cl}_n\otimes\mathrm{Cl}_8$, with the 8-tick DFT identified with the $\mathbb{Z}/8\mathbb{Z}$ grading of $\mathrm{Cl}_8$. Physically this is why the forcing chain demands period $2^3$ (T7).
The grading group is $\mathbb{Z}/8\mathbb{Z}$. The mode-to-grade map sends a DFT index $k\in\mathrm{Fin},8$ to its class in that group. The bridge structure packages three requirements: that correspondence, preservation of addition mod 8, and the period-eight identity on grades (eigenvalues of the cyclic shift).
Upstream, additive compatibility is already proved by reducing to the quotient law in $\mathbb{Z}/8\mathbb{Z}$. Spatial dimension $D=3$ appears elsewhere in the module for the $\mathrm{Cl}_3$ spinor side; this definition only handles the DFT–grade half.
proof idea
Structure literal, not a tactic proof. The mode-to-grade field is the existing map $\mathrm{Fin},8\to\mathbb{Z}/8\mathbb{Z}$. Additive compatibility is the prior theorem that this map intertwines sum-mod-8 with group addition. The period-eight field is discharged by noting $8=0$ in $\mathbb{Z}/8\mathbb{Z}$ (decide) and simplifying the mode-to-grade definition, so grade$(k)+8=$ grade$(k)$.
why it matters
This is the concrete witness that the 8-point DFT and the $\mathrm{Cl}_8$ grading share the same cyclic structure: $\omega=e^{-2\pi i/8}$ modes match grades in $\mathbb{Z}/8\mathbb{Z}$. It is the dft_bridge field of the complete 8-tick Clifford bridge, which also assembles Bott periodicity, the $\mathrm{Cl}_3\cong M_2(\mathbb{C})$ isomorphism, $\mathrm{Spin}(3)\cong\mathrm{SU}(2)$, and $D=3$ spinor uniqueness.
In the forcing chain it underwrites T7 (eight-tick octave): Bott period 8 is not an external coincidence but the algebraic reason the recognition tick must close at $2^3$. Downstream completeness of the bridge depends on this instance being available as a single object rather than three scattered lemmas.
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