cliffordPeriod
plain-language theorem explainer
Names the Bott period of real Clifford algebras as the natural number 8. Anyone citing the 8-tick–Clifford bridge, D=3 spinor uniqueness, or the T8 spinor certificate uses this constant. The body is a one-line definitional assignment.
Claim. The Clifford (Bott) period is the natural number $8$: real Clifford algebras satisfy $\mathrm{Cl}_{n+8}(\mathbb{R}) \cong \mathrm{Cl}_n(\mathbb{R}) \otimes M_{16}(\mathbb{R})$, so the fundamental period is $8$.
background
The module CliffordBridge links Recognition Science's eight-tick register to classical Clifford algebra theory. Bott periodicity states that real Clifford algebras repeat every eight dimensions: $\mathrm{Cl}_{n+8}(\mathbb{R}) \cong \mathrm{Cl}_n(\mathbb{R}) \otimes \mathrm{Cl}_8(\mathbb{R})$, with $\mathrm{Cl}8$ matrix-isomorphic to $M{16}(\mathbb{R})$. The same period-8 pattern appears in real $K$-theory ($KO(S^n)$) and in the sequence of normed division algebras $\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}$.
In RS, the forcing chain already produces an eight-tick octave (period $2^3$) and forces $D=3$ spatial dimensions (T7–T8). This definition simply records the classical Clifford period as the constant used throughout the bridge, so later statements can equate the recognition period with Bott's period without hard-coding the numeral.
proof idea
Definitional: the constant is assigned the value $8$ by :=. No lemmas or tactics. The companion theorem cliffordPeriod_eq_eight is rfl and exists only for documentation and simp-friendly equality.
why it matters
This constant is the shared numeral that makes the 8-tick–Bott identification a typed equality rather than a prose claim. Downstream, eight_tick_is_bott_period proves cliffordPeriod = 8 ∧ cliffordPeriod = 2^3; CliffordBridgeCert.verified requires cliffordPeriod = 8; D3SpinorUniqueness records that the Clifford period agrees with the recognition period; and Complete8TickCliffordBridge bundles Bott periodicity with the DFT–Clifford mode map.
In the forcing chain it feeds T8_To_CanonicalSpinor_Bridge and ultimate_inevitability_extended, which surface $\mathrm{Cl}_3 \cong M_2(\mathbb{C})$, $\mathrm{Spin}(3) \cong \mathrm{SU}(2)$, spinor dimension $2$, and Clifford dimension $2^3 = 8$ as the canonical spinor structure at $D=3$. Framework landmark: T7 (eight-tick octave) meets classical Bott period 8.
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