complete8TickCliffordBridge
plain-language theorem explainer
A single record that packages every verified link between the RS eight-tick cycle and Clifford/Bott structure: Bott period 8, the DFT–grading bridge, Cl₃ ≅ M₂(ℂ), Spin(3) ≅ SU(2), uniqueness of the D=3 spinor, and the identity 8=2³. Anyone citing the Clifford bridge as a finished object uses this witness. The body is a pure structure literal wiring already-proved component defs.
Claim. There exists a complete bridge object whose fields are: Bott periodicity of period $8$; a DFT–Clifford mode-to-grading correspondence; an isomorphism $\mathrm{Cl}_3 \cong M_2(\mathbb{C})$; an isomorphism $\mathrm{Spin}(3) \cong \mathrm{SU}(2)$; uniqueness of the $D=3$ spinor representation; and the equality $8 = 2^3$.
background
The module builds the Recognition Science claim that the forced eight-tick octave is the same mathematical period as Bott periodicity for real Clifford algebras: $\mathrm{Cl}_{n+8} \cong \mathrm{Cl}_n \otimes \mathrm{Cl}_8$. Companion facts are the DFT mode grading of the eight-tick cycle, the low-dimensional isomorphisms that supply spinors in three space dimensions, and the elementary identity $8=2^3$ that matches the T7 octave to three binary degrees of freedom.
Complete8TickCliffordBridge is the bundling structure: Bott period data, a DFT–Clifford bridge (mode-to-grading map preserving addition and carrying period-eight eigenvalues), $\mathrm{Cl}_3 \cong M_2(\mathbb{C})$, $\mathrm{Spin}(3) \cong \mathrm{SU}(2)$, and a uniqueness statement for the $D=3$ spinor. Upstream witnesses already exist as separate defs: bottPeriodicity (rank period identity and minimality of residue 8), canonicalBridge (mode grading with period-eight eigenvalues), and cl3_iso_m2c (carrier and dimension match for $\mathrm{Cl}_3$).
RS-native time is measured in ticks with $\tau_0 = 1$; one octave is eight ticks. The present definition does not re-prove those facts; it only assembles them.
proof idea
Definitional structure literal, not a tactic proof. Each field is filled by a named upstream witness: bottPeriodicity for Bott data, canonicalBridge for the DFT–grading bridge, cl3_iso_m2c for $\mathrm{Cl}_3 \cong M_2(\mathbb{C})$, spin3_iso_su2 for $\mathrm{Spin}(3) \cong \mathrm{SU}(2)$, and d3_spinor_uniqueness for the $D=3$ spinor uniqueness field. The last field eight_equals_two_cubed is discharged by rfl (definitional equality $8=2^3$). No new algebraic work occurs here.
why it matters
This is the module's packaged "bridge exists" object: one place to point when arguing that the RS eight-tick cycle is Bott's period rather than an ad hoc discrete clock. It sits on the T7 eight-tick octave and the T8 $D=3$ forcing landmarks, and it makes the spinor side of the $D=3$ story (via $\mathrm{Cl}_3$ and $\mathrm{Spin}(3)\cong\mathrm{SU}(2)$) available as a single hypothesis for later geometry or gauge constructions.
No downstream consumers are recorded yet (used_by is empty), so its role is archival and interface-level: close the Clifford-bridge chapter and expose a typed witness. The adjacent theorem comment in the file ("8-Tick Period is Bott Period") is the prose claim this bundle is meant to underwrite. It does not by itself derive particle content or coupling constants; those remain elsewhere in the forcing chain and mass ladder.
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