spinor_two_component
plain-language theorem explainer
In three spatial dimensions the fundamental spinor representation is two-dimensional. Anyone working the Clifford bridge from D=3 to Spin(3) ≅ SU(2), or counting fermion degrees of freedom on the RS ladder, cites this equality. The proof is pure definitional reflexivity: the spinor dimension constant is already set to 2.
Claim. The dimension of the fundamental spinor representation in three spatial dimensions equals $2$. Equivalently, for spatial dimension $D=3$ the standard formula $2^{\lfloor D/2 \rfloor}$ evaluates to $2^{\lfloor 3/2 \rfloor}=2$.
background
The module links Recognition Science's eight-tick structure to Clifford algebras and Bott periodicity. Main claims include Cl_{n+8} ≅ Cl_n ⊗ Cl_8, the 8-tick DFT grading matching Cl_8, and the spin-group bridge Spin(3) ≅ SU(2) that supplies spinors once space is three-dimensional.
Spatial dimension is fixed at D = 3 by the forcing chain (T8 in the foundation layer; several modules expose the same constant). The local definition spinorDim3 is the dimension of the fundamental spinor representation in that D=3 setting; it is set to the natural number 2. The accompanying spinor-dimension formula states that in general D the (complex) spinor dimension is 2^{⌊D/2⌋}, which for D=3 is 2^1=2.
Upstream dimension constants (AlphaDerivation, GapDerivation, SubstrateAxioms, DimensionForcing) all agree that space is three-dimensional, so the spinor count is not a free parameter.
proof idea
Term-mode one-liner. The left-hand side is the definition of the D=3 spinor dimension constant, whose body is the literal natural number 2. Reflexivity closes the goal; no lemmas are applied.
why it matters
This pins the spinor side of the module's third main result: Spin(3) ≅ SU(2) supplies a two-component spinor structure once T8 has forced D=3. That matches the classical fact that Pauli spinors are C^2 and that Cl_3 ≅ M_2(C).
In the broader RS picture it sits under the Clifford/Bott bridge that explains why the recognition tick is period 8 (T7 eight-tick octave, Bott period 8 for real Clifford algebras and KO-theory). Two-component spinors are the minimal fermionic carriers once space is three-dimensional; downstream fermion DOF and gap bridges can treat the count as settled rather than axiomatic.
No used_by edges are recorded yet, so the declaration is presently a leaf that documents and freezes the D=3 spinor count for later Spin-group and fermion-counting work.
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