SpectralSector
The SpectralSector inductive type enumerates four layers of the B3 symmetry decomposition of the 3-cube Q3, with constructors color, weak, hypercharge, and conjugate. Researchers deriving Standard Model gauge structure from Recognition Science's forced D=3 would cite this when decomposing vertex count into sector dimensions. The definition proceeds by direct enumeration followed by pattern-matching assignments for dim, gauge_rank, and matter_dim.
claimThe spectral sectors of the binary 3-cube are the four types color (dimension 3, corresponding to SU(3)), weak (dimension 2, SU(2)), hypercharge (dimension 1, U(1)), and conjugate (dimension 2), with total dimension summing to 8 = 2^3.
background
In the Spectral Emergence module the binary cube is introduced via the sibling definition V(D) := 2^D, so that D=3 yields exactly eight vertices. The module document states that T8 forces Q3 whose symmetry B3 = S3 ⋉ (Z/2Z)^3 decomposes into the listed sectors, producing the gauge content SU(3) × SU(2) × U(1) together with a residual conjugate sector. Upstream results supply the Dimension structure (exponents for length, time, mass) and the native-unit gauge U with c=1, tau0=1 tick.
proof idea
The declaration is an inductive definition introducing the four constructors color, weak, hypercharge, conjugate. The three auxiliary functions dim, gauge_rank, and matter_dim are each defined by exhaustive pattern matching that assigns the fixed natural-number values listed in the doc-comment.
why it matters in Recognition Science
This definition supplies the sector decomposition on which downstream results such as sector_dim_sum, gauge_generators_eq_edges, and fermion_flavors rest; those theorems establish that gauge generators equal the edge count of Q3 and that chiral fermion flavors total 24. It fills the explicit step in the module's self-consistency loop from T8 (D=3) through the eight-tick octave to the Standard Model gauge group and the phi-ladder mass hierarchy. The framework landmark is the equality |Aut(Q3)| = 48 matching the chiral fermion state count.
scope and limits
- Does not derive the B3 decomposition from the cube geometry.
- Does not assign specific particle fields or representations.
- Does not compute numerical values of couplings or masses.
- Does not address the case D ≠ 3.
formal statement (Lean)
131inductive SpectralSector where
132 | color : SpectralSector
133 | weak : SpectralSector
134 | hypercharge : SpectralSector
135 | conjugate : SpectralSector
136 deriving DecidableEq, Repr
137
138/-- Dimension of each spectral sector. -/
139def SpectralSector.dim : SpectralSector → ℕ
140 | .color => 3
141 | .weak => 2
142 | .hypercharge => 1
143 | .conjugate => 2
144
145/-- The gauge group rank (dimension of the Lie algebra generators)
146 associated with each sector. -/
147def SpectralSector.gauge_rank : SpectralSector → ℕ
148 | .color => 8 -- SU(3): 3²-1 = 8 generators
149 | .weak => 3 -- SU(2): 2²-1 = 3 generators
150 | .hypercharge => 1 -- U(1): 1 generator
151 | .conjugate => 0 -- Not a gauge sector
152
153/-- The matter representation dimension (how many components
154 a fermion field has in this sector). -/
155def SpectralSector.matter_dim : SpectralSector → ℕ
156 | .color => 3
157 | .weak => 2
158 | .hypercharge => 1
159 | .conjugate => 2
160
161/-! ### Sector Dimension Theorems -/
162
163/-- **THEOREM**: Sector dimensions sum to 8 = |V(Q₃)|.
164 The spectral decomposition accounts for every vertex. -/