SpectralSector
plain-language theorem explainer
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.
Claim. The 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
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.
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