spin1_count
plain-language theorem explainer
The spin-1 sector of the quaternion group Q3 contains exactly six elements. Particle physicists modeling electroweak symmetry breaking in Recognition Science would cite this count when assigning gauge bosons to quaternion representations. The proof is a direct decision procedure applied to the explicit list definition of the sector.
Claim. The spin-1 sector of the quaternion group $Q_3$, defined as the set of six elements $Q_3 = {±i, ±j, ±k}$, has cardinality 6.
background
The quaternion group $Q_3$ has eight elements and appears in Recognition Science as the symmetry group of the eight-tick cycle. The spin-1 sector is the explicit subset {±i, ±j, ±k} that corresponds to the gauge bosons under the electroweak breaking pattern SU(2)×U(1) → U(1). The module uses this partition to separate the three longitudinal modes from the physical Higgs and to relate their mass ratios to the Casimir eigenvalues of the two sectors.
proof idea
The proof is a one-line term that invokes the decide tactic on the explicit list definition of the spin-1 sector.
why it matters
This enumeration anchors the assignment of three degrees of freedom to the longitudinal polarizations of W± and Z within the eight-tick structure. It supports the subsequent derivation of the Higgs-to-W mass ratio from the J-cost curvature and the rung offset on the phi-ladder. The result fills the representation count needed for the electroweak sector in the Recognition Science framework.
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