q3_order
plain-language theorem explainer
The quaternion group Q₃ partitions into a two-element spin-0 sector and a six-element spin-1 sector whose lengths sum to eight. Particle physicists modeling electroweak symmetry breaking inside Recognition Science cite this count when assigning Casimir eigenvalues to the Higgs and gauge bosons. The proof is a one-line decision procedure that evaluates the explicit list definitions of the two sectors.
Claim. $|S_0| + |S_1| = 8$, where $S_0 = [+1, -1]$ is the spin-0 sector and $S_1 = [±i, ±j, ±k]$ is the spin-1 sector.
background
Recognition Science identifies the quaternion group Q₃ = {±1, ±i, ±j, ±k} with the symmetry of the eight-tick cycle that governs the recognition operator. The module splits Q₃ into a spin-0 sector {+1, -1} whose elements generate the Higgs doublet and a spin-1 sector {±i, ±j, ±k} whose elements become the longitudinal modes of W± and Z after SU(2)×U(1) → U(1) breaking. The upstream structure from UniversalForcingSelfReference supplies the meta-realization axioms that justify treating Q₃ as the fundamental period of R̂.
proof idea
The proof is a one-line wrapper that applies the decide tactic to the explicit list definitions of Spin0Sector and Spin1Sector, confirming that their lengths sum to eight.
why it matters
This cardinality statement closes the counting step required for the Casimir-eigenvalue calculation that determines the Higgs-to-W mass ratio in the Recognition Science electroweak sector. It sits inside the eight-tick octave (T7) of the forcing chain and supplies the numerical input for the spin-0 versus spin-1 offset used in the phi-ladder mass formula. No downstream theorems are recorded yet, but the result is a prerequisite for any derivation of m_H / m_W from J-cost curvature.
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