casimir_ratio_undefined
plain-language theorem explainer
The Casimir eigenvalue for the spin-zero representation of Q3 vanishes identically. Recognition Science model builders working on electroweak symmetry breaking cite this to establish that the Higgs-to-W mass ratio is fixed by potential curvature rather than representation eigenvalues. The proof is a direct one-line application of the spin-zero Casimir result.
Claim. For the spin-0 sector, the Casimir eigenvalue satisfies $C_2 = j(j+1) = 0$ when $j=0$.
background
The module treats the quaternion group Q3 as the symmetry of the eight-tick cycle under electroweak breaking SU(2)×U(1)→U(1). The four complex degrees of freedom of the Higgs doublet split into three spin-1 Goldstone modes and one spin-0 physical Higgs. The Casimir operator is defined by the standard formula $C_2(j) = j(j+1)$ for integer $j$. Upstream results supply the general definition and the explicit spin-zero case obtained by direct substitution.
proof idea
One-line wrapper that applies the spin-zero Casimir theorem, which itself reduces by simplification to the general definition $C_2(j) = j(j+1)$.
why it matters
The result shows that the mass ratio cannot be read off a Casimir ratio because the spin-0 eigenvalue is zero; the ratio instead follows from the forced quartic coupling λ = J''(1)/2 = 1/2 on the phi-ladder. It supports the mass formula yardstick · phi^(rung-8+gap) and the eight-tick octave structure of Q3. No downstream theorems are recorded yet.
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