casimir_ratio_undefined
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.
claimFor 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 in Recognition Science
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.
scope and limits
- Does not derive the numerical Higgs-to-W mass ratio.
- Does not compute the quartic coupling from the J-cost potential.
- Does not incorporate electroweak mixing or the Weinberg angle.
- Does not treat higher-spin or non-Q3 representations.
formal statement (Lean)
98theorem casimir_ratio_undefined : casimir 0 = 0 := spin0_casimir
proof body
Term-mode proof.
99
100/-! ## The Correct Mass Ratio Derivation -/
101
102/-- The φ-forced quartic coupling: λ = J″(1)/2 = 1/2.
103 J(x) = ½(x + x⁻¹) - 1 → J″(1) = 1 → λ_RS = J″(1)/2 = 1/2. -/