casimir
Casimir supplies the quadratic Casimir eigenvalue C₂ = j(j+1) for integer spin label j inside the Q₃ representation theory of the electroweak sector. Researchers tracing the Higgs-to-W mass ratio through the phi-ladder and J-cost potential cite this when separating the spin-0 singlet from the spin-1 triplet. The definition is a direct transcription of the standard angular-momentum formula into the reals with no further computation.
claimThe Casimir eigenvalue for the spin-j representation is $C_2(j) = j(j+1)$ for $j$ a natural number.
background
The Q₃ module identifies the quaternion group with the symmetry of the eight-tick cycle that forms the fundamental period of the recognition operator. Under electroweak breaking the Higgs doublet splits into a spin-1 triplet (longitudinal W±, Z) and a spin-0 singlet (physical Higgs); the module states that the mass ratio is fixed by potential curvature rather than by any Casimir ratio. The upstream Spin structure from SpinStatistics encodes the quantum number as twice an integer to keep arithmetic integral. The casimir definition supplies the eigenvalue that vanishes for the spin-0 sector and equals 2 for the spin-1 sector.
proof idea
The declaration is a direct definition that transcribes the classical quadratic Casimir formula. No lemmas are applied; the body casts the natural-number argument to reals and performs the multiplication.
why it matters in Recognition Science
This definition is the common source for the downstream theorems spin0_casimir and spin1_casimir that establish C₂(0) = 0 and C₂(1) = 2. Those theorems support the claim that the Higgs-to-W mass ratio arises from the curvature of the J-cost potential (J″(1) = 1) rather than from Casimir eigenvalues. The construction sits inside the eight-tick octave because Q₃ is the symmetry group of that cycle. It leaves open the precise numerical mapping from rung offset to observed masses once the potential curvature is inserted.
scope and limits
- Does not extend the formula to half-integer spins.
- Does not derive the quartic coupling from the second derivative of J.
- Does not compute any physical mass value.
- Does not incorporate the Weinberg angle or full electroweak mixing.
formal statement (Lean)
88noncomputable def casimir (j : ℕ) : ℝ := (j : ℝ) * ((j : ℝ) + 1)
proof body
Definition body.
89
90/-- Spin-0 Casimir eigenvalue: j=0, C₂ = 0. -/