The operator and its multiplicity

For the untwisted complex spinor Dirac operator on a round sphere SdS^d of radius RR, d≥2d\ge2, the eigenvalues and multiplicities for each sign are

λk±=±k+d/2R,md(k)=2⌊d/2⌋(k+d−1d−1),k=0,1,…. \lambda_k^\pm=\pm\frac{k+d/2}{R},\qquad m_d(k)=2^{\lfloor d/2\rfloor}\binom{k+d-1}{d-1}, \qquad k=0,1,\ldots.

The unit-radius result is Bär, The Dirac operator on space forms of positive curvature, Theorem 1; rescaling the metric supplies the displayed factor 1/R1/R.

Summing the positive branch through level k=n−1k=n-1 gives

N+(n)=∑k=0n−1md(k)=2⌊d/2⌋S(n,d). N_+(n)=\sum_{k=0}^{n-1}m_d(k) =2^{\lfloor d/2\rfloor}S(n,d).

This is a cumulative eigenstate count with multiplicity, not an individual Dirac eigenvalue. Including both signs doubles this count.

Low-dimensional examples

Sphere Spinor factor Positive cumulative count
S3S^3 22 2S(n,3)2S(n,3)
S5S^5 44 4S(n,5)4S(n,5)

At the V2 neutrino indices 10,15,2210,15,22, the round S5S^5 operator gives 8008,46512,2631208008,46512,263120 positive-branch states. Restricting to fewer components would need a specified invariant projection and its own spectrum.

At radius one on S3S^3, the eigenvalue 7/27/2 belongs to k=2k=2 and has multiplicity 1212. Multiplicity 2020 belongs to k=3k=3, whose eigenvalue is 9/29/2. The coincidence m3(3)=S(4,3)=20m_3(3)=S(4,3)=20 does not equate a shell count with a cumulative count.

Use in V2

These formulas provide a checked benchmark for the candidate sphere operators. The spinor factor counts field components; it introduces no additional physical spatial directions.

They do not select a sphere as the actual sector geometry, construct its nesting, or derive the law m=μS(n,d)m=\mu S(n,d). The mass law and the individual sphere Dirac eigenvalues remain different quantities. Existing particle depths and the neutrino mass ratios use a separately specified contact mass operator.

The mode counting checks verify the cumulative sums and the explicit S3S^3 and S5S^5 examples. They check applications of the cited spectrum, not a numerical construction of the sphere Dirac operator.