Periodic counts under two equal torus twists in the hidden (1, 5, 4) basis

Integer arithmetic in the basis (1, 5, 4) of hidden-sector multiplicities (a, b, c): dimension a + 5b + 4c, central action a + 5b − 4c, periodic count a + 5b. The central signs, the two equal torus twists and the Fourier interpretation are inputs; nothing is stated about the full orbifold or the Standard Model.

Main results

• projected_character: 2 · periodic = dim + center.
• two_equal_twists: 4 · periodic = 2 dim + 2 center.
• candidate_periodic_counts: the periodic counts 5 and 13 of the two candidate contents.
• periodic_resolves_integer_kernel: dimension, second moment and periodic count together determine the multiplicities (augmented determinant −16).

def GaugeHiggsUnification.TorusTwistBasis.dim (a b c : ℤ) : ℤ

The dimension of the content (a, b, c).

Equations

• GaugeHiggsUnification.TorusTwistBasis.dim a b c = a + 5 * b + 4 * c

def GaugeHiggsUnification.TorusTwistBasis.center (a b c : ℤ) : ℤ

The central action of the content (a, b, c).

Equations

• GaugeHiggsUnification.TorusTwistBasis.center a b c = a + 5 * b - 4 * c

def GaugeHiggsUnification.TorusTwistBasis.periodic (a b : ℤ) : ℤ

The periodic count of the content (a, b, c).

Equations

• GaugeHiggsUnification.TorusTwistBasis.periodic a b = a + 5 * b

theorem GaugeHiggsUnification.TorusTwistBasis.projected_character (a b c : ℤ) : 2 * periodic a b = dim a b c + center a b c

The projected character: twice the periodic count is dimension plus central action.

theorem GaugeHiggsUnification.TorusTwistBasis.two_equal_twists (a b c : ℤ) : 4 * periodic a b = 2 * dim a b c + 2 * center a b c

The same with two equal twists.

theorem GaugeHiggsUnification.TorusTwistBasis.candidate_periodic_counts : periodic 0 1 = 5 ∧ periodic 3 2 = 13

The periodic counts of the two candidate contents.

theorem GaugeHiggsUnification.TorusTwistBasis.augmented_determinant : 1 * (4 * 0 - 2 * 5) - 5 * (0 * 0 - 2 * 1) + 4 * (0 * 5 - 4 * 1) = -16

The determinant of the system (dimension, second moment, periodic count).

theorem GaugeHiggsUnification.TorusTwistBasis.periodic_resolves_integer_kernel (a b c : ℤ) (hd : a + 5 * b + 4 * c = 0) (h2 : 4 * b + 2 * c = 0) (hp : a + 5 * b = 0) : a = 0 ∧ b = 0 ∧ c = 0

Dimension, second moment and periodic count have trivial common integer kernel.