Function normalize_tau 

pub fn normalize_tau<S: StatisticsType>(
    tau: f64,
    beta: f64,
) -> Result<(f64, f64), Error>

Normalize τ to the range [0, β] with statistics-dependent boundary conditions

Handles boundary conditions based on statistics:

• Fermions: Anti-periodic G(τ + β) = -G(τ)
• Bosons: Periodic G(τ + β) = G(τ)

Type Parameters

• S - Statistics type (Fermionic or Bosonic)

Arguments

• tau - Imaginary time in range [-β, β]
• beta - Inverse temperature

Returns

• (tau_normalized, sign) - Normalized τ ∈ [0, β] and sign factor

Errors

• Error::InvalidParameter if beta is not positive and finite
• Error::OutOfDomain if tau is outside [-β, β] or NaN

Boundary Interpretation

• β is interpreted as β- (left limit at β): tau == beta stays in normal range
• -β is read as (-β)⁺: it wraps to 0, with a sign flip for fermions
• -0.0 (negative zero) is treated as being in the odd period for fermions

Special Cases

For Fermionic statistics:

• tau = -0.0 (negative zero) → (tau_normalized = β, sign = -1.0)
• tau = 0.0 (positive zero) → (tau_normalized = 0.0, sign = 1.0)
• tau ∈ [-β, 0) → wraps to [0, β] with sign = -1.0

For Bosonic statistics:

• tau = -0.0 (negative zero) → (tau_normalized = β, sign = 1.0) (periodic)
• tau = 0.0 (positive zero) → (tau_normalized = 0.0, sign = 1.0)
• tau ∈ [-β, 0) → wraps to [0, β] with sign = 1.0

Examples

use sparse_ir::taufuncs::normalize_tau;
use sparse_ir::traits::{Bosonic, Fermionic};

// Normal negative value
let (tau_norm, sign) = normalize_tau::<Fermionic>(-0.3, 1.0).unwrap();
assert!((tau_norm - 0.7).abs() < 1e-14);
assert_eq!(sign, -1.0);

// Negative zero
let (tau_norm, sign) = normalize_tau::<Fermionic>(-0.0, 1.0).unwrap();
assert!((tau_norm - 1.0).abs() < 1e-14);
assert_eq!(sign, -1.0);

// -β wraps to 0 (with a sign flip for fermions only)
assert_eq!(normalize_tau::<Fermionic>(-1.0, 1.0).unwrap(), (0.0, -1.0));
assert_eq!(normalize_tau::<Bosonic>(-1.0, 1.0).unwrap(), (0.0, 1.0));

// τ outside [-β, β] is an error
assert!(normalize_tau::<Fermionic>(1.5, 1.0).is_err());