Skip to main content Crate deep_causality_physics Copy item path Source pub use crate::theories ::*;beam chronometric_quantities diffraction grmhd ideal moire phase plasma polarization qgt quark_masses Light quark constituent masses for string fragmentation. ray resistive solve_gm theories Gauge Theories: Particle Physics + Gravity Implementations wrapper AbcdMatrix ABCD Matrix. $2 \times 2$ Ray Transfer Matrix. Acceleration Linear acceleration (m/s^2). AccelerationVector Acceleration vector (m/s²). Return type of momentum-equation RHS evaluators. Activity Radioactivity (Becquerels). AlfvenSpeed Alfven Speed ($v_A$). Characteristic speed of magnetic waves in plasma.
Unit: m/s. Constraint: >= 0. AmountOfSubstance Amount of Substance (Moles). Area Area (m^2). BandDrudeWeight Band Drude Weight ($D$). BeamWaist Beam Waist ($w_0$). Minimum radius of Gaussian beam.
Unit: Meters. Constraint: > 0. BerryCurvature Berry Curvature component ($Ω_{ij}$). BodyForceDensity Body force per unit volume (N/m³). CauchyStress Cauchy stress tensor (Pa). Symmetric, positive-in-tension sign convention. CentralBody Parameters describing a central gravitating body for weak-field GM recovery. ChemicalPotentialGradient Chemical Potential Gradient $\nabla μ$. ComplexBeamParameter Complex Beam Parameter ($q(z) = z + i z_R$).
Constraint: $\text{Im}(q) > 0$. Concentration Concentration field $c(\mathbf{r})$. Conductance Electrical Conductance ($G$). Conductivity Electrical Conductivity ($\sigma$).
Unit: Siemens/m (S/m). Constraint: > 0. DebyeLength Debye Length ($\lambda_D$). Screening length in plasma.
Unit: Meters (m). Constraint: > 0. Density Density (kg/m^3). Diffusivity Magnetic Diffusivity ($\eta$).
Unit: $m^2/s$. Constraint: >= 0. Displacement Displacement field $\mathbf{u}(\mathbf{r})$. EastCoastMetric A Lorentzian metric in East Coast (-+++) convention. Efficiency Thermodynamic efficiency (0.0 to 1.0). ElectricPotential Electric Potential (Volts or J/C). Energy Energy (Joules or eV). EnergyDensity Energy Density (Joules per cubic meter). Entropy Entropy (J/K). FocalLength Focal Length ($f$).
Unit: Meters. Constraint: None (can be negative for diverging lens). Force Force (N). FourMomentum Lorentz 4-momentum (E, px, py, pz) in natural units (c = 1). Frequency Frequency (Hz). Hadron A produced hadron from string fragmentation. HalfLife Half-Life (Seconds). IndexOfRefraction Index of refraction for a medium (ratio of c to phase velocity). JonesVector Jones Vector. Polarized Electric Field. Rank 1, Dim 2 Complex Tensor. KinematicViscosity Kinematic Viscosity (m^2/s). Equals dynamic viscosity divided by density. LarmorRadius Larmor Radius ($r_L$). Gyroradius of a charged particle.
Unit: Meters (m). Constraint: > 0. Length Length (m). LundParameters Configuration parameters for Lund String Fragmentation. MagneticFlux Magnetic Flux (Webers). MagneticPressure Magnetic Pressure ($P_B$). Energy density of the magnetic field.
Unit: Pascals (Pa). Constraint: >= 0. Mass Mass quantity (kg). MaxwellSolver A standard solver for Maxwell’s Equations in Geometric Algebra. Mobility Charge Carrier Mobility ($μ$). MomentOfInertia Moment of Inertia (kg·m^2). Momentum Momentum vector $\mathbf{k}$. NumericalAperture Numerical Aperture ($NA = n \sin \theta$).
Unit: Dimensionless. Constraint: > 0. OpticalPower Optical Power ($D = 1/f$).
Unit: Diopters ($m^{-1}$). Constraint: None. OrbitalAngularMomentum Orbital Angular Momentum ($L$). OrderParameter Superconducting Order Parameter ($ψ$). ParticleData Particle data from the Particle Data Group. PhaseAngle Phase Angle (Radians). PhysicalField Wrapper for CausalMultiVector representing a physical field (E, B, etc.).
Implements Default to return a zero vector. PhysicalVector PhysicsError PlasmaBeta Plasma Beta ($\beta$). Ratio of thermal to magnetic pressure.
Unit: Dimensionless. Constraint: >= 0. PlasmaFrequency Plasma Frequency ($\omega_{pe}$). Natural oscillation frequency.
Unit: Rad/s. Constraint: > 0. Pressure Pressure (Pascals). Probability Probability value [0, 1]. QuantumEigenvector Quantum Eigenvector $|u_n➢$. QuantumMetric Quantum Metric component ($g_{ij}$). QuantumVelocity Quantum Velocity vector $\partial_i H |u_n➢$. Ratio A generic dimensionless ratio. RayAngle Ray Angle ($\theta$). Angle relative to optical axis.
Unit: Radians. RayHeight Ray Height ($y$). Distance from optical axis.
Unit: Meters. ReynoldsStress Reynolds stress tensor R_ij = ⟨u'_i u'_j⟩ (Pa, after multiplication by ρ
in caller; here a kinematic Reynolds stress in m²/s² is also acceptable).
Symmetric. Diagonal entries are non-negative (variances) — not enforced
by the newtype to keep the constructor cheap; callers passing a tensor
that violates the diagonal-positivity property are responsible for
downstream interpretation. RotationRateTensor Rate-of-rotation (spin) tensor Ω = 0.5·(∇u − ∇uᵀ). Antisymmetric:
Ω_ji = −Ω_ij, with Ω_ii = 0. new checks the antisymmetry invariant
by exact equality. SpaceTimeCoordinate Represents a point in 4D Space-Time with associated kinematic and clock data. SpacetimeInterval Spacetime Interval ($s^2$). SpacetimeVector Wrapper for CausalMultiVector representing a vector in Spacetime. SpecificEnthalpy Specific Enthalpy (J/kg). Reference-state dependent; may be negative. Speed Speed quantity (scalar magnitude of velocity) (m/s). Stiffness Scalar stiffness (Young’s Modulus, etc.) (Pascals). StiffnessTensor Stiffness tensor $C_{ijkl}$ (Rank 4) used in generalized Hooke’s law. StokesVector Stokes Vector. Intensity vector $(S_0, S_1, S_2, S_3)$. Rank 1, Dim 4 Tensor.
Constraint: $S_0^2 \ge S_1^2 + S_2^2 + S_3^2$. Strain Strain tensor field $\boldsymbol{\epsilon}$ (Rank 2). StrainRateTensor Strain-rate tensor S = 0.5·(∇u + ∇uᵀ). Symmetric: S_ij = S_ji.
new checks the symmetry invariant by exact equality, matching what natural
construction 0.5·(G + Gᵀ) produces in IEEE 754. Use new_unchecked to
bypass the check in hot kernels where symmetry is guaranteed by the algebra. Stress Scalar stress (Pascals), used for simple 1D cases or invariants (Von Mises). StressTensor Cauchy stress tensor $\boldsymbol{\sigma}$ (Rank 2). Temperature Absolute Temperature (Kelvin). Time Time (Seconds). Torque Torque (N·m). TwistAngle Moiré Twist Angle ($θ$). VectorPotential Electromagnetic Vector Potential $\mathbf{A}$. Velocity3 Fluid velocity vector (m/s). VelocityGradient Velocity gradient tensor ∇u. Pinned to the Jacobian convention:
value[i][j] = ∂u_i / ∂x_j. Construction-time check is finiteness only —
any finite 3×3 matrix is a valid velocity gradient. Viscosity Dynamic Viscosity (Pa·s). ViscousStress Viscous (deviatoric) stress tensor τ (Pa). Symmetric. Distinct from the
full Cauchy stress σ = −p I + τ — only the viscous part appears in the
dissipation Φ = τ:∇u ≥ 0 and entropy-production guarantees. Volume Volume (m^3). VorticityVector Vorticity vector ω = ∇ × u (1/s). Pseudovector under spatial reflection. WallShearStress Wall Shear Stress magnitude (Pa). Stored as magnitude; sign convention is
carried by the calling context, not by this type. Wavelength Wavelength ($\lambda$).
Unit: Meters. Constraint: > 0. WestCoastMetric A Lorentzian metric in West Coast (+—) convention. Metric Defines the metric signature of the Clifford Algebra Cl(p, q, r). MetricError Errors that can occur during metric operations. PhysicsErrorEnum Detailed classification of physics-related errors. ALPHA_EM Fine structure constant α = e²/(4π) ≈ 1/137 (low-energy limit) ALPHA_EM_MZ Running fine structure constant at Z pole: α(M_Z) ≈ 1/128 ATOMIC_MASS_CONSTANT AVOGADRO_CONSTANT BOHR_MAGNETON BOHR_RADIUS BOLTZMANN_CONSTANT CLASSICAL_ELECTRON_RADIUS COSMOLOGICAL_CONSTANT Cosmological constant upper bound (in m⁻²) EARTH_ANGULAR_MOMENTUM EARTH_GM Gravity Mass Constant EARTH_GRAVITY_ACCELERATION Gravitational acceleration EARTH_J2 Earth’s J2 Oblateness coefficient (JGM-3)
https://www.sciencedirect.com/topics/engineering/oblateness EARTH_MASS_KG EARTH_RADIUS EARTH_RADIUS_EQUATORIAL Earth’s Equatorial Radius (WGS-84) EARTH_ROTATION_RATE ELECTRON_CHARGE_TO_MASS_QUOTIENT ELECTRON_COMPTON_WAVELENGTH ELECTRON_MAGNETIC_MOMENT ELECTRON_MASS ELECTRON_MUON_MASS_RATIO ELECTRON_PROTON_MASS_RATIO ELEMENTARY_CHARGE EM_COUPLING Electromagnetic coupling constant e = √(4πα) ≈ 0.303 (low-energy) EM_COUPLING_MZ Running EM coupling at Z pole e(M_Z) = √(4π α(M_Z)) ≈ 0.313 FARADAY_CONSTANT FERMI_CONSTANT Fermi coupling constant G_F in GeV⁻² FINE_STRUCTURE_CONSTANT FIRST_RADIATION_CONSTANT G GEV2_TO_NB Unit conversion factor (ℏc)² to convert GeV⁻² to nanobarns (nb) GEV2_TO_PB Unit conversion factor (ℏc)² to convert GeV⁻² to picobarns (pb) GRAPHENE_LATTICE_CONST HARTREE_ENERGY HIGGS_MASS Higgs boson mass M_H in GeV HIGGS_VEV Higgs vacuum expectation value v = (√2 G_F)^(-1/2) ≈ 246 GeV JOSEPHSON_CONSTANT MAGNETIC_FLUX_QUANTUM MINKOWSKI_4D Default 4D Minkowski spacetime (East Coast convention). MOLAR_GAS_CONSTANT MUON_ELECTRON_MASS_RATIO MUON_MAGNETIC_MOMENT NEUTRON_MAGNETIC_MOMENT NEWTONIAN_CONSTANT_OF_GRAVITATION NUCLEAR_MAGNETON PARTICLE_MINKOWSKI_4D Standard 4D Minkowski spacetime for Particle Physics. PDG_PARTICLES Common hadron database following PDG conventions. PLANCK_CONSTANT PLANCK_LENGTH Planck length in meters PLANCK_MASS Planck mass in kg PROTON_ELECTRON_MASS_RATIO PROTON_MAGNETIC_MOMENT PROTON_MASS REDUCED_PLANCK_CONSTANT RELATIVITY_MINKOWSKI_4D Standard 4D Minkowski spacetime for General Relativity. RYDBERG_CONSTANT SECOND_RADIATION_CONSTANT SIN2_THETA_W Weak mixing angle (sin²θ_W) SPEED_OF_LIGHT STEFAN_BOLTZMANN_CONSTANT THOMSON_CROSS_SECTION TOP_MASS Top quark mass m_t in GeV (heaviest Standard Model fermion) VACUUM_ELECTRIC_PERMITTIVITY VACUUM_MAGNETIC_PERMEABILITY VON_KLITZING_CONSTANT WIEN_DISPLACEMENT_LAW_CONSTANT W_MASS W boson mass in GeV Z_MASS Z boson mass in GeV Z_PARTIAL_WIDTH_EE Z boson partial width to electrons Γ_ee in GeV (PDG 2024) Z_PARTIAL_WIDTH_HAD Z boson hadronic partial width Γ_had in GeV (PDG 2024) Z_WIDTH Z boson total width Γ_Z in GeV (PDG 2024) LorentzianMetric Trait for convention-specific Lorentzian metric wrappers. QuantumGates Standard Quantum Gates interface. QuantumOps Core Quantum State Operations (Dirac Notation), parameterized over the
underlying real field $R$ so the operations can be carried out at f32,
f64, f128, or any other real-field precision. alfven_speed alfven_speed_kernel Calculates the characteristic speed of Alfven waves.
$$ v_A = \frac{B}{\sqrt{\mu_0 \rho}} $$ all_structure_constants Returns all non-zero structure constants as a list of (a, b, c, f^abc). angular_momentum Causal wrapper for kinematics::angular_momentum_kernel angular_momentum_kernel Calculates angular momentum as the outer product of radius and linear momentum: $L = r \wedge p$. apply_gate Causal wrapper for mechanics::apply_gate_kernel apply_gate_kernel Applies a quantum gate to a state: $|\psi’\rangle = U |\psi\rangle$. beam_spot_size beam_spot_size_kernel Extracts the beam spot size $w(z)$ from the complex beam parameter $q$. bernoulli_pressure Causal wrapper for mechanics::bernoulli_pressure_kernel bernoulli_pressure_kernel Calculates pressure $P_2$ using Bernoulli’s principle. bernoulli_total_head Causal wrapper for ideal_flow::bernoulli_total_head_kernel bernoulli_total_head_kernel Bernoulli total head H = p / (ρ · g) + u² / (2 · g) + h (m). binding_energy Causal wrapper for physics::binding_energy_kernel binding_energy_kernel Calculates nuclear binding energy (mass defect): $E = \Delta m c^2$. bistritzer_macdonald Wrapper for moire::bistritzer_macdonald_kernel bistritzer_macdonald_kernel Constructs the Bistritzer-MacDonald Continuum Hamiltonian for Twisted Bilayer Graphene (TBG). boltzmann_factor Causal wrapper for stats::boltzmann_factor_kernel boltzmann_factor_kernel Calculates the unnormalized Boltzmann factor: $e^{-E/k_BT}$. bond_number Causal wrapper for dimensionless::bond_number_kernel bond_number_kernel Bond (Eötvös) number Bo = ρ · g · L² / σ. Gravity to surface tension. born_probability Causal wrapper for mechanics::born_probability_kernel born_probability_kernel Calculates the Born probability: $P = |\langle \text{basis} | \text{state} \rangle|^2$. cahn_hilliard_flux Wrapper for phase::cahn_hilliard_flux_kernel cahn_hilliard_flux_kernel Calculates the Cahn-Hilliard Flux with degenerate mobility (Type B). capillary_number Causal wrapper for dimensionless::capillary_number_kernel capillary_number_kernel Capillary number Ca = μ · u / σ. Viscous to surface-tension forces. carnot_efficiency Causal wrapper for stats::carnot_efficiency_kernel carnot_efficiency_kernel Calculates Carnot Efficiency: $\eta = 1 - \frac{T_C}{T_H}$. chronometric_volume Causal wrapper for spacetime::chronometric_volume_kernel chronometric_volume_kernel Calculates Chronometric Volume (4-Volume) from 3 vectors? circulation Causal wrapper for ideal_flow::circulation_kernel circulation_kernel Circulation Γ = ∮ u · dl as a discrete line integral. commutator Causal wrapper for mechanics::commutator_kernel commutator_kernel Calculates commutator $[A, B] = AB - BA$. compressible_ns_continuity_rhs Causal wrapper for compressible_ns::compressible_ns_continuity_rhs_kernel compressible_ns_energy_rhs Causal wrapper for compressible_ns::compressible_ns_energy_rhs_kernel compressible_ns_momentum_rhs Causal wrapper for compressible_ns::compressible_ns_momentum_rhs_kernel confinement_potential_kernel Computes the linear confining potential: $V(r) = \sigma r + V_0$. continuity_rhs Causal wrapper for governing::continuity_rhs_kernel continuity_rhs_kernel Continuity equation RHS: ∂ρ/∂t = −∇·(ρu) = −(u·∇ρ + ρ · ∇·u). convective_acceleration Causal wrapper for governing::convective_acceleration_kernel convective_acceleration_kernel Convective acceleration (u·∇)u. covariant_derivative_kernel Computes the gauge covariant derivative: $D_\mu \psi = \partial_\mu \psi + i g A_\mu \psi$. debye_length debye_length_kernel Calculates the Debye Length $\lambda_D$.
$$ \lambda_D = \sqrt{\frac{\epsilon_0 k_B T_e}{n_e e^2}} $$ degree_of_polarization degree_of_polarization_kernel Calculates the Degree of Polarization (DOP) from a Stokes vector. delta_criterion Causal wrapper for coherent_structures::delta_criterion_kernel delta_criterion_kernel Δ-criterion (Chong, Perry & Cantwell 1990; generalized form): the
discriminant of the depressed velocity-gradient characteristic polynomial
Δ = (p̃/3)³ + (q̃/2)² where
p̃ = Q − P²/3, q̃ = 2P³/27 − PQ/3 + R,
and (P, Q, R) are the velocity-gradient invariants in the CPC convention. dissipation_rate Causal wrapper for turbulence::dissipation_rate_kernel dissipation_rate_kernel Dissipation rate ε = 2 ν · S':S' = 2 ν · Σ_{i,j} S'_{ij}². doppler_effect_approaching Monadic wrapper for general::doppler_effect_kernel doppler_effect_kernel Calculates the observed frequency due to the Doppler effect for longitudinal motion. dynamic_pressure Causal wrapper for ideal_flow::dynamic_pressure_kernel dynamic_pressure_kernel Dynamic pressure q = 0.5 · ρ · u² (Pa). eckert_number Causal wrapper for dimensionless::eckert_number_kernel eckert_number_kernel Eckert number Ec = u² / (c_p · ΔT). Kinetic energy to enthalpy difference. eddy_viscosity_boussinesq Causal wrapper for turbulence::eddy_viscosity_boussinesq_kernel eddy_viscosity_boussinesq_kernel Boussinesq-closure eddy viscosity ν_t. effective_band_drude_weight Wrapper for qgt::effective_band_drude_weight_kernel effective_band_drude_weight_kernel Approximates the Effective Band Drude Weight ($D$) using band curvature and quantum geometry. einstein_tensor Causal wrapper for gravity::einstein_tensor_kernel einstein_tensor_kernel Calculates the Einstein Tensor: $G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu}$. energy_density_kernel Calculates the Electromagnetic Energy Density: $u = \frac{1}{2}(E^2 + B^2)$. energy_momentum_tensor_em energy_momentum_tensor_em_kernel Calculates the electromagnetic stress-energy tensor $T^{\mu\nu}{EM}$.
$$ T^{\mu\nu} = F^{\mu\alpha}F^\nu \alpha - \frac{1}{4} g^{\mu\nu} F_{\alpha\beta}F^{\alpha\beta} $$ enstrophy_density Causal wrapper for kinematics::enstrophy_density_kernel enstrophy_density_kernel Enstrophy density ξ = 0.5 · ‖ω‖². entropy_production_rate Causal wrapper for compressible::entropy_production_rate_kernel entropy_production_rate_kernel Local entropy-production rate density (W/(m³·K)) for a Newtonian fluid:
σ = Φ/T + κ · ‖∇T‖² / T²
where Φ = τ : ∇u is viscous dissipation and κ is thermal conductivity. escape_velocity Causal wrapper for mechanics::escape_velocity_kernel escape_velocity_kernel Calculates escape velocity: $v_e = \sqrt{\frac{2GM}{r}}$. euler_momentum_rhs Causal wrapper for euler::euler_momentum_rhs_kernel expectation_value Causal wrapper for mechanics::expectation_value_kernel expectation_value_kernel Calculates the expectation value: $\langle A \rangle = \langle \psi | A | \psi \rangle$. fidelity Causal wrapper for mechanics::fidelity_kernel fidelity_kernel Calculates Quantum Fidelity: $F = |\langle \psi_{\text{ideal}} | \psi_{\text{actual}} \rangle|^2$. foppl_von_karman_strain Wrapper for moire::foppl_von_karman_strain_kernel foppl_von_karman_strain_kernel Calculates the Föppl-von Kármán stress tensor using the complete theory (Field-based). foppl_von_karman_strain_simple Wrapper for moire::foppl_von_karman_strain_simple_kernel foppl_von_karman_strain_simple_kernel Calculates the Föppl-von Kármán stress tensor (Simplified / Local). friction_velocity Causal wrapper for boundary_layer::friction_velocity_kernel friction_velocity_kernel Friction velocity u_τ = √(τ_w / ρ) (m/s). froude_number Causal wrapper for dimensionless::froude_number_kernel froude_number_kernel Froude number Fr = u / √(g · L). Inertial to gravitational forces. gaussian_q_propagation gaussian_q_propagation_kernel Propagates a Gaussian beam’s complex $q$-parameter through an ABCD optical system. gell_mann_matrices The 8 Gell-Mann matrices λ^a (a = 1..8), generators of SU(3). generalized_master_equation Causal wrapper for estimation::generalized_master_equation_kernel generalized_master_equation_kernel Generalized Master Equation Kernel. generate_schwarzschild_metric Generates a Schwarzschild-like 4D metric tensor. geodesic_deviation Causal wrapper for gravity::geodesic_deviation_kernel geodesic_deviation_kernel Calculates Geodesic Deviation acceleration: $A^\mu = -R^\mu_{\nu\sigma\rho} V^\nu n^\sigma V^\rho$. geodesic_integrator_kernel Integrates the geodesic equation using 4th-order Runge-Kutta (RK4). ginzburg_landau_free_energy Wrapper for phase::ginzburg_landau_free_energy_kernel ginzburg_landau_free_energy_kernel Calculates the Ginzburg-Landau Free Energy density. grashof_number Causal wrapper for dimensionless::grashof_number_kernel grashof_number_kernel Grashof number Gr = g · β · ΔT · L³ / ν². Buoyancy to viscous forces. grating_equation grating_equation_kernel Calculates the diffraction angle for a Grating using the Grating Equation. haruna_cz_gate Causal wrapper for mechanics::haruna_cz_gate_kernel haruna_cz_gate_kernel Implements Haruna’s Logical CZ Gate. haruna_hadamard_gate Causal wrapper for mechanics::haruna_hadamard_gate_kernel haruna_hadamard_gate_kernel Implements Haruna’s Logical Hadamard Gate. haruna_s_gate Causal wrapper for mechanics::haruna_s_gate_kernel haruna_s_gate_kernel Implements Haruna’s Logical S-Gate. haruna_t_gate Causal wrapper for mechanics::haruna_t_gate_kernel haruna_t_gate_kernel Implements Haruna’s Logical T-Gate. haruna_x_gate Causal wrapper for mechanics::haruna_x_gate_kernel haruna_x_gate_kernel Implements Haruna’s Logical X-Gate. haruna_z_gate Causal wrapper for mechanics::haruna_z_gate_kernel haruna_z_gate_kernel Implements Haruna’s Logical Z-Gate. heat_capacity Causal wrapper for stats::heat_capacity_kernel heat_capacity_kernel Calculates Heat Capacity: $C = \frac{dE}{dT}$. heat_diffusion Causal wrapper for stats::heat_diffusion_kernel heat_diffusion_kernel Calculates the Heat Equation step: $\frac{\partial u}{\partial t} = \alpha \Delta u$. helicity_density Causal wrapper for kinematics::helicity_density_kernel helicity_density_kernel Helicity density h = u · ω. hookes_law Causal wrapper for mechanics::hookes_law_kernel hookes_law_kernel Calculates generalized Hooke’s Law: $\sigma_{ij} = C_{ijkl} \epsilon_{kl}$. hydrostatic_pressure Causal wrapper for mechanics::hydrostatic_pressure_kernel hydrostatic_pressure_kernel Calculates hydrostatic pressure: $P = P_0 + \rho g h$. ideal_gas_law Causal wrapper for stats::ideal_gas_law_kernelRatio (or scalar). ideal_gas_law_kernel Calculates the Ideal Gas Constant $R$ from state variables: $R = \frac{PV}{nT}$. ideal_induction ideal_induction_kernel Calculates the time evolution of the magnetic field (Frozen-in flux).
$$ \frac{\partial \mathbf{B}}{\partial t} = \nabla \times (\mathbf{v} \times \mathbf{B}) $$ incompressible_ns_rhs Causal wrapper for incompressible_ns::incompressible_ns_rhs_kernel integral_length_scale Causal wrapper for turbulence::integral_length_scale_kernel integral_length_scale_kernel Integral length scale L = k^(3/2) / ε (m). jones_rotation jones_rotation_kernel Rotates a Jones Matrix (operator) by an angle $\phi$. kalman_filter_linear Causal wrapper for estimation::kalman_filter_linear_kernel kalman_filter_linear_kernel Standard Linear Kalman Filter Update Step. kinetic_energy Causal wrapper for kinematics::kinetic_energy_kernel kinetic_energy_density Causal wrapper for governing::kinetic_energy_density_kernel kinetic_energy_density_kernel Kinetic energy density ρ · 0.5 · ‖u‖² (J/m³). kinetic_energy_kernel Calculates translational kinetic energy: $K = \frac{1}{2} m v^2$. klein_gordon Causal wrapper for mechanics::klein_gordon_kernel klein_gordon_kernel Calculates the Klein-Gordon operator action: $(\Delta + m^2)\psi$. knudsen_number Causal wrapper for dimensionless::knudsen_number_kernel knudsen_number_kernel Knudsen number Kn = λ / L. Mean free path to characteristic length. kolmogorov_length Causal wrapper for turbulence::kolmogorov_length_kernel kolmogorov_length_kernel Kolmogorov length scale η = (ν³ / ε)^(1/4) (m). kolmogorov_time Causal wrapper for turbulence::kolmogorov_time_kernel kolmogorov_time_kernel Kolmogorov time scale τ_η = (ν / ε)^(1/2) (s). kolmogorov_velocity Causal wrapper for turbulence::kolmogorov_velocity_kernel kolmogorov_velocity_kernel Kolmogorov velocity scale u_η = (ν · ε)^(1/4) (m/s). kutta_joukowski_lift Causal wrapper for ideal_flow::kutta_joukowski_lift_kernel kutta_joukowski_lift_kernel Kutta–Joukowski lift per unit span L' = ρ · u_∞ · Γ (N/m). lagrangian_density_kernel Calculates the Electromagnetic Lagrangian Density: $\mathcal{L} = -\frac{1}{4} F_{\mu\nu} F^{\mu\nu}$. lambda2 Causal wrapper for coherent_structures::lambda2_kernel lambda2_kernel λ₂-criterion (Jeong & Hussain 1995): the second-largest (middle)
eigenvalue of the symmetric tensor M = S² + Ω² where S and Ω are
the symmetric and antisymmetric parts of ∇u. larmor_radius larmor_radius_kernel Calculates the Larmor Radius (Gyroradius).
$$ r_L = \frac{m v_\perp}{|q| B} $$ lens_maker lens_maker_kernel Calculates optical power and focal length using the Lens Maker’s Equation. lewis_number Causal wrapper for dimensionless::lewis_number_kernel lewis_number_kernel Lewis number Le = α / D. Thermal to mass diffusivity. log_law_velocity Causal wrapper for boundary_layer::log_law_velocity_kernel log_law_velocity_kernel Logarithmic law of the wall u⁺ = (1/κ) · ln(y⁺) + B. logical_cz Calculates the Logical CZ gate: $CZ(\gamma_1, \gamma_2) = \exp(i \pi a(\gamma_1) a(\gamma_2))$. logical_hadamard Calculates the Logical Hadamard gate. logical_s Calculates the Logical S gate: $S(\gamma) = \exp(i \frac{\pi}{2} a(\gamma)^2)$. logical_t Calculates the Logical T gate. logical_x Calculates the Logical X gate: $X(\tilde{\gamma}) = \exp(i \pi b(\tilde{\gamma}))$. logical_z Calculates the Logical Z gate: $Z(\gamma) = \exp(i \pi a(\gamma))$. lorentz_force Causal wrapper for forces::lorentz_force_kernel lorentz_force_kernel Calculates the magnetic force component of the Lorentz Force: $F = J \times B$. lorenz_gauge Causal wrapper for fields::lorenz_gauge_kernel lorenz_gauge_kernel Calculates the Lorenz gauge condition: $\nabla \cdot A = 0$. mach_number Causal wrapper for dimensionless::mach_number_kernel mach_number_kernel Mach number M = u / a. Flow speed in units of the local sound speed. magnetic_helicity_density Causal wrapper for fields::magnetic_helicity_density_kernel magnetic_helicity_density_kernel Calculates Magnetic Helicity Density: $h = A \cdot B$. magnetic_pressure magnetic_pressure_kernel Calculates magnetic pressure.
$$ P_B = \frac{B^2}{2\mu_0} $$ magnetic_reconnection_rate magnetic_reconnection_rate_kernel Estimates reconnection rate (Sweet-Parker model simplified).
$$ v_{in} = \frac{v_A}{\sqrt{S}} $$ maxwell_gradient Causal wrapper for fields::maxwell_gradient_kernel maxwell_gradient_kernel Calculates the Maxwell gradient (Electromagnetic Field Tensor). newtonian_viscous_stress Causal wrapper for constitutive::newtonian_viscous_stress_kernel newtonian_viscous_stress_kernel Newtonian viscous stress with Stokes hypothesis (bulk viscosity ζ = 0):
τ = 2μ S − (2/3) μ (∇·u) I. newtonian_viscous_stress_with_bulk Causal wrapper for constitutive::newtonian_viscous_stress_with_bulk_kernel newtonian_viscous_stress_with_bulk_kernel Newtonian viscous stress with bulk viscosity:
τ = 2μ S − (2/3) μ (∇·u) I + ζ (∇·u) I. nusselt_number Causal wrapper for dimensionless::nusselt_number_kernel nusselt_number_kernel Nusselt number Nu = h · L / k. Convective to conductive heat transfer. orbital_velocity Causal wrapper for mechanics::orbital_velocity_kernel orbital_velocity_kernel Calculates orbital velocity: $v = \sqrt{\frac{GM}{r}}$. parallel_transport_kernel Parallel transports a vector along a discrete path using Christoffel symbols. particle_stokes_number Causal wrapper for dimensionless::particle_stokes_number_kernel particle_stokes_number_kernel Particle Stokes number St = τ_p · u / L. partition_function Causal wrapper for stats::partition_function_kernel partition_function_kernel Calculates Partition Function: $Z = \sum e^{-E_i / k_B T}$. pdg_lookup Lookup particle data by PDG ID. pdg_mass Get particle mass by PDG ID (convenience function). peclet_number Causal wrapper for dimensionless::peclet_number_kernel peclet_number_kernel Peclet number (thermal) Pe = u · L / α. Convective to diffusive heat transport. power_law_apparent_viscosity Causal wrapper for constitutive::power_law_apparent_viscosity_kernel power_law_apparent_viscosity_kernel Power-law (Ostwald–de Waele) apparent viscosity: μ_eff = K · γ̇^(n−1). poynting_vector Causal wrapper for fields::poynting_vector_kernel poynting_vector_kernel Calculates the Poynting Vector (Energy Flux): $S = E \times B$. prandtl_number Causal wrapper for dimensionless::prandtl_number_kernel prandtl_number_kernel Prandtl number Pr = ν / α. Momentum diffusivity to thermal diffusivity. pressure_gradient_force Causal wrapper for governing::pressure_gradient_force_kernel pressure_gradient_force_kernel Pressure gradient force per unit mass −(1/ρ) · ∇p. pressure_work Causal wrapper for governing::pressure_work_kernel pressure_work_kernel Reversible pressure work p · ∇·u (W/m³). proca_equation Causal wrapper for fields::proca_equation_kernel proca_equation_kernel Calculates the Proca Equation (Massive Electromagnetism): $\delta F + m^2 A = J$. proper_time_kernel Computes the proper time along a discrete worldline. q_criterion Causal wrapper for coherent_structures::q_criterion_kernel q_criterion_kernel Q-criterion: Q = 0.5 · (‖Ω‖² − ‖S‖²) = −0.5 · tr(∇u · ∇u). quantum_geometric_tensor Wrapper for qgt::quantum_geometric_tensor_kernel quantum_geometric_tensor_kernel Calculates the Quantum Geometric Tensor (QGT) component $Q_{ij}^n(\mathbf{k})$ for band $n$. quasi_qgt Wrapper for qgt::quasi_qgt_kernel quasi_qgt_kernel Calculates the Quasi-QGT $q_{ij}^n(\mathbf{k})$. radioactive_decay Causal wrapper for physics::radioactive_decay_kernel radioactive_decay_kernel Calculates the remaining amount of a radioactive substance: $N(t) = N_0 \cdot 2^{-t / t_{1/2}}$. ray_transfer ray_transfer_kernel Applies an ABCD matrix to a ray vector. rayleigh_number Causal wrapper for dimensionless::rayleigh_number_kernel rayleigh_number_kernel Rayleigh number Ra = g · β · ΔT · L³ / (ν · α). Drives free convection. relativistic_current Wrapper for relativistic current density calculation. relativistic_current_kernel Calculates relativistic current density J^μ via covariant divergence. resistive_diffusion resistive_diffusion_kernel Calculates the diffusion term of the induction equation.
$$ \frac{\partial \mathbf{B}}{\partial t}_{diff} = \eta \nabla^2 \mathbf{B} $$ reynolds_number Causal wrapper for dimensionless::reynolds_number_kernel reynolds_number_kernel Reynolds number Re = u · L / ν. Ratio of inertial to viscous forces. reynolds_stress Causal wrapper for turbulence::reynolds_stress_kernel reynolds_stress_kernel Reynolds-stress tensor R_ij = ⟨u'_i u'_j⟩ packaged as a ReynoldsStress<R> richardson_number Causal wrapper for dimensionless::richardson_number_kernel richardson_number_kernel Richardson number Ri = g · β · ΔT · L / u². Buoyancy to shear forces. rotation_rate_tensor Causal wrapper for kinematics::rotation_rate_tensor_kernel rotation_rate_tensor_kernel Rate-of-rotation (spin) tensor Ω = 0.5 · (∇u − ∇uᵀ). rotational_kinetic_energy Causal wrapper for kinematics::rotational_kinetic_energy_kernel rotational_kinetic_energy_kernel Calculates rotational kinetic energy: $K_{rot} = \frac{1}{2} I \omega^2$. running_coupling_kernel Computes the running QCD coupling constant $\alpha_s(Q^2)$ (asymptotic freedom). scalar_advection_diffusion Causal wrapper for governing::scalar_advection_diffusion_kernel scalar_advection_diffusion_kernel Generic scalar transport RHS: −u · ∇φ + D · ∇²φ + S. schmidt_number Causal wrapper for dimensionless::schmidt_number_kernel schmidt_number_kernel Schmidt number Sc = ν / D. Momentum diffusivity to mass diffusivity. schwarzschild_radius Causal wrapper for mechanics::schwarzschild_radius_kernel schwarzschild_radius_kernel Calculates Schwarzschild radius: $r_s = \frac{2GM}{c^2}$. shannon_entropy Causal wrapper for stats::shannon_entropy_kernel shannon_entropy_kernel Calculates Shannon Entropy: $H = -\sum p_i \ln(p_i)$. single_slit_irradiance single_slit_irradiance_kernel Calculates the Single Slit Diffraction Irradiance. skin_friction_coefficient Causal wrapper for boundary_layer::skin_friction_coefficient_kernel skin_friction_coefficient_kernel Skin-friction coefficient C_f = τ_w / (0.5 · ρ · u_∞²). snells_law snells_law_kernel Calculates refracted angle using Snell’s Law or returns a Critical Angle error. solve_gm_analytical Causal wrapper for solve_gm_analytical_kernel solve_gm_analytical_kernel Recovers GM within the weak-field 1PN limit from two space-time coordinates. spacetime_interval Causal wrapper for spacetime::spacetime_interval_kernel spacetime_interval_kernel Calculates the spacetime interval $s^2 = g_{\mu\nu} x^\mu x^\nu$ (squared magnitude in metric). specific_enthalpy Causal wrapper for compressible::specific_enthalpy_kernel specific_enthalpy_kernel Specific enthalpy h = c_p · T (J/kg) for an ideal gas with constant c_p. speed_of_sound_ideal_gas Causal wrapper for compressible::speed_of_sound_ideal_gas_kernel speed_of_sound_ideal_gas_kernel Ideal-gas speed of sound a = √(γ · R_s · T). stokes_from_jones stokes_from_jones_kernel Converts a pure state Jones vector to a Stokes vector. stokes_momentum_rhs Causal wrapper for stokes::stokes_momentum_rhs_kernel strain_rate_tensor Causal wrapper for kinematics::strain_rate_tensor_kernel strain_rate_tensor_kernel Strain-rate tensor S = 0.5 · (∇u + ∇uᵀ). stream_function_2d Causal wrapper for ideal_flow::stream_function_2d_kernel stream_function_2d_kernel 2D stream-function differential update dψ = u · dy − v · dx. strouhal_number Causal wrapper for dimensionless::strouhal_number_kernel strouhal_number_kernel Strouhal number Sr = f · L / u. Oscillation frequency to flow. structure_constant Returns the totally antisymmetric SU(3) structure constants f^{abc}. swirling_strength Causal wrapper for coherent_structures::swirling_strength_kernel swirling_strength_kernel Swirling strength λ_ci (Zhou et al. 1999): the imaginary part of the
complex eigenvalue pair of ∇u, or zero when all eigenvalues are real. taylor_microscale Causal wrapper for turbulence::taylor_microscale_kernel taylor_microscale_kernel Taylor microscale λ = √(15 · ν · k / ε) (m). thermal_expansion Causal wrapper for mechanics::thermal_expansion_kernel thermal_expansion_kernel Calculates thermal expansion strain: $\epsilon = \alpha \Delta T$. time_dilation_angle Causal wrapper for spacetime::time_dilation_angle_kernel time_dilation_angle_kernel Calculates Time Dilation Angle (Rapidity $\eta$). torque Causal wrapper for kinematics::torque_kernel torque_kernel Calculates torque as the outer product of radius and force: $\tau = r \wedge F$. total_enthalpy Causal wrapper for compressible::total_enthalpy_kernel total_enthalpy_kernel Total (stagnation) enthalpy h_0 = h + 0.5 · ‖u‖² (J/kg). total_pressure_isentropic Causal wrapper for compressible::total_pressure_isentropic_kernel total_pressure_isentropic_kernel Isentropic stagnation pressure
p_0 = p · (1 + (γ−1)/2 · M²)^(γ/(γ−1)). total_temperature_isentropic Causal wrapper for compressible::total_temperature_isentropic_kernel total_temperature_isentropic_kernel Isentropic stagnation temperature
T_0 = T · (1 + (γ−1)/2 · M²). turbulent_kinetic_energy Causal wrapper for turbulence::turbulent_kinetic_energy_kernel turbulent_kinetic_energy_kernel Turbulent kinetic energy k = 0.5 · ⟨u' · u'⟩. velocity_gradient_invariants Causal wrapper for kinematics::velocity_gradient_invariants_kernel velocity_gradient_invariants_kernel Invariants (P, Q, R) of the velocity gradient tensor A = ∇u,
following the Chong–Perry–Cantwell (1990) convention used in vortex
classification: velocity_potential_2d Causal wrapper for ideal_flow::velocity_potential_2d_kernel velocity_potential_2d_kernel 2D velocity-potential differential update dφ = u · dx + v · dy. viscous_diffusion Causal wrapper for governing::viscous_diffusion_kernel viscous_diffusion_kernel Viscous diffusion acceleration ν · ∇²u. viscous_dissipation_rate Causal wrapper for governing::viscous_dissipation_rate_kernel viscous_dissipation_rate_kernel Viscous dissipation rate Φ = τ : ∇u (W/m³). viscous_length_scale Causal wrapper for boundary_layer::viscous_length_scale_kernel viscous_length_scale_kernel Viscous length scale δ_ν = ν / u_τ (m). viscous_sublayer_velocity Causal wrapper for boundary_layer::viscous_sublayer_velocity_kernel viscous_sublayer_velocity_kernel Viscous sublayer velocity profile u⁺ = y⁺ (valid for y⁺ ≲ 5). von_mises_stress Causal wrapper for mechanics::von_mises_stress_kernel von_mises_stress_kernel Calculates Von Mises Stress from a 3x3 Stress Tensor. vorticity_from_gradient Causal wrapper for kinematics::vorticity_from_gradient_kernel vorticity_from_gradient_kernel Vorticity vector ω = ∇ × u, computed from the velocity gradient. vorticity_transport Causal wrapper for governing::vorticity_transport_kernel vorticity_transport_kernel Vorticity transport RHS: −(u·∇)ω + (ω·∇)u + ν · ∇²ω. wall_shear_stress_newtonian Causal wrapper for boundary_layer::wall_shear_stress_newtonian_kernel wall_shear_stress_newtonian_kernel Newtonian wall shear stress magnitude τ_w = μ · |∂u/∂y|_wall (Pa). wave_speed Monadic wrapper for general::wave_speed_kernel wave_speed_kernel Calculates the speed of a wave given its frequency and wavelength. weber_number Causal wrapper for dimensionless::weber_number_kernel weber_number_kernel Weber number We = ρ · u² · L / σ. Inertial to surface-tension forces. wilson_loop_kernel Computes a simplified Wilson loop trace for confinement analysis. y_plus Causal wrapper for boundary_layer::y_plus_kernel y_plus_kernel Dimensionless wall distance y⁺ = y · u_τ / ν. Gate Operator ParticleMetric For Particle Physics modules: West Coast (+—) PhysicsMetric Default physics metric (configurable): East Coast. RelativityMetric For General Relativity modules: East Coast (-+++)