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Kinematic operators

124 operators in the kinematic category of the live registry. Each is a named formula you can compose inside a state contract or call directly through POST /api/zeq/compute. KO42 is always on; add up to three more per call (total ≤ 4), per the 7-step protocol.

OperatorDescriptionEquation
KO1Position vector magnitude in 3D Cartesian coordinates, giving the distance from the origin.|\vec{r}| = \sqrt{x^2 + y^2 + z^2}
KO10Centripetal acceleration directed toward the center of circular motion.a_c = \frac{v^2}{r} = r\omega^2
KO100Driven harmonic oscillator amplitude response showing resonance peak near natural frequency.A(\omega) = \frac{F_0/m}{\sqrt{(\omega_0^2-\omega^2)^2 + (2\zeta\omega_0\omega)^2}}
KO11Torque as the cross product of position vector and applied force.\tau = r \times F
KO12Angular momentum equals moment of inertia times angular velocity for rigid body rotation.L = I\omega
KO13Moment of inertia as the sum of mass elements times their squared distances from the rotation axis.I = \sum m_i r_i^2
KO14Rotational Newton's second law: net torque equals moment of inertia times angular acceleration.\tau = I\alpha
KO15Rotational kinetic energy: one-half moment of inertia times angular velocity squared.K_{rot} = \frac{1}{2}I\omega^2
KO16Angular position under constant angular acceleration as a function of time.\theta(t) = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2
KO17Angular velocity-displacement relation under constant angular acceleration.\omega^2 = \omega_0^2 + 2\alpha\Delta\theta
KO18Rolling without slipping constraint: center-of-mass speed equals angular velocity times radius.v_{cm} = \omega R
KO19Tangential acceleration component for a point on a rotating body.a_{tan} = r\alpha
KO2Velocity as the time derivative of position, defining instantaneous rate of displacement.\vec{v} = \frac{d\vec{r}}{dt}
KO20Total kinetic energy of a rolling body: translational plus rotational contributions.K_{total} = \frac{1}{2}mv_{cm}^2 + \frac{1}{2}I_{cm}\omega^2
KO21Work as the line integral of force along displacement, modulated by Zeq HulyaPulse.W = \int \vec{F} \cdot d\vec{r} \cdot [1 + \alpha \sin(2\pi \cdot 1.287t)]
KO22Kinetic energy with Zeq HulyaPulse modulation at the 1.287 Hz resonance frequency.K = \frac{1}{2}mv^2 \cdot [1 + \alpha \sin(2\pi \cdot 1.287t)]
KO23Gravitational potential energy near Earth's surface: mgh.U_g = mgh \quad (\text{gravitational PE, KO metric})
KO24Elastic potential energy stored in a spring: one-half times spring constant times displacement squared.U_s = \frac{1}{2}kx^2
KO25Total mechanical energy as kinetic plus potential, with Zeq HulyaPulse modulation.E = K + U \cdot [1 + \alpha \sin(2\pi \cdot 1.287t)]
KO26Instantaneous power as the rate of work done, or force dotted with velocity.P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}
KO27Mechanical efficiency as the ratio of useful work output to total work input.\eta = \frac{W_{out}}{W_{in}}
KO28Linear momentum with Zeq HulyaPulse resonance modulation.\vec{p} = m\vec{v} \cdot [1 + \alpha \sin(2\pi \cdot 1.287t)]
KO29Force as the time derivative of momentum (Newton's second law in general form).\vec{F} = \frac{d\vec{p}}{dt}
KO3Acceleration as the second time derivative of position, measuring rate of velocity change.\vec{a} = \frac{d\vec{v}}{dt} = \frac{d^2\vec{r}}{dt^2}
KO30Impulse-momentum theorem: impulse equals the change in momentum.J = \int \vec{F} dt = \Delta\vec{p}
KO31Conservation of linear momentum: total momentum before equals total momentum after collision.\sum \vec{p}_i = \sum \vec{p}_f
KO32Coefficient of restitution measuring the elasticity of a collision.e = \frac{v_{2f} - v_{1f}}{v_{1i} - v_{2i}}
KO33Center-of-mass velocity computed as the mass-weighted average of particle velocities.v_{cm} = \frac{\sum m_i v_i}{\sum m_i}
KO34Center-of-mass position computed as the mass-weighted average of particle positions.\vec{r}_{cm} = \frac{\sum m_i \vec{r}_i}{\sum m_i}
KO35Aerodynamic drag force proportional to velocity squared, fluid density, drag coefficient, and area.F_{drag} = \frac{1}{2}\rho v^2 C_D A
KO36Terminal velocity where drag force balances gravitational force on a falling object.v_t = \sqrt{\frac{2mg}{\rho C_D A}}
KO37Vertical position in projectile motion as a function of time under gravity.y(t) = y_0 + v_0 t - \frac{1}{2}gt^2
KO38Projectile range on level ground as a function of launch speed and angle.R = \frac{v_0^2 \sin(2\theta)}{g}
KO39Maximum height of a projectile launched at angle theta with initial speed v0.H = \frac{v_0^2 \sin^2\theta}{2g}
KO4Kinematic position equation under constant acceleration, giving position as a function of time.\vec{r}(t) = \vec{r}_0 + \vec{v}_0 t + \frac{1}{2}\vec{a}t^2
KO40Total flight time of a projectile on level ground.T_{flight} = \frac{2v_0 \sin\theta}{g}
KO41Zeq propagation time constant: the reciprocal of the empirically discovered HulyaPulse frequency.\tau_{prop} = \frac{1}{f_{CMB}} \approx 0.777s
KO42KO42 — bounded modulation convention: every result is phase-stamped to the 1.287 Hz system clock; amplitude capped at 10^-3 by construction (CONSTANTS-CHARTER.md).R(t) = S_0\,[1 + \alpha\sin(2\pi f_H t)],\; S_0 = 1,\; \alpha = 10^{-3}
KO42_1Zeq-modified spacetime metric with first-order HulyaPulse oscillation in the time component.ds^2 = g_{\mu\nu} dx^\mu dx^\nu + \alpha_1 \sin(2\pi \cdot 1.287t) dt^2
KO42_10Zeq-modified spacetime metric with Fourier series HulyaPulse summing 10 harmonics.ds^2 = g_{\mu\nu} dx^\mu dx^\nu + \alpha_{10} \sum_n \sin(2n\pi \cdot 1.287t)/n dt^2
KO42_2Zeq-modified spacetime metric with phase-shifted HulyaPulse at pi/4 offset.ds^2 = g_{\mu\nu} dx^\mu dx^\nu + \alpha_2 \sin(2\pi \cdot 1.287t + \pi/4) dt^2
KO42_3Zeq-modified spacetime metric with cosine HulyaPulse modulation.ds^2 = g_{\mu\nu} dx^\mu dx^\nu + \alpha_3 \cos(2\pi \cdot 1.287t) dt^2
KO42_4Zeq-modified spacetime metric with doubled-frequency HulyaPulse harmonic.ds^2 = g_{\mu\nu} dx^\mu dx^\nu + \alpha_4 \sin(4\pi \cdot 1.287t) dt^2
KO42_5Zeq-modified spacetime metric with exponentially damped HulyaPulse oscillation.ds^2 = g_{\mu\nu} dx^\mu dx^\nu + \alpha_5 \sin(2\pi \cdot 1.287t) \cdot e^{-\lambda t} dt^2
KO42_6Zeq-modified spacetime metric with first and second harmonic HulyaPulse superposition.ds^2 = g_{\mu\nu} dx^\mu dx^\nu + \alpha_6 [\sin(2\pi \cdot 1.287t) + \sin(4\pi \cdot 1.287t)/2] dt^2
KO42_7Zeq-modified spacetime metric with sech-envelope HulyaPulse soliton.ds^2 = g_{\mu\nu} dx^\mu dx^\nu + \alpha_7 \sin(2\pi \cdot 1.287t) \cdot \mathrm{sech}(t/\tau) dt^2
KO42_8Zeq-modified spacetime metric with Bessel function HulyaPulse for axial symmetry.ds^2 = g_{\mu\nu} dx^\mu dx^\nu + \alpha_8 J_0(2\pi \cdot 1.287t) dt^2
KO42_9Zeq-modified spacetime metric with chaotic phase HulyaPulse modulation.ds^2 = g_{\mu\nu} dx^\mu dx^\nu + \alpha_9 \sin(2\pi \cdot 1.287t + \phi_{\mathrm{chaos}}) dt^2
KO42-1Measures rate of structural awareness changeΓ_sac = dA/dt · sin(2π·1.287·t) · ∂φ/∂A
KO42-10Fourier series Zeq resonance summing 10 harmonics of the 1.287 Hz fundamental.K_int = 1/T ∫I(t)·R(t)·e^(i2π·1.287·t)dt
KO42-2Enables awareness of past awareness statesΨ_rec(t) = ∫e^(-(t-τ)/τ_c) · Ψ_rec(τ) · sin(2π·1.287·τ)dτ
KO42-3Models resonant field connecting Zeq siblingsR_sib = Σκ_k · e^(i(ω_k t + φ_k)) · δ(r - r_k)
KO42-4Describes flow of conscious statesJ_c = -D_c ∇ψ + v_c ψ + α sin(2π·1.287·t) n̂
KO42-5Damped Zeq resonance at half-power golden ratio modulation with exponential decay.G_meta = ∂²F/∂t∂φ + λ·H(F)·cos(2π·1.287·t)
KO42-6Phase-shifted Zeq resonance cosine waveform at 0.6 golden ratio power.C_phase = |1/T ∫e^(iθ(t))·e^(-i2π·1.287·t)dt|²
KO42-7Sinc-function Zeq resonance at 0.7 golden ratio power for bandwidth-limited signals.I_inv = ∂/∂t(δS/δφ) + β·sin(2π·1.287·t)·δ²S/δφ²
KO42-8Bessel-function Zeq resonance at 0.8 golden ratio power for cylindrical symmetry.ρ_q = Σp_i log p_i · (1 - e^(-t/τ_q)) · cos(2π·1.287·t)
KO42-9Complex exponential Zeq resonance at 0.9 golden ratio power with phase offset.H_temp = ∫e^(-t/τ_h)·φ(t)·sin(2π·1.287·t)dt
KO42.1Zeq sine resonance with golden ratio raised to the 1.287 power.ds² = g_μνdx^μ dx^ν + α [sin(2π·1.287·t) + 0.1 sin(4π·1.287·t)] dt²
KO42.2Zeq cosine resonance with golden ratio raised to the 1.287 power.ds² = g_μνdx^μ dx^ν + β sin(2π·1.287·t) dt²
KO42.3Automatic metric tensioning across three free harmonic frequencies - the foundation of mathematical consciousnessφ_c^42 · T_metric = ∇_μ g^μν [1.287 Hz ⊗ 0.618 Hz ⊗ 2.083 Hz] · sin(2π·1.287·t) + cos(2π·0.618·t) + exp(2π·2.083·t)
KO423Generalized Zeq operator applying the golden ratio to an arbitrary function of space and time.\mathcal{O}_{423} = \phi \cdot f(x,t)
KO43Zeq Hamiltonian coupling the golden ratio to the 1.287 Hz angular frequency.\mathcal{H} = \phi \cdot \omega_{1.287}
KO44Zeq synchronization angular frequency: 2pi times 1.287 Hz.\Omega_{sync} = 2\pi \cdot 1.287
KO45Proper time deficit: difference between proper and coordinate time in relativistic frames.\Delta\tau = \tau_{proper} - \tau_{coord}
KO46Golden ratio definition: (1+sqrt(5))/2, the fundamental constant of Zeq framework aesthetics.\Phi_{golden} = \frac{1 + \sqrt{5}}{2}
KO47CMB peak frequency derived from thermal energy of the cosmic microwave background.f_{CMB} = \frac{k_B T_{CMB}}{h}
KO48Zeq master operator combining golden ratio identity with Hamiltonian scaled by operator count.\mathcal{M} = \phi \cdot I + H \cdot \frac{N_{op}}{1204}
KO49CMB entropy parameter: ratio of CMB entropy to Boltzmann constant.\beta = \frac{S_{CMB}}{k_B}
KO5Velocity under constant acceleration: final velocity equals initial velocity plus acceleration times time.v = v_0 + at
KO50Unified wavefunction as weighted superposition of all operator eigenstates.\psi_{unified} = \sum_i w_i \Psi_i
KO51Simple harmonic motion: position as cosine function of time with amplitude and phase.x(t) = A\cos(\omega t + \phi)
KO52Natural frequency of a mass-spring system: square root of spring constant over mass.\omega = \sqrt{\frac{k}{m}}
KO53Period of a mass-spring oscillator: 2pi times square root of mass over spring constant.T = 2\pi\sqrt{\frac{m}{k}}
KO54Frequency as the reciprocal of period for periodic motion.f = \frac{1}{T}
KO55Wave speed equals frequency times wavelength, the fundamental wave relation.v_{wave} = f\lambda
KO56Traveling wave solution: sinusoidal displacement as a function of position and time.y(x,t) = A\sin(kx - \omega t)
KO57Wave number as 2pi divided by wavelength, measuring spatial frequency.k = \frac{2\pi}{\lambda}
KO58Wave intensity as power per unit area for energy transport by waves.I = \frac{P}{A}
KO59Sound intensity level in decibels with Zeq HulyaPulse modulation.\beta = 10\log_{10}\frac{I}{I_0} \cdot [1 + \alpha \sin(2\pi \cdot 1.287t)]
KO6Velocity-displacement relation under constant acceleration, eliminating time from kinematics.v^2 = v_0^2 + 2a\Delta x
KO60Beat frequency from two interfering waves with Zeq HulyaPulse modulation.f_{\mathrm{beat}} = |f_1 - f_2| \cdot [1 + \alpha \sin(2\pi \cdot 1.287t)]
KO61Circular motion position vector in complex exponential form.\vec{r}(t) = r_0 e^{i\omega t}
KO62Coriolis acceleration in a rotating reference frame, deflecting moving objects.\vec{a}_{Coriolis} = -2\vec{\omega} \times \vec{v}
KO63Centrifugal acceleration in a rotating reference frame, directed outward from the axis.\vec{a}_{cent} = -\omega^2 \vec{r}
KO64Lagrangian as kinetic minus potential energy, the basis of analytical mechanics.\mathcal{L} = T - U
KO65Euler-Lagrange equation: the equation of motion derived from the principle of least action.\frac{d}{dt}\frac{\partial\mathcal{L}}{\partial\dot{q}} - \frac{\partial\mathcal{L}}{\partial q} = 0
KO66Hamiltonian as the Legendre transform of the Lagrangian, giving total energy in phase space.H = \sum_i p_i \dot{q}_i - \mathcal{L}
KO67Poisson bracket: the fundamental algebraic structure of classical Hamiltonian mechanics.\{f, g\} = \sum_i \left(\frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}\right)
KO68Action integral: the time integral of the Lagrangian, whose extremum gives the equations of motion.S = \int \mathcal{L} dt
KO69Projectile trajectory with aerodynamic drag correction at the ninth power of velocity.x_{69}(t) = v_0 t \cos\theta - \frac{1}2c_d\rho A v^{9}t^2/m
KO7Angular velocity as the time derivative of angular position for rotational motion.\omega = \frac{d\theta}{dt}
KO70Hamilton-Jacobi equation: partial differential equation for the action function in classical mechanics.\frac{\partial S}{\partial t} + H = 0
KO71Lorentz factor gamma measuring time dilation and length contraction at relativistic speeds.\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}
KO72Lorentz transformation for time coordinate between inertial reference frames.t\prime = \gamma(t - vx/c^2)
KO73Lorentz transformation for spatial coordinate between inertial reference frames.x\prime = \gamma(x - vt)
KO74Orbital trajectory in polar coordinates with eccentricity and argument of periapsis.r_{74}(\theta) = \frac{a(1-e^2)}{1 + e\cos(\theta - \omega_{4})}
KO75Relativistic energy with HulyaPulse modulation at the Zeq 1.287 Hz resonance.E = \gamma mc^2 \cdot [1 + \alpha \sin(2\pi \cdot 1.287 t)]
KO76Energy-momentum relation with Zeq HulyaPulse resonance modulation.E^2 = (pc)^2 + (m_0 c^2)^2 \quad (\text{KO rest-mass variant})
KO77Relativistic velocity addition formula preventing superluminal speeds.u = \frac{u\prime + v}{1 + u\prime v/c^2}
KO78Time dilation: moving clocks run slower by the Lorentz factor.\Delta t\prime = \gamma \Delta t
KO79Second orbital trajectory variant with different argument of periapsis.r_{79}(\theta) = \frac{a(1-e^2)}{1 + e\cos(\theta - \omega_{9})}
KO8Angular acceleration as the time derivative of angular velocity.\alpha = \frac{d\omega}{dt}
KO80Relativistic mass increase with velocity via the Lorentz factor.m_{rel} = \gamma m_0
KO81Newton's gravitational force with Zeq HulyaPulse modulation.F = G\frac{m_1 m_2}{r^2}
KO82Gravitational field strength with Zeq HulyaPulse modulation.g_{\text{KO}} = \frac{GM}{r^2} \cdot (1 + \alpha \sin(2\pi \cdot 1.287 t))
KO83Gravitational potential energy between two masses separated by distance r.U = -G\frac{m_1 m_2}{r}
KO84Escape velocity from a gravitational well with Zeq HulyaPulse modulation.v_{\mathrm{esc}} = \sqrt{\frac{2GM}{r}} \cdot [1 + \alpha \sin(2\pi \cdot 1.287t)]
KO85Circular orbital velocity around a central mass.v_{orb} = \sqrt{\frac{GM}{r}}
KO86Kepler's third law: orbital period related to semi-major axis and central mass.T = 2\pi\sqrt{\frac{r^3}{GM}}
KO87Orbital energy: total mechanical energy of an orbiting body, negative for bound orbits.E_{orb} = -\frac{GMm}{2a}
KO88Parallel axis theorem: moment of inertia about a shifted axis plus mass times offset squared.I_{88} = \int r^2 dm + m_{8}d^2
KO89Schwarzschild radius with Zeq HulyaPulse modulation for event horizon oscillation.r_S = \frac{2GM}{c^2}
KO9Tangential speed from angular velocity: linear speed equals radius times angular velocity.v = r\omega
KO90Gravitational potential: negative GM/r for the potential energy per unit mass.\Phi = -\frac{GM}{r}
KO91Net force as the vector sum of all forces acting on a body.\vec{F}_{net} = \sum_i \vec{F}_i
KO92Normal force on an inclined plane proportional to weight times cosine of inclination angle.\vec{N} = -m\vec{g} \cos\theta
KO93Relativistic momentum with Lorentz factor correction.p_{93} = \gamma_{3} m v
KO94Static friction inequality: friction force bounded by coefficient of static friction times normal force.f_s \leq \mu_s N
KO95Hooke's law: restoring force of a spring proportional to displacement.F_{spring} = -kx
KO96Period of a simple pendulum depending on length and gravitational acceleration.T = 2\pi\sqrt{\frac{l}{g}}
KO97Damped natural frequency of an oscillator reduced by the damping ratio.\omega_d = \omega_0\sqrt{1 - \zeta^2}
KO98Damped harmonic oscillator solution with exponential decay and oscillation.x(t) = A e^{-\zeta\omega_0 t}\cos(\omega_d t + \phi)
KO99Quality factor Q of an oscillator: ratio of stored energy to energy dissipated per cycle.Q = \frac{\omega_0}{2\zeta\omega_0} = \frac{1}{2\zeta}

Compute with one of these​

curl -sS -X POST https://zeqsdk.com/api/zeq/compute \
  -H "Authorization: Bearer $ZEQ_KEY" \
  -H "Content-Type: application/json" \
  -d '{"operators":["KO1"],"inputs":{}}'

The response carries the bare physics value, its unit and uncertainty, the generated master equation, and a signed envelope you can verify on any node.

See also​