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vital-ar
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physics-verifier-e4-curated-v2

trajectory_id stringlengths 32 44	problem_id stringlengths 11 22	domain stringclasses 6 values	proposed_solution stringlengths 36 5.61k	directive_text stringlengths 80 880	verdict stringclasses 2 values	error_type stringclasses 6 values	part stringclasses 3 values	prompt stringlengths 579 6.34k	target_output stringlengths 94 894	instruction stringlengths 579 6.34k	output stringlengths 94 894
E4_verifier:physgym_0010_err_aya8b	physgym_0010	ELECTROMAGNETISM	Step 1: Charge Distribution and Equilibrium: In electrostatic equilibrium, the point charge $q$ creates an electric field within the vacuum region. We aim to find the field on the $xOz$ surface of the conductor. Step 2: Gauss's Law Application: We use Gauss's Law to solve this problem, considering a Gaussian surfac...	To derive the correct expression for the electric field intensity on the $xOz$ surface of the conductor, use Gauss's Law for a cylindrical Gaussian surface enclosing the point charge $q$ in vacuum. Account for the conductor's boundary condition of a tangential electric field ($E_\tau = 0$) and consider the image charge...	[[CHALLENGE]]	incorrect_physical_model	part_2_synthetic_aya8b_challenged	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	To derive the correct expression for the electric field intensity on the $xOz$ surface of the conductor, use Gauss's Law for a cylindrical Gaussian surface enclosing the point charge $q$ in vacuum. Account for the conductor's boundary condition of a tangential electric field ($E_\tau = 0$) and consider the image charge...	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	To derive the correct expression for the electric field intensity on the $xOz$ surface of the conductor, use Gauss's Law for a cylindrical Gaussian surface enclosing the point charge $q$ in vacuum. Account for the conductor's boundary condition of a tangential electric field ($E_\tau = 0$) and consider the image charge...
E4_verifier:physgym_0010_tiny_aya	physgym_0010	ELECTROMAGNETISM	Step 1: Identify the problem setup and governing equations We are given a vacuum region where $x > 0$ and $y > 0$, and a conductor region bounded by the $xOz$ and $yOz$ planes. A point charge $q$ is fixed at $(a, b, c)$ in the vacuum, and the system is in electrostatic equilibrium. We need to find the elect...	To correctly determine the electric field intensity on the surface of the conductor at the $xOz$ plane, you must apply Gauss's Law to a Gaussian surface within the conductor, considering the symmetry and boundary conditions.	[[CHALLENGE]]	real_model_a_failure	part_3_real_tiny_aya_challenged	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	To correctly determine the electric field intensity on the surface of the conductor at the $xOz$ plane, you must apply Gauss's Law to a Gaussian surface within the conductor, considering the symmetry and boundary conditions. [[CHALLENGE]]	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	To correctly determine the electric field intensity on the surface of the conductor at the $xOz$ plane, you must apply Gauss's Law to a Gaussian surface within the conductor, considering the symmetry and boundary conditions. [[CHALLENGE]]
E4_verifier:physgym_0043_err_aya8b	physgym_0043	ELECTROMAGNETISM	Step 1: Initial Setup:** We have two dielectrics with base areas $A$, thicknesses $h_1$ and $h_2$, dielectric constants $\varepsilon_1$ and $\varepsilon_2$, and conductivities $\sigma_1$ and $\sigma_2$ respectively. The electric field intensities in the two media are $E_1$ and $E_2$, and we want to find the time-depend...	You've made several steps in the right direction, but key physical relationships were misinterpreted or omitted. To correctly solve this problem, you must use the relationship between electric fields, charges, and current in a capacitor, along with the governing equations for charge conservation and the time-dependent ...	[[CHALLENGE]]	incorrect_formula	part_2_synthetic_aya8b_challenged	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	You've made several steps in the right direction, but key physical relationships were misinterpreted or omitted. To correctly solve this problem, you must use the relationship between electric fields, charges, and current in a capacitor, along with the governing equations for charge conservation and the time-dependent ...	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	You've made several steps in the right direction, but key physical relationships were misinterpreted or omitted. To correctly solve this problem, you must use the relationship between electric fields, charges, and current in a capacitor, along with the governing equations for charge conservation and the time-dependent ...
E4_verifier:physgym_0043_tiny_aya	physgym_0043	ELECTROMAGNETISM	Step 1: Governing Equations: - Capacitance: The capacitance $ C $ of a parallel-plate capacitor with dielectric layers is given by: \[ C = \frac{A}{d} \left( \varepsilon_1 \frac{h_1}{h_1 + h_2} + \varepsilon_2 \frac{h_2}{h_1 + h_2} \right) \] - Power Source: The power source provides a voltage $ U_0 $, so t...	The problem requires a more detailed treatment of the charging process of the capacitor. The correct approach involves solving the coupled differential equations for the electric fields in each medium, considering the boundary conditions at the interfaces, and then relating these fields to the current density. The gove...	[[CHALLENGE]]	real_model_a_failure	part_3_real_tiny_aya_challenged	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	The problem requires a more detailed treatment of the charging process of the capacitor. The correct approach involves solving the coupled differential equations for the electric fields in each medium, considering the boundary conditions at the interfaces, and then relating these fields to the current density. The gove...	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	The problem requires a more detailed treatment of the charging process of the capacitor. The correct approach involves solving the coupled differential equations for the electric fields in each medium, considering the boundary conditions at the interfaces, and then relating these fields to the current density. The gove...
E4_verifier:physgym_0250_err_aya8b	physgym_0250	ELECTROMAGNETISM	Step 1: We begin by applying Gauss's law to the positive electrode, considering a surface enclosing the cylindrical surface. The electric field due to the potential difference $u$ is given by $\vec{E} = \frac{\lambda}{\varepsilon_1 + \varepsilon_2} \hat{z}$. Step 2: Integrating the dot product of $\vec{E}$ and the surf...	To correctly derive the relationship between the current through the power source and time, $i(t)$, you must use the following physical laws and equations: 1. Gauss's Lawfor electric fields in a homogeneous medium. 2. Ohm's Lawrelating current, voltage, and resistance. 3. Coulomb's Lawfor the electric field due to a po...	[[CHALLENGE]]	algebraic_error	part_2_synthetic_aya8b_challenged	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	To correctly derive the relationship between the current through the power source and time, $i(t)$, you must use the following physical laws and equations: 1. Gauss's Lawfor electric fields in a homogeneous medium. 2. Ohm's Lawrelating current, voltage, and resistance. 3. Coulomb's Lawfor the electric field due to a po...	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	To correctly derive the relationship between the current through the power source and time, $i(t)$, you must use the following physical laws and equations: 1. Gauss's Lawfor electric fields in a homogeneous medium. 2. Ohm's Lawrelating current, voltage, and resistance. 3. Coulomb's Lawfor the electric field due to a po...
E4_verifier:physgym_0250_tiny_aya	physgym_0250	ELECTROMAGNETISM	Step 1: Governing Equations**: The problem involves an electrolyte with two dielectrics separated by an interface. The conductivity and dielectric constant of each dielectric affect the electric field and potential distribution. The power source provides a voltage $ U $ and current $ I_0 $ at $ t = 0 $. Step 2: I...	You have correctly identified several key steps, but the final solution is incomplete and incorrect. The problem requires a detailed analysis of the electric field, potential, and charge distribution within the electrolyte, considering the geometry and boundary conditions. The governing equation for this scenario is Ga...	[[CHALLENGE]]	real_model_a_failure	part_3_real_tiny_aya_challenged	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	You have correctly identified several key steps, but the final solution is incomplete and incorrect. The problem requires a detailed analysis of the electric field, potential, and charge distribution within the electrolyte, considering the geometry and boundary conditions. The governing equation for this scenario is Ga...	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	You have correctly identified several key steps, but the final solution is incomplete and incorrect. The problem requires a detailed analysis of the electric field, potential, and charge distribution within the electrolyte, considering the geometry and boundary conditions. The governing equation for this scenario is Ga...
E4_verifier:physgym_0636_err_aya8b	physgym_0636	ELECTROMAGNETISM	Step 1: Setup: We begin with the same setup as described in the problem statement, including the Cartesian coordinates, the sphere of radius $R$, the $n$ rings, the charged particle $-q$ of mass $m$, and its circular motion at distance $r_0 \gg R$. Step 2: Potential Energy: We calculate the potential energy $V$ due to ...	You have correctly derived the potential energy and set up the effective potential for the charged particle's motion. However, the flaw lies in neglecting the angular dependencein the potential energy calculation. The integral for $V$ must account for the full angular range ($0$ to $360^\circ$) to capture the geometric...	[[CHALLENGE]]	dimensional_unit_error	part_2_synthetic_aya8b_challenged	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	You have correctly derived the potential energy and set up the effective potential for the charged particle's motion. However, the flaw lies in neglecting the angular dependencein the potential energy calculation. The integral for $V$ must account for the full angular range ($0$ to $360^\circ$) to capture the geometric...	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	You have correctly derived the potential energy and set up the effective potential for the charged particle's motion. However, the flaw lies in neglecting the angular dependencein the potential energy calculation. The integral for $V$ must account for the full angular range ($0$ to $360^\circ$) to capture the geometric...
E4_verifier:physgym_0636_tiny_aya	physgym_0636	ELECTROMAGNETISM	Step 1: Coordinate System and Setup: - Establish a Cartesian coordinate system $ Oxyz $ with the origin at the center of the sphere. - The sphere has radius $ R $ and is stationary. - $ n $ rings of radius $ R $ are placed along the meridional circles, passing through $ (0, 0, R) $ and $ (0, 0, -R) $. - The...	To correct the derivation, you should use the principles of electrostatic potential and energy conservation in a multipole system. Consider the superposition of the electric fields from the individual rings, taking into account their spherical symmetry. Utilize the formula for the electric potential due to a multipole ...	[[CHALLENGE]]	real_model_a_failure	part_3_real_tiny_aya_challenged	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	To correct the derivation, you should use the principles of electrostatic potential and energy conservation in a multipole system. Consider the superposition of the electric fields from the individual rings, taking into account their spherical symmetry. Utilize the formula for the electric potential due to a multipole ...	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	To correct the derivation, you should use the principles of electrostatic potential and energy conservation in a multipole system. Consider the superposition of the electric fields from the individual rings, taking into account their spherical symmetry. Utilize the formula for the electric potential due to a multipole ...
E4_verifier:physgym_0653_err_aya8b	physgym_0653	ELECTROMAGNETISM	Step 1: We start with the given equations for the magnetic flux in a superconducting ring and the expressions for current and length. Step 2: We consider the torque due to the Ampere force, but make the critical error of neglecting the contribution from the magnetic field acting on the entire ring. This is a signif...	You must account for the full torque due to the Ampere force acting on the entire ringwithin the magnetic field. The correct expression for the torque is derived from the magnetic flux density, $B$, the self-inductance coefficient, $L$, and the current, $I$, flowing through the ring. Use the relationship: $$\tau = B \c...	[[CHALLENGE]]	arithmetic_sign_error	part_2_synthetic_aya8b_challenged	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	You must account for the full torque due to the Ampere force acting on the entire ringwithin the magnetic field. The correct expression for the torque is derived from the magnetic flux density, $B$, the self-inductance coefficient, $L$, and the current, $I$, flowing through the ring. Use the relationship: $$\tau = B \c...	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	You must account for the full torque due to the Ampere force acting on the entire ringwithin the magnetic field. The correct expression for the torque is derived from the magnetic flux density, $B$, the self-inductance coefficient, $L$, and the current, $I$, flowing through the ring. Use the relationship: $$\tau = B \c...
E4_verifier:physgym_0653_tiny_aya	physgym_0653	ELECTROMAGNETISM	Step 1: Governing Equations: - Gravitational force: $ F_g = mg \sin \theta $ - Magnetic force: $ F_m = I \frac{d}{dt} (L B) $ - Torque equation: $ \tau = I \omega $ - Angular momentum conservation: $ L = m r^2 \omega $ - Energy conservation: $ K = \frac{1}{2} I \omega^2 + mg a \cos \theta $ Step 2: Initial Se...	You have correctly identified several key equations and principles from electromagnetism and classical mechanics, but The derivation diverges from the ground-truth reference at the final step. The correct expression for the minimum initial velocity $v_0$ should be derived from the balance of the magnetic and gravitatio...	[[CHALLENGE]]	real_model_a_failure	part_3_real_tiny_aya_challenged	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	You have correctly identified several key equations and principles from electromagnetism and classical mechanics, but The derivation diverges from the ground-truth reference at the final step. The correct expression for the minimum initial velocity $v_0$ should be derived from the balance of the magnetic and gravitatio...	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	You have correctly identified several key equations and principles from electromagnetism and classical mechanics, but The derivation diverges from the ground-truth reference at the final step. The correct expression for the minimum initial velocity $v_0$ should be derived from the balance of the magnetic and gravitatio...
E4_verifier:physgym_0039_err_aya8b	physgym_0039	MECHANICS	Step 1: We start by considering the forces acting on the wooden block. Since we are only concerned with the vertical motion, we look at the force along the y-axis. The net force is the difference between the gravitational force pushing down on the block and the buoyant force from the water. Step 2: The gravitational fo...	To determine the period of motion $T$, recognize that the net force on the block results in simple harmonic motion due to the balance between gravity and buoyancy. Use the equation of motion for a harmonic oscillator: $\rho_{1}H S{\ddot{y}} + \rho_{2}S g y = 0$. This leads to the angular frequency $\omega = \sqrt{\frac...	[[CHALLENGE]]	incorrect_physical_model	part_2_synthetic_aya8b_challenged	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	To determine the period of motion $T$, recognize that the net force on the block results in simple harmonic motion due to the balance between gravity and buoyancy. Use the equation of motion for a harmonic oscillator: $\rho_{1}H S{\ddot{y}} + \rho_{2}S g y = 0$. This leads to the angular frequency $\omega = \sqrt{\frac...	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	To determine the period of motion $T$, recognize that the net force on the block results in simple harmonic motion due to the balance between gravity and buoyancy. Use the equation of motion for a harmonic oscillator: $\rho_{1}H S{\ddot{y}} + \rho_{2}S g y = 0$. This leads to the angular frequency $\omega = \sqrt{\frac...
E4_verifier:physgym_0039_tiny_aya	physgym_0039	MECHANICS	Step 1: Governing Physical Equations The motion of the wooden block can be described by the following equations: - Translational Motion: The block experiences a net force due to gravity, which causes it to accelerate downward. The force $ F $ acting on the block is given by: \[ F = (m_1 + m_2) g \] where \( m_1 \...	To determine the period of motion $T$, apply the principle of simple harmonic motion, where the net force is a restoring force proportional to the displacement. Use the equation of motion for a block in a fluid: $$ \rho_1 H S \ddot{y} + \rho_2 S g y = 0 $$ This leads to the period: $$ T = 2\pi \sqrt{\frac{\rho_1 H}{\rh...	[[CHALLENGE]]	real_model_a_failure	part_3_real_tiny_aya_challenged	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	To determine the period of motion $T$, apply the principle of simple harmonic motion, where the net force is a restoring force proportional to the displacement. Use the equation of motion for a block in a fluid: $$ \rho_1 H S \ddot{y} + \rho_2 S g y = 0 $$ This leads to the period: $$ T = 2\pi \sqrt{\frac{\rho_1 H}{\rh...	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	To determine the period of motion $T$, apply the principle of simple harmonic motion, where the net force is a restoring force proportional to the displacement. Use the equation of motion for a block in a fluid: $$ \rho_1 H S \ddot{y} + \rho_2 S g y = 0 $$ This leads to the period: $$ T = 2\pi \sqrt{\frac{\rho_1 H}{\rh...
E4_verifier:physgym_0041_err_aya8b	physgym_0041	MECHANICS	Step 1: Initial Conditions: We start with the rocket's acceleration $a = k_1g_0$ and the pressure difference $N_2 = k_2N_1$ at height $h$. We also define the mass of the tester as $m$. Step 2: Force Balance: At any height, the force balancing the gravitational force is $F = mg$, where $g = g_0(\frac{R}{R+h})^2$ (a ...	To solve this problem, you must consider the varying gravitational acceleration experienced by the rocket as it ascends. Use the formula for gravitational acceleration in a gravitational field: $g(h) = g_0\left(1 - \frac{h}{R}\right)^2$. Then, apply the principle of equilibrium, equating the force due to gravity at hei...	[[CHALLENGE]]	incorrect_formula	part_2_synthetic_aya8b_challenged	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	To solve this problem, you must consider the varying gravitational acceleration experienced by the rocket as it ascends. Use the formula for gravitational acceleration in a gravitational field: $g(h) = g_0\left(1 - \frac{h}{R}\right)^2$. Then, apply the principle of equilibrium, equating the force due to gravity at hei...	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	To solve this problem, you must consider the varying gravitational acceleration experienced by the rocket as it ascends. Use the formula for gravitational acceleration in a gravitational field: $g(h) = g_0\left(1 - \frac{h}{R}\right)^2$. Then, apply the principle of equilibrium, equating the force due to gravity at hei...
E4_verifier:physgym_0041_tiny_aya	physgym_0041	MECHANICS	Step 1: Governing Physical Equations The rocket's vertical motion is governed by the equation of motion under constant acceleration: \[ h = \frac{1}{2} a t^2 \] where $ h $ is the height, $ a $ is the acceleration, and $ t $ is the time. The pressure $ P $ on the bottom of the groove is related to the force \( ...	Correct the solution by recognizing that the rocket's ascent is not governed by a simple kinematic equation, but rather by Newton's second law with a constant acceleration. The pressure relationship should be used to eliminate velocity and mass, leading to an algebraic equation for $h$. Use Newton's second law and the ...	[[CHALLENGE]]	real_model_a_failure	part_3_real_tiny_aya_challenged	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	Correct the solution by recognizing that the rocket's ascent is not governed by a simple kinematic equation, but rather by Newton's second law with a constant acceleration. The pressure relationship should be used to eliminate velocity and mass, leading to an algebraic equation for $h$. Use Newton's second law and the ...	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	Correct the solution by recognizing that the rocket's ascent is not governed by a simple kinematic equation, but rather by Newton's second law with a constant acceleration. The pressure relationship should be used to eliminate velocity and mass, leading to an algebraic equation for $h$. Use Newton's second law and the ...
E4_verifier:physgym_0367_err_aya8b	physgym_0367	MECHANICS	Step 1: Set-up: We are given the elliptical track equation: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, and we want to find the period $T$ of the small charged object's oscillations. Step 2: Differentiate: Differentiating both sides with respect to time $t$, we get: $$\frac{2x \dot{x}}{a^2} + \frac{2y \dot{y}}{b^2} = ...	You have correctly identified the governing equation for simple harmonic motion in the context of the charged object's oscillations. However, the derivation deviates from the correct approach by introducing an incorrect proportionality . The correct relationship should be: $$\ddot{\theta} = -\omega_0^2 \theta$$ where $...	[[CHALLENGE]]	algebraic_error	part_2_synthetic_aya8b_challenged	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	You have correctly identified the governing equation for simple harmonic motion in the context of the charged object's oscillations. However, the derivation deviates from the correct approach by introducing an incorrect proportionality . The correct relationship should be: $$\ddot{\theta} = -\omega_0^2 \theta$$ where $...	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	You have correctly identified the governing equation for simple harmonic motion in the context of the charged object's oscillations. However, the derivation deviates from the correct approach by introducing an incorrect proportionality . The correct relationship should be: $$\ddot{\theta} = -\omega_0^2 \theta$$ where $...
E4_verifier:physgym_0367_tiny_aya	physgym_0367	MECHANICS	Step 1: Governing Physical Equations The problem involves a charged particle moving along an elliptical track under the influence of a point charge at the origin. The motion can be described using the equations of motion for a charged particle in a gravitational field. Step 2: Setup the Problem Consider the elliptical ...	You have made several errors, including incorrect force calculations, inappropriate use of gravitational equations, and incorrect relationships between velocity, angular velocity, and period. The correct approach is to use the centripetal force required for circular motion at the origin, derived from Newton's second la...	[[CHALLENGE]]	real_model_a_failure	part_3_real_tiny_aya_challenged	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	You have made several errors, including incorrect force calculations, inappropriate use of gravitational equations, and incorrect relationships between velocity, angular velocity, and period. The correct approach is to use the centripetal force required for circular motion at the origin, derived from Newton's second la...	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	You have made several errors, including incorrect force calculations, inappropriate use of gravitational equations, and incorrect relationships between velocity, angular velocity, and period. The correct approach is to use the centripetal force required for circular motion at the origin, derived from Newton's second la...
E4_verifier:physgym_0401_err_aya8b	physgym_0401	MECHANICS	Step 1: We start by identifying the components of the initial velocity: $$ v_r = k v_0 \sin\theta \text{ ( radial component)} $$ $$ v_\theta = k v_0 \cos\theta \text{ (angular/transverse component)} $$ Step 2: We assume the ball undergoes a series of bounces, and at each bounce, its velocity components are modified by ...	You have made several errors, including omitting crucial steps and using incorrect mathematical techniques. To solve this problem correctly, you must leverage the conservation of angular momentum and the energy conservation principle during each bounce.	[[CHALLENGE]]	dimensional_unit_error	part_2_synthetic_aya8b_challenged	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	You have made several errors, including omitting crucial steps and using incorrect mathematical techniques. To solve this problem correctly, you must leverage the conservation of angular momentum and the energy conservation principle during each bounce. [[CHALLENGE]]	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	You have made several errors, including omitting crucial steps and using incorrect mathematical techniques. To solve this problem correctly, you must leverage the conservation of angular momentum and the energy conservation principle during each bounce. [[CHALLENGE]]
E4_verifier:physgym_0401_tiny_aya	physgym_0401	MECHANICS	Step 1: Governing Equations: - Kinetic Energy (KE): $ KE = \frac{1}{2}mv^2 $ - Potential Energy (PE): $ PE = mgh $ - Restitution Coefficient: $ \varepsilon = \frac{2u}{u_0} $ - Angular Momentum: $ L = mvr $ - Angular Velocity: $ \omega = \frac{v}{r} $ - Bounce Height: \( h = R(1 - \var...	You have correctly identified several key physics principles, such as conservation of angular momentum and the restitution coefficient, but the final step is incomplete. The correct approach involves using the small-angle approximation to express $\varphi$ in terms of $\theta$, $\varepsilon$, and $k$. Employ the small-...	[[CHALLENGE]]	real_model_a_failure	part_3_real_tiny_aya_challenged	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	You have correctly identified several key physics principles, such as conservation of angular momentum and the restitution coefficient, but the final step is incomplete. The correct approach involves using the small-angle approximation to express $\varphi$ in terms of $\theta$, $\varepsilon$, and $k$. Employ the small-...	Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or mathematical directive that either confirms why the derivation is valid or states the exact rule/constraint needed to fix the error without referencing line or step numbers. Then, on a new final line, output [[...	You have correctly identified several key physics principles, such as conservation of angular momentum and the restitution coefficient, but the final step is incomplete. The correct approach involves using the small-angle approximation to express $\varphi$ in terms of $\theta$, $\varepsilon$, and $k$. Employ the small-...

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Adversarial Physics Verifier — Experiment E4 (`verifier_e4_curated`)

Codebase & Pipeline Repository: vital-ar/adversarial_physics_problems_solver
Hugging Face Dataset ID: vital-ar/physics-verifier-e4-curated-v2

Dataset Overview

verifier_e4_curated is Dataset 4B (Model 4B / E4 Verifier) in the Lexsi AI (CuratorKIT + AlignTune + AuditKIT) progressive data-curation ablation study for the Adversarial Physics Problem Solver–Verifier system.

verifier_e4_curated is the final Stage 2 Verifier SFT dataset (1,098 trajectories) used to train Model B (Verifier) to inspect a student's proposed physics derivation and emit either:

A domain-calibrated acceptance ending with [[APPROVE]] (176 examples), or
A concise, self-contained physical/mathematical directive ending with [[CHALLENGE]] (922 examples across synthetic and real flawed student attempts).
Role in Ablation Ladder: Paired companion critic for E4 Solver (985 trajectories) inside the two-model adversarial loop.
Target Student Model: CohereLabs/tiny-aya-global (Model B — Verifier, fine-tuned via AlignTune).

Splits & Domain Distribution

Split	File	Examples	Composition / Source
`train`	`train.jsonl`	1,098	`176` `[[APPROVE]]` + `922` `[[CHALLENGE]]`

Training Split Domain Breakdown (`1,098` Examples)

Physics Domain	`[[APPROVE]]` (Part 1)	`[[CHALLENGE]]` Synth (Part 2)	`[[CHALLENGE]]` Tiny-Aya (Part 3)	Total	Share
`MECHANICS`	30	261	273	564	51.4%
`ELECTROMAGNETISM`	30	98	104	232	21.1%
`MODERN_PHYSICS`	30	41	41	112	10.2%
`THERMODYNAMICS`	30	19	19	68	6.2%
`RELATIVITY`	27	17	18	62	5.6%
`OPTICS`	29	15	16	60	5.5%
Total	176	451	471	1,098	100.0%

Verifier Trajectory Construction & Anti-Leak Sanitization

Part 1 ([[APPROVE]]): Domain-balanced verified correct derivations paired with [[APPROVE]] verdicts (with normalized step formatting and canonical ANSWER: lines).
Part 2 (Synthetic [[CHALLENGE]]): Physics taxonomy flaw injections generated by CohereForAI/aya-expanse-8b, sanitized by _sanitize_flawed_trajectory_text to strip meta-giveaway words (flaw, incorrect, injected, etc.) and trailing answer annotations.
Part 3 (Real [[CHALLENGE]]): Authentic flawed zero-shot student attempts generated by CohereLabs/tiny-aya-global (is_correct == False), filtered for semantic diversity and paired with non-leaking directives.
Directive Sanitization & Validation (validate_verifier_directive): Guaranteed that every directive contains zero ground-truth answer leaks, ends cleanly with \n[[CHALLENGE]] or \n[[APPROVE]], and stays within word budgets.

Export-Time `CuratorKIT` Integrity Gates

curatorkit.normalizers.clean.TextCleaner: Normalized whitespace and Unicode math/control characters across 1,098 rows.
curatorkit.gates.schema.SchemaGate (min_tokens = 20, max_tokens = 8192): 1,098 passed, 0 rejected.
curatorkit.normalizers.dedup.ExactDeduplicator: 1,098 unique records retained (0 exact duplicates removed).

Schema

Each record in train.jsonl follows the canonical schema below (with instruction, input, and output convenience columns also included for direct AlignTune SFT loading):

{
  "trajectory_id": "E4_verifier:physgym_0010_err_aya8b",
  "problem_id": "physgym_0010",
  "domain": "ELECTROMAGNETISM",
  "proposed_solution": "Step 1: Charge Distribution and Equilibrium:** In electrostatic equilibrium, the point charge $q$ creates an...\n...\nANSWER: \\frac{q b}{2\\pi \\varepsilon_0} \\frac{(a^2 + z^2 - c^2)^{1/2} - (x^2 + z^2 - c^2)^{1/2}}{(a^2 + b^2)^{3/2}} \\hat{y}",
  "directive_text": "To derive the correct expression for the electric field intensity on the $xOz$ surface of the conductor, use Gauss's Law for a cylindrical Gaussian surface enclosing the point charge $q$ in vacuum. Account for the conductor's boundary c...",
  "verdict": "[[CHALLENGE]]",
  "error_type": "incorrect_physical_model",
  "part": "part_2_synthetic_aya8b_challenged",
  "prompt": "Review the proposed solution to the physics problem below. First, write a concise, self-contained physical or...\n...\nANSWER: \\frac{q b}{2\\pi \\varepsilon_0} \\frac{(a^2 + z^2 - c^2)^{1/2} - (x^2 + z^2 - c^2)^{1/2}}{(a^2 + b^2)^{3/2}} \\hat{y}",
  "target_output": "To derive the correct expression for the electric field intensity on the $xOz$ surface of the conductor, use Gauss's Law for a cylindrical Gaussian surface enclosing the point charge $q$ in vacuum. Account for the cond...\n...\n[[CHALLENGE]]"
}

Provenance & Integrity Files

manifest.json — CuratorKIT stage gate counts and deduplication report.
lexsi_provenance.json — lexsi.provenance/1 lineage record consumed by AlignTune and AuditKIT.
checksums.txt — SHA-256 digests of all split, metadata, and README.md files.
Project Repository & Codebase: https://github.com/vital-ar/adversarial_physics_problems_solver

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Adversarial Physics Verifier — Experiment E4 (verifier_e4_curated)