Datasets:

vital-ar
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physics-verifier-e4-curated-v3

trajectory_id stringlengths 32 40	problem_id stringlengths 11 15	domain stringclasses 6 values	proposed_solution stringlengths 509 4.99k	directive_text stringlengths 86 769	verdict stringclasses 2 values	error_type stringclasses 6 values	part stringclasses 3 values	prompt stringlengths 1.19k 5.89k	target_output stringlengths 100 783	instruction stringlengths 1.19k 5.89k	output stringlengths 100 783
E4_verifier:lit_mech_0307_err_aya8b	lit_mech_0307	MECHANICS	Derivation: Step 1: Flawed Student Derivation Step 2: 1. We start with the general equation for the center of mass: xcm = m1x1 + m2x2 + ... + mnxn Step 3: 2. Since all the masses are equal, we can factor out 'm' from each term: xcm = m(x1 + x2 + ... + xn) Step 4: 3. Now, we incorrectly assume that the sum of the po...	To correctly show that the center of mass equation simplifies to an average, apply the concept of weighted averages. Since all masses are equal, the equation becomes: xcm = (x1 + x2 + ... + xn) / n.	[[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 show that the center of mass equation simplifies to an average, apply the concept of weighted averages. Since all masses are equal, the equation becomes: xcm = (x1 + x2 + ... + xn) / n. [[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 show that the center of mass equation simplifies to an average, apply the concept of weighted averages. Since all masses are equal, the equation becomes: xcm = (x1 + x2 + ... + xn) / n. [[CHALLENGE]]
E4_verifier:lit_optics_0659_err_aya8b	lit_optics_0659	OPTICS	Derivation: Step 1: Identify the given information: The problem provides two diffraction patterns with the same red light wavelength. The top pattern has four fringes and a spacing of 5.88 mm, while the bottom pattern has five fringes with an unknown spacing Step 2: Calculate the numerical ratio: We can find the ra...	To solve this problem, you must understand the relationship between slit width (d), wavelength (λ), and diffraction angle (θ). The key equation is d * sin(θ) = m * λ, where m is an integer (order of diffraction). Since both patterns use the same wavelength, changes in θ are proportional to changes in slit width. Theref...	[[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 solve this problem, you must understand the relationship between slit width (d), wavelength (λ), and diffraction angle (θ). The key equation is d * sin(θ) = m * λ, where m is an integer (order of diffraction). Since both patterns use the same wavelength, changes in θ are proportional to changes in slit width. Theref...	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 understand the relationship between slit width (d), wavelength (λ), and diffraction angle (θ). The key equation is d * sin(θ) = m * λ, where m is an integer (order of diffraction). Since both patterns use the same wavelength, changes in θ are proportional to changes in slit width. Theref...
E4_verifier:lit_em_0021_sol1_approve	lit_em_0021	ELECTROMAGNETISM	Derivation: Step 1: The electric force is either in the same direction as the electric field or in the opposite direction, depending on whether the particles are positively or negatively charged. Step 2: Since the magnetic force is perpendicular to both v and B and the question states that the electric field is per...	Applying the governing principles, identifying the key principles at play. The electric force acts in the direction of the electric field, and the magnetic force is perpendicular to both the velocity and the magnetic field. By assuming forces in opposite directions, we can set them equal in magnitude and solve for the ...	[[APPROVE]]	null	part_1_domain_balanced_approved	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 [[...	Applying the governing principles, identifying the key principles at play. The electric force acts in the direction of the electric field, and the magnetic force is perpendicular to both the velocity and the magnetic field. By assuming forces in opposite directions, we can set them equal in magnitude and solve for 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 [[...	Applying the governing principles, identifying the key principles at play. The electric force acts in the direction of the electric field, and the magnetic force is perpendicular to both the velocity and the magnetic field. By assuming forces in opposite directions, we can set them equal in magnitude and solve for the ...
E4_verifier:lit_thermo_0725_err_aya8b	lit_thermo_0725	THERMODYNAMICS	Derivation: Step 1: Define the Given Values: We are given the following constants and values: n = 6.0 × 10²³ atoms, k = 1.38 × 10⁻²³ J/K, T = 273 K, P = 1.01 × 10⁵ Pa Step 2: Apply the Ideal Gas Law: The ideal gas law is V = nkT/P Step 3: Substitute the Values: Plugging in, we get V = (6.0 × 10²³)(1.38 × 10⁻²³)(273...	Correct the derivation by using the correct conversion factor of 1/1000 to convert from cubic meters to liters The conversion factor between cubic meters (m³) and liters (L) is 1000 L/m³.	[[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 [[...	Correct the derivation by using the correct conversion factor of 1/1000 to convert from cubic meters to liters The conversion factor between cubic meters (m³) and liters (L) is 1000 L/m³. [[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 [[...	Correct the derivation by using the correct conversion factor of 1/1000 to convert from cubic meters to liters The conversion factor between cubic meters (m³) and liters (L) is 1000 L/m³. [[CHALLENGE]]
E4_verifier:lit_em_0133_err_aya8b	lit_em_0133	ELECTROMAGNETISM	Derivation: Step 1: We are given that the energy density is proportional to the square of the electric field, so we can write: α = (E1 + E2)^2, where α is a unitless constant, just Step 2: The dot product of the two fields gives: E2·(E1 + E2) = E21 + 2E1·E2 + E22, but here we make the error of assuming this is equa...	To find the angle between the two field vectors, you must use the relationship between the dot product of the fields and their magnitudes, along with the given energy density proportionality. The correct algebraic step is to use the dot product to express one component of the field in terms of the other and the angle, ...	[[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 find the angle between the two field vectors, you must use the relationship between the dot product of the fields and their magnitudes, along with the given energy density proportionality. The correct algebraic step is to use the dot product to express one component of the field in terms of the other and the angle, ...	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 find the angle between the two field vectors, you must use the relationship between the dot product of the fields and their magnitudes, along with the given energy density proportionality. The correct algebraic step is to use the dot product to express one component of the field in terms of the other and the angle, ...
E4_verifier:lit_mech_0203_err_aya8b	lit_mech_0203	MECHANICS	Derivation: Step 1: We start with the equation relating angular displacement, angular velocity, and time: ωt = 2π (over one period) Step 2: Similarly, for linear displacement and velocity, we have kx = 2π√(t), where ω and k are constants (note the incorrect sign in the second equation) Step 3: Dividing the two equa...	To correct the derivation, use the correct relationship between angular and linear motion, and ensure consistent units. The correct equation for linear displacement is $kx = 2π$ and not $kx = 2π√(t)$. After dividing these equations, the units on both sides must match. The left side has units of $(m/\cancel{s})$ and the...	[[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 [[...	To correct the derivation, use the correct relationship between angular and linear motion, and ensure consistent units. The correct equation for linear displacement is $kx = 2π$ and not $kx = 2π√(t)$. After dividing these equations, the units on both sides must match. The left side has units of $(m/\cancel{s})$ 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 [[...	To correct the derivation, use the correct relationship between angular and linear motion, and ensure consistent units. The correct equation for linear displacement is $kx = 2π$ and not $kx = 2π√(t)$. After dividing these equations, the units on both sides must match. The left side has units of $(m/\cancel{s})$ and the...
E4_verifier:lit_mech_0194_err_aya8b	lit_mech_0194	MECHANICS	Derivation: Step 1: Flawed Student Derivation Step 2: 1. Identify the relevant information: We are given that each rectangle in both graphs has a width of 1 second and a height of 1 m/s. This means each rectangle represents a distance of 1 m. 2. Count the rectangles: In (a), there are 14 rectangles between the curv...	To solve this problem, you must consider the direction of motion represented by each rectangle/square, treating positive distances as contributions to displacement and negative distances as subtractions. Use the concept of cumulative displacement to sum the individual distances.	[[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 [[...	To solve this problem, you must consider the direction of motion represented by each rectangle/square, treating positive distances as contributions to displacement and negative distances as subtractions. Use the concept of cumulative displacement to sum the individual distances. [[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 solve this problem, you must consider the direction of motion represented by each rectangle/square, treating positive distances as contributions to displacement and negative distances as subtractions. Use the concept of cumulative displacement to sum the individual distances. [[CHALLENGE]]
E4_verifier:lit_em_0152_tiny_aya	lit_em_0152	ELECTROMAGNETISM	Derivation: Step 1: Identify the particles involved in the beta decay process The beta decay of a neutron involves the transformation of a neutron into a proton, an electron, and an antineutrino. The particles involved are: - Neutron (n) - Proton (p) - Electron (e⁻) - Antineutrino (ν̄) Step 2: Write the mass differ...	To correctly solve this problem, you must apply Einstein's mass-energy equivalence principle, which states that mass and energy are interchangeable through the equation: $$E = mc^2$$ Where: - $E$ is the energy released, - $m$ is the mass difference between the initial and final states, - $c$ is the speed of light.	[[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 solve this problem, you must apply Einstein's mass-energy equivalence principle, which states that mass and energy are interchangeable through the equation: $$E = mc^2$$ Where: - $E$ is the energy released, - $m$ is the mass difference between the initial and final states, - $c$ is the speed of light. [[CH...	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 solve this problem, you must apply Einstein's mass-energy equivalence principle, which states that mass and energy are interchangeable through the equation: $$E = mc^2$$ Where: - $E$ is the energy released, - $m$ is the mass difference between the initial and final states, - $c$ is the speed of light. [[CH...
E4_verifier:lit_mech_0574_err_aya8b	lit_mech_0574	MECHANICS	Derivation: Step 1: Flawed Student Derivation Step 2: 1. We start by recalling the definition of the dot product, as given in our textbook: A·B = \|A\| \|B\| cos(θAB). 2. To prove rotational invariance, we consider a system after a rotation. We'll denote the rotated vectors as A' and B'. 3. Naively, one might assume th...	To prove rotational invariance, recognize that the dot product depends only on the magnitudes of vectors and the cosine of the angle between them: A·B = \|A\|\|B\| cos(θAB) Rotations preserve these magnitudes and angles, ensuring the dot product remains unchanged.	[[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 prove rotational invariance, recognize that the dot product depends only on the magnitudes of vectors and the cosine of the angle between them: A·B = \|A\|\|B\| cos(θAB) Rotations preserve these magnitudes and angles, ensuring the dot product remains unchanged. [[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 prove rotational invariance, recognize that the dot product depends only on the magnitudes of vectors and the cosine of the angle between them: A·B = \|A\|\|B\| cos(θAB) Rotations preserve these magnitudes and angles, ensuring the dot product remains unchanged. [[CHALLENGE]]
E4_verifier:lit_mech_0329_err_aya8b	lit_mech_0329	MECHANICS	Derivation: Step 1: Flawed Student Derivation Step 2: 1. We start with the potential energy of the spring-mass system, given by $U = \frac{1}{2}kL^2 + mgy$, where $L$ is the displacement from equilibrium, $g$ is the acceleration due to gravity, and $k$ is the spring constant. 2. Since we're interested in the period...	To find the period of vertical oscillations, you must use the relationship between the mass, spring constant, and acceleration due to gravity, along with the conservation of mechanical energy. Use the equation for mechanical energy conservation: $U = K$, and the potential energy of the spring-mass system: $U = \frac{1}...	[[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 [[...	To find the period of vertical oscillations, you must use the relationship between the mass, spring constant, and acceleration due to gravity, along with the conservation of mechanical energy. Use the equation for mechanical energy conservation: $U = K$, and the potential energy of the spring-mass system: $U = \frac{1}...	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 find the period of vertical oscillations, you must use the relationship between the mass, spring constant, and acceleration due to gravity, along with the conservation of mechanical energy. Use the equation for mechanical energy conservation: $U = K$, and the potential energy of the spring-mass system: $U = \frac{1}...
E4_verifier:lit_thermo_0715_sol1_approve	lit_thermo_0715	THERMODYNAMICS	Derivation: Step 1: Given Information: We are told that the volume of the cylinder increases by a factor of 5.0, implying that the final volume (V2) is 5 times the initial volume (V1). The final pressure (P2) is given as 1.0 atmosphere Step 2: Ideal Gas Law: The relationship between pressure (P), volume (V), and te...	To derive the governing relation, employing the ideal gas law and the adiabatic index for polyatomic gases to establish a pressure-volume relationship. By substituting given values and applying algebraic manipulation, the initial pressure is correctly calculated as 8.5 atmospheres, adhering to the principles of thermod...	[[APPROVE]]	null	part_1_domain_balanced_approved	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 governing relation, employing the ideal gas law and the adiabatic index for polyatomic gases to establish a pressure-volume relationship. By substituting given values and applying algebraic manipulation, the initial pressure is correctly calculated as 8.5 atmospheres, adhering to the principles of thermod...	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 governing relation, employing the ideal gas law and the adiabatic index for polyatomic gases to establish a pressure-volume relationship. By substituting given values and applying algebraic manipulation, the initial pressure is correctly calculated as 8.5 atmospheres, adhering to the principles of thermod...
E4_verifier:lit_modern_0637_err_aya8b	lit_modern_0637	MODERN_PHYSICS	Derivation: Step 1: Problem Understanding: We are tasked with comparing the number of photons per second emitted by a 100-watt FM radio transmitter and a 100-watt lightbulb, both emitting 100 joules of energy Step 2: Electromagnetic Spectrum Chart: Referring to the chart, we observe that the frequency of FM radio w...	Correct the derivation by emphasizing the massive frequency differencebetween radio waves and visible light, leveraging Planck's constant (h)to show that the photon energy is significantly higher for visible light. Then, conclude that the number of photons in the radio signal is a million times greaterdue to the much l...	[[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 [[...	Correct the derivation by emphasizing the massive frequency differencebetween radio waves and visible light, leveraging Planck's constant (h)to show that the photon energy is significantly higher for visible light. Then, conclude that the number of photons in the radio signal is a million times greaterdue to the much l...	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 derivation by emphasizing the massive frequency differencebetween radio waves and visible light, leveraging Planck's constant (h)to show that the photon energy is significantly higher for visible light. Then, conclude that the number of photons in the radio signal is a million times greaterdue to the much l...
E4_verifier:lit_mech_0160_err_aya8b	lit_mech_0160	MECHANICS	Derivation: Step 1: Understanding the Problem: We are tasked with finding the acceleration of a car traveling at a constant 100 km/h for 100 seconds Step 2: Recall Velocity and Acceleration: Velocity (v) is the change in position over time, and acceleration (a) is the rate at which velocity changes Step 3: Given In...	To correct the derivation, the Solver should recognize that while the change in velocity (∆v) appears to be zero, the constant velocity does not imply the absence of acceleration. Instead, apply Newton's second law of motion, F = ma, where F is the net force acting on the car, m is its mass, and a is the acceleration. ...	[[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 [[...	To correct the derivation, the Solver should recognize that while the change in velocity (∆v) appears to be zero, the constant velocity does not imply the absence of acceleration. Instead, apply Newton's second law of motion, F = ma, where F is the net force acting on the car, m is its mass, and a is the acceleration. ...	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, the Solver should recognize that while the change in velocity (∆v) appears to be zero, the constant velocity does not imply the absence of acceleration. Instead, apply Newton's second law of motion, F = ma, where F is the net force acting on the car, m is its mass, and a is the acceleration. ...
E4_verifier:lit_rel_0703_err_aya8b	lit_rel_0703	RELATIVITY	Derivation: Step 1: Define the Variable: Let's start by defining our variable $x$ as the velocity $v$, squaring it later for simplicity Step 2: Express $\gamma$ in terms of $x$: We know that $\gamma = \frac{1}{\sqrt{1-x^2}}$ Step 3: Find the First Derivative: Differentiating $\gamma$ with respect to $x$, we get $\g...	To correctly expand the equation $K = m(G - 1)$ in a Taylor series and identify the nonrelativistic term, use the following: - Relativistic Lorentz factor:$\gamma = \frac{1}{\sqrt{1-v^2}}$, and its derivatives with respect to $v^2$. - Taylor series expansion:Utilize the power series expansion of $\gamma$ around $v^2 = ...	[[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 expand the equation $K = m(G - 1)$ in a Taylor series and identify the nonrelativistic term, use the following: - Relativistic Lorentz factor:$\gamma = \frac{1}{\sqrt{1-v^2}}$, and its derivatives with respect to $v^2$. - Taylor series expansion:Utilize the power series expansion of $\gamma$ around $v^2 = ...	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 expand the equation $K = m(G - 1)$ in a Taylor series and identify the nonrelativistic term, use the following: - Relativistic Lorentz factor:$\gamma = \frac{1}{\sqrt{1-v^2}}$, and its derivatives with respect to $v^2$. - Taylor series expansion:Utilize the power series expansion of $\gamma$ around $v^2 = ...
E4_verifier:lit_mech_0429_err_aya8b	lit_mech_0429	MECHANICS	Derivation: Step 1: Flawed Student Derivation Step 2: 1. We start by estimating the linear dimensions of a blade of grass. Let's say it's 0.1 cm thick, 1 cm wide, and 10 cm tall, giving a volume of 1 cm³. 2. Assuming it has the same density as water, we can estimate its mass. Using the formula density = mass / volu...	To correctly estimate the kinetic energy of the blade of grass, use the formula KE = (1/2)mv², where mis mass and vis velocity. Ensure you apply the correct division.	[[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 estimate the kinetic energy of the blade of grass, use the formula KE = (1/2)mv², where mis mass and vis velocity. Ensure you apply the correct division. [[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 estimate the kinetic energy of the blade of grass, use the formula KE = (1/2)mv², where mis mass and vis velocity. Ensure you apply the correct division. [[CHALLENGE]]
E4_verifier:lit_modern_0603_err_aya8b	lit_modern_0603	MODERN_PHYSICS	Derivation: Step 1: Given: We are given that the ground state of boron-10 (10B) nucleus has a spin quantum number s = 3 Step 2: Confusing Spin and Particle Mass: In classical physics, we might associate spin with the rotation of an object and thus might assume that a higher spin indicates a heavier particle Step 3:...	To correctly categorize the ground state of 10B, apply the fundamental distinction between fermions and bosons based on their spin statistics, as described by quantum mechanics. Since the given spin (s = 3) is an integer, the ground state of 10B is a boson Explain why the classical association of spin with mass is misl...	[[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 correctly categorize the ground state of 10B, apply the fundamental distinction between fermions and bosons based on their spin statistics, as described by quantum mechanics. Since the given spin (s = 3) is an integer, the ground state of 10B is a boson Explain why the classical association of spin with mass is misl...	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 categorize the ground state of 10B, apply the fundamental distinction between fermions and bosons based on their spin statistics, as described by quantum mechanics. Since the given spin (s = 3) is an integer, the ground state of 10B is a boson Explain why the classical association of spin with mass is misl...
E4_verifier:lit_mech_0513_err_aya8b	lit_mech_0513	MECHANICS	Derivation: Step 1: Problem Setup: We are given the air friction force as $F = be^{-cx}$, and we want to find the kinetic energy lost due to air friction from launch to the point of escape Step 2: Work Done by Air Friction: The work done by the air friction force is given by the integral $\int_0^\infty F \, dx$ Ste...	To correctly find the kinetic energy lost due to air friction, you must consider the work done by the force over the entire altitude range, not just the limits at $0$ and $\infty$. Use the given force law $F = be^{-cx}$ and integrate it from $0$ to $\infty$ to find the work done. The correct integral evaluation involve...	[[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 [[...	To correctly find the kinetic energy lost due to air friction, you must consider the work done by the force over the entire altitude range, not just the limits at $0$ and $\infty$. Use the given force law $F = be^{-cx}$ and integrate it from $0$ to $\infty$ to find the work done. The correct integral evaluation involve...	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 find the kinetic energy lost due to air friction, you must consider the work done by the force over the entire altitude range, not just the limits at $0$ and $\infty$. Use the given force law $F = be^{-cx}$ and integrate it from $0$ to $\infty$ to find the work done. The correct integral evaluation involve...
E4_verifier:lit_mech_0268_err_aya8b	lit_mech_0268	MECHANICS	Derivation: Step 1: We are given the equation ω = k/m, where ω is angular frequency, k is a constant of proportionality, and m is mass Step 2: Examining the units on each side, ω has rad/s, k has no units, and m has kg Step 3: On the left, ω is indeed rad/s, but we must convert m to kg for consistent units. 1 kg = ...	To correct the derivation, consider the dimensional analysis of the equation ω = k/m. The key is to recognize that the units of angular frequency, ω, are rad/s, which implies a dimensionless quantity involving only angular and temporal measurements. Since k is a constant, its units must cancel out with the units of mas...	[[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 [[...	To correct the derivation, consider the dimensional analysis of the equation ω = k/m. The key is to recognize that the units of angular frequency, ω, are rad/s, which implies a dimensionless quantity involving only angular and temporal measurements. Since k is a constant, its units must cancel out with the units of mas...	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, consider the dimensional analysis of the equation ω = k/m. The key is to recognize that the units of angular frequency, ω, are rad/s, which implies a dimensionless quantity involving only angular and temporal measurements. Since k is a constant, its units must cancel out with the units of mas...
E4_verifier:lit_mech_0161_err_aya8b	lit_mech_0161	MECHANICS	Derivation: Step 1: Identify Forces: We consider the forces acting on the elevator car: the force of gravity, $F_g = -mg$, and the maximum tension force from the cable, $F_c = T$ Step 2: Total Force: The net force, $F_{\text{net}}$, is the difference between these forces: $F_{\text{net}} = F_c - F_g = T - mg$ Step ...	To correctly solve the problem, you must consider the cable's tension as the limiting factor, not the net force. Use Newton's second law, the governing physical relation, and the fact that the cable's maximum tension is $T$ to derive: $$m = \frac{T}{a}$$.	[[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 correctly solve the problem, you must consider the cable's tension as the limiting factor, not the net force. Use Newton's second law, the governing physical relation, and the fact that the cable's maximum tension is $T$ to derive: $$m = \frac{T}{a}$$. [[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 solve the problem, you must consider the cable's tension as the limiting factor, not the net force. Use Newton's second law, the governing physical relation, and the fact that the cable's maximum tension is $T$ to derive: $$m = \frac{T}{a}$$. [[CHALLENGE]]
E4_verifier:lit_mech_0517_err_aya8b	lit_mech_0517	MECHANICS	Derivation: Step 1: Flawed Derivation Step 2: 1. We are given z = 3 + i, and we want to find arg(1/z). First, let's recall that the argument of a complex number z = a + bi is given by arctan(b/a). 2. To find arg(1/z), we need to find the argument of its real and imaginary parts separately. So, 1/z = (1/a) - (b/a)i....	To correctly determine the tension in the string, use Newton's Second Law for static friction, which states that the frictional force (µkN2) is equal to the applied force (T). Apply Newton's Second Law for static friction: T = µkN.	[[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 determine the tension in the string, use Newton's Second Law for static friction, which states that the frictional force (µkN2) is equal to the applied force (T). Apply Newton's Second Law for static friction: T = µkN. [[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 tension in the string, use Newton's Second Law for static friction, which states that the frictional force (µkN2) is equal to the applied force (T). Apply Newton's Second Law for static friction: T = µkN. [[CHALLENGE]]
E4_verifier:lit_modern_0631_err_aya8b	lit_modern_0631	MODERN_PHYSICS	Derivation: Step 1: Flawed Student Derivation Step 2: 1. We know that a nutritional "calorie" is equivalent to 4.184 joules (as per the definition of a kilocalorie and the Joule). So, 150 kilocalories is equal to 150 * 4.184 J = 627.6 J. Step 3: 2. The food in the bowl weighs approximately 0.2 kg, and assuming each...	To make an order-of-magnitude estimate of the chemical energy stored in one atom of food, use Einstein's equation E = mc²to relate mass and energy. Since the mass of a typical atom is given as 10^-26 kg, calculate the corresponding energy in joules.	[[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 make an order-of-magnitude estimate of the chemical energy stored in one atom of food, use Einstein's equation E = mc²to relate mass and energy. Since the mass of a typical atom is given as 10^-26 kg, calculate the corresponding energy in joules. [[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 make an order-of-magnitude estimate of the chemical energy stored in one atom of food, use Einstein's equation E = mc²to relate mass and energy. Since the mass of a typical atom is given as 10^-26 kg, calculate the corresponding energy in joules. [[CHALLENGE]]
E4_verifier:lit_em_0043_err_aya8b	lit_em_0043	ELECTROMAGNETISM	Derivation: Step 1: Define Inductance: We start with the definition of inductance, U = LI²/2, where U is the energy stored, L is the inductance, and I is the current Step 2: Parallel Inductors: When two identical inductors are connected in parallel, we can treat them as a single inductor with an effective inductanc...	To correct the derivation, use the principle that the total energy stored in a parallel circuit of identical inductors is twice the energy stored in one inductor, and equate this to the inductance definition. Utilize the relationship between total energy, inductance, and current in parallel circuits: U = 2 * Uone, wher...	[[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 correct the derivation, use the principle that the total energy stored in a parallel circuit of identical inductors is twice the energy stored in one inductor, and equate this to the inductance definition. Utilize the relationship between total energy, inductance, and current in parallel circuits: U = 2 * Uone, wher...	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, use the principle that the total energy stored in a parallel circuit of identical inductors is twice the energy stored in one inductor, and equate this to the inductance definition. Utilize the relationship between total energy, inductance, and current in parallel circuits: U = 2 * Uone, wher...
E4_verifier:lit_mech_0426_err_aya8b	lit_mech_0426	MECHANICS	Derivation: Step 1: 1. We begin by considering the forces acting on the board, which include the gravitational force (mg), the normal forces (N1 and N2) from the wheels, and the frictional forces (µk * N1 and µk * N2) acting in opposite directions. Step 2: 2. Assuming the board is not rotating and kinetic friction ...	To correctly characterize the board's motion, consider the coupled translational and rotational dynamics of the system. Use Newton's laws of motion and the conservation of angular momentum to account for the interaction between the rotating wheels and the board. The governing equations will include both linear and angu...	[[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 correctly characterize the board's motion, consider the coupled translational and rotational dynamics of the system. Use Newton's laws of motion and the conservation of angular momentum to account for the interaction between the rotating wheels and the board. The governing equations will include both linear and angu...	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 characterize the board's motion, consider the coupled translational and rotational dynamics of the system. Use Newton's laws of motion and the conservation of angular momentum to account for the interaction between the rotating wheels and the board. The governing equations will include both linear and angu...
E4_verifier:physgym_0072_err_aya8b	physgym_0072	THERMODYNAMICS	Derivation: Step 1: Initial Conditions: We are given the initial pressures and volumes, and the adiabatic and quasistatic nature of the problem, as described Step 2: Energy Conservation (Incorrect Application): Assuming constant acceleration of the piston, we apply Newton's second law to the entire system (gas + pi...	To correctly solve this problem, you must consider the adiabatic and quasistatic nature of the processes, apply the first law of thermodynamics, and use the ideal gas law with the adiabatic index $\gamma=5/3$. The key equations involve the conservation of energy and pressure. Use the first law of thermodynamics, the id...	[[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 correctly solve this problem, you must consider the adiabatic and quasistatic nature of the processes, apply the first law of thermodynamics, and use the ideal gas law with the adiabatic index $\gamma=5/3$. The key equations involve the conservation of energy and pressure. Use the first law of thermodynamics, the id...	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 solve this problem, you must consider the adiabatic and quasistatic nature of the processes, apply the first law of thermodynamics, and use the ideal gas law with the adiabatic index $\gamma=5/3$. The key equations involve the conservation of energy and pressure. Use the first law of thermodynamics, the id...
E4_verifier:lit_em_0066_err_aya8b	lit_em_0066	ELECTROMAGNETISM	Derivation: Step 1: Electric Field Expression: We begin with the standard expression for the electric field of a uniform ring $$ E = \frac{1}{4\pi\epsilon_0} \frac{q}{(a^2 + b^2)^{3/2}} $$ Step 2: Behavior for a = 0: When $a = 0$, we can simplify the expression as $$ E = \frac{1}{4\pi\epsilon_0} \frac{q}{b^3} $$ St...	To correctly capture the behavior for both $a = 0$ and $a \gg b$, you must consider the full expression for the electric field of a uniform ring and apply the appropriate limits. The key is to use the original formula: $$ E = \frac{1}{4\pi\epsilon_0} \frac{q}{(a^2 + b^2)^{3/2}} $$ and evaluate it for the given cases. F...	[[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 correctly capture the behavior for both $a = 0$ and $a \gg b$, you must consider the full expression for the electric field of a uniform ring and apply the appropriate limits. The key is to use the original formula: $$ E = \frac{1}{4\pi\epsilon_0} \frac{q}{(a^2 + b^2)^{3/2}} $$ and evaluate it for the given cases. F...	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 capture the behavior for both $a = 0$ and $a \gg b$, you must consider the full expression for the electric field of a uniform ring and apply the appropriate limits. The key is to use the original formula: $$ E = \frac{1}{4\pi\epsilon_0} \frac{q}{(a^2 + b^2)^{3/2}} $$ and evaluate it for the given cases. F...
E4_verifier:lit_mech_0183_err_aya8b	lit_mech_0183	MECHANICS	Derivation: Step 1: ## Flawed Student Derivation Step 2: 1. We are given that Roy's mass (m1) is 60 kg and Laurie's mass (m2) is 65 kg. They are separated by a distance (r) of 1.5 meters. Step 3: 2. Since they are on Earth, we can use Newton's Law of Universal Gravitation: F = G * (m1 * m2) / r^2. Step 4: 3. Plug...	To solve this problem, use Newton's Law of Universal Gravitation: the governing physical relation.	[[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 [[...	To solve this problem, use Newton's Law of Universal Gravitation: the governing physical relation. [[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 solve this problem, use Newton's Law of Universal Gravitation: the governing physical relation. [[CHALLENGE]]

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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-v3

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 (720 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]] (108 examples), or
A concise, self-contained physical/mathematical directive ending with [[CHALLENGE]] (612 examples across synthetic and real flawed student attempts).
Role in Ablation Ladder: Paired companion critic for E4 Solver (758 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`	720	`108` `[[APPROVE]]` + `612` `[[CHALLENGE]]`

Training Split Domain Breakdown (`720` Examples)

Physics Domain	`[[APPROVE]]` (Part 1)	`[[CHALLENGE]]` Synth (Part 2)	`[[CHALLENGE]]` Tiny-Aya (Part 3)	Total	Share
`MECHANICS`	18	345	10	373	51.8%
`ELECTROMAGNETISM`	18	115	5	138	19.2%
`RELATIVITY`	18	27	18	63	8.8%
`MODERN_PHYSICS`	18	39	3	60	8.3%
`OPTICS`	18	17	17	52	7.2%
`THERMODYNAMICS`	18	15	1	34	4.7%
Total	108	558	54	720	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 720 rows.
curatorkit.gates.schema.SchemaGate (min_tokens = 20, max_tokens = 8192): 720 passed, 0 rejected.
curatorkit.normalizers.dedup.ExactDeduplicator: 720 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:lit_mech_0481_err_aya8b",
  "problem_id": "lit_mech_0481",
  "domain": "MECHANICS",
  "proposed_solution": "Step 1: **Vertical Forces:** Similar to the reference solution, we start by considering the vertical forces on the brick, writing T sinθ = Mg.\nStep 2: **Horizontal Forces:** In the horizontal direction, we note...\n...\nANSWER: 9.8 R / cosθ",
  "directive_text": "To correctly find the speed of the brick, use the relationship between the gravitational force, tension, and the radius of the circular path: T = Mv Then, substitute v = Mg / (T tanθ), where Tis related to the tension and the angle θ.",
  "verdict": "[[CHALLENGE]]",
  "error_type": "dimensional_unit_error",
  "part": "part_2_synthetic_aya8b_challenged",
  "prompt": "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/cons...\n...\nANSWER: 9.8 R / cosθ",
  "target_output": "To correctly find the speed of the brick, use the relationship between the gravitational force, tension, and the radius of the circular path: T = Mv Then, substitute v = Mg / (T tanθ), where Tis related to the tension...\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