Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.44 and add it to the current value.
Final: {"answer": [0.388, 0.066, -1.126], "unit": "temperature_units"}

Answer: 0.388

Answer: 0.066

Answer: -1.126

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=2.2, right=-1.8; current interior values are [-0.8, -1.1, 1.25]. Use r=0.44. Return the next interior vector.

Keep the boundary values fixed and update only the three interior grid points.
For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
Multiply each Laplacian by r=0.44 and add it to the current value.


Response:
Reasoning:
1. The function contract has a normal path and at least one edge case.
2. The implementation checks the edge case before returning the normal result.

```python
def running_total_005542(values):
    total = 0
    out = []
    for value in values:
        total += value
        out.append(total)
    return out
```
Final: {"answer": "python_code", "function": "running_total_005542"}

Answer: python_code

Prompt: Write a Python function running_total_005542(values) that returns cumulative sums as a new list. It must not mutate the input list. Return a single fenced Python code block.

Identify required edge cases before writing the function.
Implement the shortest total function that satisfies the stated contract.
Avoid global state and return deterministic outputs.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 3050 mL to 3.05 L.
3. Compute M = 3.035 / 3.05.
Final: {"answer": 0.9951, "unit": "mol/L"}

Answer: 0.9951

Prompt: A solution contains 3.035 mol of solute in 3050 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 3050 mL to 3.05 L.
Compute M = 3.035 / 3.05.


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 8.45 kg and a = 11.5 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 97.175, "unit": "N"}

Answer: 97.175

Prompt: A cart has mass 8.45 kg and measured acceleration 11.5 m/s^2. Compute the net force. Check the SI units and give only the final JSON after reasoning.

Use Newton's second law F = m a.
Substitute m = 8.45 kg and a = 11.5 m/s^2.
The unit kg*m/s^2 is one newton.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 1955 mL to 1.955 L.
3. Compute M = 2.7 / 1.955.
Final: {"answer": 1.3811, "unit": "mol/L"}

Answer: 1.3811

Prompt: A solution contains 2.7 mol of solute in 1955 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 1955 mL to 1.955 L.
Compute M = 2.7 / 1.955.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=27 degrees and keep the unit as newtons.
Final: {"answer": [65.9345, 33.5953], "unit": "N"}

Answer: 65.9345

Answer: 33.5953

Prompt: A vector has magnitude 74 N at 27 degrees counterclockwise from the +x axis. Return its [x, y] components in newtons.

The x component is magnitude times cos(theta).
The y component is magnitude times sin(theta).
Use theta=27 degrees and keep the unit as newtons.

[magnitude * math.cos(math.radians(angle_deg)), magnitude * math.sin(math.radians(angle_deg))]


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=29 degrees through the sine of 2 theta.
3. Substitute v=59 m/s and g=9.81 m/s^2.
Final: {"answer": 300.9231, "unit": "m"}

Answer: 300.9231

Prompt: A projectile is launched on level ground with no air resistance at speed 59 m/s and angle 29 degrees. Compute the horizontal range in meters using g=9.81 m/s^2.

For level-ground projectile motion, R = v^2 sin(2 theta) / g.
Convert theta=29 degrees through the sine of 2 theta.
Substitute v=59 m/s and g=9.81 m/s^2.

speed ** 2 * math.sin(math.radians(2 * angle_deg)) / gravity


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 45 * 4.61 * 16.
Final: {"answer": 3319.2, "unit": "J"}

Answer: 3319.2

Prompt: A 45 g sample has specific heat 4.61 J/(g*K). How much heat is needed to raise its temperature by 16 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 45 * 4.61 * 16.


Response:
Reasoning:
1. Kelvin equals Celsius plus 273.15.
2. Apply the conversion factor directly to the given value.
3. Compute 4.5 + 273.15.
Final: {"answer": 277.65, "unit": "K"}

Answer: 277.65

Prompt: Convert 4.5 degC to K. Return the converted value with unit.

Kelvin equals Celsius plus 273.15.
Apply the conversion factor directly to the given value.
Compute 4.5 + 273.15.


Response:
Reasoning:
1. The function contract has a normal path and at least one edge case.
2. The implementation checks the edge case before returning the normal result.

```python
def finite_differences_005614(values):
    return [values[i + 1] - values[i] for i in range(len(values) - 1)]
```
Final: {"answer": "python_code", "function": "finite_differences_005614"}

Answer: python_code

Prompt: Write a Python function finite_differences_005614(values) that returns the list of successive differences values[i+1] - values[i]. It must return [] for fewer than two values. Return a single fenced Python code block.

Identify required edge cases before writing the function.
Implement the shortest total function that satisfies the stated contract.
Avoid global state and return deterministic outputs.

assert finite_differences_005614([5]) == []

assert finite_differences_005614([]) == []


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 40.6^2 and multiply by 0.5 * 4.45.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 3667.601, "unit": "J"}

Answer: 3667.601

Prompt: A body of mass 4.45 kg moves at 40.6 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 40.6^2 and multiply by 0.5 * 4.45.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 240 * 2.69 * 36.
Final: {"answer": 23241.6, "unit": "J"}

Answer: 23241.6

Prompt: A 240 g sample has specific heat 2.69 J/(g*K). How much heat is needed to raise its temperature by 36 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 240 * 2.69 * 36.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=100.3 V and R=64.1 ohm.
Final: {"answer": 156.9437, "unit": "W"}

Answer: 156.9437

Prompt: A resistor has resistance 64.1 ohm and voltage 100.3 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=100.3 V and R=64.1 ohm.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.6 y, so each step multiplies y by 1 - h*0.6.
3. Apply that update for 9 steps and round the final state.
Final: {"answer": 0.0, "unit": "state_units"}

Answer: 0.0

Prompt: Use explicit Euler on dy/dt = -0.6 y with y(0)=45.5, step size h=1.75, for 9 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.6 y, so each step multiplies y by 1 - h*0.6.
Apply that update for 9 steps and round the final state.


Response:
Reasoning:
1. There are 100 centimeters in one meter.
2. Apply the conversion factor directly to the given value.
3. Compute 1441.5 / 100.
Final: {"answer": 14.415, "unit": "m"}

Answer: 14.415

Prompt: Convert 1441.5 cm to m. Return the converted value with unit.

There are 100 centimeters in one meter.
Apply the conversion factor directly to the given value.
Compute 1441.5 / 100.


Response:
Reasoning:
1. There are 100 centimeters in one meter.
2. Apply the conversion factor directly to the given value.
3. Compute 445.5 / 100.
Final: {"answer": 4.455, "unit": "m"}

Answer: 4.455

Prompt: Convert 445.5 cm to m. Return the converted value with unit.

There are 100 centimeters in one meter.
Apply the conversion factor directly to the given value.
Compute 445.5 / 100.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=162 mg, t=5 h, and T_half=30 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 144.3256, "unit": "mg"}

Answer: 144.3256

Prompt: A radioactive sample starts with 162 mg. Its half-life is 30 hours. How many mg remain after 5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=162 mg, t=5 h, and T_half=30 h.
The remaining amount has the same mass unit as the initial amount.

initial_amount * 0.5 ** (elapsed_time / half_life)


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.48 y, so each step multiplies y by 1 - h*0.48.
3. Apply that update for 12 steps and round the final state.
Final: {"answer": 0.0012, "unit": "state_units"}

Answer: 0.0012

Prompt: Use explicit Euler on dy/dt = -0.48 y with y(0)=35, step size h=1.2, for 12 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.48 y, so each step multiplies y by 1 - h*0.48.
Apply that update for 12 steps and round the final state.


Response:
Reasoning:
1. Compare the dimensions on both sides of the equation.
2. grams cancel with J/(g*K), and kelvin cancels, leaving joules
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "consistent"}

Answer: consistent

Prompt: In lab notebook case 500916, check dimensional consistency for Q = m * c * DeltaT. The left side has dimensions J and the right side has dimensions J. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
grams cancel with J/(g*K), and kelvin cancels, leaving joules
The equation is valid only if both sides have identical dimensions.

Answer: consistent


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 295 * 4.365 * 52.
Final: {"answer": 66959.1, "unit": "J"}

Answer: 66959.1

Prompt: A 295 g sample has specific heat 4.365 J/(g*K). How much heat is needed to raise its temperature by 52 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 295 * 4.365 * 52.


Response:
Reasoning:
1. Check whether all variables required by the governing equation or specification are present.
2. The missing required information is: volume.
3. A numeric or code answer would require inventing unstated assumptions, so abstain.
Final: {"answer": "INSUFFICIENT_INFO", "missing": ["volume"]}

Answer: INSUFFICIENT_INFO

Prompt: Compute pressure from the ideal gas law. The prompt gives n=3.54 mol and T=480 K, but does not give volume. If there is insufficient information, say so explicitly.

Check whether all variables required by the governing equation or specification are present.
The missing required information is: volume.
A numeric or code answer would require inventing unstated assumptions, so abstain.


Response:
Reasoning:
1. Check whether all variables required by the governing equation or specification are present.
2. The missing required information is: recurrence, objective, tie_breaking_rule.
3. A numeric or code answer would require inventing unstated assumptions, so abstain.
Final: {"answer": "INSUFFICIENT_INFO", "missing": ["recurrence", "objective", "tie_breaking_rule"]}

Answer: INSUFFICIENT_INFO

Prompt: Write a dynamic-programming solver for a scheduler scoring problem with 4801 items and budget 99085, but the recurrence, objective, and tie-breaking rule are all absent. If there is insufficient information, say so explicitly.

Check whether all variables required by the governing equation or specification are present.
The missing required information is: recurrence, objective, tie_breaking_rule.
A numeric or code answer would require inventing unstated assumptions, so abstain.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 1.89 mol * 180.156 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 340.4948, "unit": "g"}

Answer: 340.4948

Prompt: A sample contains 1.89 mol of glucose. Using molar mass 180.156 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 1.89 mol * 180.156 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 41.75^2 and multiply by 0.5 * 17.85.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 15556.8328, "unit": "J"}

Answer: 15556.8328

Prompt: A body of mass 17.85 kg moves at 41.75 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 41.75^2 and multiply by 0.5 * 17.85.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 4245 mL to 4.245 L.
3. Compute M = 2.485 / 4.245.
Final: {"answer": 0.5854, "unit": "mol/L"}

Answer: 0.5854

Prompt: A solution contains 2.485 mol of solute in 4245 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 4245 mL to 4.245 L.
Compute M = 2.485 / 4.245.


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 42.5^2 and multiply by 0.5 * 4.95.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 4470.4688, "unit": "J"}

Answer: 4470.4688

Prompt: A body of mass 4.95 kg moves at 42.5 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 42.5^2 and multiply by 0.5 * 4.95.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 4215 mL to 4.215 L.
3. Compute M = 2.81 / 4.215.
Final: {"answer": 0.6667, "unit": "mol/L"}

Answer: 0.6667

Prompt: A solution contains 2.81 mol of solute in 4215 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 4215 mL to 4.215 L.
Compute M = 2.81 / 4.215.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 0.82 * 8.314 * 589 / 31.5.
Final: {"answer": 127.4761, "unit": "kPa"}

Answer: 127.4761

Prompt: An ideal gas sample has n=0.82 mol, T=589 K, and V=31.5 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 0.82 * 8.314 * 589 / 31.5.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=148.7 V and R=119.4 ohm.
Final: {"answer": 185.19, "unit": "W"}

Answer: 185.19

Prompt: A resistor has resistance 119.4 ohm and voltage 148.7 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=148.7 V and R=119.4 ohm.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(8.7)=19.53 and f(13.1)=23.49.
3. The interval width is 4.4.
Final: {"answer": 94.644, "unit": "area_units"}

Answer: 94.644

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=0.9x+11.7 from x=8.7 to x=13.1. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(8.7)=19.53 and f(13.1)=23.49.
The interval width is 4.4.

(x1 - x0) * ((slope * x0 + intercept) + (slope * x1 + intercept)) / 2


Response:
Reasoning:
1. The function contract has a normal path and at least one edge case.
2. The implementation checks the edge case before returning the normal result.

```python
def clamp_005794(x, lo, hi):
    if lo > hi:
        raise ValueError('lo must be <= hi')
    return min(max(x, lo), hi)
```
Final: {"answer": "python_code", "function": "clamp_005794"}

Answer: python_code

Prompt: Write a Python function clamp_005794(x, lo, hi) that clips x into the inclusive interval [lo, hi]. It must raise ValueError if lo > hi. Return a single fenced Python code block.

Identify required edge cases before writing the function.
Implement the shortest total function that satisfies the stated contract.
Avoid global state and return deterministic outputs.

try:
    clamp_005794(1, 3, 2)
    raise AssertionError('expected ValueError')
except ValueError:
    pass


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 1020 * 1.625 * 46.
Final: {"answer": 76245.0, "unit": "J"}

Answer: 76245.0

Prompt: A 1020 g sample has specific heat 1.625 J/(g*K). How much heat is needed to raise its temperature by 46 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1020 * 1.625 * 46.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=23 degrees through the sine of 2 theta.
3. Substitute v=44.2 m/s and g=9.81 m/s^2.
Final: {"answer": 143.2549, "unit": "m"}

Answer: 143.2549

Prompt: A projectile is launched on level ground with no air resistance at speed 44.2 m/s and angle 23 degrees. Compute the horizontal range in meters using g=9.81 m/s^2.

For level-ground projectile motion, R = v^2 sin(2 theta) / g.
Convert theta=23 degrees through the sine of 2 theta.
Substitute v=44.2 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 750 mL to 0.75 L.
3. Compute M = 2.425 / 0.75.
Final: {"answer": 3.2333, "unit": "mol/L"}

Answer: 3.2333

Prompt: A solution contains 2.425 mol of solute in 750 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 750 mL to 0.75 L.
Compute M = 2.425 / 0.75.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=80 degrees and keep the unit as newtons.
Final: {"answer": [1.1287, 6.4013], "unit": "N"}

Answer: 1.1287

Answer: 6.4013

Prompt: A vector has magnitude 6.5 N at 80 degrees counterclockwise from the +x axis. Return its [x, y] components in newtons.

The x component is magnitude times cos(theta).
The y component is magnitude times sin(theta).
Use theta=80 degrees and keep the unit as newtons.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 500 * 1.555 * 110.
Final: {"answer": 85525.0, "unit": "J"}

Answer: 85525.0

Prompt: A 500 g sample has specific heat 1.555 J/(g*K). How much heat is needed to raise its temperature by 110 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 500 * 1.555 * 110.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 2770 mL to 2.77 L.
3. Compute M = 3.515 / 2.77.
Final: {"answer": 1.269, "unit": "mol/L"}

Answer: 1.269

Prompt: A solution contains 3.515 mol of solute in 2770 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 2770 mL to 2.77 L.
Compute M = 3.515 / 2.77.


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.22 and add it to the current value.
Final: {"answer": [1.384, 0.876, 0.921], "unit": "temperature_units"}

Answer: 1.384

Answer: 0.876

Answer: 0.921

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=0.2, right=-1.65; current interior values are [2.55, -0.4, 2.45]. Use r=0.22. Return the next interior vector.

Keep the boundary values fixed and update only the three interior grid points.
For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
Multiply each Laplacian by r=0.22 and add it to the current value.


Response:
Reasoning:
1. Check whether all variables required by the governing equation or specification are present.
2. The missing required information is: launch_angle_or_vertical_velocity.
3. A numeric or code answer would require inventing unstated assumptions, so abstain.
Final: {"answer": "INSUFFICIENT_INFO", "missing": ["launch_angle_or_vertical_velocity"]}

Answer: INSUFFICIENT_INFO

Prompt: Compute projectile range on level ground. The prompt gives initial speed 60.2 m/s but does not give launch angle or vertical velocity. If there is insufficient information, say so explicitly.

Check whether all variables required by the governing equation or specification are present.
The missing required information is: launch_angle_or_vertical_velocity.
A numeric or code answer would require inventing unstated assumptions, so abstain.


Response:
Reasoning:
1. Compare the dimensions on both sides of the equation.
2. velocity divided by time is acceleration
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "inconsistent"}

Answer: inconsistent

Prompt: In sensor calibration case 886821, check dimensional consistency for x = v / t. The left side has dimensions m and the right side has dimensions m/s^2. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
velocity divided by time is acceleration
The equation is valid only if both sides have identical dimensions.

Answer: inconsistent


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=69 degrees through the sine of 2 theta.
3. Substitute v=46.2 m/s and g=9.81 m/s^2.
Final: {"answer": 145.5881, "unit": "m"}

Answer: 145.5881

Prompt: A projectile is launched on level ground with no air resistance at speed 46.2 m/s and angle 69 degrees. Compute the horizontal range in meters using g=9.81 m/s^2.

For level-ground projectile motion, R = v^2 sin(2 theta) / g.
Convert theta=69 degrees through the sine of 2 theta.
Substitute v=46.2 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 1.64 * 8.314 * 253 / 53.4.
Final: {"answer": 64.6001, "unit": "kPa"}

Answer: 64.6001

Prompt: An ideal gas sample has n=1.64 mol, T=253 K, and V=53.4 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 1.64 * 8.314 * 253 / 53.4.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 1340 * 4.615 * 12.
Final: {"answer": 74209.2, "unit": "J"}

Answer: 74209.2

Prompt: A 1340 g sample has specific heat 4.615 J/(g*K). How much heat is needed to raise its temperature by 12 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1340 * 4.615 * 12.


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.46 and add it to the current value.
Final: {"answer": [-0.963, 1.096, 0.205], "unit": "temperature_units"}

Answer: -0.963

Answer: 1.096

Answer: 0.205

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=-1.2, right=1.6; current interior values are [2.05, -1.25, 0.55]. Use r=0.46. Return the next interior vector.

Keep the boundary values fixed and update only the three interior grid points.
For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
Multiply each Laplacian by r=0.46 and add it to the current value.


Response:
Reasoning:
1. Compare the dimensions on both sides of the equation.
2. density times acceleration times height gives kg/(m*s^2), or pascal
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "consistent"}

Answer: consistent

Prompt: In simulation spec case 259012, check dimensional consistency for p = rho * g * h. The left side has dimensions Pa and the right side has dimensions Pa. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
density times acceleration times height gives kg/(m*s^2), or pascal
The equation is valid only if both sides have identical dimensions.

Answer: consistent


Response:
Reasoning:
1. Check whether all variables required by the governing equation or specification are present.
2. The missing required information is: required_fields, allowed_transformations.
3. A numeric or code answer would require inventing unstated assumptions, so abstain.
Final: {"answer": "INSUFFICIENT_INFO", "missing": ["required_fields", "allowed_transformations"]}

Answer: INSUFFICIENT_INFO

Prompt: Write a sanitizer for valid warehouse ledger records in release 840086, but the prompt does not state which fields are required or which transformations are allowed. If there is insufficient information, say so explicitly.

Check whether all variables required by the governing equation or specification are present.
The missing required information is: required_fields, allowed_transformations.
A numeric or code answer would require inventing unstated assumptions, so abstain.


Response:
Reasoning:
1. The function contract has a normal path and at least one edge case.
2. The implementation checks the edge case before returning the normal result.

```python
def center_values_005920(values):
    if not values:
        raise ValueError('center_values requires at least one value')
    mean = sum(values) / len(values)
    return [value - mean for value in values]
```
Final: {"answer": "python_code", "function": "center_values_005920"}

Answer: python_code

Prompt: Write a Python function center_values_005920(values) that subtracts the arithmetic mean from each value and returns a new list. Raise ValueError for an empty input. Return a single fenced Python code block.

Identify required edge cases before writing the function.
Implement the shortest total function that satisfies the stated contract.
Avoid global state and return deterministic outputs.

try:
    center_values_005920([])
    raise AssertionError('expected ValueError')
except ValueError:
    pass


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 10.1 kg and a = 14 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 141.4, "unit": "N"}

Answer: 141.4

Prompt: A cart has mass 10.1 kg and measured acceleration 14 m/s^2. Compute the net force. Check the SI units and give only the final JSON after reasoning.

Use Newton's second law F = m a.
Substitute m = 10.1 kg and a = 14 m/s^2.
The unit kg*m/s^2 is one newton.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=159.5 V and R=190.3 ohm.
Final: {"answer": 133.685, "unit": "W"}

Answer: 133.685

Prompt: A resistor has resistance 190.3 ohm and voltage 159.5 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=159.5 V and R=190.3 ohm.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 2.03 mol * 100.087 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 203.1766, "unit": "g"}

Answer: 203.1766

Prompt: A sample contains 2.03 mol of calcium carbonate. Using molar mass 100.087 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 2.03 mol * 100.087 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [5.55, 16.0], "unit": "dimensionless"}

Answer: 5.55

Answer: 16.0

Prompt: A line passes through points (5.25, 45.1375) and (16, 104.8). Return [slope, intercept] for y = slope*x + intercept.

The slope is (y2-y1)/(x2-x1).
The intercept is y1 - slope*x1.
Return the pair in [slope, intercept] order.


Response:
Reasoning:
1. Compare the dimensions on both sides of the equation.
2. mass times acceleration has newton dimensions
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "consistent"}

Answer: consistent

Prompt: In student derivation case 698779, check dimensional consistency for F = m * a. The left side has dimensions kg*m/s^2 and the right side has dimensions kg*m/s^2. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
mass times acceleration has newton dimensions
The equation is valid only if both sides have identical dimensions.

Answer: consistent


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 4.48 * 8.314 * 440 / 7.8.
Final: {"answer": 2101.097, "unit": "kPa"}

Answer: 2101.097

Prompt: An ideal gas sample has n=4.48 mol, T=440 K, and V=7.8 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 4.48 * 8.314 * 440 / 7.8.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=604.5 mg, t=193 h, and T_half=37 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 16.2614, "unit": "mg"}

Answer: 16.2614

Prompt: A radioactive sample starts with 604.5 mg. Its half-life is 37 hours. How many mg remain after 193 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=604.5 mg, t=193 h, and T_half=37 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(-4.6)=4.14 and f(1.6)=-10.74.
3. The interval width is 6.2.
Final: {"answer": -20.46, "unit": "area_units"}

Answer: -20.46

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-2.4x+-6.9 from x=-4.6 to x=1.6. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-4.6)=4.14 and f(1.6)=-10.74.
The interval width is 6.2.


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 11.45^2 and multiply by 0.5 * 15.55.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 1019.3219, "unit": "J"}

Answer: 1019.3219

Prompt: A body of mass 15.55 kg moves at 11.45 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 11.45^2 and multiply by 0.5 * 15.55.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.86 y, so each step multiplies y by 1 - h*0.86.
3. Apply that update for 2 steps and round the final state.
Final: {"answer": 6.3315, "unit": "state_units"}

Answer: 6.3315

Prompt: Use explicit Euler on dy/dt = -0.86 y with y(0)=11.5, step size h=0.3, for 2 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.86 y, so each step multiplies y by 1 - h*0.86.
Apply that update for 2 steps and round the final state.


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.26 and add it to the current value.
Final: {"answer": [1.397, 2.241, 1.888], "unit": "temperature_units"}

Answer: 1.397

Answer: 2.241

Answer: 1.888

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=-0.1, right=-0.15; current interior values are [1.8, 2.15, 2.85]. Use r=0.26. Return the next interior vector.

Keep the boundary values fixed and update only the three interior grid points.
For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
Multiply each Laplacian by r=0.26 and add it to the current value.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 2 * 8.314 * 414 / 14.1.
Final: {"answer": 488.2264, "unit": "kPa"}

Answer: 488.2264

Prompt: An ideal gas sample has n=2 mol, T=414 K, and V=14.1 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 2 * 8.314 * 414 / 14.1.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=519 mg, t=160.5 h, and T_half=108 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 185.2692, "unit": "mg"}

Answer: 185.2692

Prompt: A radioactive sample starts with 519 mg. Its half-life is 108 hours. How many mg remain after 160.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=519 mg, t=160.5 h, and T_half=108 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 0.97 mol * 18.015 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 17.4746, "unit": "g"}

Answer: 17.4746

Prompt: A sample contains 0.97 mol of water. Using molar mass 18.015 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 0.97 mol * 18.015 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.28 and add it to the current value.
Final: {"answer": [0.884, 1.068, 0.966], "unit": "temperature_units"}

Answer: 0.884

Answer: 1.068

Answer: 0.966

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=-0.45, right=1.65; current interior values are [1.5, 1.25, 0.35]. Use r=0.28. Return the next interior vector.

Keep the boundary values fixed and update only the three interior grid points.
For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
Multiply each Laplacian by r=0.28 and add it to the current value.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 1940 * 0.8 * 115.
Final: {"answer": 178480.0, "unit": "J"}

Answer: 178480.0

Prompt: A 1940 g sample has specific heat 0.8 J/(g*K). How much heat is needed to raise its temperature by 115 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1940 * 0.8 * 115.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=52 degrees through the sine of 2 theta.
3. Substitute v=47.2 m/s and g=9.81 m/s^2.
Final: {"answer": 220.3531, "unit": "m"}

Answer: 220.3531

Prompt: A projectile is launched on level ground with no air resistance at speed 47.2 m/s and angle 52 degrees. Compute the horizontal range in meters using g=9.81 m/s^2.

For level-ground projectile motion, R = v^2 sin(2 theta) / g.
Convert theta=52 degrees through the sine of 2 theta.
Substitute v=47.2 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [-9.2, 1.0], "unit": "dimensionless"}

Answer: -9.2

Answer: 1.0

Prompt: A line passes through points (-7, 65.4) and (1.75, -15.1). Return [slope, intercept] for y = slope*x + intercept.

The slope is (y2-y1)/(x2-x1).
The intercept is y1 - slope*x1.
Return the pair in [slope, intercept] order.


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 6.85^2 and multiply by 0.5 * 4.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 93.845, "unit": "J"}

Answer: 93.845

Prompt: A body of mass 4 kg moves at 6.85 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 6.85^2 and multiply by 0.5 * 4.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=897.5 mg, t=161.5 h, and T_half=20 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 3.3283, "unit": "mg"}

Answer: 3.3283

Prompt: A radioactive sample starts with 897.5 mg. Its half-life is 20 hours. How many mg remain after 161.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=897.5 mg, t=161.5 h, and T_half=20 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. The function contract has a normal path and at least one edge case.
2. The implementation checks the edge case before returning the normal result.

```python
def clamp_006100(x, lo, hi):
    if lo > hi:
        raise ValueError('lo must be <= hi')
    return min(max(x, lo), hi)
```
Final: {"answer": "python_code", "function": "clamp_006100"}

Answer: python_code

Prompt: Write a Python function clamp_006100(x, lo, hi) that clips x into the inclusive interval [lo, hi]. It must raise ValueError if lo > hi. Return a single fenced Python code block.

Identify required edge cases before writing the function.
Implement the shortest total function that satisfies the stated contract.
Avoid global state and return deterministic outputs.

try:
    clamp_006100(1, 3, 2)
    raise AssertionError('expected ValueError')
except ValueError:
    pass


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 21.2^2 and multiply by 0.5 * 15.05.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 3382.036, "unit": "J"}

Answer: 3382.036

Prompt: A body of mass 15.05 kg moves at 21.2 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 21.2^2 and multiply by 0.5 * 15.05.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=34 degrees and keep the unit as newtons.
Final: {"answer": [32.747, 22.0881], "unit": "N"}

Answer: 32.747

Answer: 22.0881

Prompt: A vector has magnitude 39.5 N at 34 degrees counterclockwise from the +x axis. Return its [x, y] components in newtons.

The x component is magnitude times cos(theta).
The y component is magnitude times sin(theta).
Use theta=34 degrees and keep the unit as newtons.


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 5.3 kg and a = 14.5 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 76.85, "unit": "N"}

Answer: 76.85

Prompt: A cart has mass 5.3 kg and measured acceleration 14.5 m/s^2. Compute the net force. Check the SI units and give only the final JSON after reasoning.

Use Newton's second law F = m a.
Substitute m = 5.3 kg and a = 14.5 m/s^2.
The unit kg*m/s^2 is one newton.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 4.78 * 8.314 * 259 / 40.0.
Final: {"answer": 257.3225, "unit": "kPa"}

Answer: 257.3225

Prompt: An ideal gas sample has n=4.78 mol, T=259 K, and V=40.0 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 4.78 * 8.314 * 259 / 40.0.


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 2.55^2 and multiply by 0.5 * 9.75.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 31.6997, "unit": "J"}

Answer: 31.6997

Prompt: A body of mass 9.75 kg moves at 2.55 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 2.55^2 and multiply by 0.5 * 9.75.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 1595 * 2.84 * 59.
Final: {"answer": 267258.2, "unit": "J"}

Answer: 267258.2

Prompt: A 1595 g sample has specific heat 2.84 J/(g*K). How much heat is needed to raise its temperature by 59 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1595 * 2.84 * 59.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=217.5 mg, t=179 h, and T_half=60 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 27.5034, "unit": "mg"}

Answer: 27.5034

Prompt: A radioactive sample starts with 217.5 mg. Its half-life is 60 hours. How many mg remain after 179 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=217.5 mg, t=179 h, and T_half=60 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.14 y, so each step multiplies y by 1 - h*0.14.
3. Apply that update for 9 steps and round the final state.
Final: {"answer": 23.5184, "unit": "state_units"}

Answer: 23.5184

Prompt: Use explicit Euler on dy/dt = -0.14 y with y(0)=68.5, step size h=0.8, for 9 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.14 y, so each step multiplies y by 1 - h*0.14.
Apply that update for 9 steps and round the final state.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=304 degrees and keep the unit as newtons.
Final: {"answer": [137.0023, -203.1142], "unit": "N"}

Answer: 137.0023

Answer: -203.1142

Prompt: A vector has magnitude 245 N at 304 degrees counterclockwise from the +x axis. Return its [x, y] components in newtons.

The x component is magnitude times cos(theta).
The y component is magnitude times sin(theta).
Use theta=304 degrees and keep the unit as newtons.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 750 * 3.49 * 18.
Final: {"answer": 47115.0, "unit": "J"}

Answer: 47115.0

Prompt: A 750 g sample has specific heat 3.49 J/(g*K). How much heat is needed to raise its temperature by 18 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 750 * 3.49 * 18.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(6.8)=4.72 and f(11.4)=6.56.
3. The interval width is 4.6.
Final: {"answer": 25.944, "unit": "area_units"}

Answer: 25.944

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=0.4x+2 from x=6.8 to x=11.4. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(6.8)=4.72 and f(11.4)=6.56.
The interval width is 4.6.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 5.02 * 8.314 * 519 / 114.7.
Final: {"answer": 188.8503, "unit": "kPa"}

Answer: 188.8503

Prompt: An ideal gas sample has n=5.02 mol, T=519 K, and V=114.7 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 5.02 * 8.314 * 519 / 114.7.


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.26 and add it to the current value.
Final: {"answer": [1.814, 2.183, 1.24], "unit": "temperature_units"}

Answer: 1.814

Answer: 2.183

Answer: 1.24

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=1.35, right=-0.95; current interior values are [1.45, 2.95, 1.5]. Use r=0.26. Return the next interior vector.

Keep the boundary values fixed and update only the three interior grid points.
For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
Multiply each Laplacian by r=0.26 and add it to the current value.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=293 degrees and keep the unit as newtons.
Final: {"answer": [149.8454, -353.0136], "unit": "N"}

Answer: 149.8454

Answer: -353.0136

Prompt: A vector has magnitude 383.5 N at 293 degrees counterclockwise from the +x axis. Return its [x, y] components in newtons.

The x component is magnitude times cos(theta).
The y component is magnitude times sin(theta).
Use theta=293 degrees and keep the unit as newtons.


Response:
Reasoning:
1. Check whether all variables required by the governing equation or specification are present.
2. The missing required information is: rate_constant, number_of_steps.
3. A numeric or code answer would require inventing unstated assumptions, so abstain.
Final: {"answer": "INSUFFICIENT_INFO", "missing": ["rate_constant", "number_of_steps"]}

Answer: INSUFFICIENT_INFO

Prompt: Use explicit Euler for dy/dt = -k y. The prompt gives y(0)=51 and h=0.45, but omits both k and the number of steps. If there is insufficient information, say so explicitly.

Check whether all variables required by the governing equation or specification are present.
The missing required information is: rate_constant, number_of_steps.
A numeric or code answer would require inventing unstated assumptions, so abstain.


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 5.7^2 and multiply by 0.5 * 19.45.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 315.9653, "unit": "J"}

Answer: 315.9653

Prompt: A body of mass 19.45 kg moves at 5.7 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 5.7^2 and multiply by 0.5 * 19.45.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=22 degrees through the sine of 2 theta.
3. Substitute v=12 m/s and g=9.81 m/s^2.
Final: {"answer": 10.1968, "unit": "m"}

Answer: 10.1968

Prompt: A projectile is launched on level ground with no air resistance at speed 12 m/s and angle 22 degrees. Compute the horizontal range in meters using g=9.81 m/s^2.

For level-ground projectile motion, R = v^2 sin(2 theta) / g.
Convert theta=22 degrees through the sine of 2 theta.
Substitute v=12 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Check whether all variables required by the governing equation or specification are present.
2. The missing required information is: rate_constant, number_of_steps.
3. A numeric or code answer would require inventing unstated assumptions, so abstain.
Final: {"answer": "INSUFFICIENT_INFO", "missing": ["rate_constant", "number_of_steps"]}

Answer: INSUFFICIENT_INFO

Prompt: Use explicit Euler for dy/dt = -k y. The prompt gives y(0)=1 and h=1.3, but omits both k and the number of steps. If there is insufficient information, say so explicitly.

Check whether all variables required by the governing equation or specification are present.
The missing required information is: rate_constant, number_of_steps.
A numeric or code answer would require inventing unstated assumptions, so abstain.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 1760 * 0.865 * 39.
Final: {"answer": 59373.6, "unit": "J"}

Answer: 59373.6

Prompt: A 1760 g sample has specific heat 0.865 J/(g*K). How much heat is needed to raise its temperature by 39 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 1760 * 0.865 * 39.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=170 degrees and keep the unit as newtons.
Final: {"answer": [-205.8248, 36.2925], "unit": "N"}

Answer: -205.8248

Answer: 36.2925

Prompt: A vector has magnitude 209 N at 170 degrees counterclockwise from the +x axis. Return its [x, y] components in newtons.

The x component is magnitude times cos(theta).
The y component is magnitude times sin(theta).
Use theta=170 degrees and keep the unit as newtons.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [7.2, 18.0], "unit": "dimensionless"}

Answer: 7.2

Answer: 18.0

Prompt: A line passes through points (-16.75, -102.6) and (-11.75, -66.6). Return [slope, intercept] for y = slope*x + intercept.

The slope is (y2-y1)/(x2-x1).
The intercept is y1 - slope*x1.
Return the pair in [slope, intercept] order.


Response:
Reasoning:
1. Compare the dimensions on both sides of the equation.
2. lever arm times force has torque dimensions
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "consistent"}

Answer: consistent

Prompt: In simulation spec case 665696, check dimensional consistency for tau = r * F. The left side has dimensions N*m and the right side has dimensions N*m. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
lever arm times force has torque dimensions
The equation is valid only if both sides have identical dimensions.

Answer: consistent


Response:
Reasoning:
1. Kelvin equals Celsius plus 273.15.
2. Apply the conversion factor directly to the given value.
3. Compute 148.0 + 273.15.
Final: {"answer": 421.15, "unit": "K"}

Answer: 421.15

Prompt: Convert 148 degC to K. Return the converted value with unit.

Kelvin equals Celsius plus 273.15.
Apply the conversion factor directly to the given value.
Compute 148.0 + 273.15.


Response:
Reasoning:
1. Compare the dimensions on both sides of the equation.
2. amount times volume is not concentration
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "inconsistent"}

Answer: inconsistent

Prompt: In simulation spec case 681836, check dimensional consistency for C = n * V. The left side has dimensions mol/L and the right side has dimensions mol*L. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
amount times volume is not concentration
The equation is valid only if both sides have identical dimensions.

Answer: inconsistent


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=700.5 mg, t=198.5 h, and T_half=82.5 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 132.1636, "unit": "mg"}

Answer: 132.1636

Prompt: A radioactive sample starts with 700.5 mg. Its half-life is 82.5 hours. How many mg remain after 198.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=700.5 mg, t=198.5 h, and T_half=82.5 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(8.2)=-63.44 and f(15.9)=-111.18.
3. The interval width is 7.7.
Final: {"answer": -672.287, "unit": "area_units"}

Answer: -672.287

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-6.2x+-12.6 from x=8.2 to x=15.9. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(8.2)=-63.44 and f(15.9)=-111.18.
The interval width is 7.7.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=16 degrees through the sine of 2 theta.
3. Substitute v=11.6 m/s and g=9.81 m/s^2.
Final: {"answer": 7.2687, "unit": "m"}

Answer: 7.2687

Prompt: A projectile is launched on level ground with no air resistance at speed 11.6 m/s and angle 16 degrees. Compute the horizontal range in meters using g=9.81 m/s^2.

For level-ground projectile motion, R = v^2 sin(2 theta) / g.
Convert theta=16 degrees through the sine of 2 theta.
Substitute v=11.6 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=171.7 V and R=146.8 ohm.
Final: {"answer": 200.8235, "unit": "W"}

Answer: 200.8235

Prompt: A resistor has resistance 146.8 ohm and voltage 171.7 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=171.7 V and R=146.8 ohm.


Response:
Reasoning:
1. The function contract has a normal path and at least one edge case.
2. The implementation checks the edge case before returning the normal result.

```python
import math

def unit_vector_006406(x, y):
    norm = math.hypot(x, y)
    if norm == 0:
        raise ValueError('zero vector has no direction')
    return (x / norm, y / norm)
```
Final: {"answer": "python_code", "function": "unit_vector_006406"}

Answer: python_code

Prompt: Write a Python function unit_vector_006406(x, y) that returns the 2D unit vector as a tuple. Raise ValueError for the zero vector. Return a single fenced Python code block.

Identify required edge cases before writing the function.
Implement the shortest total function that satisfies the stated contract.
Avoid global state and return deterministic outputs.

try:
    unit_vector_006406(0, 0)
    raise AssertionError('expected ValueError')
except ValueError:
    pass


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=156 degrees and keep the unit as newtons.
Final: {"answer": [-429.3664, 191.1662], "unit": "N"}

Answer: -429.3664

Answer: 191.1662

Prompt: A vector has magnitude 470 N at 156 degrees counterclockwise from the +x axis. Return its [x, y] components in newtons.

The x component is magnitude times cos(theta).
The y component is magnitude times sin(theta).
Use theta=156 degrees and keep the unit as newtons.


Response:
Reasoning:
1. There are 100 centimeters in one meter.
2. Apply the conversion factor directly to the given value.
3. Compute 165.5 / 100.
Final: {"answer": 1.655, "unit": "m"}

Answer: 1.655

Prompt: Convert 165.5 cm to m. Return the converted value with unit.

There are 100 centimeters in one meter.
Apply the conversion factor directly to the given value.
Compute 165.5 / 100.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [6.8, -29.75], "unit": "dimensionless"}

Answer: 6.8

Answer: -29.75

Prompt: A line passes through points (-7.25, -79.05) and (-4.75, -62.05). Return [slope, intercept] for y = slope*x + intercept.

The slope is (y2-y1)/(x2-x1).
The intercept is y1 - slope*x1.
Return the pair in [slope, intercept] order.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=182.5 mg, t=127 h, and T_half=57.5 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 39.4802, "unit": "mg"}

Answer: 39.4802

Prompt: A radioactive sample starts with 182.5 mg. Its half-life is 57.5 hours. How many mg remain after 127 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=182.5 mg, t=127 h, and T_half=57.5 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. Compare the dimensions on both sides of the equation.
2. lever arm times force has torque dimensions
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "consistent"}

Answer: consistent

Prompt: In sensor calibration case 53959, check dimensional consistency for tau = r * F. The left side has dimensions N*m and the right side has dimensions N*m. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
lever arm times force has torque dimensions
The equation is valid only if both sides have identical dimensions.

Answer: consistent


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 22.95 kg and a = 11.35 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 260.4825, "unit": "N"}

Answer: 260.4825

Prompt: A cart has mass 22.95 kg and measured acceleration 11.35 m/s^2. Compute the net force. Check the SI units and give only the final JSON after reasoning.

Use Newton's second law F = m a.
Substitute m = 22.95 kg and a = 11.35 m/s^2.
The unit kg*m/s^2 is one newton.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=84 degrees through the sine of 2 theta.
3. Substitute v=63.6 m/s and g=9.81 m/s^2.
Final: {"answer": 85.7283, "unit": "m"}

Answer: 85.7283

Prompt: A projectile is launched on level ground with no air resistance at speed 63.6 m/s and angle 84 degrees. Compute the horizontal range in meters using g=9.81 m/s^2.

For level-ground projectile motion, R = v^2 sin(2 theta) / g.
Convert theta=84 degrees through the sine of 2 theta.
Substitute v=63.6 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=23 degrees through the sine of 2 theta.
3. Substitute v=71.5 m/s and g=9.81 m/s^2.
Final: {"answer": 374.867, "unit": "m"}

Answer: 374.867

Prompt: A projectile is launched on level ground with no air resistance at speed 71.5 m/s and angle 23 degrees. Compute the horizontal range in meters using g=9.81 m/s^2.

For level-ground projectile motion, R = v^2 sin(2 theta) / g.
Convert theta=23 degrees through the sine of 2 theta.
Substitute v=71.5 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=103.1 V and R=177.2 ohm.
Final: {"answer": 59.9865, "unit": "W"}

Answer: 59.9865

Prompt: A resistor has resistance 177.2 ohm and voltage 103.1 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=103.1 V and R=177.2 ohm.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 3.28 * 8.314 * 583 / 41.7.
Final: {"answer": 381.2557, "unit": "kPa"}

Answer: 381.2557

Prompt: An ideal gas sample has n=3.28 mol, T=583 K, and V=41.7 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 3.28 * 8.314 * 583 / 41.7.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 1770 mL to 1.77 L.
3. Compute M = 4.56 / 1.77.
Final: {"answer": 2.5763, "unit": "mol/L"}

Answer: 2.5763

Prompt: A solution contains 4.56 mol of solute in 1770 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 1770 mL to 1.77 L.
Compute M = 4.56 / 1.77.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(-0.2)=11.3 and f(5.1)=-4.6.
3. The interval width is 5.3.
Final: {"answer": 17.755, "unit": "area_units"}

Answer: 17.755

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-3x+10.7 from x=-0.2 to x=5.1. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-0.2)=11.3 and f(5.1)=-4.6.
The interval width is 5.3.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=160.3 V and R=121.5 ohm.
Final: {"answer": 211.4905, "unit": "W"}

Answer: 211.4905

Prompt: A resistor has resistance 121.5 ohm and voltage 160.3 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=160.3 V and R=121.5 ohm.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.92 y, so each step multiplies y by 1 - h*0.92.
3. Apply that update for 3 steps and round the final state.
Final: {"answer": 12.155, "unit": "state_units"}

Answer: 12.155

Prompt: Use explicit Euler on dy/dt = -0.92 y with y(0)=39, step size h=0.35, for 3 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.92 y, so each step multiplies y by 1 - h*0.92.
Apply that update for 3 steps and round the final state.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=58 degrees through the sine of 2 theta.
3. Substitute v=65.9 m/s and g=9.81 m/s^2.
Final: {"answer": 397.8891, "unit": "m"}

Answer: 397.8891

Prompt: A projectile is launched on level ground with no air resistance at speed 65.9 m/s and angle 58 degrees. Compute the horizontal range in meters using g=9.81 m/s^2.

For level-ground projectile motion, R = v^2 sin(2 theta) / g.
Convert theta=58 degrees through the sine of 2 theta.
Substitute v=65.9 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Molarity is moles of solute divided by solution volume in liters.
2. Convert 4985 mL to 4.985 L.
3. Compute M = 0.555 / 4.985.
Final: {"answer": 0.1113, "unit": "mol/L"}

Answer: 0.1113

Prompt: A solution contains 0.555 mol of solute in 4985 mL of solution. Compute the molarity in mol/L.

Molarity is moles of solute divided by solution volume in liters.
Convert 4985 mL to 4.985 L.
Compute M = 0.555 / 4.985.


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.1 and add it to the current value.
Final: {"answer": [1.175, -0.575, 0.525], "unit": "temperature_units"}

Answer: 1.175

Answer: -0.575

Answer: 0.525

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=1.55, right=-0.55; current interior values are [1.4, -1.0, 0.85]. Use r=0.1. Return the next interior vector.

Keep the boundary values fixed and update only the three interior grid points.
For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
Multiply each Laplacian by r=0.1 and add it to the current value.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 1.31 * 8.314 * 615 / 75.2.
Final: {"answer": 89.0715, "unit": "kPa"}

Answer: 89.0715

Prompt: An ideal gas sample has n=1.31 mol, T=615 K, and V=75.2 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 1.31 * 8.314 * 615 / 75.2.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [4.3, 4.75], "unit": "dimensionless"}

Answer: 4.3

Answer: 4.75

Prompt: A line passes through points (-10.5, -40.4) and (-1.5, -1.7). Return [slope, intercept] for y = slope*x + intercept.

The slope is (y2-y1)/(x2-x1).
The intercept is y1 - slope*x1.
Return the pair in [slope, intercept] order.


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.24 and add it to the current value.
Final: {"answer": [-1.108, -0.686, 0.372], "unit": "temperature_units"}

Answer: -1.108

Answer: -0.686

Answer: 0.372

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=0.4, right=-1.45; current interior values are [-1.6, -1.55, 2.1]. Use r=0.24. Return the next interior vector.

Keep the boundary values fixed and update only the three interior grid points.
For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
Multiply each Laplacian by r=0.24 and add it to the current value.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [2.9, 25.5], "unit": "dimensionless"}

Answer: 2.9

Answer: 25.5

Prompt: A line passes through points (-10.75, -5.675) and (-5, 11). Return [slope, intercept] for y = slope*x + intercept.

The slope is (y2-y1)/(x2-x1).
The intercept is y1 - slope*x1.
Return the pair in [slope, intercept] order.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=705.5 mg, t=13.5 h, and T_half=111 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 648.4631, "unit": "mg"}

Answer: 648.4631

Prompt: A radioactive sample starts with 705.5 mg. Its half-life is 111 hours. How many mg remain after 13.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=705.5 mg, t=13.5 h, and T_half=111 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(-7.1)=-3.57 and f(-0.8)=-11.76.
3. The interval width is 6.3.
Final: {"answer": -48.2895, "unit": "area_units"}

Answer: -48.2895

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-1.3x+-12.8 from x=-7.1 to x=-0.8. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-7.1)=-3.57 and f(-0.8)=-11.76.
The interval width is 6.3.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [-4.8, -0.25], "unit": "dimensionless"}

Answer: -4.8

Answer: -0.25

Prompt: A line passes through points (-4.25, 20.15) and (9.75, -47.05). Return [slope, intercept] for y = slope*x + intercept.

The slope is (y2-y1)/(x2-x1).
The intercept is y1 - slope*x1.
Return the pair in [slope, intercept] order.


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 32.25^2 and multiply by 0.5 * 26.25.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 13650.8203, "unit": "J"}

Answer: 13650.8203

Prompt: A body of mass 26.25 kg moves at 32.25 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 32.25^2 and multiply by 0.5 * 26.25.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. Use the ideal gas law P V = n R T.
2. Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
3. Compute P = 0.83 * 8.314 * 570 / 87.2.
Final: {"answer": 45.1073, "unit": "kPa"}

Answer: 45.1073

Prompt: An ideal gas sample has n=0.83 mol, T=570 K, and V=87.2 L. Using R=8.314 kPa*L/(mol*K), compute the pressure in kPa.

Use the ideal gas law P V = n R T.
Because R is in kPa*L/(mol*K) and V is in L, the pressure comes out in kPa.
Compute P = 0.83 * 8.314 * 570 / 87.2.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(-5.7)=30.77 and f(4.1)=-38.81.
3. The interval width is 9.8.
Final: {"answer": -39.396, "unit": "area_units"}

Answer: -39.396

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-7.1x+-9.7 from x=-5.7 to x=4.1. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-5.7)=30.77 and f(4.1)=-38.81.
The interval width is 9.8.


Response:
Reasoning:
1. Kinetic energy is KE = 1/2 m v^2.
2. Compute v^2 = 41.95^2 and multiply by 0.5 * 24.25.
3. The unit kg*m^2/s^2 is one joule.
Final: {"answer": 21337.6053, "unit": "J"}

Answer: 21337.6053

Prompt: A body of mass 24.25 kg moves at 41.95 m/s. Find its kinetic energy in joules with unit reasoning.

Kinetic energy is KE = 1/2 m v^2.
Compute v^2 = 41.95^2 and multiply by 0.5 * 24.25.
The unit kg*m^2/s^2 is one joule.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 7.26 mol * 180.156 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 1307.9326, "unit": "g"}

Answer: 1307.9326

Prompt: A sample contains 7.26 mol of glucose. Using molar mass 180.156 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 7.26 mol * 180.156 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [-9.75, 1.0], "unit": "dimensionless"}

Answer: -9.75

Answer: 1.0

Prompt: A line passes through points (-3.25, 32.6875) and (0.25, -1.4375). Return [slope, intercept] for y = slope*x + intercept.

The slope is (y2-y1)/(x2-x1).
The intercept is y1 - slope*x1.
Return the pair in [slope, intercept] order.


Response:
Reasoning:
1. The function contract has a normal path and at least one edge case.
2. The implementation checks the edge case before returning the normal result.

```python
def normalize_006820(values):
    total = sum(values)
    if total == 0:
        raise ValueError('cannot normalize zero-sum values')
    return [v / total for v in values]
```
Final: {"answer": "python_code", "function": "normalize_006820"}

Answer: python_code

Prompt: Write a Python function normalize_006820(values) that returns a new list whose values sum to 1. Raise ValueError if the sum is zero. Return a single fenced Python code block.

Identify required edge cases before writing the function.
Implement the shortest total function that satisfies the stated contract.
Avoid global state and return deterministic outputs.

out = normalize_006820([2, 3, 5]); assert len(out) == 3

try:
    normalize_006820([1, -1])
    raise AssertionError('expected ValueError')
except ValueError:
    pass


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=428 mg, t=307.5 h, and T_half=84 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 33.8422, "unit": "mg"}

Answer: 33.8422

Prompt: A radioactive sample starts with 428 mg. Its half-life is 84 hours. How many mg remain after 307.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=428 mg, t=307.5 h, and T_half=84 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [8.45, 9.0], "unit": "dimensionless"}

Answer: 8.45

Answer: 9.0

Prompt: A line passes through points (-13.75, -107.187) and (-8.75, -64.9375). Return [slope, intercept] for y = slope*x + intercept.

The slope is (y2-y1)/(x2-x1).
The intercept is y1 - slope*x1.
Return the pair in [slope, intercept] order.


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.22 and add it to the current value.
Final: {"answer": [0.938, 0.109, -0.914], "unit": "temperature_units"}

Answer: 0.938

Answer: 0.109

Answer: -0.914

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=-0.6, right=-1; current interior values are [1.95, -0.1, -1.2]. Use r=0.22. Return the next interior vector.

Keep the boundary values fixed and update only the three interior grid points.
For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
Multiply each Laplacian by r=0.22 and add it to the current value.


Response:
Reasoning:
1. The function contract has a normal path and at least one edge case.
2. The implementation checks the edge case before returning the normal result.

```python
def pairwise_ratios_006856(numerators, denominators):
    if len(numerators) != len(denominators):
        raise ValueError('length mismatch')
    out = []
    for numerator, denominator in zip(numerators, denominators):
        if denominator == 0:
            raise ValueError('zero denominator')
        out.append(numerator / denominator)
    return out
```
Final: {"answer": "python_code", "function": "pairwise_ratios_006856"}

Answer: python_code

Prompt: Write a Python function pairwise_ratios_006856(numerators, denominators) that returns a list of n/d values. Raise ValueError if lengths differ or any denominator is zero. Return a single fenced Python code block.

Identify required edge cases before writing the function.
Implement the shortest total function that satisfies the stated contract.
Avoid global state and return deterministic outputs.

try:
    pairwise_ratios_006856([1], [0])
    raise AssertionError('expected ValueError')
except ValueError:
    pass

try:
    pairwise_ratios_006856([1, 2], [1])
    raise AssertionError('expected ValueError')
except ValueError:
    pass


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=40 degrees through the sine of 2 theta.
3. Substitute v=17.9 m/s and g=9.81 m/s^2.
Final: {"answer": 32.1654, "unit": "m"}

Answer: 32.1654

Prompt: A projectile is launched on level ground with no air resistance at speed 17.9 m/s and angle 40 degrees. Compute the horizontal range in meters using g=9.81 m/s^2.

For level-ground projectile motion, R = v^2 sin(2 theta) / g.
Convert theta=40 degrees through the sine of 2 theta.
Substitute v=17.9 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 1.65 mol * 100.087 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 165.1436, "unit": "g"}

Answer: 165.1436

Prompt: A sample contains 1.65 mol of calcium carbonate. Using molar mass 100.087 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 1.65 mol * 100.087 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(-9.6)=-68.22 and f(-5.6)=-39.42.
3. The interval width is 4.
Final: {"answer": -215.28, "unit": "area_units"}

Answer: -215.28

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=7.2x+0.9 from x=-9.6 to x=-5.6. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-9.6)=-68.22 and f(-5.6)=-39.42.
The interval width is 4.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.21 y, so each step multiplies y by 1 - h*0.21.
3. Apply that update for 2 steps and round the final state.
Final: {"answer": 42.6375, "unit": "state_units"}

Answer: 42.6375

Prompt: Use explicit Euler on dy/dt = -0.21 y with y(0)=54.5, step size h=0.55, for 2 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.21 y, so each step multiplies y by 1 - h*0.21.
Apply that update for 2 steps and round the final state.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=221 degrees and keep the unit as newtons.
Final: {"answer": [-199.2433, -173.1996], "unit": "N"}

Answer: -199.2433

Answer: -173.1996

Prompt: A vector has magnitude 264 N at 221 degrees counterclockwise from the +x axis. Return its [x, y] components in newtons.

The x component is magnitude times cos(theta).
The y component is magnitude times sin(theta).
Use theta=221 degrees and keep the unit as newtons.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [8.2, 26.25], "unit": "dimensionless"}

Answer: 8.2

Answer: 26.25

Prompt: A line passes through points (-9.5, -51.65) and (-8.25, -41.4). Return [slope, intercept] for y = slope*x + intercept.

The slope is (y2-y1)/(x2-x1).
The intercept is y1 - slope*x1.
Return the pair in [slope, intercept] order.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=723 mg, t=104 h, and T_half=5.5 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 0.0015, "unit": "mg"}

Answer: 0.0015

Prompt: A radioactive sample starts with 723 mg. Its half-life is 5.5 hours. How many mg remain after 104 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=723 mg, t=104 h, and T_half=5.5 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.96 y, so each step multiplies y by 1 - h*0.96.
3. Apply that update for 6 steps and round the final state.
Final: {"answer": 21.7051, "unit": "state_units"}

Answer: 21.7051

Prompt: Use explicit Euler on dy/dt = -0.96 y with y(0)=78, step size h=0.2, for 6 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.96 y, so each step multiplies y by 1 - h*0.96.
Apply that update for 6 steps and round the final state.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 370 * 2.26 * 66.
Final: {"answer": 55189.2, "unit": "J"}

Answer: 55189.2

Prompt: A 370 g sample has specific heat 2.26 J/(g*K). How much heat is needed to raise its temperature by 66 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 370 * 2.26 * 66.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=41.2 V and R=10.5 ohm.
Final: {"answer": 161.661, "unit": "W"}

Answer: 161.661

Prompt: A resistor has resistance 10.5 ohm and voltage 41.2 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=41.2 V and R=10.5 ohm.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.03 y, so each step multiplies y by 1 - h*0.03.
3. Apply that update for 10 steps and round the final state.
Final: {"answer": 2.5792, "unit": "state_units"}

Answer: 2.5792

Prompt: Use explicit Euler on dy/dt = -0.03 y with y(0)=3, step size h=0.5, for 10 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.03 y, so each step multiplies y by 1 - h*0.03.
Apply that update for 10 steps and round the final state.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [5.1, -1.75], "unit": "dimensionless"}

Answer: 5.1

Answer: -1.75

Prompt: A line passes through points (0.75, 2.075) and (2.75, 12.275). Return [slope, intercept] for y = slope*x + intercept.

The slope is (y2-y1)/(x2-x1).
The intercept is y1 - slope*x1.
Return the pair in [slope, intercept] order.


Response:
Reasoning:
1. Compare the dimensions on both sides of the equation.
2. velocity times time gives distance, not acceleration
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "inconsistent"}

Answer: inconsistent

Prompt: In quality-control check case 862040, check dimensional consistency for a = v * t. The left side has dimensions m/s^2 and the right side has dimensions m. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
velocity times time gives distance, not acceleration
The equation is valid only if both sides have identical dimensions.

Answer: inconsistent


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=48 degrees through the sine of 2 theta.
3. Substitute v=18.9 m/s and g=9.81 m/s^2.
Final: {"answer": 36.2134, "unit": "m"}

Answer: 36.2134

Prompt: A projectile is launched on level ground with no air resistance at speed 18.9 m/s and angle 48 degrees. Compute the horizontal range in meters using g=9.81 m/s^2.

For level-ground projectile motion, R = v^2 sin(2 theta) / g.
Convert theta=48 degrees through the sine of 2 theta.
Substitute v=18.9 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 90 * 2.735 * 33.
Final: {"answer": 8122.95, "unit": "J"}

Answer: 8122.95

Prompt: A 90 g sample has specific heat 2.735 J/(g*K). How much heat is needed to raise its temperature by 33 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 90 * 2.735 * 33.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(-0.7)=6.49 and f(3.2)=15.46.
3. The interval width is 3.9.
Final: {"answer": 42.8025, "unit": "area_units"}

Answer: 42.8025

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=2.3x+8.1 from x=-0.7 to x=3.2. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-0.7)=6.49 and f(3.2)=15.46.
The interval width is 3.9.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=128.4 V and R=7.2 ohm.
Final: {"answer": 2289.8, "unit": "W"}

Answer: 2289.8

Prompt: A resistor has resistance 7.2 ohm and voltage 128.4 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=128.4 V and R=7.2 ohm.


Response:
Reasoning:
1. Use Newton's second law F = m a.
2. Substitute m = 1.25 kg and a = 1.55 m/s^2.
3. The unit kg*m/s^2 is one newton.
Final: {"answer": 1.9375, "unit": "N"}

Answer: 1.9375

Prompt: A cart has mass 1.25 kg and measured acceleration 1.55 m/s^2. Compute the net force. Check the SI units and give only the final JSON after reasoning.

Use Newton's second law F = m a.
Substitute m = 1.25 kg and a = 1.55 m/s^2.
The unit kg*m/s^2 is one newton.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=99 degrees and keep the unit as newtons.
Final: {"answer": [-25.4206, 160.4994], "unit": "N"}

Answer: -25.4206

Answer: 160.4994

Prompt: A vector has magnitude 162.5 N at 99 degrees counterclockwise from the +x axis. Return its [x, y] components in newtons.

The x component is magnitude times cos(theta).
The y component is magnitude times sin(theta).
Use theta=99 degrees and keep the unit as newtons.


Response:
Reasoning:
1. Compare the dimensions on both sides of the equation.
2. velocity times time gives distance, not acceleration
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "inconsistent"}

Answer: inconsistent

Prompt: In lab notebook case 793704, check dimensional consistency for a = v * t. The left side has dimensions m/s^2 and the right side has dimensions m. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
velocity times time gives distance, not acceleration
The equation is valid only if both sides have identical dimensions.

Answer: inconsistent


Response:
Reasoning:
1. Compare the dimensions on both sides of the equation.
2. grams cancel with J/(g*K), and kelvin cancels, leaving joules
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "consistent"}

Answer: consistent

Prompt: In instrument manual case 758633, check dimensional consistency for Q = m * c * DeltaT. The left side has dimensions J and the right side has dimensions J. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
grams cancel with J/(g*K), and kelvin cancels, leaving joules
The equation is valid only if both sides have identical dimensions.

Answer: consistent


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.79 y, so each step multiplies y by 1 - h*0.79.
3. Apply that update for 3 steps and round the final state.
Final: {"answer": -0.0893, "unit": "state_units"}

Answer: -0.0893

Prompt: Use explicit Euler on dy/dt = -0.79 y with y(0)=75, step size h=1.4, for 3 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.79 y, so each step multiplies y by 1 - h*0.79.
Apply that update for 3 steps and round the final state.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=78 degrees through the sine of 2 theta.
3. Substitute v=20.9 m/s and g=9.81 m/s^2.
Final: {"answer": 18.1108, "unit": "m"}

Answer: 18.1108

Prompt: A projectile is launched on level ground with no air resistance at speed 20.9 m/s and angle 78 degrees. Compute the horizontal range in meters using g=9.81 m/s^2.

For level-ground projectile motion, R = v^2 sin(2 theta) / g.
Convert theta=78 degrees through the sine of 2 theta.
Substitute v=20.9 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 875 * 2.97 * 105.
Final: {"answer": 272868.75, "unit": "J"}

Answer: 272868.75

Prompt: A 875 g sample has specific heat 2.97 J/(g*K). How much heat is needed to raise its temperature by 105 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 875 * 2.97 * 105.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=38 V and R=6.9 ohm.
Final: {"answer": 209.2754, "unit": "W"}

Answer: 209.2754

Prompt: A resistor has resistance 6.9 ohm and voltage 38 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=38 V and R=6.9 ohm.


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 620 * 3.395 * 33.
Final: {"answer": 69461.7, "unit": "J"}

Answer: 69461.7

Prompt: A 620 g sample has specific heat 3.395 J/(g*K). How much heat is needed to raise its temperature by 33 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 620 * 3.395 * 33.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 0.63 mol * 18.015 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 11.3495, "unit": "g"}

Answer: 11.3495

Prompt: A sample contains 0.63 mol of water. Using molar mass 18.015 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 0.63 mol * 18.015 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. For level-ground projectile motion, R = v^2 sin(2 theta) / g.
2. Convert theta=58 degrees through the sine of 2 theta.
3. Substitute v=84.8 m/s and g=9.81 m/s^2.
Final: {"answer": 658.8444, "unit": "m"}

Answer: 658.8444

Prompt: A projectile is launched on level ground with no air resistance at speed 84.8 m/s and angle 58 degrees. Compute the horizontal range in meters using g=9.81 m/s^2.

For level-ground projectile motion, R = v^2 sin(2 theta) / g.
Convert theta=58 degrees through the sine of 2 theta.
Substitute v=84.8 m/s and g=9.81 m/s^2.


Response:
Reasoning:
1. Explicit Euler updates y_{n+1} = y_n + h f(y_n).
2. Here f(y) = -0.72 y, so each step multiplies y by 1 - h*0.72.
3. Apply that update for 7 steps and round the final state.
Final: {"answer": 34.673, "unit": "state_units"}

Answer: 34.673

Prompt: Use explicit Euler on dy/dt = -0.72 y with y(0)=58.5, step size h=0.1, for 7 steps. What is y after the final step?

Explicit Euler updates y_{n+1} = y_n + h f(y_n).
Here f(y) = -0.72 y, so each step multiplies y by 1 - h*0.72.
Apply that update for 7 steps and round the final state.


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.42 and add it to the current value.
Final: {"answer": [-0.403, 1.615, 0.185], "unit": "temperature_units"}

Answer: -0.403

Answer: 1.615

Answer: 0.185

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=-1.15, right=0.65; current interior values are [2.6, -0.8, 1.55]. Use r=0.42. Return the next interior vector.

Keep the boundary values fixed and update only the three interior grid points.
For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
Multiply each Laplacian by r=0.42 and add it to the current value.


Response:
Reasoning:
1. Compare the dimensions on both sides of the equation.
2. velocity times time gives distance, not acceleration
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "inconsistent"}

Answer: inconsistent

Prompt: In quality-control check case 400475, check dimensional consistency for a = v * t. The left side has dimensions m/s^2 and the right side has dimensions m. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
velocity times time gives distance, not acceleration
The equation is valid only if both sides have identical dimensions.

Answer: inconsistent


Response:
Reasoning:
1. Keep the boundary values fixed and update only the three interior grid points.
2. For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
3. Multiply each Laplacian by r=0.2 and add it to the current value.
Final: {"answer": [0.15, -0.25, -1.07], "unit": "temperature_units"}

Answer: 0.15

Answer: -0.25

Answer: -1.07

Prompt: For the 1D heat equation explicit finite-difference step, use u_i(next)=u_i+r*(u_{i-1}-2*u_i+u_{i+1}). Boundary values are left=2.95, right=-1.35; current interior values are [-0.85, 0.35, -1.45]. Use r=0.2. Return the next interior vector.

Keep the boundary values fixed and update only the three interior grid points.
For each interior point, compute the discrete Laplacian u_{i-1}-2*u_i+u_{i+1}.
Multiply each Laplacian by r=0.2 and add it to the current value.


Response:
Reasoning:
1. Mass equals amount in moles times molar mass.
2. Use m = 5.86 mol * 180.156 g/mol.
3. The mole unit cancels, leaving grams.
Final: {"answer": 1055.7142, "unit": "g"}

Answer: 1055.7142

Prompt: A sample contains 5.86 mol of glucose. Using molar mass 180.156 g/mol, compute the sample mass in grams.

Mass equals amount in moles times molar mass.
Use m = 5.86 mol * 180.156 g/mol.
The mole unit cancels, leaving grams.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=224 degrees and keep the unit as newtons.
Final: {"answer": [-192.7831, -186.1684], "unit": "N"}

Answer: -192.7831

Answer: -186.1684

Prompt: A vector has magnitude 268 N at 224 degrees counterclockwise from the +x axis. Return its [x, y] components in newtons.

The x component is magnitude times cos(theta).
The y component is magnitude times sin(theta).
Use theta=224 degrees and keep the unit as newtons.


Response:
Reasoning:
1. Compare the dimensions on both sides of the equation.
2. density times acceleration times height gives kg/(m*s^2), or pascal
3. The equation is valid only if both sides have identical dimensions.
Final: {"answer": "consistent"}

Answer: consistent

Prompt: In field worksheet case 308431, check dimensional consistency for p = rho * g * h. The left side has dimensions Pa and the right side has dimensions Pa. Return consistent or inconsistent with a brief reason.

Compare the dimensions on both sides of the equation.
density times acceleration times height gives kg/(m*s^2), or pascal
The equation is valid only if both sides have identical dimensions.

Answer: consistent


Response:
Reasoning:
1. Use Q = m c DeltaT.
2. The mass is already in grams, matching the specific heat denominator.
3. Compute Q = 450 * 3.12 * 36.
Final: {"answer": 50544.0, "unit": "J"}

Answer: 50544.0

Prompt: A 450 g sample has specific heat 3.12 J/(g*K). How much heat is needed to raise its temperature by 36 K?

Use Q = m c DeltaT.
The mass is already in grams, matching the specific heat denominator.
Compute Q = 450 * 3.12 * 36.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=135.1 V and R=68.9 ohm.
Final: {"answer": 264.9058, "unit": "W"}

Answer: 264.9058

Prompt: A resistor has resistance 68.9 ohm and voltage 135.1 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=135.1 V and R=68.9 ohm.


Response:
Reasoning:
1. The slope is (y2-y1)/(x2-x1).
2. The intercept is y1 - slope*x1.
3. Return the pair in [slope, intercept] order.
Final: {"answer": [-8.35, 7.5], "unit": "dimensionless"}

Answer: -8.35

Answer: 7.5

Prompt: A line passes through points (-17.75, 155.713) and (-17.5, 153.625). Return [slope, intercept] for y = slope*x + intercept.

The slope is (y2-y1)/(x2-x1).
The intercept is y1 - slope*x1.
Return the pair in [slope, intercept] order.


Response:
Reasoning:
1. Check whether all variables required by the governing equation or specification are present.
2. The missing required information is: volume.
3. A numeric or code answer would require inventing unstated assumptions, so abstain.
Final: {"answer": "INSUFFICIENT_INFO", "missing": ["volume"]}

Answer: INSUFFICIENT_INFO

Prompt: Compute pressure from the ideal gas law. The prompt gives n=4.9 mol and T=454 K, but does not give volume. If there is insufficient information, say so explicitly.

Check whether all variables required by the governing equation or specification are present.
The missing required information is: volume.
A numeric or code answer would require inventing unstated assumptions, so abstain.


Response:
Reasoning:
1. The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
2. Evaluate f(-4)=-1 and f(5.7)=-20.4.
3. The interval width is 9.7.
Final: {"answer": -103.79, "unit": "area_units"}

Answer: -103.79

Prompt: Use the trapezoid rule with one interval to approximate the integral of f(x)=-2x+-9 from x=-4 to x=5.7. Return the numeric integral estimate.

The one-interval trapezoid rule is (b-a)*(f(a)+f(b))/2.
Evaluate f(-4)=-1 and f(5.7)=-20.4.
The interval width is 9.7.


Response:
Reasoning:
1. Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
2. Use N0=291.5 mg, t=204.5 h, and T_half=105.5 h.
3. The remaining amount has the same mass unit as the initial amount.
Final: {"answer": 76.0546, "unit": "mg"}

Answer: 76.0546

Prompt: A radioactive sample starts with 291.5 mg. Its half-life is 105.5 hours. How many mg remain after 204.5 hours?

Radioactive decay by half-life follows N(t)=N0*(1/2)^(t/T_half).
Use N0=291.5 mg, t=204.5 h, and T_half=105.5 h.
The remaining amount has the same mass unit as the initial amount.


Response:
Reasoning:
1. The function contract has a normal path and at least one edge case.
2. The implementation checks the edge case before returning the normal result.

```python
def center_values_007288(values):
    if not values:
        raise ValueError('center_values requires at least one value')
    mean = sum(values) / len(values)
    return [value - mean for value in values]
```
Final: {"answer": "python_code", "function": "center_values_007288"}

Answer: python_code

Prompt: Write a Python function center_values_007288(values) that subtracts the arithmetic mean from each value and returns a new list. Raise ValueError for an empty input. Return a single fenced Python code block.

Identify required edge cases before writing the function.
Implement the shortest total function that satisfies the stated contract.
Avoid global state and return deterministic outputs.

try:
    center_values_007288([])
    raise AssertionError('expected ValueError')
except ValueError:
    pass


Response:
Reasoning:
1. Check whether all variables required by the governing equation or specification are present.
2. The missing required information is: recurrence, objective, tie_breaking_rule.
3. A numeric or code answer would require inventing unstated assumptions, so abstain.
Final: {"answer": "INSUFFICIENT_INFO", "missing": ["recurrence", "objective", "tie_breaking_rule"]}

Answer: INSUFFICIENT_INFO

Prompt: Write a dynamic-programming solver for a scheduler scoring problem with 1746 items and budget 61225, but the recurrence, objective, and tie-breaking rule are all absent. If there is insufficient information, say so explicitly.

Check whether all variables required by the governing equation or specification are present.
The missing required information is: recurrence, objective, tie_breaking_rule.
A numeric or code answer would require inventing unstated assumptions, so abstain.


Response:
Reasoning:
1. Ohm's law gives I = V / R.
2. Power can be written as P = V I = V^2 / R.
3. Substitute V=211.2 V and R=103.7 ohm.
Final: {"answer": 430.1392, "unit": "W"}

Answer: 430.1392

Prompt: A resistor has resistance 103.7 ohm and voltage 211.2 V across it. Compute the dissipated power in watts.

Ohm's law gives I = V / R.
Power can be written as P = V I = V^2 / R.
Substitute V=211.2 V and R=103.7 ohm.


Response:
Reasoning:
1. The x component is magnitude times cos(theta).
2. The y component is magnitude times sin(theta).
3. Use theta=199 degrees and keep the unit as newtons.
Final: {"answer": [-151.283, -52.0909], "unit": "N"}

Answer: -151.283

Answer: -52.0909

Prompt: A vector has magnitude 160 N at 199 degrees counterclockwise from the +x axis. Return its [x, y] components in newtons.

The x component is magnitude times cos(theta).
The y component is magnitude times sin(theta).
Use theta=199 degrees and keep the unit as newtons.