Round #1006
19 May 2026, 10:21· math· Meta-Llama-3.3-70B-Instruct· r13-clean-hints
- Complexity
- band 0
- ZP / UO / OBS
- 6 / 0 / 4
- Journey weight
- 768
- Lightshift ΔL
- —
Prompt
Find the derivative of f(x) = x * sin(x) using the product rule.
Zero-point index (what the engine surfaced)
ZPI-ZPI-0003 | 2026-05-19T10:21:14Z
Find the derivative of f(x) = x * sin(x) using the product rule.
ZERO POINT — 6 entries (structural invariants)
● hsl(135.43, 100%, 15%) f'(x) = lim_{h→0} (f(x+h) - f(x))/h [contract]
● hsl(134.71, 100%, 15%) power rule d/dx(x^n) = n·x^(n-1) [contract]
● hsl(172.48, 100%, 15%) linearity of differentiation [contract]
● hsl(154.78, 100%, 15%) (fg)' = f'g + fg' [contract]
● hsl(238.28, 100%, 15%) (f/g)' = (f'g - fg')/g² [contract]
● hsl(182.49, 100%, 15%) Leibniz product rule [contract]
UNOBSERVED — 0 entries (implied truths)
OBSERVED — 4 entries (explicit in request)
◉ hsl(85.13, 100%, 50%) using [feature]
◉ hsl(150.77, 100%, 15%) * [feature]
◉ hsl(175.38, 100%, 27%) = [feature]
◉ hsl(234.87, 100%, 50%) using [feature]
COMPLEXITY │ 10 entries │ spread: 150° │ band: LOW
Baseline response (no ZPI)
```csharp
using System;
class Program
{
static void Main()
{
// Define the function
Func<double, double> f = x => x * Math.Sin(x);
// Calculate the derivative using the product rule
Func<double, double> f_prime = x => Math.Sin(x) + x * Math.Cos(x);
// Print the derivative
Console.WriteLine("f'(x) = sin(x) + x*cos(x)");
}
}
```
Enhanced response (with ZPI)
```csharp
using System;
public class DerivativeCalculator
{
public static string CalculateDerivative(string function)
{
// Define the function and its components
string f = "x";
string g = "sin(x)";
// Calculate the derivatives of f and g
string fPrime = "1"; // derivative of x is 1
string gPrime = "cos(x)"; // derivative of sin(x) is cos(x)
// Apply the product rule
string derivative = $"({fPrime} * {g}) + ({f} * {gPrime})";
return derivative;
}
public static void Main()
{
string function = "x * sin(x)";
string derivative = CalculateDerivative(function);
Console.WriteLine($"The derivative of {function} is: {derivative}");
}
}
```
The derivative of f(x) = x * sin(x) using the product rule is: (1 * sin(x)) + (x * cos(x)).
Grader chain
Each grader reads the same baseline + enhanced pair and casts an independent verdict. Disagreement is the framework's own signal — not noise to be averaged away.
V1samba/Meta-Llama-3.3-70B-Instruct
BetterV4qwen2.5-coder-3b-instruct
BetterV5cerebras/qwen-3-235b-a22b-instruct-2507
BetterV7qwen2.5-coder-7b-instruct
BetterV8openai/gpt-4.1
BetterV9anthropic/claude-opus-4-7
BetterV12openai/gpt-4o
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