Round #1066
19 May 2026, 18:19· math· Meta-Llama-3.3-70B-Instruct· r13-clean-hints
- Complexity
- band 0
- ZP / UO / OBS
- 3 / 0 / 2
- Journey weight
- 410
- Lightshift ΔL
- —
Prompt
Compute the sum of the infinite geometric series 1 + 1/2 + 1/4 + 1/8 + ... Give the exact value.
Zero-point index (what the engine surfaced)
ZPI-ZPI-0025 | 2026-05-19T18:19:40Z
Compute the sum of the infinite geometric series 1 + 1/2 + 1/4 + 1/8 + ... Give the exact value.
ZERO POINT — 3 entries (structural invariants)
● hsl(288.85, 100%, 15%) geometric series sum a/(1-r) for |r| < 1 [contract]
● hsl(125.07, 100%, 15%) 0.999... = 9/10 + 9/100 + ... = 1 [contract]
● hsl(29.83, 100%, 15%) convergence vs divergence test (ratio / comparison) [contract]
UNOBSERVED — 0 entries (implied truths)
OBSERVED — 2 entries (explicit in request)
◉ hsl(92.31, 100%, 22%) value [feature]
◉ hsl(148.72, 100%, 22%) + [feature]
COMPLEXITY │ 5 entries │ spread: 56° │ band: LOW
Baseline response (no ZPI)
```csharp
using System;
class Program
{
static void Main()
{
// Define the first term and common ratio
double a = 1;
double r = 1/2;
// Calculate the sum of the infinite geometric series
double sum_series = a / (1 - r);
Console.WriteLine(sum_series);
}
}
```
Enhanced response (with ZPI)
```csharp
public class GeometricSeriesCalculator
{
public static double CalculateGeometricSeriesSum(double a, double r)
{
if (Math.Abs(r) >= 1)
{
throw new ArgumentException("The series diverges because |r| is not less than 1.");
}
return a / (1 - r);
}
public static void Main(string[] args)
{
double a = 1; // first term
double r = 0.5; // common ratio
double sum = CalculateGeometricSeriesSum(a, r);
System.Console.WriteLine(sum);
}
}
```
The sum of the infinite geometric series 1 + 1/2 + 1/4 + 1/8 + ... is 2.
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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