# E080: Chebyshev bias: leader fraction vs. x.

```{figure} ../_static/experiments/e080_hero.png
:width: 80%
:alt: Preview figure for E080
```

**Tags:** `number-theory`, `quantitative-exploration`, `visualization`, `prime-races`

## Highlights
- Estimates the fraction of x where one residue class leads another.
- Plots a running bias estimate over log-sampled x values.
- Detects coarse sign-change neighborhoods for the race difference.

## What this experiment does

For the mod 4 race D(x)=π(x;4,3)-π(x;4,1), define the empirical leader fraction:

The implementation focuses on a compact, reproducible numerical workflow: deterministic parameter defaults, structured output folders, and one or more figures saved for the gallery.

## Outputs

This experiment writes into `out/e080/`:

- `figures/fig_01_bias_fraction.png`

## How to run

```bash
make run EXP=e080
```

## Notes

- The gallery preview figure shipped with the documentation uses conservative cutoffs so builds stay fast. If you run the experiment locally, increase the cutoffs to see the asymptotic regime more clearly.
- Prime-race plots depend on the chosen sampling of `x` (linear vs. log grid). The qualitative “who leads” picture can change when you zoom in.

## Published run snapshot

If this experiment is included in the docs gallery, include the published snapshot (report + params).

```{include} ../reports/e080.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

::: {dropdown} params.json (snapshot)
:open:

```{literalinclude} ../params/e080.json
:language: json
```

:::

## References

{cite:t}`granvillemartin2006primenumberraces, rubinsteinsarnak1994chebyshevsbias`

## Related experiments

- {doc}`e118` (E118: Chebyshev bias: lead-time statistics)
- {doc}`e076` (Chebyshev θ(x;q,a): weighted prime counts in progressions.)
- {doc}`e061` (Chebyshev ψ(x) and prime powers)
- {doc}`e103` (E103: Chebyshev ψ(x): prime powers drive jumps)
- {doc}`e072` (Prime race mod 4: pi(x;4,3) vs. pi(x;4,1).)