# Godel Threshold Verification Artifact

Artifact ID: `godel-threshold-verification-001`

Status: first public validation artifact for the Computational Relativity Lab.

## Purpose

This artifact verifies a known result for the normalized Godel chronology threshold used by the public Godel CTC Visualizer. It is a reproducibility check for the site pipeline, not a new physics claim.

The checked angular metric component is:

```text
g_phi_phi(r) = sinh^2(r) - sinh^4(r)
```

The known critical radius is:

```text
r_c = ln(1 + sqrt(2)) = 0.881373587020...
```

At this radius, `sinh(r_c) = 1`, so:

```text
g_phi_phi(r_c) = 1 - 1 = 0
```

For `r < r_c`, `g_phi_phi` is positive in the sample below. For `r > r_c`, `g_phi_phi` is negative in the sample below. In this circular azimuthal test, the negative sign marks the chronology-violating side of the known Godel threshold.

## Verification Formula

```text
f(r) = sinh(r)^2 - sinh(r)^4
r_c = ln(1 + sqrt(2))
```

## Sample Values

| Case | r | f(r) | Sign | Interpretation |
| --- | ---: | ---: | --- | --- |
| Below threshold | 0.800000000000 | 0.166633696127 | positive | Outside the chronology-violating side for this circular test. |
| At threshold | 0.881373587020 | 0.000000000000 | zero | Critical radius in this normalization. |
| Above threshold | 0.950000000000 | -0.252490673321 | negative | Beyond the threshold; circular azimuthal curves are timelike in the known Godel result. |

## Minimal Reproduction

```js
const rc = Math.log(1 + Math.SQRT2);

function gPhiPhi(r) {
  const s = Math.sinh(r);
  return s * s - s * s * s * s;
}

for (const r of [0.8, rc, 0.95]) {
  console.log(r.toFixed(12), gPhiPhi(r).toFixed(12));
}
```

Expected output:

```text
0.800000000000 0.166633696127
0.881373587020 0.000000000000
0.950000000000 -0.252490673321
```

## Scope and Caution

- This artifact verifies a known Godel threshold relation used for site validation.
- It is the first validation artifact behind Yuvan Raam Chandra's Computational Relativity Lab.
- It does not claim new physical results.
- It does not compute Christoffel symbols, curvature tensors, Einstein tensor components, or energy-density quantities.
- Future artifacts should include explicit conventions, source references, scripts or notebooks, and expected outputs.