# C559 — Locate the Joseph cyclic symmetry

## Question

Does a three-cycle act consistently on the complete component family?

## Inputs

```json
{
  "rotation": "(a,b,c)→(b,c,a)"
}
```

## Sources

```json
[
  "File62 §§6.1–6.4; C558"
]
```

## Opened utc

2026-09-28T00:14:08.923110+00:00

## Results

```json
{
  "states": 9,
  "component_displays": 27,
  "example": [
    [
      4,
      23,
      83
    ],
    [
      23,
      83,
      4
    ],
    [
      83,
      4,
      23
    ]
  ],
  "action_domain": "ordered duration triples"
}
```

## Finding

A genuine C3 action rotates duration components and returns after three applications. It does not establish a permutation of historical people or complete chronology states.

## Reassessment

Test how whole-year and exact phase choices change crossed containers.

## Checks

```json
{
  "all_rotations": true,
  "third_power_identity": true
}
```

## Closed utc

2026-09-28T00:14:08.923700+00:00

## Record provenance

Exact record hash: `a22fb0384fb192c1b75b6ee07ee84c20b8fb963fa4a8e3f94493c549c335dd5a`

Source journal: `Research_Cycles/C0532_C0631/journal.json`