# C558 — Generate the fixed-Joseph comparison family

## Question

Can one flank-transfer formula reproduce all nine comparison states?

## Inputs

```json
{
  "regular_Joseph": [
    1915,
    1805
  ],
  "phases": [
    0,
    3,
    "7/2"
  ],
  "clutch": [
    0,
    7,
    14
  ]
}
```

## Sources

```json
[
  "File62 §§6.1–6.4"
]
```

## Opened utc

2026-09-28T00:13:53.938889+00:00

## Results

```json
{
  "partitions": [
    [
      18,
      9,
      83
    ],
    [
      11,
      16,
      83
    ],
    [
      4,
      23,
      83
    ],
    [
      "43/2",
      "11/2",
      83
    ],
    [
      "29/2",
      "25/2",
      83
    ],
    [
      "15/2",
      "39/2",
      83
    ],
    [
      21,
      6,
      83
    ],
    [
      14,
      13,
      83
    ],
    [
      7,
      20,
      83
    ]
  ],
  "formula": [
    "18+p−d",
    "9−p+d",
    "83"
  ]
}
```

## Finding

The clutch transfers duration between two flanks while preserving their27 total and the83 overlap. All nine states follow without moving regular Joseph1915–1805.

## Reassessment

Test whether the reported cyclic symmetry acts on dates or on duration components.

## Checks

```json
{
  "nine": true,
  "totals": true
}
```

## Closed utc

2026-09-28T00:13:53.939439+00:00

## Record provenance

Exact record hash: `b91c3ca9bc4a64f1989f0bd86f9b584dec686a7ba001f9a48b4ae95796aa0f35`

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