# C552 — Generate the cumulative root family

## Question

Does one affine rule reconstruct all six source rails?

## Inputs

```json
{
  "base": [
    2435,
    2260,
    2080,
    1933,
    1796,
    1663,
    1526
  ],
  "phases": [
    0,
    "7/2"
  ],
  "clutch": [
    0,
    7,
    14
  ]
}
```

## Sources

```json
[
  "File62 §§1.2–1.4; File61 §§7.3A–7.6"
]
```

## Opened utc

2026-09-28T00:12:31.275020+00:00

## Results

```json
{
  "rails": [
    [
      2435,
      2260,
      2080,
      1933,
      1796,
      1663,
      1526
    ],
    [
      2428,
      2253,
      2073,
      1926,
      1789,
      1656,
      1519
    ],
    [
      2421,
      2246,
      2066,
      1919,
      1782,
      1649,
      1512
    ],
    [
      "4877/2",
      "4527/2",
      "4167/2",
      "3873/2",
      "3599/2",
      "3333/2",
      "3059/2"
    ],
    [
      "4863/2",
      "4513/2",
      "4153/2",
      "3859/2",
      "3585/2",
      "3319/2",
      "3045/2"
    ],
    [
      "4849/2",
      "4499/2",
      "4139/2",
      "3845/2",
      "3571/2",
      "3305/2",
      "3031/2"
    ]
  ],
  "coordinate_matches": 42
}
```

## Finding

C(p,d)=C0+p−d reconstructs all42 source coordinates. This is one family generated from seven inputs and two local state parameters, not42 independent successes.

## Reassessment

Establish why the source clutch is14 and whether its three envelopes are independent.

## Checks

```json
{
  "six_rails": true,
  "invariant_spans": true
}
```

## Closed utc

2026-09-28T00:12:31.275664+00:00

## Record provenance

Exact record hash: `ed4d4ba1528f03351c874e6224383840e9da8b1b703b22a8c50818d742eda54a`

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