# C1039 — An explicit Covenant dependency circuit

## Question

Is the household-middle equality independent of the other source conditions?

## Inputs

```json
{}
```

## Sources

```json
[
  "C1035–C1038"
]
```

## Opened utc

2026-09-28T04:04:10.364331+00:00

## Results

```json
{
  "dependency": "A9 = −A2 + A3 + A4 + A5",
  "coefficient_residual": [
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0
  ]
}
```

## Finding

The household-middle equality is forced by the second147 rail, equal lives, twin death arms and Joseph’s double span. It adds no separate constraint after those are retained.

## Reassessment

Derive the common-Key landing condition without using its299/276 outputs as inputs.

## Checks

```json
{
  "full_coefficient_identity": true,
  "nontrivial_circuit": true
}
```

## Closed utc

2026-09-28T04:04:10.365715+00:00

## Record provenance

Exact record hash: `9077235cff22b1c3611d63caf46ccc7f96c50ef126f428e7733246504b3ce587`

Source journal: `Research_Cycles/C0932_C1131/journal.json`