# C1049 — Minimal clutch observation basis

## Question

Which observed coordinates recover the unrestricted array?

## Inputs

```json
{}
```

## Sources

```json
[
  "C1048"
]
```

## Opened utc

2026-09-28T04:06:10.774293+00:00

## Results

```json
{
  "basis_indices": [
    0,
    1,
    2,
    3,
    4,
    5,
    6,
    21,
    7
  ],
  "basis_determinant": -1,
  "incomplete_observation_ranks": {
    "no_Aaron": 8,
    "no_shift": 8,
    "one_row_absent": 8
  }
}
```

## Finding

Seven unshifted Nisan coordinates, one unshifted Aaron coordinate and one half-clutch coordinate form an integer observation basis. Losing a whole phase, all shifts, or an entire source row leaves one parameter undetermined.

## Reassessment

Impose exact source phase and repeated-lifespan equalities separately.

## Checks

```json
{
  "unimodular": true,
  "all_rank_eight": true
}
```

## Closed utc

2026-09-28T04:06:10.780793+00:00

## Record provenance

Exact record hash: `c8a828e2e93ef2bb227323d1d70b4352e1272e6a17404399769678784e033813`

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