# C971 — Bidirectional cap equation packet

## Question

How much independent information do forward and inverse reconstructions require?

## Inputs

```json
{
  "variables": [
    "bJ",
    "bE",
    "bM",
    "bL",
    "lifeJ",
    "lifeM",
    "lifeL",
    "k"
  ]
}
```

## Sources

```json
[
  "C964",
  "C965",
  "C970"
]
```

## Opened utc

2026-09-28T03:49:35.209792+00:00

## Results

```json
{
  "equations": [
    [
      1,
      1,
      1,
      1,
      -1,
      0,
      0,
      1
    ],
    [
      0,
      0,
      1,
      1,
      0,
      -1,
      0,
      1
    ],
    [
      0,
      0,
      0,
      1,
      0,
      0,
      -1,
      1
    ]
  ],
  "rank": 3,
  "unconstrained_dimension": 5,
  "after_k_and_Enoch_held": 3
}
```

## Finding

The late-row relation has three independent equations on eight variables. Holding k and Enoch leaves equivalent three-number choices: the changed begettings or the capped lives. This is bidirectional constraint, not double evidence.

## Reassessment

Propagate the cap-derived reductions into the cumulative field.

## Checks

```json
{
  "unique_life_columns": true,
  "dimension": true
}
```

## Closed utc

2026-09-28T03:49:35.210019+00:00

## Record provenance

Exact record hash: `18f9402a8cb0d54db1c43591f00741da5a12074576156ff93e4888f59c086fb2`

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