# C1307 — Register Key compatibility with complete path sums

## Question

When does applyingKeys before oraftergrouping give thesamefield?

## Inputs

```json
{}
```

## Sources

```json
[
  "C1306",
  "C984–C1007 inherited",
  "inherited/model/completed_path_interface_ledger.json"
]
```

## Opened utc

2026-09-28T06:10:27.058589+00:00

## Results

```json
{
  "Key_sum_relation": {
    "path": "model/relation_Key_sums.json",
    "sha256": "ba7b84e3f9838473c66fb0ecd9dfd3cb849acc3fa0a8048d0e8ad48449a34954",
    "bytes": 1035
  }
}
```

## Finding

UniformKeys commute withsumming aselectedpath. DifferentcomponentKeys commute withblockgroupingonlyunder aconstant-factorcondition withinthatblock, so retainedcutsremain explanatorydata.

## Reassessment

Register anchoredcomposition andintegerintermediate-domain conditions.

## Checks

```json
{
  "bound_evidence": true,
  "three_limits": true
}
```

## Closed utc

2026-09-28T06:10:27.061608+00:00

## Record provenance

Exact record hash: `938fb61b3523e5340588102c78b22635fe7b45b9b396bdb6ab0f5ae611f6ff9f`

Source journal: `Research_Cycles/C1132_C1431_Recovered/journal.json`