# C993 — Universal coarsening criterion

## Question

When can a componentwise conversion be replaced by one factor per merged block?

## Inputs

```json
{
  "coarsening": "sum fine components into declared blocks",
  "conversion": "diagonal factors k_i"
}
```

## Sources

```json
[
  "C991",
  "C992",
  "File63 retained partitions"
]
```

## Opened utc

2026-09-28T03:54:01.790588+00:00

## Results

```json
{
  "necessity": "Test each unit vector e_i: the fine factor k_i must equal its containing coarse-block factor.",
  "sufficiency": "If factors are constant within each block, factoring that common value out of the block sum proves equality.",
  "criterion": "C·diag(k_i)=diag(k_B)·C iff k_i=k_B within every block"
}
```

## Finding

Coarsening commutes with conversion for every vector exactly when the factors are constant inside each merged block. Equality on one supplied total is weaker.

## Reassessment

Exhibit the information hidden by one equal-total partial/whole comparison.

## Checks

```json
{
  "uniform_example": true,
  "mixed_not_universal": true
}
```

## Closed utc

2026-09-28T03:54:01.791014+00:00

## Record provenance

Exact record hash: `e154b01f56e7b52fcf5a1193c7078496c77ac689a15b3616b00dbaced8f3cb8f`

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