# C752 — Derive the finite rounding defect law

## Question

Can one residue table explain every parts-versus-total defect?

## Inputs

```json
{}
```

## Sources

```json
[
  "C749–750; File51a rounding rules"
]
```

## Opened utc

2026-09-28T01:53:30.703808+00:00

## Results

```json
{
  "table_b_rows_r_columns": [
    [
      0,
      0,
      0,
      0,
      0
    ],
    [
      0,
      0,
      -5,
      0,
      0
    ],
    [
      0,
      -5,
      -5,
      0,
      0
    ],
    [
      0,
      0,
      0,
      5,
      5
    ],
    [
      0,
      0,
      0,
      5,
      0
    ]
  ],
  "formula": "K(b,r)=Q(b)+Q(r)-Q(b+r)",
  "periodicity": "Q(x+5k)=Q(x)+5k"
}
```

## Finding

Every compatible integer row belongs to one of 25 residue classes; defects are only −5,0,+5. Methuselah and Reu share class(2,2).

## Reassessment

Check whether the three traditions add new defect classes within the admitted domains.

## Checks

```json
{
  "values": true,
  "periodic_exhaustion": true
}
```

## Closed utc

2026-09-28T01:53:30.706032+00:00

## Record provenance

Exact record hash: `5ccd38e85638d691f9012d5c0547fe166baa524b185c348fa0797ba2e05ac2e1`

Source journal: `Research_Cycles/C0732_C0831/journal.json`