# C755 — Invert and classify the MT residual field

## Question

Which entire interval families survive rounding exactly?

## Inputs

```json
{}
```

## Sources

```json
[
  "C754"
]
```

## Opened utc

2026-09-28T01:54:10.979733+00:00

## Results

```json
{
  "recovered_e": [
    0,
    -2,
    0,
    0,
    0,
    -2,
    0,
    1,
    -2,
    0,
    0,
    2,
    2,
    1,
    1,
    1,
    0,
    2,
    0,
    0,
    0,
    -2,
    -2,
    2,
    -2,
    0
  ],
  "equal_residual_classes": {
    "-4": [
      "Terah",
      "Abraham",
      "Isaac",
      "Jacob"
    ],
    "-2": [
      "Serug",
      "Nahor",
      "Levi",
      "Amram"
    ],
    "-1": [
      "Reu"
    ],
    "0": [
      "Adam",
      "Seth",
      "Peleg",
      "Kohath",
      "Moses",
      "terminal1406"
    ],
    "1": [
      "Eber"
    ],
    "2": [
      "Enosh",
      "Kenan",
      "Mahalalel",
      "Jared"
    ],
    "3": [
      "Lamech",
      "Shelah"
    ],
    "4": [
      "Enoch",
      "Methuselah"
    ],
    "5": [
      "Noah",
      "Shem",
      "Arphaxad"
    ]
  },
  "interval_rule": "rounded(i,j)-actual(i,j)=D_i-D_j"
}
```

## Finding

Every pair within one residual class preserves its interval. These are consequences of one field, not independent numerical hits. Adjacent differences recover every row residual.

## Reassessment

Transfer the full residual operator to LXX and isolate changed row support.

## Checks

```json
{
  "inverse": true,
  "interval_identity": true
}
```

## Closed utc

2026-09-28T01:54:10.980078+00:00

## Record provenance

Exact record hash: `b6b9a4bf00039ec2e00aea2aa885f410548a951dd6d84a46c3b6972ca86ae596`

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