# C513 — One rule generates the14726 family

## Question

Can the complete set of Creation-held results be compressed into one affine formula?

## Inputs

```json
{
  "inverse_coordinates": {
    "Mahalalel": 1731,
    "Shem": 5526,
    "Shelah": 2421,
    "Eber": 9321,
    "Terah": 3226,
    "Abraham": 2076,
    "Moses": 1616,
    "Flood_countdown": 8516,
    "Affliction_hub": 1846
  },
  "held_anchor": 14726
}
```

## Sources

```json
[
  "File52c §§3.7.3,3.11.4",
  "C509,C512"
]
```

## Opened utc

2026-09-27T22:48:23.882482+00:00

## Results

```json
{
  "normal_form": "X=6+115k;14726=6+115*128;Creation-held result=125k−1274;Nativity-held result=125k+6",
  "members": {
    "Mahalalel": {
      "k": 15,
      "complement_k": 113,
      "Creation_held": 601,
      "Nativity_held": 1881
    },
    "Shem": {
      "k": 48,
      "complement_k": 80,
      "Creation_held": 4726,
      "Nativity_held": 6006
    },
    "Shelah": {
      "k": 21,
      "complement_k": 107,
      "Creation_held": 1351,
      "Nativity_held": 2631
    },
    "Eber": {
      "k": 81,
      "complement_k": 47,
      "Creation_held": 8851,
      "Nativity_held": 10131
    },
    "Terah": {
      "k": 28,
      "complement_k": 100,
      "Creation_held": 2226,
      "Nativity_held": 3506
    },
    "Abraham": {
      "k": 18,
      "complement_k": 110,
      "Creation_held": 976,
      "Nativity_held": 2256
    },
    "Moses": {
      "k": 14,
      "complement_k": 114,
      "Creation_held": 476,
      "Nativity_held": 1756
    },
    "Flood_countdown": {
      "k": 74,
      "complement_k": 54,
      "Creation_held": 7976,
      "Nativity_held": 9256
    },
    "Affliction_hub": {
      "k": 16,
      "complement_k": 112,
      "Creation_held": 726,
      "Nativity_held": 2006
    }
  }
}
```

## Finding

Nine outputs follow from one formula and the supplied member coefficients. The1280 anchor-change difference is uniform. This compresses the calculation, while the nine source selections and their k values remain inputs needing explanation.

## Reassessment

Test a whole three-node relation under the formula: the Abraham–affliction–Moses midpoint.

## Checks

```json
{
  "integral_k": true,
  "formula": true,
  "shift": true
}
```

## Closed utc

2026-09-27T22:48:23.882801+00:00

## Record provenance

Exact record hash: `792c142e2edb2594537b5cae429bc1067f054069231af78c09516262886a34b1`

Source journal: `Research_Cycles/C0482_C0531/journal.json`