# C501 — The three-Key integral domain

## Question

Why does the middle member admit all three integral calendar outputs?

## Inputs

```json
{
  "Jared": 8372,
  "Noah_G1": 8970,
  "Flood_G2": 9568,
  "Creation": 7912
}
```

## Sources

```json
[
  "File52c §3.15",
  "C500"
]
```

## Opened utc

2026-09-27T22:43:53.341661+00:00

## Results

```json
{
  "common_input_lattice": 897,
  "rows": {
    "Jared": {
      "E": 9100,
      "P": "25480/3",
      "J": 8400,
      "all_integral": false
    },
    "Noah_G1": {
      "E": 9750,
      "P": 9100,
      "J": 9000,
      "all_integral": true
    },
    "Flood_G2": {
      "E": 10400,
      "P": "29120/3",
      "J": 9600,
      "all_integral": false
    },
    "Creation": {
      "E": 8600,
      "P": "24080/3",
      "J": "103200/13",
      "all_integral": false
    }
  },
  "JNF_mod3_coefficients": [
    2,
    0,
    1
  ]
}
```

## Finding

The common integral span domain is897Z, since897=lcm(23,69,299). For598m, E and J are integral for all integer m; P additionally needs3|m. Of14,15,16 only Noah’s15 qualifies. This explains the three-Key middle role from the existing progression.

## Reassessment

Use those outputs to derive the shared2926 endpoint and assess which placement assumptions remain.

## Checks

```json
{
  "lattice": true,
  "middle_only": true
}
```

## Closed utc

2026-09-27T22:43:53.341906+00:00

## Record provenance

Exact record hash: `3ad039dfc81a6b222c3776f8f9384647015ccaf55a0585aaa31e4e508fecbee3`

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