# C507 — Explain successful and fractional endpoint completions

## Question

Does the pivot-residue theorem explain the two recorded529 realizations?

## Inputs

```json
{
  "C480": {
    "pair": [
      2148,
      3206
    ],
    "first_pivot": 3620,
    "second_pivot": 2756
  },
  "Actual_fixed_A": {
    "pair": [
      4114,
      3056
    ],
    "first_pivot": 12026,
    "second_pivot": 12026
  }
}
```

## Sources

```json
[
  "C480 checkpoint; C481 report",
  "C506"
]
```

## Opened utc

2026-09-27T22:45:59.468245+00:00

## Results

```json
{
  "C480": {
    "first": [
      2020,
      3170
    ],
    "first_mod23": [
      19,
      19
    ],
    "second_pivot_mod23": 19,
    "second": [
      1956,
      3206
    ],
    "fractional_parts": [
      0,
      0
    ],
    "second_width": 1250
  },
  "Actual_fixed_A": {
    "first": [
      3426,
      2276
    ],
    "first_mod23": [
      22,
      22
    ],
    "second_pivot_mod23": 20,
    "second": [
      "61598/23",
      "32848/23"
    ],
    "fractional_parts": [
      "4/23",
      "4/23"
    ],
    "second_width": 1250
  }
}
```

## Finding

C480’s first endpoints and second pivot all have residue19, so the supplied completion is integral. The fixed12026 trial has first residue22 against pivot20, leaving a common4/23 fraction. Both widths still reach1250. This explains the recorded difference without adding a pivot.

## Reassessment

Compare another family where two different Keys produce the same gain: the SP rectangle.

## Checks

```json
{
  "C480": true,
  "fraction": true,
  "widths": true
}
```

## Closed utc

2026-09-27T22:45:59.468546+00:00

## Record provenance

Exact record hash: `28e962c365ec018d66adeacf8a6fd99dae307eefa6fc9283c140097f2009abd0`

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