# C609 — Generalize allocation endpoint equivalence

## Question

How large is the junction discrepancy between whole-Key and terminal-E routes?

## Inputs

```json
{
  "examples": [
    [
      483,
      "1/6",
      "70/69"
    ],
    [
      12558,
      "1/26",
      "300/299"
    ]
  ]
}
```

## Sources

```json
[
  "C568–C571; algebraic path comparison"
]
```

## Opened utc

2026-09-28T00:27:38.215901+00:00

## Results

```json
{
  "comparisons": [
    {
      "S": 483,
      "prefix": "805/2",
      "suffix": "161/2",
      "total": 490,
      "whole_total": 490,
      "junction_gap": "35/6"
    },
    {
      "S": 12558,
      "prefix": 12075,
      "suffix": 483,
      "total": 12600,
      "whole_total": 12600,
      "junction_gap": "525/13"
    }
  ],
  "general_condition": "k=1+f(E−1)",
  "junction_gap_formula": "(k−1)(1−f)S"
}
```

## Finding

For both source partitions the endpoint agreement follows from one gain-allocation law, while the interior discrepancy is fixed by(k−1)(1−f)S. Retaining the path reveals exactly what the total-span comparison forgets.

## Reassessment

Test why the Rounded inverse module must preserve component boundaries.

## Checks

```json
{
  "endpoints": true,
  "gaps": true
}
```

## Closed utc

2026-09-28T00:27:38.216246+00:00

## Record provenance

Exact record hash: `7f35356e54ac66cba7786f922af6e959e6a4780608e91cee1d35582d3b880bd6`

Source journal: `Research_Cycles/C0532_C0631/journal.json`