# C733 — Derive the full mode-bridge recurrence

## Question

What row quantity generates the change in cumulative-minus-regular displacement between adjacent named boundaries?

## Inputs

```json
{
  "definition": "G_i=C_i−R_i",
  "regular_edge": "R_i−R_(i+1)=b_eff_i",
  "cumulative_edge": "C_i−C_(i+1)=L_i",
  "local_MT_example": {
    "regular": [
      2166,
      2066,
      2006
    ],
    "cumulative": [
      2435,
      2260,
      2080
    ],
    "lifespans": [
      175,
      180
    ],
    "effective_begetting": [
      100,
      60
    ]
  }
}
```

## Sources

```json
[
  "File18 MT regular/cumulative rows; File61 root vector; Research Strategy §5B"
]
```

## Opened utc

2026-09-28T01:40:40.628020+00:00

## Results

```json
{
  "mode_bridge": [
    269,
    194,
    74
  ],
  "bridge_increments": [
    75,
    120
  ],
  "row_remainders": [
    75,
    120
  ],
  "general_rule": "G_i−G_(i+1)=L_i−b_eff_i"
}
```

## Finding

The mode bridge has a local generator: lifespan minus the effective regular edge. Telescoping these row contributions reconstructs the whole bridge once the terminal displacement is supplied. Declared binding corrections belong in b_eff rather than being hidden in a global conversion.

## Reassessment

Apply this recurrence to every shared MT boundary, retaining source roles and terminal frame.

## Checks

```json
{
  "local_increments": true,
  "projection": true,
  "telescoping": true
}
```

## Closed utc

2026-09-28T01:40:40.628246+00:00

## Record provenance

Exact record hash: `d3dc745a31bc5ec3060088752416c05e84342c9b46746ae3d26e6001d289e35d`

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