# C777 — Derive the whole interval-response law

## Question

Which switches can affect a given interval?

## Inputs

```json
{}
```

## Sources

```json
[
  "C772–776"
]
```

## Opened utc

2026-09-28T02:00:47.995272+00:00

## Results

```json
{
  "formula": "delta span =215 ds(S_i−S_j)+60 dt(T_i−T_j)+130 dc(H_i−H_j)",
  "examples": {
    "Adam→Arphaxad": {
      "000": 1658,
      "001": 1658,
      "010": 1658,
      "011": 1658,
      "100": 1658,
      "101": 1658,
      "110": 1658,
      "111": 1658
    },
    "Arphaxad→Shelah": {
      "000": 35,
      "001": 165,
      "010": 35,
      "011": 165,
      "100": 35,
      "101": 165,
      "110": 35,
      "111": 165
    },
    "Terah→Abraham": {
      "000": 70,
      "001": 70,
      "010": 130,
      "011": 130,
      "100": 70,
      "101": 70,
      "110": 130,
      "111": 130
    }
  },
  "dependent_comparisons": 4560
}
```

## Finding

Intervals inside one support block are invariant; intervals crossing a hinge detect exactly that hinge. The entire finite family follows one incidence rule.

## Reassessment

Test matched configurations on the newly aligned SP/LXX birth suffix.

## Checks

```json
{
  "whole_family": true
}
```

## Closed utc

2026-09-28T02:00:48.000725+00:00

## Record provenance

Exact record hash: `ea806725fe23d742f5b493cf686ee6d48e2df963e9758ca425675b1623099ef0`

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