> ## Documentation Index
> Fetch the complete documentation index at: https://docs.aui.io/llms.txt
> Use this file to discover all available pages before exploring further.

# Derivations

> Facts worked out from other facts — where your business arithmetic, thresholds, and rules-of-thumb live, deterministically.

Some facts nobody supplies — they're **worked out**: an order's total, a claim's size band, whether a customer qualifies. `derivations.yaml` is where those live. Each derivation names one fact, the facts it's computed from, and exactly one way to compute it — and the runtime recomputes it **every turn from live inputs**, so a derived value can never go stale mid-conversation.

```yaml theme={"dark"}
derivations:
  - id: order.total
    means: What the order comes to.
    given: [order.subtotal, order.tax]
    value: order.subtotal + order.tax
```

* **`id`** — the fact this row produces (declared in the [vocabulary](/overview/vocabulary) with `known_from: derived`).
* **`means`** — what the value is, in words a caller could hear.
* **`given`** — the input facts; until they're established, the derived fact is simply unknown.
* **One body** — how it's computed. There are four kinds:

## The four kinds

**A formula (`value:`)** — arithmetic and logic over other facts. Computed exactly (decimal, never float):

```yaml theme={"dark"}
- id: order.total
  given: [order.subtotal, order.tax]
  value: order.subtotal + order.tax
```

**A ladder (`decide:`)** — ordered conditions where the *first* match wins, so the order is the meaning. `otherwise:` is mandatory — every ladder says what happens when nothing matches:

```yaml theme={"dark"}
- id: claim.band
  means: How large the claim is for routing purposes.
  given: [claim.amount]
  decide:
    - when: claim.amount >= 10000
      then: large
    - when: claim.amount >= 1000
      then: medium
  otherwise: small
```

**A table (`table:`)** — a lookup over one or two enum axes. Because the axes are enums, a missing cell is a build error, not a runtime surprise:

```yaml theme={"dark"}
- id: rate.pct
  means: The commission rate for this rep.
  table:
    rows: rep.region
    cols: rep.segment
    values:
      emea: { enterprise: 12, smb: 7 }
      amer: { enterprise: 11, smb: 6 }
```

**A model instruction (`model:`)** — for the rare value no formula can produce, like a search query built from a caller's request. This is the one kind computed once and kept, rather than recomputed every turn.

## Why derivations matter

Derivations are where policy text becomes computation. A threshold from your methodology, a fee schedule, an eligibility rule — each becomes a row that always computes the same answer from the same inputs, shows its work in the trace, and can name the exact provision it implements with `encodes:`:

```yaml theme={"dark"}
- id: refund.fee_waived
  encodes: "Returns policy §4.2"
  means: Whether the return fee is waived.
  given: [customer.tier, order.total]
  value: customer.tier == "premium" or order.total > 200
```

[Capabilities](/overview/capabilities-and-policies) list derived facts in their `needs`, gates test them in `requires`, and replies read them back — so the arithmetic your business runs on is written once, in one file, and enforced everywhere.