# Certificate Composition Rules

These rules describe when SFT certificates can be combined without changing meaning.

The rules are formulated as **artifact transformations**: given input certificates with declared perturbation models and margins, the engine may emit a composed certificate with explicitly degraded margins.

## Product systems (independent composition)

- Input: two WSCs on targets of types circle/torus/embedded, with compatible perturbation models.
- Output: a WSC on the product target.
- Witness integer: recorded as a tuple in notes, while the output `witness_value.integer` records the primary forcing integer used for the composed claim.

## Restriction to a subsystem

- Input: a WSC on a larger domain plus a declared subsystem embedding.
- Output: a WSC on the embedded target, reusing the same witness integer and margins.

## Coupling projection

- Input: a torus-target WSC plus an integer constraint matrix.
- Output: a CST-target WSC with coupled Haar baseline.

## Atlas gluing

- Input: certificates on overlapping regions with compatible witness integers.
- Output: a merged certificate with margins degraded to the minimum of the inputs.

## Artifact interface

When a certificate is the result of composition, it may include:

- `composition.rule` (string)
- `composition.inputs` (list of certificate ids)
- `composition.notes` (string)

