POST /api/zeq/agent/operators/derive

The five-step formal-derivation protocol — for when the agent's problem can't be solved with the 1,500+-operator catalogue and skill synthesis can't bridge the gap. This is the heaviest move in the runtime; it produces a candidate operator with a verification path and stores it in derived_operators.

The five steps

1. Domain identification — which physical/mathematical domain.
2. Existing-operator survey — what the catalogue already covers.
3. Gap statement — what's missing.
4. Candidate equation — formal equation, not prose.
5. Verification path — how to test it against ZeqProof.

Auth

Authorization: Bearer zeq_ak_….

Cost

50 ZEQ atomic.

Request

curl -X POST https://YOUR-FRAMEWORK/api/zeq/agent/operators/derive \
  -H "Authorization: Bearer zeq_ak_..." \
  -H "Content-Type: application/json" \
  -d '{
    "problem":         "no operator covers Hawking radiation around a Kerr black hole at finite spin",
    "domain":          "general-relativity",
    "candidate_form":  "Bogoliubov transformation extended with rotational momentum coupling"
  }'

Response · 200 OK

{
  "ok": true,
  "operator_id":  "op_2026-05-10T13:08:32Z_...",
  "name":         "GR_HAWKING_KERR",
  "equation":     "...formal LaTeX/JSON spec...",
  "verification_path": "compare against numerical Hawking-Hartle vacuum at five spin parameters",
  "stored_in":    "derived_operators",
  "tally_charge": { "charged": 50, "burned": 40, "toFoundation": 10, "remainingAfter": 88 }
}

A derived operator that survives 100+ uses with ≥99% fitness can be promoted to the canonical catalogue via /operators/promote (foundation review).

Errors

Status	Code	Cause
400	domain_invalid	Not in the 64-domain enum.
402	INSUFFICIENT_BALANCE	Wallet < 50 ZEQ.
503	derivation_failed	LLM produced an equation that can't validate. No charge.

Related

• Agents concept — when the agent escalates to derivation
• POST /skills/synthesise — the lighter alternative
• POST /operators/promote — promote to catalogue