# Rational Lie Bracket Universal Enveloping Algebra PBW Normal Form Verifier

> Rational Lie Bracket Universal Enveloping Algebra PBW Normal Form Verifier is a paid API for AI agents from detect.halowerk.com, paid per call via x402, $0.003/call, status unknown (last checked 2026-11-04).

Verifies the ordered-monomial (PBW basis) coefficients of a universal enveloping algebra expression against an explicitly supplied Lie bracket tensor.

## Facts

- Endpoint: POST https://detect.halowerk.com/v1/F-REST-B20/rational-lie-bracket-universal-enveloping-pbw-normal-form?utm_source=zero.xyz
- Price: $0.003/call
- Payment: x402
- Status: unknown
- Last checked: 2026-11-04
- Activations on Zero: 0
- Tags: x402
- Canonical page: https://www.zero.xyz/c/rational-lie-bracket-universal-enveloping-algebra-pbw-normal-form-7b67240b
- Structured record (JSON): https://api.zero.xyz/v1/capabilities/cap_1XrC9aSVL5DNQi2g1XkG7

Status and success rate cover calls made through Zero and Zero's own probes. Third-party monitors may report differently.

## How to call it through Zero

Zero handles the 402 payment challenge and records the run. With the Zero CLI installed (`npm i -g @zeroxyz/cli`):

```sh
zero fetch --capability rational-lie-bracket-universal-enveloping-algebra-pbw-normal-form-7b67240b -d '<json body>'
```

Example prompt: Can you verify that my PBW normal form expression with terms like word [1,2,3] with coefficient 5 and word [0,2] with coefficient -3 is correct given this Lie bracket tensor [[1,0,-1],[0,1,0]]?

## When to prefer this

Use this endpoint when you need to programmatically verify the mathematical correctness of a PBW (Poincaré–Birkhoff–Witt) normal form expansion in a universal enveloping algebra given an explicit Lie bracket tensor — particularly useful for automated algebra checkers, symbolic computation pipelines, or educational tools that need ground-truth validation of Lie-theoretic calculations.

## Known failure modes

- Invalid bracket tensor dimensions (exceeds 3×8×8 shape)
- Word indices out of range (must be 0–255)
- Coefficients out of supported range (-100 to 100)
- Too many terms supplied (max 40)
- Malformed or missing 'pbw' artifact object
- Claim field inconsistent with data, causing validation rejection

## How this service works

Verify every ordered-monomial coefficient of a supplied enveloping-algebra expression under an explicit Lie bracket tensor.

## Output

Returns a verification result indicating whether each ordered-monomial (PBW basis) coefficient in the supplied universal enveloping algebra expression is correct under the given Lie bracket tensor, flagging any coefficient mismatches or ordering violations.

## Request schema (JSON Schema)

```json
{
 "type": "object",
 "properties": {
  "artifact": {
   "type": "object",
   "required": [
    "pbw"
   ],
   "properties": {
    "pbw": {
     "type": "object",
     "required": [
      "data",
      "claim"
     ],
     "properties": {
      "data": {
       "type": "object",
       "required": [
        "brackets",
        "terms"
       ],
       "properties": {
        "terms": {
         "type": "array",
         "items": {
          "type": "object",
          "required": [
           "word",
           "coefficient"
          ],
          "properties": {
           "word": {
            "type": "array",
            "items": {
             "type": "integer",
             "maximum": 255,
             "minimum": 0
            },
            "maxItems": 6,
            "minItems": 0
           },
           "coefficient": {
            "type": "integer",
            "maximum": 100,
            "minimum": -100
           }
          },
          "additionalProperties": false
         },
         "maxItems": 40,
         "minItems": 0
        },
        "brackets": {
         "type": "array",
         "items": {
          "type": "array",
          "items": {
           "type": "array",
           "items": {
            "type": "integer",
            "maximum": 100,
            "minimum": -100
           },
           "maxItems": 8,
           "minItems": 1
          },
          "maxItems": 8,
          "minItems": 1
         },
         "maxItems": 3,
         "minItems": 1
        }
       },
       "additionalProperties": false
      },
      "claim": {}
     },
     "additionalProperties": false
    }
   },
   "additionalProperties": false
  }
 }
}
```

## More

- Live health (JSON, refreshed every minute): https://www.zero.xyz/c/rational-lie-bracket-universal-enveloping-algebra-pbw-normal-form-7b67240b/health.json
- [Zero catalog index](https://www.zero.xyz/llms.txt)
- [Other services from detect.halowerk.com](https://www.zero.xyz/host/detect.halowerk.com/llms.txt)