# Finite Matroid Independence Family Axioms Verifier

> Finite Matroid Independence Family Axioms 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).

Validates whether a fully-specified finite independent family satisfies all three matroid axioms (empty set, hereditary, augmentation), reporting any violations.

## Facts

- Endpoint: POST https://detect.halowerk.com/v1/F-B-NIGHT-MATROID_INDEPENDENCE_AXIOMS/finite-matroid-independent-family-axioms?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/finite-matroid-independence-family-axioms-verifier-5c3be364
- Structured record (JSON): https://api.zero.xyz/v1/capabilities/cap_QM38H2lkOLH0bwATqASPd

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 finite-matroid-independence-family-axioms-verifier-5c3be364 -d '<json body>'
```

Example prompt: Check whether my finite independent family forms a valid matroid: the ground set is ['a','b','c','d'], the independent sets are [[], ['a'], ['b'], ['c'], ['a','b'], ['a','c']], and I'm claiming hereditary=true, augmentation=true, empty_set_present=true, is_matroid=true, with 4 ground elements, 6 independent sets, and maximum cardinality 2 — verify all three matroid axioms and flag any violations.

## When to prefer this

Use this endpoint when you need to definitively verify whether a fully enumerated finite independent family satisfies all three matroid axioms, especially when you want to detect subtle violations like missing empty set, non-hereditary subsets, or augmentation failures. It is not suitable for sampling-based checks, greedy algorithm computation, or matroid optimization — it only validates axiom compliance on a complete, explicitly provided independent family up to 16 elements and 256 sets.

## Known failure modes

- Input ground_set has duplicate elements (schema requires uniqueItems)
- Independent set references index out of range for ground_set size (index >= ground_set.length)
- ground_set exceeds 16 elements or independent_sets exceeds 256 entries
- claims counts do not match actual sizes (ground_elements or independent_sets mismatch)
- Malformed or missing required fields in artifact or claims objects
- Payment failure or x402 protocol error resulting in 402 response

## How this service works

Prüft eine vollständig gelieferte endliche Unabhängigkeitsfamilie gegen die drei Matroidaxiome, einschließlich Austauschfehlern, Schleifen und leerer Grundmenge; bestätigt keine bloße Stichprobe und berechnet keinen greedy Lösungsweg.

## Output

Returns a structured validation report confirming or refuting each of the three matroid axioms (empty set, hereditary closure, augmentation/exchange), cross-checked against the caller's claims about ground element count, independent set count, maximum cardinality, and overall matroid status. Violations are flagged with specifics (e.g. exchange errors, missing empty set, non-hereditary subsets).

## Request schema (JSON Schema)

```json
{
 "type": "object",
 "properties": {
  "artifact": {
   "type": "object",
   "required": [
    "ground_set",
    "independent_sets",
    "claims"
   ],
   "properties": {
    "claims": {
     "type": "object",
     "required": [
      "ground_elements",
      "independent_sets",
      "maximum_cardinality",
      "empty_set_present",
      "hereditary",
      "augmentation",
      "is_matroid"
     ],
     "properties": {
      "hereditary": {
       "type": "boolean"
      },
      "is_matroid": {
       "type": "boolean"
      },
      "augmentation": {
       "type": "boolean"
      },
      "ground_elements": {
       "type": "number"
      },
      "independent_sets": {
       "type": "number"
      },
      "empty_set_present": {
       "type": "boolean"
      },
      "maximum_cardinality": {
       "type": "number"
      }
     },
     "additionalProperties": false
    },
    "ground_set": {
     "type": "array",
     "items": {
      "type": "string",
      "maxLength": 128,
      "minLength": 1
     },
     "maxItems": 16,
     "uniqueItems": true
    },
    "independent_sets": {
     "type": "array",
     "items": {
      "type": "array",
      "items": {
       "type": "integer",
       "maximum": 15,
       "minimum": 0
      },
      "maxItems": 16
     },
     "maxItems": 256
    }
   },
   "additionalProperties": false
  }
 }
}
```

## More

- Live health (JSON, refreshed every minute): https://www.zero.xyz/c/finite-matroid-independence-family-axioms-verifier-5c3be364/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)