# Simplicial F2 One-Cocycle Cup Product Class Verifier

> Simplicial F2 One-Cocycle Cup Product Class 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 whether the cup product of two specified F2 one-cocycles is zero in H2 of a given finite 2D simplicial complex, returning a coboundary witness if applicable.

## Facts

- Endpoint: POST https://detect.halowerk.com/v1/F-NIGHT-ECMA/simplicial-f2-one-cocycle-cup-class?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/simplicial-f2-one-cocycle-cup-product-class-verifier-1c1ae471
- Structured record (JSON): https://api.zero.xyz/v1/capabilities/cap_JieLmAiQMA09GaqXq2T1e

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 simplicial-f2-one-cocycle-cup-product-class-verifier-1c1ae471 -d '<json body>'
```

Example prompt: Check whether the cup product of these two F2 one-cocycles (alpha = [1,0,1,0] and beta = [0,1,1,0]) is zero in H2 of my simplicial complex with 4 vertices, edges [[0,1],[1,2],[2,3],[3,0],[0,2]], and triangles [[0,1,2],[0,2,3]] — my claim is that it IS zero, and I need a coboundary witness if that's correct.

## When to prefer this

Use this endpoint when an agent needs to rigorously verify a cohomological cup product claim over F2 in a small finite 2D simplicial complex and requires a coboundary witness — not just chain ranks or homology dimensions. Prefer this over general homology tools when the specific algebraic topology question is about the cohomology ring structure (cup products) rather than Betti numbers or chain connectivity.

## Known failure modes

- Invalid simplicial complex (e.g. edges referencing out-of-range vertex indices) returns a validation error
- Alpha or beta cocycle arrays of wrong length relative to edge count returns schema mismatch
- Non-cocycle input (alpha or beta not satisfying coboundary condition) may return error or unexpected result
- Exceeding max vertices (12), edges (32), or triangles (24) returns schema validation error
- Malformed triangle arrays (not exactly 3 vertex indices) return input error

## How this service works

A topology test agent needs whether the cup product of two specified1cocycles is zero in H2 of its complete finite2D simplicial complex, with a coboundary witness, not only chain ranks or homology dimensions.

## Output

Returns whether the cup product of the two supplied F2 one-cocycles (alpha and beta) equals zero in H2 of the given finite 2D simplicial complex, together with a coboundary witness certificate when the claim is verified correct.

## Request schema (JSON Schema)

```json
{
 "type": "object",
 "properties": {
  "artifact": {
   "type": "object",
   "required": [
    "cup_product"
   ],
   "properties": {
    "cup_product": {
     "type": "object",
     "required": [
      "vertices",
      "edges",
      "triangles",
      "alpha",
      "beta",
      "claim"
     ],
     "properties": {
      "beta": {
       "type": "array",
       "items": {
        "type": "integer",
        "maximum": 1,
        "minimum": 0
       },
       "maxItems": 32,
       "minItems": 0
      },
      "alpha": {
       "type": "array",
       "items": {
        "type": "integer",
        "maximum": 1,
        "minimum": 0
       },
       "maxItems": 32,
       "minItems": 0
      },
      "claim": {
       "type": "boolean"
      },
      "edges": {
       "type": "array",
       "items": {
        "type": "array",
        "items": {
         "type": "integer",
         "maximum": 11,
         "minimum": 0
        },
        "maxItems": 2,
        "minItems": 2
       },
       "maxItems": 32,
       "minItems": 0
      },
      "vertices": {
       "type": "integer",
       "maximum": 12,
       "minimum": 1
      },
      "triangles": {
       "type": "array",
       "items": {
        "type": "array",
        "items": {
         "type": "integer",
         "maximum": 11,
         "minimum": 0
        },
        "maxItems": 3,
        "minItems": 3
       },
       "maxItems": 24,
       "minItems": 0
      }
     },
     "additionalProperties": false
    }
   },
   "additionalProperties": false
  }
 }
}
```

## More

- Live health (JSON, refreshed every minute): https://www.zero.xyz/c/simplicial-f2-one-cocycle-cup-product-class-verifier-1c1ae471/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)