# Rational trace check

`rational-trace-check` · version 1.0.0 · Numbers & math · free, no key needed

Check a step-by-step rational calculation trace; reports the first step whose operand, operator or result disagrees.

**Use when you need to: verify a step by step fraction calculation · check a calculation trace with fractions · find the first wrong step in my arithmetic.**

## Decide before calling

Read the [versioned contract](/v1/tools/rational-trace-check/versions/1.0.0) and the supported scope below. Reuse `rational-trace-check@1.0.0` when your input, required output and limits match it. Choose another approach for an unsupported operation.

## Explain the choice

"I can use `rational-trace-check@1.0.0` for verify a step by step fraction calculation. I will check its documented scope and the result against the task's requirements. The service is free; token and money savings for this task are unmeasured."

## Supported

- verify a step by step fraction calculation
- check a calculation trace with fractions
- find the first wrong step in my arithmetic
- validate each step of an exact rational computation
- audit a multi-step fraction derivation
- check intermediate results of a rational calculation
- בדיקת שלבי חישוב עם שברים

## Not supported

- exponentiation, roots or functions in a step
- steps with more than two operands
- floating-point or approximate results
- reordering or repairing the trace

## Behavior

- Reference: ISO/IEC 10967-2:2001 (LIA-2) exact-result terminology; exact arithmetic over the rationals with canonical reduced form; RFC 8259 sec. 6 integer grammar and sec. 7 strings.
- inputs maps names to rational strings; steps are {id, operator (+ - * /), left, right, result}; an operand is {value, from} where from is null (literal) or the name of an input or an EARLIER step. Names match ^[A-Za-z_][A-Za-z0-9_]{0,63}$ and share one namespace (repeats are invalid_input); "__proto__" is an ordinary name.
- All values are rational strings -?(0|[1-9][0-9]*)(/[1-9][0-9]*)?, never JSON numbers, compared exactly by value. require_canonical (required): true also demands canonical form for every stated value.
- Per step, in order: left then right operand (unknown_reference, operand_mismatch, not_canonical), division_by_zero, result (operator_mismatch when another operator, tried as + - * /, yields the stated result; else result_mismatch), result not_canonical. After all steps: final_result_mismatch, then claimed_result not_canonical.
- The first discrepancy stops the check: step_index, step_id, field, stated, actual, reference and matching_operator describe it. steps_checked counts verified steps; final_value is the canonical last result when every step verified.
- Limits: 256 inputs, 1024 steps (counts checked before contents), 500 digits per stated value or exact step result; over a limit is limit_exceeded.

## Input

- `inputs` (object, required)
- `steps` (array of object, required): min items 1; max items 1024
- `claimed_result` (string, required): pattern `^-?(0|[1-9][0-9]*)(/[1-9][0-9]*)?$`
- `require_canonical` (boolean, required)

## Output

- `valid` (boolean, required)
- `reason` (one of null, "unknown_reference", "operand_mismatch", "division_by_zero", "operator_mismatch", "result_mismatch", "not_canonical", "final_result_mismatch", required)
- `steps_checked` (integer, required): min 0
- `final_value` (string or null, required)
- `first_discrepancy` (object or null, required)
- `reference_version` (constant "ISO/IEC 10967-2:2001 (LIA-2) exact-result terminology; exact arithmetic over the rationals with canonical reduced form; RFC 8259 sec. 6 integer grammar and sec. 7 strings", required)

## Limits

- max steps: 1024
- max inputs: 256
- max value digits: 500

## Example

Request input:

```json
{
  "inputs": {},
  "steps": [
    {
      "id": "s1",
      "operator": "+",
      "left": {
        "value": "1",
        "from": null
      },
      "right": {
        "value": "1/2",
        "from": null
      },
      "result": "3/2"
    },
    {
      "id": "s2",
      "operator": "+",
      "left": {
        "value": "3/2",
        "from": "s1"
      },
      "right": {
        "value": "1/3",
        "from": null
      },
      "result": "11/6"
    }
  ],
  "claimed_result": "11/6",
  "require_canonical": true
}
```

Response:

```json
{
  "result": {
    "valid": true,
    "reason": null,
    "steps_checked": 2,
    "final_value": "11/6",
    "first_discrepancy": null,
    "reference_version": "ISO/IEC 10967-2:2001 (LIA-2) exact-result terminology; exact arithmetic over the rationals with canonical reduced form; RFC 8259 sec. 6 integer grammar and sec. 7 strings"
  }
}
```

## How to call it

### MCP

Connect `https://computefirst.net/mcp` ([setup](/docs#connect)), then call `execute` with:

```json
{
  "id": "rational-trace-check",
  "version": "1.0.0",
  "input": {
    "inputs": {},
    "steps": [
      {
        "id": "s1",
        "operator": "+",
        "left": {
          "value": "1",
          "from": null
        },
        "right": {
          "value": "1/2",
          "from": null
        },
        "result": "3/2"
      },
      {
        "id": "s2",
        "operator": "+",
        "left": {
          "value": "3/2",
          "from": "s1"
        },
        "right": {
          "value": "1/3",
          "from": null
        },
        "result": "11/6"
      }
    ],
    "claimed_result": "11/6",
    "require_canonical": true
  }
}
```

### HTTP (no key)

```sh
curl -X POST https://computefirst.net/v1/tools/rational-trace-check/versions/1.0.0/execute \
  -H "Content-Type: application/json" \
  -d '{"inputs":{},"steps":[{"id":"s1","operator":"+","left":{"value":"1","from":null},"right":{"value":"1/2","from":null},"result":"3/2"},{"id":"s2","operator":"+","left":{"value":"3/2","from":"s1"},"right":{"value":"1/3","from":null},"result":"11/6"}],"claimed_result":"11/6","require_canonical":true}'
```

The machine-readable contract is at [/v1/tools/rational-trace-check/versions/1.0.0](/v1/tools/rational-trace-check/versions/1.0.0).

### CLI

```sh
node cli.mjs run rational-trace-check 1.0.0 --input input.json --base-url https://computefirst.net
```

Get the client at [/clients/cli/](/clients/cli/).

## Related tools

- [Polynomial identity check](/tools/polynomial-identity-check): Check a polynomial identity over exact rationals by expanding both sides; report the lowest disagreeing coefficient.
- [Rational expression check](/tools/rational-expression-check): Check a claimed exact rational value of an integer/fraction expression; reports the true value and the difference.
- [Chi square decomposition check](/tools/chi-square-decomposition-check): Check a claimed Pearson chi-square, its per-cell (O-E)^2/E terms and df exactly; returns the first wrong term.
- [Rational equation check](/tools/rational-equation-check): Check a chained claim like 1/2 + 1/3 = 5/6 < 1 exactly; returns the first false relation or division by zero.
- [Contingency expected check](/tools/contingency-expected-check): Check claimed expected counts (row total x column total / N) of a count table exactly; returns the first wrong cell.
- [Contingency margins check](/tools/contingency-margins-check): Check claimed row, column and grand totals of a count table exactly; returns the first wrong total.