Source: https://guthlabs.ai/docs/instruments/set-cardinality-power-set-generator/

instrument / verified-clean

# Set Cardinality & Power Set Generator

The current ledger marks this instrument verified clean within the source-bound verification corpus.

Status: **VERIFIED_CLEAN**Lifecycle: **verified-clean**

## Inspectable records

[01**Overview**Purpose and lifecycle](<https://guthlabs.ai/docs/instruments/set-cardinality-power-set-generator/>)[02**Mechanism**Method and assumptions](<https://guthlabs.ai/docs/instruments/set-cardinality-power-set-generator/mechanism/>)[03**Implementation**Binding and validation](<https://guthlabs.ai/docs/instruments/set-cardinality-power-set-generator/implementation/>)[04**Qualification**Evidence and errata](<https://guthlabs.ai/docs/instruments/set-cardinality-power-set-generator/qualification/>)

## Ledger state

The current ledger marks this instrument verified clean within the source-bound verification corpus.

**Area**: Mathematics — arithmetic through advanced (algebra, matrices, calculus, sequences/series, number theory, combinatorics, graphing, complex numbers, geometry, logic, sets)

**Tranche**: tranche-4

**Evidence ID**: t4-201

**Ledger rows**: 1

No ledger note is attached.

## Review findings

No incorrect arithmetic formula was found, but the specification is incomplete for implementation verification. Define finite-set normalization, element equality, subset/superset semantics, Cartesian-product arity, output ordering, and one exact element cap before labeling the instrument fully verified.

### Formula and scope findings

- |P(S)|=2^|S| is correct only for finite sets whose elements are distinct; the finite-set constraint is not stated explicitly.

- Cartesian product enumeration is named but no formal definition, operand arity, or cardinality formula is provided. For finite A and B, the standard definition is A×B={(a,b):a∈A,b∈B} and |A×B|=|A|×|B|.

- Subset and superset checks have no stated formulas or equality semantics. They should define X⊆Y as every element of X being in Y, and X⊇Y as Y⊆X.

- The specification does not say whether duplicate input values are normalized as sets or treated as separate list entries.

- The power-set formula determines cardinality but not the ordering of the enumerated power-set list.

### Unit assumptions

- Set elements are mathematical values with exact equality; there are no physical units and no unit conversion.

- Inputs are finite sets represented as arrays, with duplicates removed before cardinality, subset, superset, or power-set operations.

- For the expected enumerated lists below, subsets and Cartesian-product pairs are ordered deterministically using the input-array order and increasing bit-mask order.

- The subset check is X⊆Y, and the superset check is Y⊇X.

- The Cartesian-product tests use two operands, A and B; an empty operand produces an empty Cartesian product.

## Boundary hardening

**BREAK_CLAIMED** · SELF_FLAGGED

The instrument has a documented practical break above approximately 12 elements: a 13-element set mathematically requires 8192 power-set entries and should be rejected, while an unguarded extreme input with 1024 elements causes direct JavaScript Number evaluation of 2^1024 to become Infinity.

## Errata and corrections

No applied erratum is attached to this unique-name record.

## Inspectable example

4 review vectors are recorded. One source-bound sample is shown below.

````
{
  "inputs": {
    "S": [],
    "A": [],
    "B": [
      "x"
    ],
    "subset_check": {
      "X": [],
      "Y": [
        "x"
      ]
    }
  },
  "expected_outputs": {
    "cardinality": 0,
    "power_set": [
      []
    ],
    "cartesian_product": [],
    "cartesian_product_cardinality": 0,
    "subset": true,
    "superset": true
  },
  "working": "|S|=0. Therefore |P(S)|=2^0=1, and the only subset is the empty set: P(S)=[∅]. For the product, |A×B|=|A|×|B|=0×1=0, so A×B=∅. Every element of X=∅ is in Y={x}, so X⊆Y is true; equivalently Y⊇X is true."
}
````

## Related workflows

No source-defined multi-instrument workflow membership is recorded.

## Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

- `VERIFIED_LEDGER.json · VERIFIED_CLEAN`

- `evidence-tranche4/tranche4-results.json`

- `BOUNDARY_HAZARD_CATALOG.json`