gaia.core.kan_verification module

Module: kan_verification Implements Kan complex conditions verification for GAIA framework.

Following Section 3.2 of the theoretical framework, this implements: 1. Kan complex conditions verification 2. Horn filling algorithm robustness checks 3. Lifting property verification 4. Fibration conditions 5. Homotopy extension properties

This ensures that the simplicial structure satisfies the necessary categorical conditions for proper horn extension and lifting problems.

class gaia.core.kan_verification.KanConditionType(value)

Bases: Enum

Types of Kan conditions to verify.

INNER_HORN_FILLING = 'inner_horn_filling'

OUTER_HORN_FILLING = 'outer_horn_filling'

LIFTING_PROPERTY = 'lifting_property'

FIBRATION_CONDITION = 'fibration_condition'

HOMOTOPY_EXTENSION = 'homotopy_extension'

classmethod __contains__(member)

Return True if member is a member of this enum raises TypeError if member is not an enum member

note: in 3.12 TypeError will no longer be raised, and True will also be returned if member is the value of a member in this enum

classmethod __getitem__(name)

Return the member matching name.

classmethod __iter__()

Return members in definition order.

classmethod __len__()

Return the number of members (no aliases)

class gaia.core.kan_verification.KanConditionResult(condition_type, simplex_id, simplex_name, horn_index, satisfied, confidence, error_message=None, verification_data=<factory>)

Bases: object

Result of a Kan condition verification.

condition_type: KanConditionType

simplex_id: UUID

simplex_name: str

horn_index: int | None

satisfied: bool

confidence: float

error_message: str | None = None

verification_data: Dict[str, Any]

__init__(condition_type, simplex_id, simplex_name, horn_index, satisfied, confidence, error_message=None, verification_data=<factory>)

class gaia.core.kan_verification.HornFillingVerifier(simplicial_functor)

Bases: object

Verifies horn filling conditions for inner and outer horns.

This checks that horn filling algorithms can successfully complete partial simplicial structures.

__init__(simplicial_functor)

verify_inner_horn_filling(simplex_id, horn_index, tolerance=1e-3)

Verify that inner horn Λⁿₖ (1 ≤ k ≤ n-1) can be filled.

Parameters:

• simplex_id (UUID) – ID of simplex with missing face

• horn_index (int) – Index of missing face (inner horn)

• tolerance (float) – Numerical tolerance for verification

Returns:

Verification result

Return type:

KanConditionResult

verify_outer_horn_filling(simplex_id, horn_index, tolerance=1e-3)

Verify that outer horn Λⁿₖ (k = 0 or k = n) can be filled.

Parameters:

• simplex_id (UUID) – ID of simplex with missing face

• horn_index (int) – Index of missing face (outer horn)

• tolerance (float) – Numerical tolerance for verification

Returns:

Verification result

Return type:

KanConditionResult

class gaia.core.kan_verification.LiftingPropertyVerifier(simplicial_functor)

Bases: object

Verifies lifting properties for fibrations and cofibrations.

This checks the homotopy lifting property and other categorical lifting conditions.

__init__(simplicial_functor)

verify_lifting_property(base_simplex_id, fiber_simplex_id, tolerance=1e-3)

Verify lifting property between two simplices.

This checks if there exists a lift for morphisms between the simplices.

class gaia.core.kan_verification.FibrationVerifier(simplicial_functor)

Bases: object

Verifies fibration conditions for simplicial maps.

This checks if maps between simplicial sets satisfy the fibration property.

__init__(simplicial_functor)

verify_fibration_condition(map_source_id, map_target_id)

Verify that a map satisfies fibration conditions.

A map is a fibration if it has the right lifting property with respect to all horn inclusions.

class gaia.core.kan_verification.KanComplexVerifier(simplicial_functor)

Bases: object

Complete Kan complex verification system.

This orchestrates all Kan condition verifications and provides a comprehensive assessment of the simplicial structure.

__init__(simplicial_functor)

verify_all_conditions(tolerance=1e-3)

Verify all Kan conditions for the entire simplicial complex.

Returns comprehensive verification report.

get_kan_complex_status()

Get overall Kan complex status.

suggest_improvements()

Suggest improvements to achieve Kan complex conditions.

gaia.core.kan_verification.verify_model_kan_conditions(model, tolerance=1e-3)

Verify Kan conditions for a GAIA model.

Parameters:

• model – GAIA model with simplicial structure

• tolerance (float) – Numerical tolerance for verification

Returns:

Comprehensive verification report

Return type:

Dict[str, Any]

gaia.core.kan_verification.create_kan_verification_summary(report)

Create human-readable summary of Kan verification results.