nf.derivation.ignf

Isomorphism Gauge Normal Form (IGNF)

Contract ID: nf.derivation.ignf | Family: derivation | Index Status: draft | AuthorityEffect: None

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1. Executive Summary & Metadata

Isomorphism Gauge Normal Form (IGNF) enforces structural normalization and explicit defect routing in the derivation regime.

Property	Value
Canonical ID	nf.derivation.ignf
Short Name	IGNF
Full Name	Isomorphism Gauge Normal Form
Family	derivation (MSS-N Tower Algebraic Derivation, Split Extension-Retraction & Obstruction Contracts)
Qualification	Extended Form Contract (Normal Form Contract)
Data / Math Posture	at_rest
Index Status	draft
Theory Status	established
Verification Status	absent
Contract Version	v0
Governance Fence	AuthorityEffect: None (certified_implies_authorized: false)

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2. Visual Architecture & Normalization Diagram

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3. Concrete Worked Example & Mechanism

Visual Transition Summary

Stage	Carrier / Operator	Concrete State / Invariant
1. Before / Candidate Input	Input Carrier (IGNF Candidate)	Anomaly risk: permutation gauge ; redundancy anomaly where identical ; relational topologies are...
2. Normalization / Gate	IGNF Verification Gate	every finite relation is... (certified_implies_authorized: false)
3. After / Certified Output	Certified IGNF Output	Invariant: ['carrier_size', ; 'canonical_relation_key', ; 'automorphism_subgroup',...

Contract-Declared Worked Cases

- ref: https://github.com/dbccpage/19L57G1/blob/f1d56c010e6ba10a3b9418bd7cc5953c067250f2/lean4/Derivation/RenamingStructure.lean
  section: lean4/Derivation/RenamingStructure.lean
  note: canonical key minimization and Aut(r) computation

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4. The Four-Pillar Quadrilateral & Non-Claims

Every contract in simplex-normal-forms is evaluated against the Four-Pillar Quadrilateral defined in contracts/_golden_nfs/standard.yaml:

Pillar	Requirement	Contract Specification for nf.derivation.ignf
Pillar I: Excludes Anomaly	Name and eliminate a specific class of data or state defects.	permutation gauge redundancy anomaly where identical relational topologies are represented redundantly across symmetric group orbits
Pillar II: Preserves Invariant	State the structural invariant and protected skeleton preserved across transitions.	Invariant: every finite relation is canonicalized to its unique lexicographically minimal representative under the full symmetric group action, with preserved automorphism stabilizer Protected Skeleton: ['carrier_size', 'canonical_relation_key', 'automorphism_subgroup', 'orbit_size']
Pillar III: Lossless Reconstruction	State the non-loss reconstruction operator (join, direct sum, pullback, or replay).	Lossless reconstruction under declared projection, direct sum, or pullback operators.
Pillar IV: Syntactic Decidability	State how conformance is decided from declared constraints without unbudgeted scans.	Decidable from declared schema and witness metadata within bounded evaluation budgets.

Explicit Non-Claims (certified_implies_authorized: false)

• does not claim polynomial-time canonization for unbounded graph sizes
• does not derive continuous gauge theories or Lie algebras
• does not assert physical identity beyond relational isomorphism
• does not imply operational authority

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5. Operational Verification & Terminal Routing

5.1 Required Declarations

• carrier_size
• input_relation
• canonical_key_algorithm
• finite_labeled_relation_carrier
• permutation_action
• exact_or_bounded_decision_mode
• transport_permutation

5.2 Required Evidence

• canonical_representative_tuple
• automorphism_generators
• coset_transport_permutation

5.3 Validation Criteria

Criterion ID	Name	Condition
criterion	Check	the output relation matches the minimal sorted image across all permutations in S_n
criterion	Check	the automorphism group stabilizes the relation identically
criterion	Check	orbit size satisfies orbit-stabilizer theorem
criterion	Check	the retained transport is a bijection of the complete declared vertex carrier; its inverse reconstructs the labeled relation
criterion	Check	automorphism groups are conjugated by transport; label-specific queries use the transport
criterion	Check	resource exhaustion yields Undetermined, not a certified canonical minimum

5.4 Terminal Route Map

Every evaluation terminates in a declared local route mapped to a canonical terminal in standard.yaml:

Local Terminal Route	Canonical Terminal
AcceptCanonicalRepresentative	Preserve
RejectNonCanonicalPresentation	Recompute
InvalidateNonIsomorphicTransform	Invalidate
Undetermined	Abstain

5.5 Declared Failure Modes

• symptom: a relation is presented in non-minimal form within its isomorphism orbit, diagnosis: arbitrary labeling was not canonicalized modulo S_n, terminal: RejectNonCanonicalPresentation
• symptom: a claimed isomorphism transformation alters edge cardinality or connectivity, diagnosis: invalid mapping failed to preserve relational structure, terminal: InvalidateNonIsomorphicTransform

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6. Epistemic Claim Hygiene Quadrant

In accordance with the repository's epistemic claim discipline:

1. What it supports:
   • Supports syntactic and operational checking of nf.derivation.ignf (IGNF) against its declared schema, boundary constraints, and witness obligations in this model.
   • Supports deterministic terminal routing (AcceptCanonicalRepresentative, RejectNonCanonicalPresentation, InvalidateNonIsomorphicTransform, Undetermined) when required declarations or invariants fail.
2. What it does not support:
   • Does not confer operational authority (AuthorityEffect: None, certified_implies_authorized: false).
   • Does not claim semantic truth of undeclared external processes or unverified upstream inputs beyond the declaredprotected skeleton.
3. The narrowest defensible claim:
   • Within the declared substrate at_rest, this contract enforces that permutation gauge redundancy anomaly where identical relational topologies are represented redundantly across symmetric group orbits is either excluded under every finite relation is canonicalized to its unique lexicographically minimal representative under the full symmetric group action, with preserved automorphism stabilizer or routed to an explicit refusal/recovery terminal.
4. The next test needed:
   • Property-based falsification and mutation testing across boundary edge cases and composed multi-step pipelines for nf.derivation.ignf.

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7. Canonical YAML Contract (contracts/nf.derivation.ignf.yaml)

Click to expand full canonical YAML contract (contracts/nf.derivation.ignf.yaml)

# Draft declaration: no bound local contract evaluator is supplied; upstream mathematics and schema checks are separate evidence.
schema_version: smplx.nf.contract.v0
contract_kind: form_contract
identity:
  id: nf.derivation.ignf
  name: Isomorphism Gauge Normal Form
  short_name: IGNF
  version: v0
  status: draft
  family: derivation
  theory_status: established
  contract_status: draft
  verification_status: absent
classification:
  hierarchy_level: form
  subject_kind:
  - relational_carrier
  - permutation_orbit
  - gauge_invariant_observable
  data_posture: at_rest
  mathematical_posture: formal_algebraic
purpose:
  excludes_anomaly: permutation gauge redundancy anomaly where identical relational topologies are represented
    redundantly across symmetric group orbits
  preserves_invariant: every finite relation is canonicalized to its unique lexicographically minimal
    representative under the full symmetric group action, with preserved automorphism stabilizer
  protected_skeleton:
  - carrier_size
  - canonical_relation_key
  - automorphism_subgroup
  - orbit_size
  erased_structure:
  - arbitrary_node_labeling_permutations
  non_claims:
  - does not claim polynomial-time canonization for unbounded graph sizes
  - does not derive continuous gauge theories or Lie algebras
  - does not assert physical identity beyond relational isomorphism
  - does not imply operational authority
quadrilateral:
  qualifies_as_true_normal_form: false
  qualification_gap: no_project_join_theorem
  anomaly_exclusion: redundant isomorphic representations of relational states causing combinatorial search
    tree explosion
  lossless_join:
    status: orbit_coset_reconstruction
    statement: the original labeled relation is recovered exactly from the canonical representative, the
      automorphism stabilizer, and the coset permutation
  dependency_preservation:
    status: graph_structure_preserved
    statement: edge adjacency and node degrees are preserved identically under the canonicalization map
  syntactic_decision:
    status: declaration_decidable
    statement: decidable via exhaustive or canonical permutation search over S_n on finite carriers
declarations_required:
- carrier_size
- input_relation
- canonical_key_algorithm
- finite_labeled_relation_carrier
- permutation_action
- exact_or_bounded_decision_mode
- transport_permutation
evidence_required:
- canonical_representative_tuple
- automorphism_generators
- coset_transport_permutation
validation_criteria:
- the output relation matches the minimal sorted image across all permutations in S_n
- the automorphism group stabilizes the relation identically
- orbit size satisfies orbit-stabilizer theorem |Orb(r)| * |Aut(r)| == n!
- the retained transport is a bijection of the complete declared vertex carrier; its inverse reconstructs
  the labeled relation
- automorphism groups are conjugated by transport; label-specific queries use the transport
- resource exhaustion yields Undetermined, not a certified canonical minimum
terminal_routes:
- AcceptCanonicalRepresentative
- RejectNonCanonicalPresentation
- InvalidateNonIsomorphicTransform
- Undetermined
terminal_route_map:
  AcceptCanonicalRepresentative: Preserve
  RejectNonCanonicalPresentation: Recompute
  InvalidateNonIsomorphicTransform: Invalidate
  Undetermined: Abstain
certification:
  certificate_required: true
  certificate_kind: gauge_canon_certificate
  obstruction_class_required: false
  loss_ledger_required: false
  verifier_required: true
authorization:
  authorized_by: null
  certified_implies_authorized: false
failure_modes:
- symptom: a relation is presented in non-minimal form within its isomorphism orbit
  diagnosis: arbitrary labeling was not canonicalized modulo S_n
  terminal: RejectNonCanonicalPresentation
- symptom: a claimed isomorphism transformation alters edge cardinality or connectivity
  diagnosis: invalid mapping failed to preserve relational structure
  terminal: InvalidateNonIsomorphicTransform
examples:
- ref: https://github.com/dbccpage/19L57G1/blob/f1d56c010e6ba10a3b9418bd7cc5953c067250f2/lean4/Derivation/RenamingStructure.lean
  section: lean4/Derivation/RenamingStructure.lean
  note: canonical key minimization and Aut(r) computation
references:
- https://github.com/dbccpage/19L57G1/blob/f1d56c010e6ba10a3b9418bd7cc5953c067250f2/docs/GENESIS_CLOSEOUT.md
- ./README.md