nf.ddc.hodge_canonical

Hodge Canonical Normal Form (HCNF)

Contract ID: nf.ddc.hodge_canonical | Family: shape | Index Status: active | AuthorityEffect: None

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1. Executive Summary & Metadata

Hodge Canonical Normal Form (HCNF) enforces structural normalization and explicit defect routing in the shape regime.

Property	Value
Canonical ID	nf.ddc.hodge_canonical
Short Name	HCNF
Full Name	Hodge Canonical Normal Form
Family	shape (Relational, Algebraic & Cochain Shape Normal Forms (Data at Rest & Structural Decompositions))
Qualification	Extended Form Contract (Normal Form Contract)
Data / Math Posture	at_rest
Index Status	active
Theory Status	established
Verification Status	executable
Contract Version	v0
Governance Fence	AuthorityEffect: None (certified_implies_authorized: false)

Foundational Treatise Specification

• Family: shape
• Runes: ['E', 'R']
• Target Runtime Law: Law 7 (Orthogonal Repair)
• Domain: Discrete 1-cochains c in C^1(K, Q) over cell complexes.
• Anomaly Excluded: Conflating exact gradient adjustments with non-trivial harmonic loops or boundary curls.
• Preserved Invariant: Orthogonal direct sum decomposition C^1 = im d_0 \oplus im d_1^* \oplus H^1.
• Reconstruction Theorem: c = d_0 p + h + d_1^* q without rational error.

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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 (HCNF Candidate)	Anomaly risk: unseparated cochain ; defects where exact coboundary ; tension and irreducible harmoni...
2. Normalization / Gate	HCNF Verification Gate	orthogonal direct sum... (certified_implies_authorized: false)
3. After / Certified Output	Certified HCNF Output	Invariant: ['carrier_2_complex', ; 'coboundary_operators_d0_d1', ; 'exact_component_x_E',...

Contract-Declared Worked Cases

- ref: omega_engine_education/documentation/14_discrete_deformation_calculus_ddc/00_discrete_deformation_calculus_foundations.md
  section: '2.2'

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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.ddc.hodge_canonical
Pillar I: Excludes Anomaly	Name and eliminate a specific class of data or state defects.	unseparated cochain defects where exact coboundary tension and irreducible harmonic cycle circulation are entangled without orthogonal decomposition
Pillar II: Preserves Invariant	State the structural invariant and protected skeleton preserved across transitions.	Invariant: orthogonal direct sum splitting C¹(K; ℚ) = im(d₀) ⊕ im(d₁ᵀ) ⊕ ℋ¹ with exact rational coordinates and zero defect curl check Protected Skeleton: ['carrier_2_complex', 'coboundary_operators_d0_d1', 'exact_component_x_E', 'harmonic_component_x_R', 'potential_gauge_witness_phi']
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 non-exact defects can be repaired by gauge integration
• does not claim exact potential is unique without designating a gauge root per connected component
• does not imply operational authority; authority_effect is unconditionally None

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5. Operational Verification & Terminal Routing

5.1 Required Declarations

• carrier_complex
• cochain_vector
• exact_potential_witness
• cycle_coordinates

5.2 Required Evidence

None declared.

5.3 Validation Criteria

Criterion ID	Name	Condition
criterion	Check	nilpotence d₁ ∘ d₀ = 0 holds identically over the complex
criterion	Check	exact component x_E equals d₀(phi) for declared rational potential phi
criterion	Check	exact, coexact, and harmonic components are pairwise orthogonal

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
AdmitHodgeDecomposition	Preserve
ExecuteExactRepair	Repair
RejectNonCurlFree	Refuse
RejectNonOrthogonal	Reject

5.5 Declared Failure Modes

• symptom: d₁ x != 0 during attempted exact repair, diagnosis: cochain is not curl-free; exact repair inadmissible in this model, violates Law 7, terminal: RejectNonCurlFree
• symptom: sum of components does not equal input cochain x, diagnosis: projection error or non-orthogonal basis, terminal: RejectNonOrthogonal

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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.ddc.hodge_canonical (HCNF) against its declared schema, boundary constraints, and witness obligations in this model.
   • Supports deterministic terminal routing (AdmitHodgeDecomposition, ExecuteExactRepair, RejectNonCurlFree, RejectNonOrthogonal) 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 unseparated cochain defects where exact coboundary tension and irreducible harmonic cycle circulation are entangled without orthogonal decomposition is either excluded under orthogonal direct sum splitting C¹(K; ℚ) = im(d₀) ⊕ im(d₁ᵀ) ⊕ ℋ¹ with exact rational coordinates and zero defect curl check 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.ddc.hodge_canonical.

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7. Canonical YAML Contract (contracts/nf.ddc.hodge_canonical.yaml)

Click to expand full canonical YAML contract (contracts/nf.ddc.hodge_canonical.yaml)

schema_version: smplx.nf.contract.v0
schema_ref: ./normal_form_contract_schema.yaml
contract_kind: normal_form_contract
identity:
  id: nf.ddc.hodge_canonical
  name: Hodge Canonical Normal Form
  short_name: HCNF
  version: v0
  theory_status: established
  contract_status: active
  status: active
  verification_status: executable
  family: shape
classification:
  hierarchy_level: normal_form
  subject_kind:
    - 1_cochain_defect
    - discrete_hodge_decomposition
    - coboundary_potential
  data_posture: at_rest
  mathematical_posture: operational
purpose:
  excludes_anomaly: unseparated cochain defects where exact coboundary tension and irreducible harmonic cycle circulation are entangled without orthogonal decomposition
  preserves_invariant: orthogonal direct sum splitting C¹(K; ℚ) = im(d₀) ⊕ im(d₁ᵀ) ⊕ ℋ¹ with exact rational coordinates and zero defect curl check
  protected_skeleton:
    - carrier_2_complex
    - coboundary_operators_d0_d1
    - exact_component_x_E
    - harmonic_component_x_R
    - potential_gauge_witness_phi
  erased_structure:
    - gauge_choice_offset
    - intermediate_tree_search_order
  non_claims:
    - does not claim non-exact defects can be repaired by gauge integration
    - does not claim exact potential is unique without designating a gauge root per connected component
    - does not imply operational authority; authority_effect is unconditionally None
definition:
  source: omega_engine_education/documentation/14_discrete_deformation_calculus_ddc/normal_forms/README.md
  section: '2.0'
  formal_statement: >
    On a finite 2-complex with rational coboundaries d₀ and d₁, Hodge Canonical Normal Form (HCNF)
    decomposes any 1-cochain defect x into unique orthogonal components x = x_E + x_C + x_R.
    Exact repair rune E is lawful if and only if curl d₁ x == 0 and x_E lies strictly in im(d₀).
    Harmonic residuals x_R cannot be eliminated by local node potentials.
  quadrilateral:
    qualifies_as_true_normal_form: false
    qualification_gap: direct_sum_decomposition_not_relational_project_join
    anomaly_exclusion: non-curl-free cochains submitted to exact repair, or harmonic residuals treated as exact coboundaries
    lossless_join:
      status: direct_sum_reconstruction
      statement: x = x_E + x_C + x_R is uniquely and losslessly reconstructed from orthogonal Hodge projectors over ℚ
    dependency_preservation:
      status: coboundary_chain_preserved
      statement: boundary condition d₁ ∘ d₀ = 0 is verified locally on 2-cells and edges without global joins
    syntactic_decision:
      status: exact_rational_decidable
      statement: membership in im(d₀) is decidable by BFS potential integration and checking d₁x == 0
declarations_required:
  - carrier_complex
  - cochain_vector
  - exact_potential_witness
  - cycle_coordinates
evidence_required: []
validation_criteria:
  - nilpotence d₁ ∘ d₀ = 0 holds identically over the complex
  - exact component x_E equals d₀(phi) for declared rational potential phi
  - exact, coexact, and harmonic components are pairwise orthogonal
terminal_routes:
  - AdmitHodgeDecomposition
  - ExecuteExactRepair
  - RejectNonCurlFree
  - RejectNonOrthogonal
terminal_route_map:
  AdmitHodgeDecomposition: Preserve
  ExecuteExactRepair: Repair
  RejectNonCurlFree: Refuse
  RejectNonOrthogonal: Reject
certification:
  certificate_required: false
  certificate_kind: null
  obstruction_class_required: true
  loss_ledger_required: false
  verifier_required: true
  loss_ledger_trigger_id: null
authorization:
  authorized_by: null
  certified_implies_authorized: false
failure_modes:
  - symptom: d₁ x != 0 during attempted exact repair
    diagnosis: cochain is not curl-free; exact repair inadmissible in this model, violates Law 7
    terminal: RejectNonCurlFree
  - symptom: sum of components does not equal input cochain x
    diagnosis: projection error or non-orthogonal basis
    terminal: RejectNonOrthogonal
examples:
  - ref: omega_engine_education/documentation/14_discrete_deformation_calculus_ddc/00_discrete_deformation_calculus_foundations.md
    section: '2.2'
references:
  - omega_engine_governance/contracts/contracts_omega_engine_crates/mathematics/discrete_deformation_algebra/exact_repair_operator.contract.yaml
  - omega_engine_governance/contracts/contracts_omega_engine_crates/mathematics/discrete_deformation_algebra/quotient_space.contract.yaml