nf.ddc.dual_witness

Dual Witness Normal Form (DWNF)

Contract ID: nf.ddc.dual_witness | Family: evidence | Index Status: active | AuthorityEffect: None

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1. Executive Summary & Metadata

Dual Witness Normal Form (DWNF) enforces structural normalization and explicit defect routing in the evidence regime.

Property	Value
Canonical ID	nf.ddc.dual_witness
Short Name	DWNF
Full Name	Dual Witness Normal Form
Family	evidence (Cryptographic Receipts, Dual Witnesses, Loss Ledgers, CDC Logs & Schema Totality)
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: evidence
• Runes: ['R', 'E']
• Target Runtime Law: Dual Certification Law
• Domain: Optimization problems and defect minimization on 1-complexes.
• Anomaly Excluded: Unilateral primal optimality assertions and heuristic defect claims lacking certificates.
• Preserved Invariant: Simultaneous emission of primal flow and dual circulation witness with zero duality gap and Xi tie-break.
• Reconstruction Theorem: Dual certificate independently verifies primal minimizer without re-running search.

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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 (DWNF Candidate)	Anomaly risk: primal-only defect ; claims without dual circulation ; certificates, or unprincipled...
2. Normalization / Gate	DWNF Verification Gate	exact rational dual... (certified_implies_authorized: false)
3. After / Certified Output	Certified DWNF Output	Invariant: ; ['primal_defect_vector', ; 'dual_flow_witness_g3',...

Contract-Declared Worked Cases

- ref: omega_engine_education/documentation/14_discrete_deformation_calculus_ddc/00_discrete_deformation_calculus_foundations.md
  section: '2.8'

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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.dual_witness
Pillar I: Excludes Anomaly	Name and eliminate a specific class of data or state defects.	primal-only defect claims without dual circulation certificates, or unprincipled tie-breaking overriding primary defect worsening
Pillar II: Preserves Invariant	State the structural invariant and protected skeleton preserved across transitions.	Invariant: exact rational dual circulation flow certificate g₃ certifying Φ₁([x]) and strict primary equality precondition on secondary metric Ξ Protected Skeleton: ['primal_defect_vector', 'dual_flow_witness_g3', 'spanning_forest_tight_potentials', 'xi_tie_precondition']
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 permit secondary metric Ξ to break ties when Φ₁ or Γ₂ differ between candidates
• does not assert quotient norm without certifying both primal bound and dual witness
• does not confer operational authority; authority_effect is unconditionally None

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5. Operational Verification & Terminal Routing

5.1 Required Declarations

• phi1_value
• dual_witness_g3
• xi_precondition
• tie_break_certificate

5.2 Required Evidence

None declared.

5.3 Validation Criteria

Criterion ID	Name	Condition
criterion	Check	dual flow g₃ satisfies capacity constraints and vertex conservation laws
criterion	Check	primal cost matches dual flow objective in exact rational arithmetic
criterion	Check	secondary metric tie-break is invoked only when candidate Φ₁ and Γ₂ are identical

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
AdmitDualWitness	Preserve
RejectNonZeroDualityGap	Invalidate
RejectUncertifiedPrimal	Refuse
RejectInvalidTieBreak	Reject

5.5 Declared Failure Modes

• symptom: primal objective exceeds dual circulation flow value, diagnosis: duality gap non-zero; certificate invalid, terminal: RejectNonZeroDualityGap
• symptom: candidate selected by Ξ metric when Φ₁ or Γ₂ are unequal, diagnosis: secondary metric overrode primary defect worsening; violates tie-break invariant, terminal: RejectInvalidTieBreak

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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.dual_witness (DWNF) against its declared schema, boundary constraints, and witness obligations in this model.
   • Supports deterministic terminal routing (AdmitDualWitness, RejectNonZeroDualityGap, RejectUncertifiedPrimal, RejectInvalidTieBreak) 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 primal-only defect claims without dual circulation certificates, or unprincipled tie-breaking overriding primary defect worsening is either excluded under exact rational dual circulation flow certificate g₃ certifying Φ₁([x]) and strict primary equality precondition on secondary metric Ξ 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.dual_witness.

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

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

schema_version: smplx.nf.contract.v0
schema_ref: ./normal_form_contract_schema.yaml
contract_kind: normal_form_contract
identity:
  id: nf.ddc.dual_witness
  name: Dual Witness Normal Form
  short_name: DWNF
  version: v0
  theory_status: established
  contract_status: active
  status: active
  verification_status: executable
  family: evidence
classification:
  hierarchy_level: normal_form
  subject_kind:
    - dual_circulation_flow
    - phi1_certificate
    - xi_tie_break
  data_posture: at_rest
  mathematical_posture: operational
purpose:
  excludes_anomaly: primal-only defect claims without dual circulation certificates, or unprincipled tie-breaking overriding primary defect worsening
  preserves_invariant: exact rational dual circulation flow certificate g₃ certifying Φ₁([x]) and strict primary equality precondition on secondary metric Ξ
  protected_skeleton:
    - primal_defect_vector
    - dual_flow_witness_g3
    - spanning_forest_tight_potentials
    - xi_tie_precondition
  erased_structure:
    - dual_solver_pivot_sequence
  non_claims:
    - does not permit secondary metric Ξ to break ties when Φ₁ or Γ₂ differ between candidates
    - does not assert quotient norm without certifying both primal bound and dual witness
    - does not confer 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 1-cochain complex with boundary coboundaries over ℚ, Dual Witness Normal Form (DWNF)
    certifies the quotient residual norm Φ₁([x]) by producing an exact rational dual circulation
    flow g₃ such that the duality gap is certified zero. When multiple repair moves compete,
    secondary harmonic metric Ξ is admissible if and only if primary metrics Φ₁ and Γ₂ are
    strictly equal across competing candidates.
  quadrilateral:
    qualifies_as_true_normal_form: false
    qualification_gap: dual_witness_certification_not_relational_project_join
    anomaly_exclusion: uncertified quotient claims, inadequate detector masking, or secondary metric overriding primary defect worsening
    lossless_join:
      status: primal_dual_duality_gap_zero
      statement: the exact rational duality gap between primal circulation cost and dual flow witness is certified identically zero
    dependency_preservation:
      status: circulation_cycle_preserved
      statement: cycle flow g₃ is verified locally against edge capacities and flow conservation at internal vertices
    syntactic_decision:
      status: rational_duality_decidable
      statement: membership is decidable by checking primal-dual gap and evaluating XiTiePrecondition.is_satisfied()
declarations_required:
  - phi1_value
  - dual_witness_g3
  - xi_precondition
  - tie_break_certificate
evidence_required: []
validation_criteria:
  - dual flow g₃ satisfies capacity constraints and vertex conservation laws
  - primal cost matches dual flow objective in exact rational arithmetic
  - secondary metric tie-break is invoked only when candidate Φ₁ and Γ₂ are identical
terminal_routes:
  - AdmitDualWitness
  - RejectNonZeroDualityGap
  - RejectUncertifiedPrimal
  - RejectInvalidTieBreak
terminal_route_map:
  AdmitDualWitness: Preserve
  RejectNonZeroDualityGap: Invalidate
  RejectUncertifiedPrimal: Refuse
  RejectInvalidTieBreak: Reject
certification:
  certificate_required: true
  certificate_kind: dual_circulation_receipt
  obstruction_class_required: false
  loss_ledger_required: false
  verifier_required: true
  loss_ledger_trigger_id: null
authorization:
  authorized_by: null
  certified_implies_authorized: false
failure_modes:
  - symptom: primal objective exceeds dual circulation flow value
    diagnosis: duality gap non-zero; certificate invalid
    terminal: RejectNonZeroDualityGap
  - symptom: candidate selected by Ξ metric when Φ₁ or Γ₂ are unequal
    diagnosis: secondary metric overrode primary defect worsening; violates tie-break invariant
    terminal: RejectInvalidTieBreak
examples:
  - ref: omega_engine_education/documentation/14_discrete_deformation_calculus_ddc/00_discrete_deformation_calculus_foundations.md
    section: '2.8'
references:
  - omega_engine_governance/contracts/contracts_omega_engine_crates/mathematics/discrete_deformation_algebra/quotient_mass.contract.yaml
  - omega_engine_governance/contracts/contracts_omega_engine_crates/mathematics/discrete_deformation_algebra/tie_r.contract.yaml