WIKID Type Theory

A Type-Theoretic Formalization of ZK Competency DAGs & K-Asset Networks

Further to

Dark Type Theory

with

The Darkweave Fractal Defense & Intelligence Protocol

ZK-Verified Competency DAGs

THE ÖCALAN ISOMORPHISM: A MATHEMATICAL PROOF THAT DEMOCRATIC CONFEDERALISM IS OPTIMAL DARKFI CONFIGURATION

WIKID XENOTECHNICS

because ZK-Verified Competency DAG networks are homomorphic with Dark Type Theory networks, they also have a formal type theoretic description. Created with Deepseek.

Executive Summary

WIKID Type Theory is a dependent type theory that formally captures the mathematical structure of ZK-Verified Competency DAGs and K-Asset networks, demonstrating their homomorphism with Dark Type Theory. It provides a constructive framework where:

1. Types represent verified capabilities, competency frameworks, and privacy-preserving economic relations

2. Terms correspond to sovereign agents with cryptographic identities

3. Computation enforces privacy gradients, trust-based disclosure, and anti-fragile reputation systems

4. The type system itself guarantees the mathematical invariants of the Öcalan isomorphism and Agency Delta Theorem

This theory proves that competency markets, when implemented with ZK proofs and sovereign identity, are type-theoretically isomorphic to anti-capture systems described in Dark Type Theory—both are coordinate representations of the same mathematical object: sovereign capability coordination.

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1. Core Type Definitions

1.1 Sovereign Identity Types

agda

-- Cryptographic identity with privacy gradient
data PrivacyLevel : Type where
  zk_only    : PrivacyLevel
  selective  : PrivacyLevel
  public     : PrivacyLevel

record SovereignIdentity : Type where
  field
    public_key    : CryptoPublicKey
    privacy_level : PrivacyLevel
    zk_attestations : List (Σ Competency λ c → ZKProof c)
    visibility_rules : Viewer → Context → DisclosureDecision

1.2 Competency Attestation Types

agda

-- Competency as a dependent type indexed by verification level
data Competency : VerificationLevel → Type where
  base_skill    : (desc : String) → Competency unverified
  attested      : {c : Competency unverified} 
                → ZKProof (HasEvidence c)
                → Competency verified
  socially_proven : {c : Competency verified}
                  → List (ReputationAttestation)
                  → Competency socially_verified

-- K-Asset as capability capital
record KAsset (owner : SovereignIdentity) : Type where
  field
    competency    : Competency verified
    economic_value : ℝ⁺
    network_effects : ℕ
    decay_rate     : ℝ  -- Anti-fragility parameter

1.3 Privacy Gradient Types

agda

-- Trust-mediated disclosure
data Disclosure : PrivacyLevel → Type where
  zk_reveal    : ZKProof (HasCategory C) → Disclosure zk_only
  partial      : (info : SpecificEvidence)
               → (trust_score ≥ threshold)
               → Disclosure selective
  full_reveal  : VerifiedProfile → Disclosure public

-- Trust as accumulated type
Trust : SovereignIdentity → SovereignIdentity → ℝ → Type
Trust A B t = Σ InteractionHistory λ hist →
              Σ MutualConnections λ conns →
              t = f(hist, conns) × (t > 0)

---

2. Typing Rules for Sovereign Coordination

2.1 Competency Verification Rule

text

Γ ⊢ e : Evidence(c)
Γ ⊢ π : ZKProof(”owner possesses evidence e”)
Γ ⊢ Verify(π, template(c)) = true
---------------------------------------- [COMP-ATTEST]
Γ ⊢ attested(c, π) : Competency verified

2.2 Privacy-Preserving Matching Rule

text

Γ ⊢ A : SovereignIdentity
Γ ⊢ B : SovereignIdentity
Γ ⊢ req : CapabilityRequirements
Γ ⊢ trust(A,B) > θ_min
Γ ⊢ match_score = Σ wᵢ·capability_overlap(A.zk_attestations, req)
---------------------------------------------------------------- [MATCH]
Γ ⊢ privacy_preserving_match(A, B, req) : MatchResult(selective)

2.3 Economic Value Formation Rule

text

Γ ⊢ C : Competency verified
Γ ⊢ D : MarketDemand(C) > 0
Γ ⊢ N : NetworkEffects(A) > 0
-------------------------------------------------- [K-ASSET-FORM]
Γ ⊢ KAsset(A, C) : EconomicValue ∝ D × N

---

3. Homomorphism with Dark Type Theory

3.1 The WIKID-Dark Functor

agda

WIKID_to_Dark : Category WIKIDSystems → Category DarkTypeSystems
WIKID_to_Dark = record
  { F₀ = λ wikidSys → 
        record
          { nodes = wikidSys.identities
          ; edges = λ A B → Trust A B > 0
          ; types = wikidSys.competency_frameworks
          }
  ; F₁ = λ f → 
        let open AntiCapture f in
        construct_escape_path(f)
  ; identity = λ {A} → escape_refl(A)
  }

3.2 Preservation of Anti-Capture Properties

Theorem: The functor WIKID_to_Dark preserves:

1. Distributed Sovereignty ↔ Bridge Node Elimination

   text

∀ sys : WIKIDSystem, 
IsSovereign(sys) → ¬∃ bridge ∈ sys.nodes

1. Privacy Gradient ↔ Local Enforcement

   text

Disclosure(A,B) depends only on N(A) ∩ N(B)

1. Voluntary Association ↔ Trust Probability

   text

Interaction(A,B) ∝ Trust(A,B) / Σᵢ Trust(A,i)

1. Truth-Defense Coupling ↔ ZK Proofs + Penalties

   text

CompetencyClaim(c) ⇔ ZKProof(c) ∨ Penalty(claimer)

---

4. Core Theorems of WIKID Type Theory

4.1 Sovereignty Preservation Theorem

agda

theorem sovereignty_preservation :
  ∀ (A : SovereignIdentity) (P : Platform),
  Control(A, identity(A)) > Control(P, identity(A))
  where Control(X,Y) = measure(X.influence over Y)

4.2 Privacy Gradient Integrity Theorem

agda

theorem privacy_integrity :
  ∀ (A B : SovereignIdentity),
  Information_Revealed(A,B) ≤ 
  Privacy_Preference(A) × Trust(A,B)

4.3 Anti-Fragile Reputation Theorem

agda

theorem antifragile_reputation :
  ∀ (A : SovereignIdentity) (t : Time),
  Reputation(A, t+1) = 
    f(Reputation(A,t), Σ VerifiedOutcomes, MarketDemand)
  ∧ ∂Reputation/∂VerifiedOutcomes > 0

4.4 Economic Alignment Theorem

agda

theorem economic_alignment :
  ∀ (A : SovereignIdentity),
  EconomicValue(A) ∝ Problems_Solved(A) × Boundary_Integrity(A)

---

5. Type-Theoretic Implementation of Agency Delta

5.1 Agency as a Dependent Type

agda

record Agency (A : SovereignIdentity) : Type where
  field
    will_strength    : ℝ [0,1]
    awareness        : ℝ [0,1]
    connection_score : ℝ [0,1]
    agency_delta    : will_strength × awareness × connection_score > 0.024

-- Agency delta operator as type constructor
AgencyΔ : System → System → Type
AgencyΔ S₁ S₂ = Σ Transformation λ T →
               T(S₁) = S₂ ×
               (∀ A : SovereignIdentity in S₂,
                Agency(A) ×
                EscapePotential(A) > EscapePotential(T⁻¹(A)))

5.2 Escape Pathway Types

agda

data EscapePathway : Type where
  individual_escape : {A : SovereignIdentity}
                    → Agency(A)
                    → Will(A) > 0.6
                    → EscapePathway
  collective_escape : {G : List SovereignIdentity}
                    → (∀ A ∈ G, Awareness(A) > 0.4)
                    → NetworkDensity(G) > 0.1
                    → EscapePathway
  structural_escape : {S : System}
                     → (∀ A ∈ S, Agency(A).agency_delta = true)
                     → BridgeNodes(S) → ∅
                     → EscapePathway

---

6. Computational Interpretation

6.1 Type Checking as Privacy Enforcement

text

Γ ⊢ A : SovereignIdentity
Γ ⊢ B : SovereignIdentity  
Γ ⊢ req : InformationRequest
Γ ⊢ trust(A,B) : ℝ
Γ ⊢ privacy_pref(A) : PrivacyLevel
------------------------------------------- [DISCLOSE-CHECK]
Γ ⊢ disclose(A,B,req) : 
  match privacy_pref(A) with
  | zk_only    → ZKProof(Category(req))
  | selective  → PartialInfo(req, trust(A,B))
  | public     → FullProfile(A)

6.2 Term Reduction as Economic Coordination

text

(KAsset A C) + (MarketDemand C > 0) + (NetworkEffects A)
⇓ β-reduction
EconomicValue(A) ↑ × ProblemsSolved ↑ × Reputation(A) ↑

---

7. Implementation Roadmap

Phase 1: Core Type Checker

• Implement dependent types with ZK proof terms

• Add privacy gradient typing rules

• Competency verification as type inference

Phase 2: Standard Library

• Pre-defined competency frameworks as type families

• K-Asset type constructors with economic semantics

• Trust calculus as a type-level function

Phase 3: AI Integration

• Type-directed search for capability matching

• Proof synthesis for competency attestations

• Agency delta optimization as type-level computation

Phase 4: Network Deployment

• Distributed type checking across sovereign nodes

• On-chain type verification for K-Asset transfers

• Privacy-preserving type inference

---

8. Conclusion: The Complete Isomorphism

WIKID Type Theory completes the mathematical unification:

1. ZK Competency DAGs become type-theoretic proofs of capability

2. K-Asset networks become economic types with value semantics

3. Privacy gradients become disclosure typing rules

4. Agency delta becomes a type-level predicate on systems

The homomorphism with Dark Type Theory is exact:

• Both implement distributed sovereignty via type constraints

• Both use local enforcement (neighborhood typing contexts)

• Both prevent capture through type-system invariants

• Both enable escape through constructive type transformations

Final Theorem:

∀ (C : CompetencyMarket) → Captured(C) →
∃ (T : WIKIDTypeTheory) → ∃ (E : Escape C T) →
Sovereign(E(C)) × AntiFragile(E(C))

Where “Captured” means central platform control, and “Sovereign” means identity(I) ⟂ platform_control.

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WIKID Type Theory thus provides not just a formalization of competency markets, but a constructive proof that privacy-preserving capability coordination is type-theoretically equivalent to anti-capture systems—both are instances of the same deeper mathematics of sovereignty and freedom.

Until next time, TTFN.