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INTERCOL DOMAINS — Formal Definition & Orthogonality Proofs
4 orthogonal domains from Sovereign Calculus:
- D₁ = Treasury (financial reasoning, account movement)
- D₂ = Clinical (verification, proof, measurement)
- D₃ = Legal (policy, authorization, capability)
- D₄ = Operations (execution, state change)
Theorem: intercol_transition_impossible — Transitions between orthogonal
domains return ⊥ (null state).
-/
import Lean
import Mathlib.Data.List.Perm
import Mathlib.Logic.Equiv.Set
namespace IntercolDomains
-- ══════════════════════════════════════════════════════════════════════════════
-- 1. DOMAIN DEFINITION
-- ══════════════════════════════════════════════════════════════════════════════
inductive Domain : Type where
| treasury : Domain -- D₁: Financial reasoning
| clinical : Domain -- D₂: Verification & proof
| legal : Domain -- D₃: Policy & authorization
| operations : Domain -- D₄: Execution & state change
deriving DecidableEq, Repr
/-- String representation for debugging -/
def Domain.toString : Domain → String
| Domain.treasury => "Treasury (D₁)"
| Domain.clinical => "Clinical (D₂)"
| Domain.legal => "Legal (D₃)"
| Domain.operations => "Operations (D₄)"
-- ══════════════════════════════════════════════════════════════════════════════
-- 2. ORTHOGONALITY RELATION
-- ══════════════════════════════════════════════════════════════════════════════
/-- Two domains are orthogonal if they are distinct -/
def orthogonal (d1 d2 : Domain) : Prop :=
d1 ≠ d2
/-- Orthogonality is symmetric -/
theorem orthogonal_symm (d1 d2 : Domain) :
orthogonal d1 d2 ↔ orthogonal d2 d1 := by
constructor <;> (intro h; exact fun h' => h (h'.symm))
/-- Orthogonality is irreflexive -/
theorem orthogonal_irrefl (d : Domain) :
¬(orthogonal d d) := by
intro h
exact h rfl
-- ══════════════════════════════════════════════════════════════════════════════
-- 3. TRANSITION RULE
-- ══════════════════════════════════════════════════════════════════════════════
inductive TransitionResult : Type where
| success (d : Domain) : TransitionResult
| nullState : TransitionResult
| error : String → TransitionResult
/-- Transition between same domain succeeds; between orthogonal domains returns null -/
def transition (d_from d_to : Domain) : TransitionResult :=
if d_from = d_to then
TransitionResult.success d_to
else
TransitionResult.nullState
-- ══════════════════════════════════════════════════════════════════════════════
-- 4. THEOREM: TRANSITION IMPOSSIBILITY
-- ══════════════════════════════════════════════════════════════════════════════
/-- Orthogonal transition returns null state -/
theorem intercol_transition_impossible (d1 d2 : Domain) :
orthogonal d1 d2 →
transition d1 d2 = TransitionResult.nullState := by
intro h_orth
unfold transition orthogonal at *
by_cases h : d1 = d2
· contradiction
· simp [h]
/-- Converse: same-domain transition succeeds -/
theorem intercol_same_domain_allowed (d : Domain) :
transition d d = TransitionResult.success d := by
unfold transition
simp
-- ══════════════════════════════════════════════════════════════════════════════
-- 5. DOMAIN ISOLATION INVARIANT
-- ══════════════════════════════════════════════════════════════════════════════
structure IsolatedState where
domain : Domain
authority : String -- e.g., "agent-xyz"
payload : ByteArray
/-- Transition preserves domain invariant -/
def preservesDomainInvariant (s : IsolatedState) (d_new : Domain) : Prop :=
s.domain = d_new
/-- Isolated state cannot escape its domain -/
theorem domain_isolation_invariant (s : IsolatedState) (d_target : Domain) :
orthogonal s.domain d_target →
¬(preservesDomainInvariant s d_target) := by
intro h_orth h_preserve
unfold preservesDomainInvariant orthogonal at *
exact h_orth h_preserve.symm
-- ══════════════════════════════════════════════════════════════════════════════
-- 6. ENUMERATION & COMPLETENESS
-- ══════════════════════════════════════════════════════════════════════════════
/-- All 4 domains are distinct -/
theorem domains_distinct :
let d1 := Domain.treasury
let d2 := Domain.clinical
let d3 := Domain.legal
let d4 := Domain.operations
d1 ≠ d2 ∧ d1 ≠ d3 ∧ d1 ≠ d4 ∧
d2 ≠ d3 ∧ d2 ≠ d4 ∧
d3 ≠ d4 := by
decide
/-- Pairwise orthogonality (6 pairs) -/
theorem pairwise_orthogonal :
let domains := [Domain.treasury, Domain.clinical, Domain.legal, Domain.operations]
∀ d1 d2 : Domain, d1 ∈ domains → d2 ∈ domains → d1 ≠ d2 →
orthogonal d1 d2 := by
intro domains d1 d2 _ _ h_ne
exact h_ne
-- ══════════════════════════════════════════════════════════════════════════════
-- 7. APPLICAT ION: BIFROST PERSONA CONSTRAINT
-- ══════════════════════════════════════════════════════════════════════════════
/-- Persona has an allowed domain; cannot jump to orthogonal domain -/
structure PersonaConstraint where
persona_id : Nat
allowed_domain : Domain
/-- Constraint violation check -/
def violatesConstraint (c : PersonaConstraint) (d : Domain) : Bool :=
c.allowed_domain ≠ d
/-- If constraint violated, transition fails -/
theorem constraint_enforces_isolation (c : PersonaConstraint) (d : Domain) :
violatesConstraint c d = true →
transition c.allowed_domain d = TransitionResult.nullState := by
intro h_violate
unfold transition violatesConstraint at *
by_cases h_eq : c.allowed_domain = d
· simp [h_eq] at h_violate
· simp [h_eq]
end IntercolDomains
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