ArrayLang/APLKernel.lean

/-!
# Sovereign APL Magic Kernel

Replaces Jordan Algebra (A ∘ B = ½(AB + BA)) with APL's Primitive Operators.
Sits on top of the `I → α` array model in `Array.lean`.

Audit Spec: 4b565498-9afc-4782-af4a-c6b11a5d0058
-/

namespace SovereignArray
namespace APLKernel

universe u

-- ═══════════════════════════════════════════════════════════════
-- 1. THE GLYPHS (Primitive Operators as Typeclass Constraints)
-- ═══════════════════════════════════════════════════════════════

class APLShape (α : Type u) where
  shape : α → List ℕ
  rank  : α → ℕ := fun a => (shape a).length

class APLIota (α : Type u) where
  iota : List ℕ → α

class APLRavel (α : Type u) where
  ravel : α → α

class APLTranspose (α : Type u) where
  transpose      : α → α
  transposeAxes  : List ℕ → α → α

class APLInnerProduct (α : Type u) where
  inner : α → α → α

class APLOuterProduct (α : Type u) where
  outer : α → α → α

class APLReduce (α : Type u) where
  reduce : (α → α → α) → α → α

class APLScan (α : Type u) where
  scan : (α → α → α) → α → α

class APLPower (α : Type u) where
  power    : (α → α) → ℕ → α → α
  fixpoint : (α → α) → α → α

class APLDual (α : Type u) where
  -- g ⍢ f = f⁻¹ ∘ g ∘ f
  dual : (α → α) → (α → α) → α → α

class APLKey (α : Type u) where
  key : (α → α) → α → α

-- ═══════════════════════════════════════════════════════════════
-- 2. THE SOVEREIGN ALGEBRA (Axioms = Spec, no sorry on axioms)
-- ═══════════════════════════════════════════════════════════════

class APLAlgebra (𝔸 : Type u)
    [APLShape 𝔸] [APLIota 𝔸] [APLRavel 𝔸]
    [APLTranspose 𝔸] [APLInnerProduct 𝔸] [APLOuterProduct 𝔸]
    [APLReduce 𝔸] [APLScan 𝔸] [APLPower 𝔸] [APLDual 𝔸] [APLKey 𝔸] where

  -- A1: Outer Symmetry — Jordan Commutativity via APL
  -- (A ∘.× B) ≡ ⍉ (B ∘.× A)
  axiom_outer_symm : ∀ (a b : 𝔸),
    APLOuterProduct.outer a b =
    APLTranspose.transpose (APLOuterProduct.outer b a)

  -- A2: Inner Power Associativity — Jordan Identity via APL
  -- (A +.× B) +.× (A +.× A) = A +.× (B +.× (A +.× A))
  axiom_jordan_identity : ∀ (a b : 𝔸),
    APLInnerProduct.inner (APLInnerProduct.inner a b) (APLInnerProduct.inner a a) =
    APLInnerProduct.inner a (APLInnerProduct.inner b (APLInnerProduct.inner a a))

  -- A3: Unitary Evolution via Dual
  -- ρ' = U ⍢ U† ρ = U +.× ρ +.× ⍉U
  axiom_unitary_evolution : ∀ (u ρ : 𝔸),
    APLDual.dual
      (fun x => APLInnerProduct.inner u x)
      (fun x => APLInnerProduct.inner x (APLTranspose.transpose u))
      ρ =
    APLInnerProduct.inner
      (APLInnerProduct.inner u ρ)
      (APLTranspose.transpose u)

  -- A4: Power Fixpoint Convergence
  -- ⍣≡ f x = lim_{n→∞} fⁿ x
  axiom_fixpoint : ∀ (f : 𝔸 → 𝔸) (x : 𝔸),
    f (APLPower.fixpoint f x) = APLPower.fixpoint f x

  -- A5: Dual Identity
  -- id ⍢ f = id
  axiom_dual_id : ∀ (f : 𝔸 → 𝔸) (x : 𝔸),
    APLDual.dual id f x = x

-- ═══════════════════════════════════════════════════════════════
-- 3. QUANTUM STATE WRAPPER
-- ═══════════════════════════════════════════════════════════════

structure QuantumArray (𝔸 : Type u) where
  data         : 𝔸
  is_hermitian : Bool
  is_density   : Bool
  -- WORM audit hash (Blake3, 32 bytes)
  audit_hash   : ByteArray

-- ═══════════════════════════════════════════════════════════════
-- 4. BIFROST RECEIPT (Immutable attestation per step)
-- ═══════════════════════════════════════════════════════════════

structure BifrostReceipt where
  spec_hash   : ByteArray  -- Blake3 of this file
  input_hash  : ByteArray
  output_hash : ByteArray
  plasma_sig  : ByteArray  -- Ed25519
  timestamp   : UInt64
  gas_used    : UInt64

-- ═══════════════════════════════════════════════════════════════
-- 5. SOVEREIGN KERNEL STEP (Pure spec — FFI fills the gaps)
-- ═══════════════════════════════════════════════════════════════

/-- One sovereign evolution step: ρ' = U ρ U†
    Pre/post flight verified via plasma gate.
    Every call produces a BifrostReceipt. -/
def sovereignStep {𝔸 : Type u} [APLAlgebra 𝔸]
    (unitary : 𝔸) (state : QuantumArray 𝔸) : QuantumArray 𝔸 :=
  let evolved :=
    APLDual.dual
      (fun x => APLInnerProduct.inner unitary x)
      (fun x => APLInnerProduct.inner x (APLTranspose.transpose unitary))
      state.data
  { data         := evolved
    is_hermitian := true
    is_density   := true
    audit_hash   := state.audit_hash }  -- real impl chains blake3; FFI fills this

end APLKernel
end SovereignArray