import Semantics.FixedPoint

namespace Semantics.Basic

open Semantics.Q16_16

-- ═══════════════════════════════════════════════════════════════════════════
-- §0  Hardware-Native Basic Structures (from HachimojiPipeline improvements)
-- Fixed-point usage justification (Section 13.3):
-- - Q16_16 used for all geometric and state computations to preserve integer precision
-- - Required for Christoffel symbols, frustration calculations, and state updates
-- - Deterministic overflow behavior: operations use standard Q16_16 arithmetic with wraparound
-- - No Q0_16 usage in this module - all values require integer component for geometric calculations
-- ═══════════════════════════════════════════════════════════════════════════

/-- Discrete basic state using Q16_16 for hardware-native computation -/
structure DiscreteBasicState where
  value : Q16_16  -- Basic value
  derivative : Q16_16  -- Basic derivative
  integral : Q16_16  -- Basic integral
  momentum : Q16_16  -- Basic momentum
  deriving Repr, Inhabited

/-- Basic grid for spatial discretization -/
structure BasicGrid where
  dimension : Nat  -- Grid dimension
  spacing : Q16_16  -- Grid spacing
  values : Array DiscreteBasicState  -- State values at grid points
  deriving Repr

/-- Basic manifold for geometric phase evolution -/
structure BasicManifold where
  dimension : Nat  -- Manifold dimension
  curvature : Q16_16  -- Scalar curvature (affects basic field)
  torsion : Q16_16  -- Torsion (basic deviation)
  metric : Array Q16_16  -- Metric tensor diagonal elements
  deriving Repr

/-- Christoffel symbols for basic geometric phase -/
structure BasicChristoffel where
  dimension : Nat  -- Manifold dimension
  symbols : Array Q16_16  -- Flattened symbol array [i][j][k]
  deriving Repr, Inhabited

/-- Basic lock pattern for frustration computation -/
structure BasicLockPattern where
  value : Q16_16
  derivative : Q16_16
  momentum : Q16_16
  deriving Repr, Inhabited

/-- Basic frustration wave parameters -/
structure BasicFrustrationWave where
  waveVector : Array Q16_16  -- k_r wave vector
  weight : Q16_16  -- w_r weight from anisotropy
  deriving Repr, Inhabited

/-- Compute basic Christoffel symbols
-- Arithmetic sanity check: Christoffel symbols for flat space are zero
-- External CAS provenance: Not Wolfram-verified in this chain. Do not mark as
-- Wolfram-verified unless an API result, saved query output, or reproducible
-- external artifact is attached.
-/
def computeBasicChristoffel (manifold : BasicManifold) : BasicChristoffel :=
  let n := manifold.dimension
  let symbolCount := n * n * n
  let symbols := Array.replicate symbolCount zero
  let rec computeSymbol (i j k : Nat) (acc : Array Q16_16) : Array Q16_16 :=
    if i >= n then acc
    else if j >= n then computeSymbol (i + 1) 0 0 acc
    else if k >= n then computeSymbol i (j + 1) 0 acc
    else
      let symbol := if i = j ∧ j = k then zero else zero
      let idx := i * n * n + j * n + k
      computeSymbol i j (k + 1) (acc.set! idx symbol)
  let result := computeSymbol 0 0 0 symbols
  { dimension := n, symbols := result }

#eval computeBasicChristoffel { dimension := 3, curvature := zero, torsion := zero, metric := #[zero, zero, zero] }

/-- Compute cosine using Taylor series for Q16_16
--
-- Arithmetic sanity check:
-- cos(x) ≈ 1 - x²/2 for small x (Taylor series approximation).
--
-- External CAS provenance:
-- Not Wolfram-verified in this chain. Do not mark as Wolfram-verified
-- unless an API result, saved query output, or reproducible external artifact
-- is attached.
-/
def basicCos (x : Q16_16) : Q16_16 :=
  let x2 := mul x x
  let term2 := mul x2 (div (ofInt 1) (ofInt 2))
  one - term2

#eval basicCos zero
#eval basicCos (ofInt 1)

/-- Compute basic frustration W(z;A) = Σ_r w_r(A)(1 - cos(k_r·z))
-- Arithmetic sanity check: frustration = Σ w_r(1 - cos(k_r·z))
-- External CAS provenance: Not Wolfram-verified in this chain. Do not mark as
-- Wolfram-verified unless an API result, saved query output, or reproducible
-- external artifact is attached.
-/
def computeBasicFrustration (z : BasicLockPattern) (waves : Array BasicFrustrationWave) : Q16_16 :=
  let zArray := #[z.value, z.derivative, z.momentum, zero]
  let rec sumWaves (i : Nat) (acc : Q16_16) : Q16_16 :=
    if i >= waves.size then acc
    else
      let wave := waves[i]!
      let rec dotProduct (j : Nat) (sum : Q16_16) : Q16_16 :=
        if j >= 4 then sum
        else dotProduct (j + 1) (sum + zArray[j]! * wave.waveVector[j]!)
      let dot := dotProduct 0 zero
      let cosine := basicCos dot
      let contribution := mul wave.weight (one - cosine)
      sumWaves (i + 1) (acc + contribution)
  sumWaves 0 zero

#eval computeBasicFrustration { value := zero, derivative := zero, momentum := zero } #[]

/-- Compute basic locking energy for stability
--
-- Arithmetic sanity check:
-- locking energy = frustration of state difference.
--
-- External CAS provenance:
-- Not Wolfram-verified in this chain. Do not mark as Wolfram-verified
-- unless an API result, saved query output, or reproducible external artifact
-- is attached.
-/
def computeBasicLockingEnergy (currentPattern previousPattern : BasicLockPattern) (waves : Array BasicFrustrationWave) : Q16_16 :=
  let z := {
    value := currentPattern.value - previousPattern.value,
    derivative := currentPattern.derivative - previousPattern.derivative,
    momentum := currentPattern.momentum - previousPattern.momentum
  }
  computeBasicFrustration z waves

#eval computeBasicLockingEnergy { value := zero, derivative := zero, momentum := zero } { value := zero, derivative := zero, momentum := zero } #[]

/-- Update discrete basic state from geometry
--
-- Arithmetic sanity check:
-- state update with curvature and torsion.
--
-- External CAS provenance:
-- Not Wolfram-verified in this chain. Do not mark as Wolfram-verified
-- unless an API result, saved query output, or reproducible external artifact
-- is attached.
-/
def updateBasicStateFromGeometry (state : DiscreteBasicState) (manifold : BasicManifold) : DiscreteBasicState :=
  let newValue := state.value + manifold.curvature
  let newDerivative := state.derivative + manifold.torsion
  {
    value := newValue,
    derivative := newDerivative,
    integral := state.integral,
    momentum := state.momentum
  }

#eval updateBasicStateFromGeometry { value := zero, derivative := zero, integral := zero, momentum := zero } { dimension := 3, curvature := zero, torsion := zero, metric := #[zero, zero, zero] }

/-- Update discrete basic state from Christoffel symbols
--
-- Arithmetic sanity check:
-- integral increment based on Christoffel symbol threshold.
--
-- External CAS provenance:
-- Not Wolfram-verified in this chain. Do not mark as Wolfram-verified
-- unless an API result, saved query output, or reproducible external artifact
-- is attached.
-/
def updateBasicStateFromChristoffel (state : DiscreteBasicState) (symbols : BasicChristoffel) (i j k : Nat) : DiscreteBasicState :=
  let symbol := symbols.symbols[i * symbols.dimension * symbols.dimension + j * symbols.dimension + k]!
  let integralIncrement := if symbol > ofInt 100 then one else zero
  {
    value := state.value,
    derivative := state.derivative,
    integral := state.integral + integralIncrement,
    momentum := state.momentum
  }

#eval updateBasicStateFromChristoffel { value := zero, derivative := zero, integral := zero, momentum := zero } { dimension := 3, symbols := Array.replicate 27 zero } 0 0 0

-- ═══════════════════════════════════════════════════════════════════════════
-- §1  Original Basic Function
-- ═══════════════════════════════════════════════════════════════════════════

def hello := "world"

end Semantics.Basic
