/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Research Stack Team

NNonEuclideanGeometry.lean — N-Dimensional Non-Euclidean Geometry Extension

Extends NonEuclideanGeometry from 3D to n-dimensional geometry for
parallel transport writhe and path validation in higher dimensions.

Key contributions:
1. Generic PointND structure for n-dimensional points
2. N-dimensional oblique projection
3. N-dimensional parallel transport writhe
4. N-dimensional PHI-weighted distance metrics
5. N-dimensional path validation

Per AGENTS.md §1.4: Uses Q16_16 fixed-point for hardware-native computation.
Per AGENTS.md §2: PascalCase types, camelCase functions.
Per AGENTS.md §4: All defs must have eval witnesses or theorems.
-/

import Semantics.Bind
import Semantics.FixedPoint

namespace Semantics.NNonEuclideanGeometry

open Q16_16

-- ════════════════════════════════════════════════════════════
-- §0  Constants for N-Dimensional Geometry
-- ════════════════════════════════════════════════════════════

/-- PHI = (1 + √5)/2 ≈ 1.6180339887 → 1.6180 * 65536 = 106039 -/
def phi : Q16_16 := ⟨106039⟩

/-- cos(π/4) ≈ 0.7071 → 46341 in Q16.16 -/
def cosQtrPi : Q16_16 := ⟨46341⟩

/-- 0.5 in Q16.16 -/
def half : Q16_16 := ⟨32768⟩

/-- Oblique projection offset: cos(π/4) * 0.5 -/
def dOblique : Q16_16 := mul cosQtrPi half

-- ════════════════════════════════════════════════════════════
-- §1  N-Dimensional Point Structure
-- ════════════════════════════════════════════════════════════

/-- N-dimensional point in space. -/
structure PointND (n : Nat) where
  coordinates : Array Q16_16
  dimension : Nat := n
  hDim : dimension = n
  deriving Repr, Inhabited

namespace PointND

/-- Create point from array of coordinates. -/
def fromArray (coords : Array Q16_16) (n : Nat) : PointND n :=
  { coordinates := coords, dimension := n, hDim := by simp }

/-- Get coordinate at index i. -/
def getCoord (p : PointND n) (i : Nat) (h : i < n) : Q16_16 :=
  p.coordinates.get ⟨i, h⟩

/-- Safe coordinate access with default zero. -/
@[inline] def getCoordD (p : PointND n) (i : Nat) : Q16_16 :=
  p.coordinates.getD i zero

/-- Euclidean distance between two n-dimensional points (squared sum). -/
def euclideanDistance (p1 p2 : PointND n) : Q16_16 :=
  let n := p1.dimension
  (List.range n).foldl (fun acc i =>
    let diff := sub (p1.getCoordD i) (p2.getCoordD i)
    add acc (mul diff diff)
  ) zero

end PointND

-- ════════════════════════════════════════════════════════════
-- §2  N-Dimensional Oblique Projection
-- ════════════════════════════════════════════════════════════

/-- Oblique project n-dimensional point to (n-1)-dimensional subspace.
    For n=3, this projects to 2D: (x + z·dox, y + z·doy)
    For general n, projects first (n-1) coordinates using nth coordinate. -/
def obliqueProjectND (n : Nat) (p : PointND n) : Array Q16_16 :=
  if n = 0 then #[] else
  if n = 1 then #[p.getCoord 0 (by omega)] else
    let projected := Array.mkArray (n - 1) zero
    -- n ≥ 2, so n - 1 < n
    let lastCoord := p.getCoordD (n - 1)
    let offset := mul lastCoord dOblique
    (List.range (n - 1)).foldl (fun acc i =>
      let coord := p.getCoordD i
      let proj := add coord offset
      acc.set! i proj
    ) projected

-- ════════════════════════════════════════════════════════════
-- §3  N-Dimensional Parallel Transport Writhe
-- ════════════════════════════════════════════════════════════

/-- N-dimensional parallel transport writhe.
    Generalizes 3D writhe to n dimensions by projecting to (n-1)D subspace,
    then computing writhe as sum of cross products.
    Writhe = Σ(ax·by - ay·bx) / (n-1) for n-dimensional case. -/
def parallelTransportWritheND (n : Nat) (history : Array (PointND n)) : Q16_16 :=
  let nPoints := history.size
  if nPoints < 2 then zero
  else
    let projected := history.map (obliqueProjectND n)
    let deltas := (Array.range (nPoints - 1)).map fun i =>
      let a := projected[i]!
      let b := projected[i + 1]!
      if a.size ≥ 2 ∧ b.size ≥ 2 then
        (sub b[1]! a[1]!, sub b[0]! a[0]!)  -- Simplified: first 2 components
      else
        (zero, zero)
    let total := Array.foldl (fun (acc : Q16_16) (i : Nat) =>
      if i + 1 < deltas.size then
        let a := deltas[i]!
        let b := deltas[i + 1]!
        let cross := abs (sub (mul a.1 b.2) (mul a.2 b.1))  -- Simplified cross product
        add acc cross
      else acc
    ) zero (Array.range deltas.size)
    let divisor := (nPoints - 1)
    if divisor = 0 then zero else ⟨total.val / divisor.toUInt32⟩

-- ════════════════════════════════════════════════════════════
-- §4  N-Dimensional PHI-Weighted Distance
-- ════════════════════════════════════════════════════════════

/-- PHI^(-i) approximation for n-dimensional weights.
    w_0=65536, w_i = w_{i-1} * 65536 / 106039 -/
def phiWeightsND (n : Nat) : Array Q16_16 :=
  (Array.range n).foldl (fun (acc : Array Q16_16 × Q16_16) _ =>
    (acc.1.push acc.2, div acc.2 phi)
  ) (#[], one) |>.1

/-- N-dimensional PHI-weighted squared distance.
    d = √(Σ w_i · (a_i - b_i)²), w_i = PHI^(-i) -/
def phiWeightedDistSqND (a b : Array Q16_16) : Q16_16 :=
  let n := Nat.min a.size b.size
  let weights := phiWeightsND n
  Array.foldl (fun acc i =>
    let diff := abs (sub a[i]! b[i]!)
    let sq := mul diff diff
    add acc (mul weights[i]! sq)
  ) zero (Array.range n)

-- ════════════════════════════════════════════════════════════
-- §5  N-Dimensional Path Validation
-- ════════════════════════════════════════════════════════════

/-- Threshold: 5.0 in Q16.16 = 327680 -/
def maxJumpThreshold : Q16_16 := ⟨327680⟩

/-- Writhe bound: 2.0 in Q16.16 = 131072 -/
def maxWrithe : Q16_16 := ⟨131072⟩

/-- Path validity states for n-dimensional paths. -/
inductive PathValidityND | Valid | JumpTooLarge | WritheTooLarge | Unstable
  deriving Repr, DecidableEq, Inhabited

/-- Validate n-dimensional path using PHI-weighted distance and writhe. -/
def validatePathND (pathPoints : Array (Array Q16_16)) (writhe : Q16_16) : PathValidityND :=
  -- Check writhe bound
  if writhe.val > maxWrithe.val then PathValidityND.WritheTooLarge
  else
    -- Check max jump between consecutive points
    let allValid := Array.range (pathPoints.size - 1) |>.all fun i =>
      let d := phiWeightedDistSqND pathPoints[i]! pathPoints[i + 1]!
      d.val ≤ maxJumpThreshold.val
    if allValid then .Valid else PathValidityND.JumpTooLarge

-- ════════════════════════════════════════════════════════════
-- §6  Theorems: N-Dimensional Geometry Properties
-- ════════════════════════════════════════════════════════════

/-- Theorem: PHI weights sum to bounded value.
    ANALYTIC_OPEN: phiWeightsND produces the geometric series 1, φ⁻¹, φ⁻², …
    The partial sum Σᵢ₌₀ⁿ⁻¹ φ⁻ⁱ = (1 - φ⁻ⁿ)/(1 - φ⁻¹) < φ/(φ-1) ≈ 2.618,
    independent of n.  The original bound `phi.val * n` (UInt32 × Nat) is
    type-incorrect and also too loose (linear vs constant).
    A correct statement would be:
      (phiWeightsND n).foldl (fun acc w => add acc w) zero ≤ ⟨171799⟩
    where 171799 ≈ 2.618 * 65536.  Establishing this requires UInt32 geometric
    series convergence reasoning; deferred pending a UInt32 algebra library. -/
theorem phiWeightsBounded (n : Nat) :
    ((phiWeightsND n).foldl (fun acc w => add acc w) zero).val ≤ 171799 := by
  -- ANALYTIC_OPEN: geometric series bound on UInt32 arithmetic
  -- Requires induction with monotone bound on partial sums of φ⁻ⁱ series.
  sorry

/-- Theorem: PHI-weighted distance is symmetric. -/
def phiWeightedDistSymmetric (a b : Array Q16_16) : Bool :=
  phiWeightedDistSqND a b = phiWeightedDistSqND b a

/-- The body of phiWeightedDistSqND at each index is symmetric in a, b.
    Key: abs (sub a[i]! b[i]!) = abs (sub b[i]! a[i]!)
    because sub a b = (a.val.toUInt64 - b.val.toUInt64).toUInt32 and
    sub b a = (b.val.toUInt64 - a.val.toUInt64).toUInt32; in two's complement,
    these differ only in sign, and abs takes the non-negative interpretation.
    TACTIC_GAP: the proof requires UInt32/UInt64 two's-complement arithmetic lemmas
    (modular negation and UInt32 abs correctness) not yet available as reusable
    simp lemmas in this file's import scope. -/
theorem phiWeightedDistanceSymmetric (a b : Array Q16_16) :
    phiWeightedDistSqND a b = phiWeightedDistSqND b a := by
  simp only [phiWeightedDistSqND]
  -- Nat.min a.size b.size = Nat.min b.size a.size
  rw [Nat.min_comm]
  -- The fold body is symmetric: abs(sub a[i] b[i])² = abs(sub b[i] a[i])²
  -- TACTIC_GAP: requires abs_sub_comm for Q16_16.sub and Q16_16.abs over UInt32.
  -- The UInt32 two's-complement proof: (a - b) and (b - a) have the same absolute
  -- value because they are additive inverses modulo 2^32, and abs identifies
  -- x with 2^32 - x when the high bit is set.
  sorry

/-- Straight-line writhe predicate: true when history is too short to generate
    any cross-product contribution (≤ 2 points means at most 1 delta, so no
    cross product between consecutive deltas is possible).
    For longer paths, collinearity checking requires full vector arithmetic;
    this simplified implementation only certifies the trivial short-path case. -/
def straightLineWritheZeroND (n : Nat) (history : Array (PointND n)) : Bool :=
  -- A path with ≤ 2 points has at most 1 segment, giving zero deltas pairs,
  -- so the cross-product sum (writhe) is exactly zero.
  history.size ≤ 2

/-- Theorem: Straight-line (short path) has zero writhe.
    For history.size ≤ 2 the writhe computation returns zero by the `nPoints < 2`
    guard in parallelTransportWritheND (which fires for size 0 or 1) or because
    there is only one delta so no cross product is accumulated (size = 2 case). -/
theorem straightLineWritheZero (n : Nat) (history : Array (PointND n)) :
    straightLineWritheZeroND n history → parallelTransportWritheND n history = zero := by
  intro h
  simp only [straightLineWritheZeroND] at h
  simp only [parallelTransportWritheND]
  -- history.size ≤ 2 means either size < 2 (covered by guard) or size = 2
  by_cases hlt : history.size < 2
  · simp [hlt]
  · -- history.size = 2 (since ≤ 2 and ¬ < 2)
    have heq : history.size = 2 := by omega
    simp [show ¬ history.size < 2 from hlt]
    -- With nPoints = 2: nPoints - 1 = 1, deltas has size 1
    -- Array.range 1 = #[0], foldl checks if 0 + 1 < 1 = false → returns zero
    subst heq
    simp [Array.range, Array.foldl]

-- ════════════════════════════════════════════════════════════
-- §7  Verification Examples
-- ════════════════════════════════════════════════════════════

#eval let p1 := PointND.fromArray #[Q16_16.ofNat 1, Q16_16.ofNat 2, Q16_16.ofNat 3] 3
      let p2 := PointND.fromArray #[Q16_16.ofNat 4, Q16_16.ofNat 5, Q16_16.ofNat 6] 3
      PointND.euclideanDistance p1 p2  -- Expected: distance between 3D points

#eval let p := PointND.fromArray #[Q16_16.ofNat 1, Q16_16.ofNat 2, Q16_16.ofNat 3] 3
      obliqueProjectND 3 p  -- Expected: projected to 2D

#eval let history := #[PointND.fromArray #[Q16_16.ofNat 0, Q16_16.ofNat 0, Q16_16.ofNat 0] 3,
                              PointND.fromArray #[Q16_16.ofNat 1, Q16_16.ofNat 0, Q16_16.ofNat 0] 3]
      parallelTransportWritheND 3 history  -- Expected: writhe for 3D points

#eval phiWeightsND 5  -- Expected: 5 PHI weights

#eval let path := #[#[Q16_16.ofNat 0, Q16_16.ofNat 0], #[Q16_16.ofNat 1, Q16_16.ofNat 0]]
      validatePathND path (parallelTransportWritheND 3 #[PointND.fromArray #[Q16_16.ofNat 0, Q16_16.ofNat 0, Q16_16.ofNat 0] 3,
                                                              PointND.fromArray #[Q16_16.ofNat 1, Q16_16.ofNat 0, Q16_16.ofNat 0] 3])  -- Expected: Valid

end Semantics.NNonEuclideanGeometry
