/-
  UnitDistCandidateGen.lean — Unit-Distance Point Sets → ManifoldShortcut

  The Erdős unit-distance problem: what is the maximum number of unit
  distances among N points in the plane? The known bounds are:
    u(N) ≤ O(N^{4/3}) (Spencer-Szemerédi-Trotter bound)
    u(N) = N^{1 + c/log log N} (best known lower bound)

  For a dense grid of point configurations, the ManifoldShortcut uses
  golden-angle perturbation to walk the configuration space efficiently.
  Each perturbation moves one point along the golden spiral (φ⁻¹ contraction,
  Kähler gate from GoldenSpiral.lean).

  The ManifoldEquation for a point set:
    - delta = optimal bound - achieved (gap to SST bound)
    - spectral = 1/rank (sparser configuration = better)
    - program = encoding cost of N points
    - coherence = angular uniformity of the unit-distance graph
    - rank = number of distinct distance classes

  AngrySphinx bounds the search: each point perturbation costs 2^depth.
  The NaN boundary terminates when the configuration converges.
-/

import Mathlib.Data.Real.Basic
import SilverSight.FixedPoint
import SilverSight.PIST.ManifoldShortcut
import SilverSight.AngrySphinx
import SilverSight.GoldenSpiral

namespace SilverSight.PIST.UnitDistCandidateGen

open SilverSight.FixedPoint
open SilverSight.FixedPoint.Q16_16
open SilverSight.PIST.ManifoldShortcut
open SilverSight.AngrySphinx
open SilverSight.GoldenSpiral

/-! ## §1  Point Set as ManifoldEquation

  A configuration of N points in R² with unit distances has:
  - u(N) = number of unit-distance pairs
  - SST bound: u(N) ≤ c·N^{4/3} (the optimal for this N)
  - The coherence is the angular uniformity of the distance graph
  - Spectral cost ~ 1/u(N) (more distances = sparser structure)
  - Program cost = 2*N*log₂(N) (encoding the point coordinates)
-/

/-- A point set configuration with its unit-distance count. -/
structure PointSet where
  numPoints : Nat
  unitDistances : Nat
  coherence : Q16_16
  -- Angular uniformity measure of the unit-distance graph
  rank : Nat
  -- Number of distinct distance classes (unit distance = 1 class)
  deriving Repr, Inhabited

/-- SST bound in Q16_16: u(N) ≤ c·N^{4/3}.
    c is the constant from the Spencer-Szemerédi-Trotter theorem.
    Approximated as c ≈ 2.5 for the Q16_16 encoding. -/
def sstBound (n : Nat) : Q16_16 :=
  -- Placeholder: c * n^(4/3) in Q16_16
  -- For simplicity: bound = c * n  (linear proxy until real irrational exponent)
  Q16_16.ofNat (2 * n)

/-- Wrap a point set as a ManifoldEquation.

    The Lagrangian terms:
    - delta: gap between u(N) and SST bound (how close to optimal)
    - spectral: 1/u(N) (more distances = sparser = easier)
    - program: 2*N cost to encode coordinates
    - coherence: angular uniformity of the distance graph
    - rank: 1 (all distances are unit = single class)
    - kData: SST bound (the proven upper bound)
-/
def pointsToEquation (ps : PointSet) : ManifoldEquation :=
  let u := ps.unitDistances
  let n := ps.numPoints
  let bound := sstBound n
  { deltaCost :=
      -- delta = sstBound - u (how far from optimal, as fraction of bound)
      if Q16_16.lt (Q16_16.ofNat u) bound then
        Q16_16.sub bound (Q16_16.ofNat u)
      else zero
    spectralCost := Q16_16.ofRatio 1 (max u 1)
    -- 1/u: more distances = sparser structure = lower spectral cost
    programCost := Q16_16.ofRatio (2 * n) n
    -- Encoding cost ~ 2 (constant per point for coordinates)
    coherence := ps.coherence
    -- Angular uniformity of unit-distance graph
    rank := ps.rank
    -- Number of distance classes (1 for unit distance problem)
    kData := bound
    -- SST bound = the theoretical maximum
  }

/-! ## §2  Golden-Angle Perturbation

  The golden angle θ_g = 2π/φ² is maximally irrational. Perturbing
  a point along the golden spiral ensures no alignment with any rational
  symmetry axis — each configuration is maximally different from all
  previous ones (observerless).

  This uses GoldenSpiral.lean's phi_inv contraction.
-/

/-- Perturb one point in the set by the golden angle.
    For a point at (r,θ) in polar coordinates, the perturbed point is
    (r, θ + θ_g) if expanding, or (r/φ, θ + θ_g) if contracting.

    Since we work in Lean with Q16_16, we approximate the perturbation
    as a rotation by the golden angle fraction. -/
def goldenPerturb (idx : Nat) (ps : PointSet) : PointSet :=
  -- Placeholder: actual perturbation would modify point coordinates
  -- For the search framework, track the perturbation index
  { ps with
    coherence := ps.coherence  -- Would recompute from perturbed coordinates
    -- After perturbation, recalculate unit distances
  }

/-! ## §3  Unit-Distance Search State

  The search perturbs one point at a time, tracking the best
  unit-distance count found. GoldenSpiral.lean's convergence
  guarantee (φ⁻¹ contraction) ensures the perturbation space
  is bounded and the search converges.
-/

structure UnitDistSearchState where
  /-- Current point set configuration -/
  pointSet : PointSet
  /-- The ShortcutSearchState tracking the best shortcut -/
  shortcutState : ShortcutSearchState
  /-- Which point to perturb next -/
  perturbIdx : Nat
  /-- Number of perturbations applied -/
  steps : Nat
  /-- Maximum steps before termination -/
  maxSteps : Nat
  deriving Repr, Inhabited

/-- N points in a regular polygon (trivial configuration). -/
def regularPolygon (n : Nat) (hN : n ≥ 3) : PointSet :=
  -- Regular N-gon inscribed in unit circle: N*(N-1)/2 unit distances
  { numPoints := n
    unitDistances := n * (n - 1) / 2
    coherence := Q16_16.one  -- Perfect angular uniformity
    rank := 1  -- All points on unit circle = single distance class
  }

/-- Initialize unit-distance search with regular polygon. -/
def initUnitDistSearch (n : Nat) (hN : n ≥ 3) (maxSteps : Nat) : UnitDistSearchState :=
  { pointSet := regularPolygon n hN
    shortcutState := initSearch (Q16_16.ofNat n)
    perturbIdx := 0
    steps := 0
    maxSteps := maxSteps
  }

/-- Check if more perturbations are available. -/
def unitDistHasNext (state : UnitDistSearchState) : Bool :=
  state.steps < state.maxSteps

/-- Evaluate a single point set configuration. -/
def unitDistEvaluateCandidate (state : UnitDistSearchState)
    (alpha beta epsilon : Q16_16) : UnitDistSearchState :=
  let eq : ManifoldEquation := pointsToEquation state.pointSet
  let newSS := evaluateCandidate state.shortcutState eq alpha beta epsilon
  { state with
    shortcutState := newSS
    steps := state.steps + 1
  }

/-! ## §4  Unit-Distance Search Loop

  Bounded by AngrySphinx and GoldenSpiral convergence.
  Each perturbation costs 2^depth; the golden contraction (φ⁻¹ < 1)
  guarantees the search space is bounded.

  From GoldenSpiral.lean: cost_outpaces_convergence — the
  AngrySphinx cost (2^depth) grows faster than the golden contraction
  (φ^{-k}) shrinks. The search always terminates.
-/

def unitDistSearchLoop (state : UnitDistSearchState) (fuel : Nat)
    (maxDepth : Nat) (alpha beta epsilon : Q16_16) : UnitDistSearchState :=
  if fuel = 0 then state
  else if ¬ unitDistHasNext state then
    state
  else if ¬ canContinue state.shortcutState maxDepth then
    state
  else
    let nextState := unitDistEvaluateCandidate state alpha beta epsilon
    let perturbed := goldenPerturb nextState.perturbIdx nextState.pointSet
    let nextPerturbIdx := (state.perturbIdx + 1) % nextState.pointSet.numPoints
    unitDistSearchLoop
      { nextState with
        pointSet := perturbed
        perturbIdx := nextPerturbIdx
      }
      (fuel - 1) maxDepth alpha beta epsilon

/-! ## §5  Result Extraction -/

structure UnitDistSearchResult where
  bestNumPoints : Nat
  bestUnitDistances : Nat
  bestRatio : Q16_16
  bestLagrangian : Q16_16
  bestQuality : Q16_16
  totalSteps : Nat
  receipt : String
  deriving Repr

def unitDistExtractResult (state : UnitDistSearchState) (alpha beta : Q16_16)
    : UnitDistSearchResult :=
  let ps := state.pointSet
  let ss := state.shortcutState
  { bestNumPoints := ps.numPoints
    bestUnitDistances := ps.unitDistances
    bestRatio := Q16_16.ofRatio ps.unitDistances ps.numPoints
    bestLagrangian := ss.bestLagrangian
    bestQuality := match ss.bestEquation with
      | none => Q16_16.zero
      | some eq => shortcutQuality eq alpha beta
    totalSteps := state.steps
    receipt := ""
  }

/-! ## §6  Evaluation Witnesses -/

-- Regular pentagon: 5 points, 10 unit distances
#eval let ps := regularPolygon 5 (by omega)
  (ps.numPoints, ps.unitDistances, ps.coherence)

-- ManifoldEquation for regular pentagon
#eval let eq := pointsToEquation (regularPolygon 5 (by omega))
  (eq.deltaCost, eq.spectralCost, eq.programCost, eq.coherence, eq.rank)

-- Lagrangian for regular pentagon
#eval lagrangian (pointsToEquation (regularPolygon 5 (by omega)))
  (Q16_16.ofRawInt 32768) (Q16_16.ofRawInt 32768)

end SilverSight.PIST.UnitDistCandidateGen
