/-
  CMYKColoringCore.lean — 4-Channel Coloring Candidate Generator

  Reimagines the CMYK OISC concept within the current SilverSight framework:
  - 4 channels (C, M, Y, K) encode color assignments for boundary points
  - Each channel is a Q16_16 value representing a color class
  - Forward encoding: boundary point → color assignment packet
  - Inverse decoding: color packet → boundary point coloring
  - Proven roundtrip correctness
  - Integrates with ManifoldShortcut/Lagrangian optimization
  - Uses AngrySphinx to bound the chromatic number search

  The conflict graph chromatic number χ is encoded as:
  - 4 channels = 4 potential colors (testing χ ≤ 4, below de Grey's 5 ≤ χ(ℝ²))
  - Each boundary point pᵢ gets a ColoringPacket assigning it to color classes
  - The Lagrangian measures coloring quality:
    * deltaCost = number of conflicts (unit-distance pairs with same color)
    * spectralCost = 1/χ (fewer colors = sparser = better)
    * programCost = description length of the coloring
    * coherence = how well the coloring respects the Sidon structure

  Connection to Sidon-Sofa Coloring (docs/research/SIDON_SOFA_COLORING.md):
  - Direction B: chromatic number of Gerver's sofa
  - The CMYK generator produces coloring candidates for Γ_γ^T
  - ManifoldShortcut finds the minimum-Lagrangian coloring
  - AngrySphinx bounds the search depth

  This is NOT the old FPGA/hardware path. This is the CMYK OISC as a
  software abstraction that fits the current PIST pipeline.
-/

import SilverSight.FixedPoint
import SilverSight.PIST.ManifoldShortcut
import SilverSight.AngrySphinx
import Mathlib.Tactic

namespace SilverSight.PIST.CMYKColoringCore

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

/-! §1 The 4-Channel Coloring Packet

  Each boundary point is assigned a coloring via 4 Q16_16 channels.
  The channel values encode color class membership:
  - C channel: primary color assignment (Q16_16 in [0, 1))
  - M channel: secondary color assignment
  - Y channel: tertiary color assignment
  - K channel: key/black channel (used for conflict resolution)

  A point belongs to color class c if channel c has the maximum value.
  This is a soft assignment: the channels can represent uncertainty or
  mixed colorings that the search will resolve.
-/

/-- Four-channel coloring packet for a boundary point. -/
structure ColoringPacket where
  cChannel : Q16_16  -- Cyan channel
  mChannel : Q16_16  -- Magenta channel
  yChannel : Q16_16  -- Yellow channel
  kChannel : Q16_16  -- Key channel
  deriving Repr, DecidableEq, Inhabited

/-- Extract the dominant color class from a packet.
    Returns 0 (C), 1 (M), 2 (Y), or 3 (K) based on max channel. -/
def dominantColor (p : ColoringPacket) : Fin 4 :=
  let vals := [p.cChannel, p.mChannel, p.yChannel, p.kChannel]
  let maxVal := vals.foldl max zero
  if p.cChannel = maxVal then ⟨0, by decide⟩
  else if p.mChannel = maxVal then ⟨1, by decide⟩
  else if p.yChannel = maxVal then ⟨2, by decide⟩
  else ⟨3, by decide⟩

/-- Check if a coloring packet is valid (all channels in [0, 1)). -/
def isValidColoring (p : ColoringPacket) : Bool :=
  Q16_16.le zero p.cChannel && Q16_16.lt p.cChannel one &&
  Q16_16.le zero p.mChannel && Q16_16.lt p.mChannel one &&
  Q16_16.le zero p.yChannel && Q16_16.lt p.yChannel one &&
  Q16_16.le zero p.kChannel && Q16_16.lt p.kChannel one

/-! §2 Forward/Inverse Encoding with Roundtrip

  The encoding maps a boundary point index to a coloring packet.
  The decoding recovers the index from the packet.
  The roundtrip theorem proves correctness.

  Encoding scheme (nibble-based):
  - Point index i (0 ≤ i < 16) → 4-bit nibble
  - High 2 bits = quandary state (0=REJECT, 1=ACCEPT, 2=HOLD, 3=QUARANTINE)
  - Low 2 bits = channel selector (0=C, 1=M, 2=Y, 3=K)
  - Each channel gets the nibble value scaled to [0, 1) in Q16_16
-/

/-- Scale factor: 1/16 in Q16_16 = 4096 (since 65536/16 = 4096). -/
def nibbleScale : Q16_16 := ofRawInt 4096

/-- Encode a point index (0-15) as a coloring packet.
    The high 2 bits (nibble / 4) determine the base value.
    The low 2 bits (nibble % 4) determine the dominant channel. -/
def encodeColoring (pointIdx : Fin 16) : ColoringPacket :=
  let nibble := pointIdx.val
  let group := nibble / 4  -- high 2 bits (0-3)
  let baseVal := mul nibbleScale (ofNat group)
  let dominant := nibble % 4  -- low 2 bits
  -- Boost the dominant channel by 0.5 (32768 in Q16_16)
  let boost := ofRawInt 32768
  { cChannel := if dominant = 0 then add baseVal boost else baseVal
    mChannel := if dominant = 1 then add baseVal boost else baseVal
    yChannel := if dominant = 2 then add baseVal boost else baseVal
    kChannel := if dominant = 3 then add baseVal boost else baseVal }

/-- Decode a coloring packet back to a point index (0-15).
    Extracts the dominant channel (low 2 bits) and group value (high 2 bits). -/
def decodeColoring (p : ColoringPacket) : Option (Fin 16) :=
  if !isValidColoring p then none
  else
    let dom := dominantColor p
    let channelVal := match dom with
      | ⟨0, _⟩ => p.cChannel
      | ⟨1, _⟩ => p.mChannel
      | ⟨2, _⟩ => p.yChannel
      | ⟨3, _⟩ => p.kChannel
    -- Subtract the boost (32768) and divide by nibbleScale (4096) to get group
    let unboosted := sub channelVal (ofRawInt 32768)
    let groupDiv := div unboosted nibbleScale
    let groupNat := groupDiv.toInt / q16Scale
    -- Reconstruct nibble: group * 4 + dominant
    let domNat := dom.val
    let nibble := groupNat.toNat * 4 + domNat
    if h : nibble < 16 then some ⟨nibble, h⟩ else none

/-- Roundtrip theorem: decoding an encoded point returns the original. -/
theorem decodeColoring_roundtrip (i : Fin 16) : decodeColoring (encodeColoring i) = some i := by
  have hall : ∀ (j : Fin 16), decodeColoring (encodeColoring j) = some j := by
    decide
  exact hall i

/-! §3 Coloring as ManifoldEquation

  A coloring candidate is wrapped as a ManifoldEquation for the
  ManifoldShortcut search. The Lagrangian measures coloring quality:

  - deltaCost: number of conflicts (unit-distance pairs with same color)
    Lower = better (fewer conflicts = closer to valid coloring)

  - spectralCost: 1/χ where χ is the number of colors used
    Lower = better (fewer colors = sparser structure)

  - programCost: description length of the coloring
    For n points: log₂(16) * n = 4n bits (each point is a nibble)

  - coherence: how well the coloring respects the Sidon structure
    1.0 = perfect (all Sidon pairs get different colors)
    0.0 = worst (Sidon pairs collide)

  - rank: number of distinct colors used (≤ 4 for CMYK)

  - kData: the theoretical minimum χ for this conflict graph
    (de Grey's bound: χ ≥ 5 for ℝ², so kData ≥ 5 for nontrivial cases)
-/

/-- A coloring candidate with its conflict graph statistics. -/
structure ColoringCandidate where
  coloring : Array ColoringPacket  -- one packet per boundary point
  numPoints : Nat
  numColors : Nat                -- χ: number of distinct colors used
  numConflicts : Nat             -- unit-distance pairs with same color
  sidonRespect : Q16_16          -- fraction of Sidon pairs with different colors
  deriving Repr, Inhabited

/-- Wrap a coloring candidate as a ManifoldEquation. -/
def coloringToEquation (cand : ColoringCandidate) : ManifoldEquation :=
  let n := cand.numPoints
  let chi := cand.numColors
  let conflicts := cand.numConflicts
  { deltaCost := ofNat conflicts
    spectralCost := ofRatio 1 (max chi 1)
    programCost := ofRatio (4 * n) 1
    coherence := cand.sidonRespect
    rank := chi
    kData := ofNat 5 }

/-- Compute the Lagrangian for a coloring candidate.
    L = conflicts + α/χ + β·4n -/
def coloringLagrangian (cand : ColoringCandidate) (alpha beta : Q16_16) : Q16_16 :=
  lagrangian (coloringToEquation cand) alpha beta

/-! §4 Search Integration with ManifoldShortcut

  The CMYK coloring search uses the ManifoldShortcut framework:
  - ShortcutSearchState tracks the best coloring found
  - evaluateCandidate checks if a new coloring is better
  - AngrySphinx bounds the search depth
  - The NaN boundary terminates when frustration → 0
-/

/-- Initialize the CMYK coloring search. -/
def initColoringSearch : ShortcutSearchState :=
  initSearch (ofNat 5)  -- kData = 5 (de Grey's bound)

/-- Evaluate a coloring candidate in the search. -/
def evaluateColoring (state : ShortcutSearchState)
    (cand : ColoringCandidate) (alpha beta epsilon : Q16_16) : ShortcutSearchState :=
  evaluateCandidate state (coloringToEquation cand) alpha beta epsilon

/-- Check if the coloring search has found a valid χ-coloring. -/
def isValidChiColoring (cand : ColoringCandidate) (chi : Nat) : Bool :=
  cand.numConflicts = 0 && cand.numColors ≤ chi

/-! §5 Conflict Detection

  Two boundary points conflict if:
  1. They are at unit distance (‖pᵢ - pⱼ‖ = 1)
  2. They are assigned the same color class
-/

/-- Count conflicts in a coloring: unit-distance pairs with same color. -/
def countConflicts (coloring : Array ColoringPacket)
    (unitDistPairs : List (Nat × Nat)) : Nat :=
  unitDistPairs.foldl (fun acc (i, j) =>
    if h_i : i < coloring.size then
      if h_j : j < coloring.size then
        let c_i := dominantColor coloring[i]
        let c_j := dominantColor coloring[j]
        if c_i = c_j then acc + 1 else acc
      else acc
    else acc) 0

/-! §6 Sidon Respect Metric

  The coloring should respect the Sidon structure: pairs of boundary
  points that form a Sidon pair (unique midpoint) should ideally get
  different colors.

  sidonRespect = (Sidon pairs with different colors) / (total Sidon pairs)
-/

/-- Compute the Sidon respect metric for a coloring. -/
def sidonRespect (coloring : Array ColoringPacket)
    (sidonPairs : List (Nat × Nat)) : Q16_16 :=
  if sidonPairs.isEmpty then one
  else
    let total := sidonPairs.length
    let respected := sidonPairs.foldl (fun acc (i, j) =>
      if h_i : i < coloring.size then
        if h_j : j < coloring.size then
          let c_i := dominantColor coloring[i]
          let c_j := dominantColor coloring[j]
          if c_i ≠ c_j then acc + 1 else acc
        else acc
      else acc) 0
    ofRatio respected total

/-! §7 SIMD/GPU Adaptation

  The CMYK abstraction is designed for SIMD parallelism on GPU compute:
  - ColoringPacket = 4 × Q16_16 = 128 bits (perfect for SIMD lanes)
  - dominantColor = max4 + compare (single GPU instruction)
  - countConflicts = parallel fold over pairs (one thread per pair)
  - Lagrangian = pure Q16_16 arithmetic (no branches, warp-level execution)

  For GPU efficiency, we provide:
  - Batch encoding (encode many points at once)
  - SOA (Structure of Arrays) layout for memory coalescing
  - Vectorized conflict counting (process pairs in parallel)
-/--/

/-- Batch encode: encode multiple point indices at once.
    GPU: launch one thread per index, write to output array. -/
def batchEncodeColoring (indices : Array (Fin 16)) : Array ColoringPacket :=
  indices.map encodeColoring

/-- SOA layout: separate arrays for each channel (better memory coalescing).
    GPU: each channel is a contiguous array, enabling coalesced loads. -/
structure ColoringSOA where
  cChannels : Array Q16_16
  mChannels : Array Q16_16
  yChannels : Array Q16_16
  kChannels : Array Q16_16
  size : Nat
  deriving Repr, Inhabited

/-- Convert AOS (Array of Structures) to SOA (Structure of Arrays).
    GPU: AOS has poor coalescing (strided access), SOA has perfect coalescing. -/
def toSOA (coloring : Array ColoringPacket) : ColoringSOA :=
  { cChannels := coloring.map (·.cChannel)
    mChannels := coloring.map (·.mChannel)
    yChannels := coloring.map (·.yChannel)
    kChannels := coloring.map (·.kChannel)
    size := coloring.size }

/-- Vectorized dominant color from SOA (GPU-friendly).
    GPU: load 4 channels in parallel, compute max, compare. -/
def dominantColorSOA (soa : ColoringSOA) (idx : Nat)
    (h : idx < soa.size) : Fin 4 :=
  let c := soa.cChannels.getD idx zero
  let m := soa.mChannels.getD idx zero
  let y := soa.yChannels.getD idx zero
  let k := soa.kChannels.getD idx zero
  let maxVal := Q16_16.max (Q16_16.max c m) (Q16_16.max y k)
  if c = maxVal then ⟨0, by decide⟩
  else if m = maxVal then ⟨1, by decide⟩
  else if y = maxVal then ⟨2, by decide⟩
  else ⟨3, by decide⟩

/-- Vectorized conflict counting from SOA (GPU kernel).
    GPU: one thread per pair, atomic add to shared counter.
    Memory: 4 coalesced loads per point (c, m, y, k channels). -/
def countConflictsSOA (soa : ColoringSOA)
    (unitDistPairs : Array (Nat × Nat)) : Nat :=
  unitDistPairs.foldl (fun acc (i, j) =>
    if h_i : i < soa.size then
      if h_j : j < soa.size then
        let c_i := dominantColorSOA soa i h_i
        let c_j := dominantColorSOA soa j h_j
        if c_i = c_j then acc + 1 else acc
      else acc
    else acc) 0

/-! §8 Warp-Level Reduction (Conceptual)

  GPU warps (32 threads) can perform parallel reductions efficiently.
  The Lagrangian evaluation is a perfect candidate:
  - Each thread computes L for one candidate
  - Warp shuffle to find minimum L
  - Thread 0 writes the result

  This is conceptual: the Lean formalization captures the algorithm,
  not the GPU-specific implementation.
-/--/

/-- Batch Lagrangian evaluation (GPU warp-level).
    GPU: one thread per candidate, warp shuffle to find minimum. -/
def batchLagrangian (candidates : Array ColoringCandidate)
    (alpha beta : Q16_16) : Array Q16_16 :=
  candidates.map (coloringLagrangian · alpha beta)

/-- Find the candidate with minimum Lagrangian (GPU warp reduction).
    GPU: parallel min reduction across warp, thread 0 returns index. -/
def findMinimumLagrangian (lagrangians : Array Q16_16) : Option (Nat × Q16_16) :=
  if h : lagrangians.isEmpty then none
  else
    have hsz : 0 < lagrangians.size := by
      apply Nat.pos_of_ne_zero
      intro hzero
      apply h
      rw [Array.isEmpty_iff_size_eq_zero]
      exact hzero
    let initVal := lagrangians[0]'(by omega)
    let idxVals := lagrangians.toList
    let result := (List.range idxVals.length).foldl (fun (accIdx, accVal) i =>
      let val := idxVals.getD i (ofRawInt 0)
      if Q16_16.lt val accVal then (i, val) else (accIdx, accVal)) (0, initVal)
    some result

end SilverSight.PIST.CMYKColoringCore
