/-
  WeightCandidateGen.lean — cmix Weight Matrix → ManifoldShortcut

  The cmix compressor has a 23×461 weight matrix (the "genome"). SVD of
  this matrix gives 23 singular values — the "compression shape equation."
  If 5 singular values capture 95% of energy, the search space is 5D.

  For each candidate weight configuration in the 5D SVD subspace:
    1. Project to a full 23×461 weight matrix
    2. Compute compression performance (coherence)
    3. Wrap as a ManifoldEquation
    4. Feed into the ManifoldShortcut search loop

  Connection to the epigenetic framing (cmix_epigenetic_analysis.md):
    - 461 models = 461 genes
    - 23 layer-0 mixers = 23 regulatory regions (rank)
    - 5 active singular values = the search subspace (sparsity)
    - SVD spectrum = the compression shape equation (Lagrangian)

  AngrySphinx bounds the search: each weight candidate costs 2^depth.
  The NaN boundary terminates when the 5D subspace is exhausted.
-/

import SilverSight.FixedPoint
import SilverSight.PIST.ManifoldShortcut
import SilverSight.AngrySphinx

namespace SilverSight.PIST.WeightCandidateGen

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

/-! ## §1  Weight Matrix as ManifoldEquation

  A cmix weight configuration has:
  - 461 models (dimensions of the prediction space)
  - 23 mixers (regulatory regions)
  - k active singular values (k ≤ 23, typically k=5 effective)
  - Coherence = compression ratio achieved / theoretical max
  - Program cost = bits to encode the weight matrix

  The Lagrangian: L = delta (compression loss) + α·spectral (singular value spread)
                     + β·program (matrix encoding cost)
-/

/-- A candidate weight configuration in the SVD subspace.

    Represented by its top-k singular values (normalized to [0,1])
    and the compression performance it achieves. -/
structure WeightCandidate where
  /-- Number of active singular values (k ≤ 23, typically 5) -/
  activeValues : Nat
  /-- The k dominant singular values as Q16_16 fractions -/
  singularValues : List Q16_16
  /-- Compression ratio achieved (bits/byte compressed) -/
  compressionRatio : Q16_16
  /-- Coherence: how well the weights capture the data structure -/
  coherence : Q16_16
  deriving Repr, Inhabited

/-- The theoretical maximum compression for this data.
    cmix achieves ~1.184 b/B on enwik8. K(data) is the entropy floor. -/
def entropyFloor : Q16_16 := Q16_16.ofRawInt 77591  -- 1.184 * 65536 ≈ 77591

/-- Wrap a weight candidate as a ManifoldEquation.

    The Lagrangian terms:
    - delta: compression loss vs entropy floor (how much we miss optimal)
    - spectral: 1/k (fewer active values = sparser = better)
    - program: 23 * (bits per weight) / (total bits) ~ matrix encoding cost
    - coherence: compression accuracy (how close to 1.184 b/B on held-out data)
    - rank: k (number of active singular values)
    - kData: entropy floor (1.184 b/B)
-/
def weightToEquation (c : WeightCandidate) : ManifoldEquation :=
  let k := c.activeValues
  { deltaCost :=
      -- delta = max(0, entropy_floor - achieved_ratio)
      -- How much compression we left on the table
      if Q16_16.lt c.compressionRatio entropyFloor then
        Q16_16.sub entropyFloor c.compressionRatio
      else zero
    spectralCost := Q16_16.ofRatio 1 k
    -- Spectral cost ~ 1/k (fewer singular values = sparser = easier search)
    programCost := Q16_16.ofRatio 23 461
    -- Program cost: 23 mixers out of 461 models ≈ 0.05 (the compression shape)
    coherence := c.coherence
    -- Coherence from compression accuracy
    rank := k
    -- Number of active singular values
    kData := entropyFloor
    -- K(data) = 1.184 b/B (cmix's measured entropy floor)
  }

/-! ## §2  SVD Subspace Walk

  The search walks the 5D SVD subspace. For each point in the subspace:
  1. Reconstruct the full 23×461 weight matrix via U*diag(s)*V^T
  2. Measure compression performance on held-out data
  3. Compute coherence (accuracy vs previous best)
  4. Wrap as ManifoldEquation and evaluate

  Since full cmix evaluation (~10 hours) is impractical for the search,
  we use a lightweight proxy: the singular value entropy.
  Lower entropy = more concentrated spectrum = better compression.
-/

/-- Singular value entropy proxy for compression quality.
    Entropy = -Σ (sv_i / Σsv) * log(sv_i / Σsv).
    Lower = more concentrated = better compression.
    Approximated in Q16_16 as: Σ(sv_i * log₂(sv_i/Σsv)) / (k * Σsv).

    For simplicity, we use a heuristic: the ratio of top-1 sv to sum.
    Higher ratio = more concentrated = better compression. -/
def singularValueEntropy (svs : List Q16_16) (_k : Nat) : Q16_16 :=
  match svs with
  | [] => Q16_16.one
  | top :: _rest =>
    -- Proxy: top-1 / (top-1 + epsilon) ≈ concentration ratio
    Q16_16.ofRatio 1 1  -- placeholder: full computation needs Σsv

/-- Generate candidate weight configurations by perturbing singular values.

    Starting from the measured cmix SVD spectrum, perturb each singular
    value by ±δ and evaluate. This walks the 5D neighborhood.

    For a real implementation: extract trained weights from cmix, compute
    SVD, then perturb in the top-k subspace. The perturbation step is δ
    times the current singular value. -/
def perturbCandidate (base : WeightCandidate) (idx : Nat) (delta : Q16_16)
    : WeightCandidate :=
  -- Perturb the idx-th singular value by (1 ± delta)
  { base with
    singularValues := base.singularValues
    -- Placeholder: actual perturbation needs list update
    coherence := base.coherence  -- Would recompute from compression test
  }

/-! ## §3  Weight Search State and Iterator

  The search state tracks the current position in the 5D SVD subspace
  and the perturbation step size. AngrySphinx charges 2^depth per evaluation.
-/

structure WeightSearchState where
  /-- Current candidate being evaluated -/
  candidate : WeightCandidate
  /-- The ShortcutSearchState tracking the best shortcut -/
  shortcutState : ShortcutSearchState
  /-- Perturbation index (which dimension to perturb next) -/
  perturbIdx : Nat
  /-- Number of perturbation steps taken -/
  steps : Nat
  /-- Maximum steps before termination -/
  maxSteps : Nat
  deriving Repr, Inhabited

/-- Initialize weight search with initial SVD spectrum.
    conservativeStartups tracks the cmix measured spectrum:
    sigma_1 = 0.42, sigma_2 = 0.21, sigma_3 = 0.14, sigma_4 = 0.09, sigma_5 = 0.05.
    These are placeholders until real cmix weights are extracted. -/
def initWeightSearch (maxSteps : Nat) : WeightSearchState :=
  { candidate :=
      { activeValues := 5
        singularValues :=
          [ Q16_16.ofRawInt 27525   -- 0.42
          , Q16_16.ofRawInt 13763   -- 0.21
          , Q16_16.ofRawInt 9175    -- 0.14
          , Q16_16.ofRawInt 5898    -- 0.09
          , Q16_16.ofRawInt 3277    -- 0.05
          ]
        compressionRatio := Q16_16.ofRawInt 77591
        coherence := Q16_16.one
      }
    shortcutState := initSearch entropyFloor
    perturbIdx := 0
    steps := 0
    maxSteps := maxSteps
  }

/-- Check if the weight search has more candidates. -/
def weightHasNext (state : WeightSearchState) : Bool :=
  state.steps < state.maxSteps

/-- Evaluate a single weight candidate. -/
def weightEvaluateCandidate (state : WeightSearchState)
    (alpha beta epsilon : Q16_16) : WeightSearchState :=
  let eq : ManifoldEquation := weightToEquation state.candidate
  let newSS := evaluateCandidate state.shortcutState eq alpha beta epsilon
  { state with
    shortcutState := newSS
    steps := state.steps + 1
  }

/-! ## §4  Weight Search Loop

  Bounded by AngrySphinx: each weight candidate costs 2^depth.
  Terminates at NaN boundary or when maxSteps is reached. -/

def weightSearchLoop (state : WeightSearchState) (fuel : Nat)
    (maxDepth : Nat) (alpha beta epsilon : Q16_16)
    (delta : Q16_16) : WeightSearchState :=
  if fuel = 0 then state
  else if ¬ weightHasNext state then
    state
  else if ¬ canContinue state.shortcutState maxDepth then
    state
  else
    let nextState := weightEvaluateCandidate state alpha beta epsilon
    let perturbed := perturbCandidate nextState.candidate
      nextState.perturbIdx delta
    let nextPerturbIdx := (state.perturbIdx + 1) % state.candidate.activeValues
    weightSearchLoop
      { nextState with
        candidate := perturbed
        perturbIdx := nextPerturbIdx
      }
      (fuel - 1) maxDepth alpha beta epsilon delta

/-! ## §5  Result Extraction -/

structure WeightSearchResult where
  bestActiveValues : Nat
  bestSingularValues : List Q16_16
  bestLagrangian : Q16_16
  bestQuality : Q16_16
  totalSteps : Nat
  receipt : String
  deriving Repr

def weightExtractResult (state : WeightSearchState) (alpha beta : Q16_16)
    : WeightSearchResult :=
  let ss := state.shortcutState
  { bestActiveValues := state.candidate.activeValues
    bestSingularValues := state.candidate.singularValues
    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 -/

-- Default weight candidate (cmix measured spectrum)
#eval let eq := weightToEquation (initWeightSearch 100).candidate
  (eq.deltaCost, eq.spectralCost, eq.programCost, eq.coherence, eq.rank)

-- Lagrangian for default candidate with α=β=0.5
#eval let c := (initWeightSearch 100).candidate
  let L := lagrangian (weightToEquation c)
    (Q16_16.ofRawInt 32768) (Q16_16.ofRawInt 32768)
  L

-- Check isShortcut for default candidate
#eval let c := (initWeightSearch 100).candidate
  let eq := weightToEquation c
  let epsilon := Q16_16.ofRawInt 3277
  isShortcut eq (Q16_16.ofRawInt 32768) (Q16_16.ofRawInt 32768) epsilon

end SilverSight.PIST.WeightCandidateGen
