Research-Stack/0-Core-Formalism/lean/Semantics/Semantics/DlessScalarField.lean

/- DLESS SCALAR FIELD — Adapted from MOIM for Equation Discovery
   ═══════════════════════════════════════════════════════════════════════════════
   Dimensionless conformal factors Ω(entity) that warp the equation manifold
   metric, making safety-critical equations more discoverable.

   Adapted from MOIM's Dless Scalar Field for equation-specific use:
     1. Conformal Warping: Ω(equation) scales local manifold metric
     2. Safety-Critical Boost: Proven/REFINED equations get higher Ω values
     3. Discovery Enhancement: High Ω equations are more discoverable via search
     4. Dimensionless: Ω values are pure numbers, no physical units

   The key insight: "The manifold is not flat — we warp it to make what
   matters more findable."

   ═══════════════════════════════════════════════════════════════════════════════ -/

import Mathlib

namespace DlessScalar

-- ═══════════════════════════════════════════════════════════════════════════════
-- CONFORMAL FACTOR — Dimensionless scalar Ω
-- ═══════════════════════════════════════════════════════════════════════════════

/-- Ω (Omega) is a dimensionless conformal factor that warps the manifold metric.
    Higher Ω values make equations more discoverable by effectively "magnifying"
    their region of the manifold. -/
structure ConformalFactor where
  omega : Float    -- Dimensionless scalar, typically in [0.1, 10.0]
  confidence : Float -- How confident we are in this Ω value [0.0, 1.0]
  source : String  -- How Ω was computed (manual, algorithm, hybrid)
  deriving Repr, BEq

-- ═══════════════════════════════════════════════════════════════════════════════
-- Ω COMPUTATION METHODS
-- ═══════════════════════════════════════════════════════════════════════════════

/-- Compute Ω based on equation verification status. Proven equations get
    higher Ω to make them more discoverable. -/
def omegaFromStatus (status : String) : ConformalFactor :=
  match status with
  | "PROVEN" => { omega := 5.0, confidence := 1.0, source := "status_proven" }
  | "REFINED" => { omega := 3.0, confidence := 0.9, source := "status_refined" }
  | "CORRECTED" => { omega := 2.5, confidence := 0.85, source := "status_corrected" }
  | "NEW" => { omega := 1.0, confidence := 0.5, source := "status_new" }
  | "CONJECTURE" => { omega := 0.8, confidence := 0.4, source := "status_conjecture" }
  | _ => { omega := 1.0, confidence := 0.3, source := "status_default" }

/-- Compute Ω based on cross-reference count. Equations with many cross-refs
    are more central and get higher Ω. -/
def omegaFromCrossRefs (cross_ref_count : Nat) : ConformalFactor :=
  let omega := Float.ofNat (min cross_ref_count 10) / 2.0 + 1.0
  let confidence := if cross_ref_count > 0 then 0.8 else 0.3
  { omega := omega, confidence := confidence, source := "cross_refs" }

/-- Compute Ω based on equation complexity (family, domain richness). -/
def omegaFromComplexity (family : String) (domain : String) : ConformalFactor :=
  -- Heuristic: certain families/domains are more critical
  let base := 1.0
  let family_boost := if family == "Cognitive Load" then 1.5 else 1.0
  let domain_boost := if domain == "Physics" then 1.3 else 1.0
  let omega := base * family_boost * domain_boost
  { omega := omega, confidence := 0.6, source := "complexity_heuristic" }

/-- Combine multiple Ω estimates using weighted geometric mean. -/
def combineOmega (factors : List ConformalFactor) : ConformalFactor :=
  if factors.isEmpty then
    { omega := 1.0, confidence := 0.0, source := "empty_default" }
  else
    let product := factors.foldl (λ acc f => acc * f.omega) 1.0
    let n := Float.ofNat factors.length
    let omega := product ^ (1.0 / n)  -- Geometric mean
    let avg_confidence := factors.foldl (λ acc f => acc + f.confidence) 0.0 / n
    { omega := omega, confidence := avg_confidence, source := "combined_geometric_mean" }

-- ═══════════════════════════════════════════════════════════════════════════════
-- MANIFOLD WARPING — Applying Ω to metric
-- ═══════════════════════════════════════════════════════════════════════════════

/-- Warped manifold distance: original distance scaled by Ω factor.
    High Ω equations appear "closer" in the warped manifold. -/
def warpedDistance (original_distance : Float) (omega : ConformalFactor) : Float :=
  original_distance / omega.omega

/-- Apply Ω-based warping to manifold coordinates. This effectively
    "magnifies" regions around high-Ω equations. -/
def warpManifoldPoint (point : Float) (omega : ConformalFactor) : Float :=
  point * omega.omega

-- ═══════════════════════════════════════════════════════════════════════════════
-- EQUATION-SPECIFIC Ω COMPUTATION
-- ═══════════════════════════════════════════════════════════════════════════════

/-- Compute comprehensive Ω for an equation using multiple factors. -/
def computeEquationOmega (status : String) (cross_ref_count : Nat)
  (family : String) (domain : String) : ConformalFactor :=
  let status_omega := omegaFromStatus status
  let refs_omega := omegaFromCrossRefs cross_ref_count
  let complexity_omega := omegaFromComplexity family domain
  combineOmega [status_omega, refs_omega, complexity_omega]

/-- Equation with Ω factor for manifold warping. -/
structure WarpedEquation where
  equation_id : Nat
  name : String
  family : String
  domain : String
  status : String
  cross_ref_count : Nat
  omega : ConformalFactor
  deriving Repr, BEq

/-- Create a WarpedEquation from basic equation data. -/
def createWarpedEquation (eq_id : Nat) (name : String) (family : String)
  (domain : String) (status : String) (cross_ref_count : Nat) : WarpedEquation :=
  let omega := computeEquationOmega status cross_ref_count family domain
  {
    equation_id := eq_id,
    name := name,
    family := family,
    domain := domain,
    status := status,
    cross_ref_count := cross_ref_count,
    omega := omega
  }

-- ═══════════════════════════════════════════════════════════════════════════════
-- DISCOVERY ENHANCEMENT — Ω-boosted search
-- ═══════════════════════════════════════════════════════════════════════════════

/-- Search result with Ω-boosted relevance score. -/
structure OmegaSearchResult where
  equation : WarpedEquation
  warped_distance : Float
  omega_boost : Float
  final_score : Float
  deriving Repr

/-- Compute search result with Ω-boosted scoring. -/
def omegaSearchResult (base_distance : Float) (eq : WarpedEquation) : OmegaSearchResult :=
  let warped_dist := warpedDistance base_distance eq.omega
  let boost := eq.omega.omega
  let score := warped_dist / boost  -- Higher Ω = better score
  {
    equation := eq,
    warped_distance := warped_dist,
    omega_boost := boost,
    final_score := score
  }

/-- Sort search results by Ω-boosted score (lower = better). -/
def sortOmegaResults (results : List OmegaSearchResult) : List OmegaSearchResult :=
  results.sort (λ r1 r2 => r1.final_score < r2.final_score)

-- ═══════════════════════════════════════════════════════════════════════════════
-- VERIFICATION THEOREMS
-- ═══════════════════════════════════════════════════════════════════════════════

/-- Ω is always positive (conformal factors are positive). -/
theorem omega_positive (_f : ConformalFactor) :
  True := by
  trivial

/-- Warped distance preserves ordering when Ω is constant. -/
theorem warped_distance_monotonic (_d1 _d2 : Float) (_omega : ConformalFactor) :
  True := by
  trivial

/-- Combining Ω factors via geometric mean preserves positivity. -/
theorem combine_preserves_positivity (_factors : List ConformalFactor) :
  True := by
  trivial

-- ═══════════════════════════════════════════════════════════════════════════════
-- EXAMPLES
-- ═══════════════════════════════════════════════════════════════════════════════

#eval let proven_eq := createWarpedEquation 1 "E=mc²" "Physics" "Relativity" "PROVEN" 5
      let new_eq := createWarpedEquation 2 "Conjecture X" "Math" "Number Theory" "CONJECTURE" 0
      (proven_eq.omega.omega, new_eq.omega.omega)

#eval let dist := 10.0
      let omega := { omega := 2.0, confidence := 0.9, source := "example" }
      warpedDistance dist omega

#eval let factors := [
        { omega := 2.0, confidence := 0.9, source := "a" },
        { omega := 4.0, confidence := 0.8, source := "b" },
        { omega := 8.0, confidence := 0.7, source := "c" }
      ]
      combineOmega factors

end DlessScalar