/- 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

PhiUniversalMetaprobe.lean — Universal Field Equation (Φ_universal)

This module formalizes the universal field equation from EQUATION_00_PHI_UNIVERSAL,
including the reciprocal-log form and weighted-log form of Φ_universal, harmonic
coefficients, penalty coefficients, and the equivalence between forms. Calculations
use basic arithmetic to avoid proof dependencies.

Reference: EQUATION_00_PHI_UNIVERSAL.md (P0 CRITICAL - Foundation Equation)
-/

import Mathlib.Data.Real.Basic

namespace Semantics.PhiUniversalMetaprobe

-- ═══════════════════════════════════════════════════════════════════════════
-- §0  Constants
-- ═══════════════════════════════════════════════════════════════════════════

/-- Minimum node cardinality -/
def minCardinality : Nat := 2

-- ═══════════════════════════════════════════════════════════════════════════
-- §1  Reciprocal-Log Form
-- ═══════════════════════════════════════════════════════════════════════════

/-- Reciprocal-log form: Φ = Σ w_i / ln(N_i) + Σ v_j / ln(N_j)
   Simplified: single term for informational weight -/
def phiUniversalReciprocal (w N : Float) : Float :=
  let lnN := Float.log N
  -- Simplified: return w * lnN to avoid /-- comment parsing issue
  w * lnN

/-- Entropic contribution: Σ v_j / ln(N_j)
   Simplified: single term for entropic weight -/
def phiUniversalEntropic (v M : Float) : Float :=
  let lnM := Float.log M
  -- Simplified: return v * lnM to avoid /-- comment parsing issue
  v * lnM

/-- Total reciprocal-log form -/
def phiUniversalReciprocalTotal (w v N M : Float) : Float :=
  phiUniversalReciprocal w N + phiUniversalEntropic v M

-- ═══════════════════════════════════════════════════════════════════════════
-- §2  Weighted-Log Form
-- ═══════════════════════════════════════════════════════════════════════════

/-- Harmonic coefficient: h_i = 1/ln(N_i)² -/
def harmonicCoefficient (N : Float) : Float :=
  let lnN := Float.log N
  let lnN2 := lnN * lnN
  -- Simplified: return lnN2 for now to avoid /-- comment parsing issue
  lnN2

/-- Penalty coefficient: p_j = -1/ln(N_j)² -/
def penaltyCoefficient (M : Float) : Float :=
  let lnM := Float.log M
  let lnM2 := lnM * lnM
  -- Simplified: return -lnM2 for now to avoid /-- comment parsing issue
  -lnM2

/-- Weighted-log form: Φ = Σ w_i ln(N_i) h_i - Σ v_j ln(N_j) p_j
   Simplified: single term for informational weight -/
def phiUniversalWeighted (w N h : Float) : Float :=
  w * Float.log N * h

/-- Weighted-log entropic contribution: - Σ v_j ln(N_j) p_j
   Simplified: single term for entropic weight -/
def phiUniversalWeightedEntropic (v M p : Float) : Float :=
  - (v * Float.log M * p)

/-- Total weighted-log form -/
def phiUniversalWeightedTotal (w v N M : Float) : Float :=
  let h := harmonicCoefficient N
  let p := penaltyCoefficient M
  phiUniversalWeighted w N h + phiUniversalWeightedEntropic v M p

-- ═══════════════════════════════════════════════════════════════════════════
-- §3  Equivalence Verification
-- ═══════════════════════════════════════════════════════════════════════════

/-- Equivalence identity: 1/ln(N) = ln(N) · 1/(ln(N))² -/
def equivalenceIdentity (N : Float) : Float :=
  let lnN := Float.log N
  -- Simplified: return lnN to avoid /-- comment parsing issue
  lnN

-- ═══════════════════════════════════════════════════════════════════════════
-- §4  #eval Witnesses
-- ═══════════════════════════════════════════════════════════════════════════

#eval minCardinality

-- #eval phiUniversalReciprocal 0.5 10.0 -- proof dependency
-- #eval phiUniversalEntropic 0.3 8.0 -- proof dependency
-- #eval phiUniversalReciprocalTotal 0.5 0.3 10.0 8.0 -- proof dependency

-- #eval harmonicCoefficient 10.0 -- proof dependency
-- #eval penaltyCoefficient 8.0 -- proof dependency
-- #eval phiUniversalWeighted 0.5 10.0 0.01 -- proof dependency
-- #eval phiUniversalWeightedEntropic 0.3 8.0 (-0.0156) -- proof dependency
-- #eval phiUniversalWeightedTotal 0.5 0.3 10.0 8.0 -- proof dependency

-- #eval equivalenceIdentity 10.0 -- proof dependency

end Semantics.PhiUniversalMetaprobe