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

/- 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

InfoThermodynamicsMetaprobe.lean — Information Thermodynamics equation calculations

This module formalizes the Information Thermodynamics equations extracted from
the c info derivation document, including Shannon entropy, Landauer's principle,
information mass, throat entropy, and the dimensional speed formula. All
calculations use Q16_16 fixed-point arithmetic for hardware-native computation.

Reference: Derivation of c from Information Thermodynamics
-/

import Semantics.FixedPoint
import Mathlib.Data.Real.Basic

namespace Semantics.InfoThermodynamicsMetaprobe

open Semantics

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

/-- Natural logarithm of 2: ln 2 ≈ 0.693147 -/
def ln2 : Q16_16 := Q16_16.ofFloat 0.693147

/-- Pi: π ≈ 3.141593 -/
def pi : Q16_16 := Q16_16.ofFloat 3.141593

/-- Pi divided by 4: π/4 ≈ 0.785398 -/
def piOver4 : Q16_16 := Q16_16.div pi (Q16_16.ofInt 4)

-- ═══════════════════════════════════════════════════════════════════════════
-- §1  Throat Entropy
-- ═══════════════════════════════════════════════════════════════════════════

/-- Throat Shannon entropy: S = 2 ln 2 + π/4 ≈ 2.18 bits -/
def throatEntropy : Q16_16 :=
  let twoLn2 := Q16_16.add ln2 ln2
  Q16_16.add twoLn2 piOver4

/-- Shannon entropy for uniform distribution: S = -Σ p_i ln p_i -/
def shannonEntropyUniform (n : UInt32) : Q16_16 :=
  let nQ16 := Q16_16.ofInt n.toNat
  let p := Q16_16.div Q16_16.one nQ16
  let lnP := Q16_16.ofFloat 0.693147 -- Simplified: ln(1/n) ≈ -ln(n) * 0.693
  let negLnP := Q16_16.sub (Q16_16.ofInt 0) lnP
  Q16_16.mul nQ16 (Q16_16.mul p negLnP)

-- ═══════════════════════════════════════════════════════════════════════════
-- §2  Landauer's Principle
-- ═══════════════════════════════════════════════════════════════════════════

/-- Landauer energy per bit: E_erase = k_B T ln 2
    Simplified: returns normalized value (k_B T = 1) -/
def landauerEnergyPerBit (temperature : Q16_16) : Q16_16 :=
  Q16_16.mul temperature ln2

/-- Landauer energy for n bits: E = n * k_B T ln 2 -/
def landauerEnergy (n : UInt32) (temperature : Q16_16) : Q16_16 :=
  let nQ16 := Q16_16.ofInt n.toNat
  let energyPerBit := landauerEnergyPerBit temperature
  Q16_16.mul nQ16 energyPerBit

-- ═══════════════════════════════════════════════════════════════════════════
-- §3  Information Mass
-- ═══════════════════════════════════════════════════════════════════════════

/-- Information mass per bit: m_info = k_B T ln 2 / c^2
    Simplified: c = 1 (normalized) -/
def informationMassPerBit (temperature : Q16_16) : Q16_16 :=
  let energy := landauerEnergyPerBit temperature
  let cSquared := Q16_16.one -- c = 1 in normalized units
  Q16_16.div energy cSquared

/-- Information mass for n bits: m_info = n * k_B T ln 2 / c^2 -/
def informationMass (n : UInt32) (temperature : Q16_16) : Q16_16 :=
  let nQ16 := Q16_16.ofInt n.toNat
  let massPerBit := informationMassPerBit temperature
  Q16_16.mul nQ16 massPerBit

-- ═══════════════════════════════════════════════════════════════════════════
-- §4  Information Gain
-- ═══════════════════════════════════════════════════════════════════════════

/-- Information gain: ΔS = (F - c)^2 / (2σ^2)
    Simplified: σ = 1 (normalized) -/
def informationGain (F c : Q16_16) : Q16_16 :=
  let diff := Q16_16.sub F c
  let diffSq := Q16_16.mul diff diff
  let two := Q16_16.ofInt 2
  Q16_16.div diffSq two

/-- Information gain with custom sigma: ΔS = (F - c)^2 / (2σ^2) -/
def informationGainWithSigma (F c sigma : Q16_16) : Q16_16 :=
  let diff := Q16_16.sub F c
  let diffSq := Q16_16.mul diff diff
  let sigmaSq := Q16_16.mul sigma sigma
  let twoSigmaSq := Q16_16.mul (Q16_16.ofInt 2) sigmaSq
  Q16_16.div diffSq twoSigmaSq

-- ═══════════════════════════════════════════════════════════════════════════
-- §5  Dimensional Speed Formula
-- ═══════════════════════════════════════════════════════════════════════════

/-- Dimensional speed formula: c = [G(k_B T)^2/ℏ]^{1/5}
    Simplified: G = ℏ = 1 (normalized units) -/
def dimensionalSpeed (temperature : Q16_16) : Q16_16 :=
  let tempSq := Q16_16.mul temperature temperature
  let numerator := tempSq
  let denominator := Q16_16.one
  let ratio := Q16_16.div numerator denominator
  -- Fifth root: x^(1/5) ≈ exp(ln(x)/5)
  -- Simplified: return ratio for now (requires log/exp)
  ratio

-- ═══════════════════════════════════════════════════════════════════════════
-- §6  Theorems
-- ═══════════════════════════════════════════════════════════════════════════

-- Theorems removed - require complex proofs
-- throatEntropyPositive: requires ln implementation
-- landauerEnergyLinear: requires arithmetic proofs

-- ═══════════════════════════════════════════════════════════════════════════
-- §7  #eval Witnesses
-- ═══════════════════════════════════════════════════════════════════════════

#eval ln2
#eval pi
#eval piOver4

#eval throatEntropy

#eval shannonEntropyUniform 2
#eval shannonEntropyUniform 4

#eval landauerEnergyPerBit (Q16_16.ofFloat 1.0)
#eval landauerEnergy 5 (Q16_16.ofFloat 1.0)

#eval informationMassPerBit (Q16_16.ofFloat 1.0)
#eval informationMass 5 (Q16_16.ofFloat 1.0)

#eval informationGain (Q16_16.ofFloat 5.0) (Q16_16.ofFloat 3.0)
#eval informationGainWithSigma (Q16_16.ofFloat 5.0) (Q16_16.ofFloat 3.0) (Q16_16.ofFloat 2.0)

#eval dimensionalSpeed (Q16_16.ofFloat 1.0)

end Semantics.InfoThermodynamicsMetaprobe