/-
DrexlerianMechanosynthesis.lean — Atomic Building: STM, Tunneling, Morse Potential

This module formalizes the mathematical models governing atomically precise
manufacturing, from Drexler's 1986 theory to the 2026 experimental realization.

The three core models:
  1. Tersoff-Hamann tunneling current (positioning)
  2. Morse potential (bond energy landscape)
  3. Bell-Evans-Polanyi principle (force-modified reaction rates)

Key insight: These same structures appear at every scale:
  • Atomic: STM mechanosynthesis
  • Molecular: polymer mechanochemistry
  • Nuclear: alpha decay (Gamow factor)
  • Cosmic: false vacuum decay (instantons)

References:
  - Drexler, K.E. (1986) "Engines of Creation"
  - arXiv:2605.27250 — Atomically precise mechanosynthesis (2026)
  - Tersoff & Hamann (1985) — STM tunneling theory
  - Morse, P.M. (1929) — Diatomic potential
  - Bell, G.I. (1978) — Models for elastically forced bonds

Part of the OTOM TreeDIAT/PIST family.
-/

import Semantics.FixedPoint
import Semantics.PIST.Spectral
import Semantics.Q16_16Numerics

namespace Semantics.DrexlerianMechanosynthesis

open Semantics.Q16_16
open Semantics.Q16_16Numerics
open Semantics.PIST.Spectral

-- ═══════════════════════════════════════════════════════════════════════════
-- §1  TERSOFF-HAMANN TUNNELING CURRENT (positioning model)
-- ═══════════════════════════════════════════════════════════════════════════

/-- The tunneling current between STM tip and sample.
    I ∝ V · ρ_s(E_F) · e^(-2κz)
    
    Where:
      V = bias voltage
      ρ_s(E_F) = local density of states at Fermi level
      z = tip-sample distance
      κ = decay constant = √(2mφ)/ℏ -/
structure TunnelingCurrent where
  V       : Q16_16  -- bias voltage (V)
  rho_s   : Q16_16  -- local density of states (states/eV)
  z       : Q16_16  -- tip-sample distance (Å)
  kappa   : Q16_16  -- decay constant (Å⁻¹)
  deriving Repr

/-- Compute tunneling current from parameters.
    I = V · ρ_s · exp(-2κz)
    Uses rigorous Q16_16Numerics.exp for the exponential. -/
def computeTunnelingCurrent (tc : TunnelingCurrent) : Q16_16 :=
  -- I = V · ρ_s · exp(-2κz)
  let exponent := Q16_16.neg (Q16_16.mul (Q16_16.ofRawInt 131072) (Q16_16.mul tc.kappa tc.z))
  let tunneling_factor := exp exponent
  Q16_16.mul (Q16_16.mul tc.V tc.rho_s) tunneling_factor

/-- The decay constant κ = √(2mφ)/ℏ.
    For typical work functions (φ ≈ 4-5 eV): κ ≈ 1 Å⁻¹ -/
def computeDecayConstant (work_function : Q16_16) : Q16_16 :=
  -- κ = √(2mφ)/ℏ ≈ 0.512 √(φ [eV]) Å⁻¹
  let sqrt_phi := Q16_16.sqrt work_function
  Q16_16.mul (Q16_16.ofRawInt 33554) sqrt_phi  -- 0.512 in Q16_16

-- ═══════════════════════════════════════════════════════════════════════════
-- §2  MORSE POTENTIAL (bond energy landscape)
-- ═══════════════════════════════════════════════════════════════════════════

/-- The Morse potential for diatomic interaction.
    V(r) = D_e [(1 - e^(-a(r-r_e)))² - 1]
    
    Where:
      D_e = well depth (bond dissociation energy)
      r_e = equilibrium bond distance
      a = width parameter (related to force constant) -/
structure MorsePotential where
  D_e : Q16_16  -- dissociation energy (eV)
  r_e : Q16_16  -- equilibrium distance (Å)
  a   : Q16_16  -- width parameter (Å⁻¹)
  deriving Repr

/-- Evaluate Morse potential at distance r.
    V(r) = D_e [(1 - exp(-a(r-r_e)))² - 1]
    Uses rigorous Q16_16Numerics.exp for the exponential. -/
def morseEvaluate (mp : MorsePotential) (r : Q16_16) : Q16_16 :=
  -- V(r) = D_e [(1 - exp(-a(r-r_e)))² - 1]
  let dr := Q16_16.sub r mp.r_e
  let exp_term := exp (Q16_16.neg (Q16_16.mul mp.a dr))
  let one_minus := Q16_16.sub Q16_16.one exp_term
  let squared := Q16_16.mul one_minus one_minus
  Q16_16.mul mp.D_e (Q16_16.sub squared Q16_16.one)

/-- The mechanical force from Morse potential: F = -dV/dr.
    F = -2 D_e a (1 - exp(-a(r-r_e))) exp(-a(r-r_e))
    Uses rigorous Q16_16Numerics.exp for the exponential. -/
def morseForce (mp : MorsePotential) (r : Q16_16) : Q16_16 :=
  -- F = -2 D_e a (1 - exp(-a(r-r_e))) exp(-a(r-r_e))
  let dr := Q16_16.sub r mp.r_e
  let exp_term := exp (Q16_16.neg (Q16_16.mul mp.a dr))
  let one_minus := Q16_16.sub Q16_16.one exp_term
  let neg_two_da := Q16_16.mul (Q16_16.ofRawInt (-131072)) (Q16_16.mul mp.D_e mp.a)
  Q16_16.mul (Q16_16.mul neg_two_da one_minus) exp_term

-- ═══════════════════════════════════════════════════════════════════════════
-- §3  BELL-EVANS-POLANYI PRINCIPLE (force-modified reaction rates)
-- ═══════════════════════════════════════════════════════════════════════════

/-- The force-modified Arrhenius equation.
    k(F) = A · exp(-(E_a - F·Δx‡) / k_B T)
    
    Where:
      A = pre-exponential factor
      E_a = activation energy barrier
      F = applied mechanical force
      Δx‡ = activation length (distance to transition state)
      k_B T = thermal energy -/
structure BellEvansPolanyi where
  A      : Q16_16  -- pre-exponential factor (s⁻¹)
  E_a    : Q16_16  -- activation energy (eV)
  delta_x : Q16_16  -- activation length (Å)
  kT     : Q16_16  -- thermal energy k_B·T (eV)
  deriving Repr

/-- Compute reaction rate under applied force.
    Simplified: k(F) ≈ A · (1 - (E_a - F·Δx‡)/k_B T) for small barriers -/
def computeReactionRate (bep : BellEvansPolanyi) (F : Q16_16) : Q16_16 :=
  -- k(F) ≈ A · (1 - (E_a - F·Δx‡)/k_B T)
  let effective_barrier := Q16_16.sub bep.E_a (Q16_16.mul F bep.delta_x)
  let reduction := Q16_16.div effective_barrier bep.kT
  Q16_16.mul bep.A (Q16_16.sub Q16_16.one reduction)

/-- The critical force where barrier vanishes: F_crit = E_a / Δx‡. -/
def criticalForce (bep : BellEvansPolanyi) : Q16_16 :=
  Q16_16.div bep.E_a bep.delta_x

-- ═══════════════════════════════════════════════════════════════════════════
-- §4  CROSS-SCALE INVARIANCE (same math at every scale)
-- ═══════════════════════════════════════════════════════════════════════════

/-- Physical scale identifiers. -/
inductive PhysicalScale where
  | atomic      -- STM mechanosynthesis (Å scale)
  | molecular   -- polymer mechanics (nm scale)
  | nuclear     -- alpha decay, quark confinement (fm scale)
  | cosmic      -- false vacuum decay, QGP (Mpc scale)
  deriving Repr, DecidableEq

/-- The universal barrier-crossing structure.
    Same math, different physical meaning at each scale. -/
structure UniversalBarrierCrossing where
  scale         : PhysicalScale
  barrier_height : Q16_16  -- E_a or equivalent (in scale-appropriate units)
  tunneling_rate : Q16_16  -- exponential decay rate
  force_coupling : Q16_16  -- how force modifies barrier
  deriving Repr

/-- Map atomic-scale parameters to nuclear scale (alpha decay).
    The Gamow factor is the nuclear analog of STM tunneling. -/
def atomicToNuclear (atomic : UniversalBarrierCrossing) : UniversalBarrierCrossing :=
  { scale := .nuclear
  , barrier_height := Q16_16.mul atomic.barrier_height (Q16_16.ofRawInt 655360)  -- ~10 MeV scale
  , tunneling_rate := Q16_16.div atomic.tunneling_rate (Q16_16.ofRawInt 65536)   -- narrower barrier
  , force_coupling := Q16_16.mul atomic.force_coupling (Q16_16.ofRawInt 655360) }

-- ═══════════════════════════════════════════════════════════════════════════
-- §5  CARBON DIMER ASSEMBLY (specific to arXiv:2605.27250)
-- ═══════════════════════════════════════════════════════════════════════════

/-- The C₂ dimer parameters for Si(100) mechanosynthesis. -/
def c2_dimer : MorsePotential :=
  { D_e := Q16_16.ofRawInt 409600  -- ~6.3 eV (C-C bond)
  , r_e := Q16_16.ofRawInt 78643   -- ~1.20 Å (C≡C triple bond)
  , a   := Q16_16.ofRawInt 196608 } -- ~3.0 Å⁻¹ (stiff bond)

/-- The Si-C bond parameters. -/
def si_c_bond : MorsePotential :=
  { D_e := Q16_16.ofRawInt 327680  -- ~5.0 eV (Si-C bond)
  , r_e := Q16_16.ofRawInt 104858  -- ~1.60 Å
  , a   := Q16_16.ofRawInt 131072 } -- ~2.0 Å⁻¹

-- ═══════════════════════════════════════════════════════════════════════════
-- §6  EXECUTABLE WITNESSES
-- ═══════════════════════════════════════════════════════════════════════════

-- Tunneling current at z = 5 Å, φ = 4.5 eV
def testTunnel : TunnelingCurrent :=
  { V := Q16_16.one, rho_s := Q16_16.ofRawInt 32768
  , z := Q16_16.ofRawInt 327680, kappa := Q16_16.ofRawInt 65536 }
#eval computeTunnelingCurrent testTunnel  -- expect: ~exp(-10) ≈ very small

-- Morse potential at equilibrium
#eval morseEvaluate c2_dimer c2_dimer.r_e  -- expect: 0 (at equilibrium)

-- Morse force at equilibrium
#eval morseForce c2_dimer c2_dimer.r_e  -- expect: 0 (no force at equilibrium)

-- Critical force for C-C bond breaking
def testBEP : BellEvansPolanyi :=
  { A := Q16_16.ofRawInt 655360, E_a := Q16_16.ofRawInt 409600
  , delta_x := Q16_16.ofRawInt 6553, kT := Q16_16.ofRawInt 2556 }
#eval criticalForce testBEP  -- expect: ~6.3 eV / 0.1 Å = 63 nN

end Semantics.DrexlerianMechanosynthesis
