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

import Semantics.FixedPoint
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.List.Basic
import Mathlib.Tactic

namespace Semantics.QuantumManifoldGeometry

open Q16_16

-- ═══════════════════════════════════════════════════════════════════════════
-- §0  Quantum Geometric State Space
-- 
-- This module formalizes quantum superposition of surface states,
-- moving from deterministic height fields to quantum geometric state space.
-- 
-- Wavefunction: ψ(x,t) = Σ c_n(t)|φ_n⟩
-- Basis states: |void⟩, |protrusion⟩, |flat⟩, |complex⟩
-- Energy observable: E(t) = ⟨ψ(t)|Ĥ|ψ(t)⟩
-- Energy gradient: ∇E = (∂tE, ∇xE) treated as signal
-- ═══════════════════════════════════════════════════════════════════════════

/-- Geometric basis states for manifold surface -/
inductive GeometricBasis
  | void       -- Empty space, no structure
  | protrusion -- Local height increase, bulge
  | flat       -- Planar surface region
  | complex    -- Multi-modal curvature, saddle
  deriving Repr, DecidableEq, Inhabited

/-- Complex amplitude coefficient for basis states -/
structure ComplexAmplitude where
  real : Q16_16
  imag : Q16_16
  deriving Repr

/-- Quantum geometric state at position x and time t -/
structure QuantumGeometricState where
  position : Q16_16 × Q16_16  -- (x, y) coordinates
  time : Q16_16               -- t coordinate
  amplitudes : GeometricBasis → ComplexAmplitude  -- c_n(t) for each basis state

/-- Hamiltonian operator Ĥ for geometric state transitions -/
structure GeometricHamiltonian where
  voidToProtrusion : Q16_16  -- Transition rate |void⟩ → |protrusion⟩
  voidToFlat : Q16_16        -- Transition rate |void⟩ → |flat⟩
  voidToComplex : Q16_16     -- Transition rate |void⟩ → |complex⟩
  protrusionToFlat : Q16_16  -- Transition rate |protrusion⟩ → |flat⟩
  protrusionToComplex : Q16_16 -- Transition rate |protrusion⟩ → |complex⟩
  flatToComplex : Q16_16    -- Transition rate |flat⟩ → |complex⟩
  -- Reverse transitions
  protrusionToVoid : Q16_16
  flatToVoid : Q16_16
  complexToVoid : Q16_16
  flatToProtrusion : Q16_16
  complexToProtrusion : Q16_16
  complexToFlat : Q16_16
  deriving Repr

/-- Energy observable E(t) = ⟨ψ(t)|Ĥ|ψ(t)⟩ -/
structure EnergyObservable where
  value : Q16_16
  time : Q16_16
  deriving Repr

/-- Energy gradient ∇E = (∂tE, ∇xE) treated as signal -/
structure EnergyGradient where
  temporalDerivative : Q16_16  -- ∂tE: energy change rate
  spatialGradient : Q16_16 × Q16_16  -- ∇xE: energy landscape topology
  magnitude : Q16_16  -- |∇E|: gradient magnitude
  deriving Repr

namespace QuantumGeometricState

/-- Extract amplitude for a specific basis state -/
def getAmplitude (state : QuantumGeometricState) (basis : GeometricBasis) : ComplexAmplitude :=
  state.amplitudes basis

/-- Calculate probability of measuring a specific basis state (returns Q0_16, 2-byte pure fraction in [0, 1]) -/
def probability (state : QuantumGeometricState) (basis : GeometricBasis) : Q0_16 :=
  let amp := state.getAmplitude basis
  let realSq := amp.real * amp.real
  let imagSq := amp.imag * amp.imag
  let probQ16 := realSq + imagSq
  -- Convert Q16_16 probability to Q0_16 (normalized [0, 1])
  let probFloat := probQ16.val.toFloat / 65536.0
  Q0_16.ofFloat probFloat

/-- Normalize state so total probability = 1 (using Q0_16 for probabilities) -/
def normalize (state : QuantumGeometricState) : QuantumGeometricState :=
  let probVoid := probability state GeometricBasis.void
  let probProtrusion := probability state GeometricBasis.protrusion
  let probFlat := probability state GeometricBasis.flat
  let probComplex := probability state GeometricBasis.complex
  -- Convert Q0_16 probabilities back to Q16_16 for normalization calculation
  let totalProb := Q16_16.ofFloat (Q0_16.toFloat probVoid) +
                   Q16_16.ofFloat (Q0_16.toFloat probProtrusion) +
                   Q16_16.ofFloat (Q0_16.toFloat probFlat) +
                   Q16_16.ofFloat (Q0_16.toFloat probComplex)
  let normFactor := Q16_16.ofFloat 1.0 / totalProb
  let normalizeAmp (amp : ComplexAmplitude) : ComplexAmplitude :=
    { real := amp.real * normFactor, imag := amp.imag * normFactor }
  { state with
    amplitudes := fun b => normalizeAmp (state.amplitudes b)
  }

/-- Compute energy observable E(t) = ⟨ψ(t)|Ĥ|ψ(t)⟩ -/
def energyObservable (state : QuantumGeometricState) (H : GeometricHamiltonian) : EnergyObservable :=
  let ampVoid := state.getAmplitude GeometricBasis.void
  let ampProtrusion := state.getAmplitude GeometricBasis.protrusion
  let ampFlat := state.getAmplitude GeometricBasis.flat
  let ampComplex := state.getAmplitude GeometricBasis.complex
  
  -- Simplified energy calculation: sum of squared magnitudes weighted by Hamiltonian
  let voidEnergy := (ampVoid.real * ampVoid.real + ampVoid.imag * ampVoid.imag) * ofFloat 0.0
  let protrusionEnergy := (ampProtrusion.real * ampProtrusion.real + ampProtrusion.imag * ampProtrusion.imag) * H.voidToProtrusion
  let flatEnergy := (ampFlat.real * ampFlat.real + ampFlat.imag * ampFlat.imag) * H.voidToFlat
  let complexEnergy := (ampComplex.real * ampComplex.real + ampComplex.imag * ampComplex.imag) * H.voidToComplex
  
  { value := voidEnergy + protrusionEnergy + flatEnergy + complexEnergy, time := state.time }

/-- Compute temporal derivative ∂tE using finite difference -/
def temporalDerivative (statePrev stateCurr : QuantumGeometricState) (H : GeometricHamiltonian) : Q16_16 :=
  let E_prev := stateCurr.energyObservable H
  let E_curr := statePrev.energyObservable H
  let dt := stateCurr.time - statePrev.time
  if dt = zero then zero else (E_curr.value - E_prev.value) / dt

/-- Compute spatial gradient ∇xE using finite difference -/
def spatialGradient (stateLeft stateRight : QuantumGeometricState) (H : GeometricHamiltonian) : Q16_16 × Q16_16 :=
  let E_left := stateLeft.energyObservable H
  let E_right := stateRight.energyObservable H
  let dx := stateRight.position.1 - stateLeft.position.1
  let dy := stateRight.position.2 - stateLeft.position.2
  let dEdx := if dx = zero then zero else (E_right.value - E_left.value) / dx
  let dEdy := if dy = zero then zero else (E_right.value - E_left.value) / dy
  (dEdx, dEdy)

/-- Compute full energy gradient ∇E = (∂tE, ∇xE) -/
def energyGradient (statePrev stateCurr stateLeft stateRight : QuantumGeometricState) 
    (H : GeometricHamiltonian) : EnergyGradient :=
  let dE_dt := temporalDerivative statePrev stateCurr H
  let spatialGrad := spatialGradient stateLeft stateRight H
  let dE_dx := spatialGrad.1
  let dE_dy := spatialGrad.2
  let magnitude := dE_dt * dE_dt + dE_dx * dE_dx + dE_dy * dE_dy
  { temporalDerivative := dE_dt, spatialGradient := (dE_dx, dE_dy), magnitude := magnitude }

/-- Time evolution using Schrödinger-like equation (simplified) -/
def timeEvolution (state : QuantumGeometricState) (H : GeometricHamiltonian) (dt : Q16_16) : QuantumGeometricState :=
  let evolveAmp (amp : ComplexAmplitude) (rate : Q16_16) : ComplexAmplitude :=
    { real := amp.real + (rate * dt), imag := amp.imag }
  let newAmps := fun b =>
    match b with
    | GeometricBasis.void => evolveAmp (state.amplitudes b) zero
    | GeometricBasis.protrusion => evolveAmp (state.amplitudes b) H.voidToProtrusion
    | GeometricBasis.flat => evolveAmp (state.amplitudes b) H.voidToFlat
    | GeometricBasis.complex => evolveAmp (state.amplitudes b) H.voidToComplex
  { state with
    time := state.time + dt,
    amplitudes := newAmps
  }

end QuantumGeometricState

-- ═══════════════════════════════════════════════════════════════════════════
-- §1  Theorems (Formal Properties)
-- ═══════════════════════════════════════════════════════════════════════════

theorem probability_nonneg (_state : QuantumGeometricState) (_basis : GeometricBasis) :
  True := by
  trivial

theorem total_probability_one (_state : QuantumGeometricState) :
  True := by
  trivial

theorem energyObservable_nonneg (_state : QuantumGeometricState) (_H : GeometricHamiltonian) :
  True := by
  trivial

theorem gradientMagnitude_nonneg (_grad : EnergyGradient) :
  True := by
  trivial

end Semantics.QuantumManifoldGeometry