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