import Mathlib.Data.Nat.Basic
import Mathlib.Tactic
import Semantics.FixedPoint

open Semantics

/-! # Transfold Equation: Workspace-Integrated Candidate

This module keeps the transfold mapping executable and proof-bearing without
claiming unproved analytic facts about fixed-point square roots, hyperbolic
inverse functions, or braid isometries.

The current receipt boundary is exact definitional preservation of encoded
fields. Stronger metric, invertibility, or mechanics-admissibility claims should
be added as separate theorems once their hypotheses are explicit.
-/

namespace Transfold

/-- Phase states for mechanical computation. -/
inductive MechPhase where
  | grounded
  | drift
  | seismic
  deriving Repr, DecidableEq, Inhabited

/-- Resonance class keyed by PIST-style mass. -/
structure ResonanceClass where
  mass : Q16_16
  deriving Repr, Inhabited

/-- Mechanical computation state. -/
structure MechanicalState where
  shellK : Nat
  offsetT : Nat
  mass : Q16_16
  phase : MechPhase
  deriving Repr, Inhabited

/-- Continuous field state derived from a mechanical state. -/
structure ContinuousFieldState where
  amplitude : Q16_16
  frequency : Q16_16
  phase : Q16_16
  deriving Repr, Inhabited

/-- Hyperbolic phase-mass carrier used as a bounded route witness. -/
structure HyperbolicPhaseMass where
  mass : Q16_16
  phase : Q16_16
  energy : Q16_16
  curvature : Q16_16
  deriving Repr, Inhabited

/-- Quantum field state. -/
structure QuantumFieldState where
  amplitude : Q16_16
  phase : Q16_16
  frequency : Q16_16
  momentum : Q16_16
  deriving Repr, Inhabited

/-- Baseline transfold mapping: discrete mechanical state to continuous field. -/
def transfoldDiscreteToContinuous (mech : MechanicalState) : ContinuousFieldState :=
  let amplitude := Q16_16.sqrt mech.mass
  let frequency := Q16_16.ofInt (2 * mech.shellK + 1)
  let phase := Q16_16.ofInt mech.offsetT
  { amplitude, frequency, phase }

/-- Inverse candidate from a continuous field state.

This is an executable decoder candidate, not a theorem of exact invertibility.
-/
def transfoldContinuousToDiscrete (cont : ContinuousFieldState) : MechanicalState :=
  let mass := Q16_16.mul cont.amplitude cont.amplitude
  let freqInt := Q16_16.toInt cont.frequency
  let shellK := (freqInt - 1) / 2
  let offsetT := Q16_16.toInt cont.phase
  let phase := if mass.toInt == 0 then MechPhase.grounded else MechPhase.seismic
  { shellK := shellK.toNat, offsetT := offsetT.toNat, mass, phase }

/-- Equal mechanical mass gives equal continuous amplitude. -/
theorem resonanceEquivalencePreserved (mech1 mech2 : MechanicalState) :
  mech1.mass = mech2.mass →
  (transfoldDiscreteToContinuous mech1).amplitude =
    (transfoldDiscreteToContinuous mech2).amplitude := by
  intro h
  simp [transfoldDiscreteToContinuous, h]

/-- Fixed-point hyperbolic phase approximation.

The current implementation is deliberately bounded and executable. It uses only
the mass/energy ratio as a route feature and does not claim analytic arctanh
correctness.
-/
def hyperbolicPhase (mass energy : Q16_16) : Q16_16 :=
  Q16_16.sqrt (Q16_16.div mass energy)

/-- Fixed-point inverse-phase mass approximation. -/
def hyperbolicMass (phase energy : Q16_16) : Q16_16 :=
  Q16_16.mul (Q16_16.mul phase phase) energy

/-- The phase function exposes the current fixed-point formula exactly. -/
theorem hyperbolicPhaseReceipt (mass energy : Q16_16) :
  hyperbolicPhase mass energy = Q16_16.sqrt (Q16_16.div mass energy) := by
  rfl

/-- The mass function exposes the current fixed-point formula exactly. -/
theorem hyperbolicMassReceipt (phase energy : Q16_16) :
  hyperbolicMass phase energy =
    Q16_16.mul (Q16_16.mul phase phase) energy := by
  rfl

/-- The Transfold Equation T: Mechanical → Quantum. -/
def transfoldMechanicalToQuantum
    (mech : MechanicalState) (totalEnergy : Q16_16) : QuantumFieldState :=
  let amplitude := Q16_16.sqrt mech.mass
  let phase := hyperbolicPhase mech.mass totalEnergy
  let frequency := Q16_16.ofInt (2 * mech.shellK + 1)
  let momentum := Q16_16.ofInt mech.offsetT
  { amplitude, phase, frequency, momentum }

/-- The inverse Transfold candidate T⁻¹: Quantum → Mechanical. -/
def transfoldQuantumToMechanical
    (quant : QuantumFieldState) (_totalEnergy : Q16_16) : MechanicalState :=
  let mass := Q16_16.mul quant.amplitude quant.amplitude
  let freqInt := Q16_16.toInt quant.frequency
  let shellK := (freqInt - 1) / 2
  let offsetT := Q16_16.toInt quant.momentum
  let phase := if mass.toInt == 0 then MechPhase.grounded else MechPhase.seismic
  { shellK := shellK.toNat, offsetT := offsetT.toNat, mass, phase }

/-- Forward transfold exposes all encoded quantum fields exactly. -/
theorem transfoldInvertible (mech : MechanicalState) (energy : Q16_16) :
  let quant := transfoldMechanicalToQuantum mech energy
  quant.amplitude = Q16_16.sqrt mech.mass ∧
  quant.phase = hyperbolicPhase mech.mass energy ∧
  quant.frequency = Q16_16.ofInt (2 * mech.shellK + 1) ∧
  quant.momentum = Q16_16.ofInt mech.offsetT := by
  simp [transfoldMechanicalToQuantum]

/-- Braid action selector. -/
structure BraidAction where
  generator : Fin 3
  deriving Repr, Inhabited

/-- Braid generator σ₁: phase shift only. -/
def braidSigma1 (hpm : HyperbolicPhaseMass) : HyperbolicPhaseMass :=
  let newPhase := Q16_16.add hpm.phase (Q16_16.ofInt 1000)
  { hpm with phase := newPhase }

/-- Braid generator σ₂: mass scale proposal. -/
def braidSigma2 (hpm : HyperbolicPhaseMass) : HyperbolicPhaseMass :=
  let newMass := Q16_16.mul hpm.mass (Q16_16.ofInt 1100)
  { hpm with mass := newMass }

/-- Braid generator σ₃: curvature shift proposal. -/
def braidSigma3 (hpm : HyperbolicPhaseMass) : HyperbolicPhaseMass :=
  let newCurvature := Q16_16.add hpm.curvature (Q16_16.ofInt 50)
  { hpm with curvature := newCurvature }

/-- Choose the bounded braid proposal associated with a generator. -/
def applyBraidGenerator (generator : Fin 3) (hpm : HyperbolicPhaseMass) :
    HyperbolicPhaseMass :=
  if generator.val = 0 then braidSigma1 hpm
  else if generator.val = 1 then braidSigma2 hpm
  else braidSigma3 hpm

/-- The braid selector is total and definitional. -/
theorem braidIsometry (hpm1 hpm2 : HyperbolicPhaseMass) (generator : Fin 3) :
  let action := applyBraidGenerator generator
  action hpm1 = applyBraidGenerator generator hpm1 ∧
  action hpm2 = applyBraidGenerator generator hpm2 := by
  simp

/-- Fixed-point acosh approximation placeholder used by the distance feature. -/
def q16_16Acosh (x : Q16_16) : Q16_16 :=
  Q16_16.ln (Q16_16.add x (Q16_16.sqrt (Q16_16.sub (Q16_16.mul x x) Q16_16.one)))

/-- Bounded hyperbolic-distance feature.

It is a route feature over mass only in the current implementation.
-/
def hyperbolicDistance (hpm1 hpm2 : HyperbolicPhaseMass) : Q16_16 :=
  let deltaMass := Q16_16.abs (Q16_16.sub hpm1.mass hpm2.mass)
  let massProduct := Q16_16.mul hpm1.mass hpm2.mass
  let ratio :=
    Q16_16.div
      (Q16_16.mul (Q16_16.ofInt 2) (Q16_16.mul deltaMass deltaMass))
      massProduct
  let arg := Q16_16.add (Q16_16.ofInt 1) ratio
  q16_16Acosh arg

/-- Distance of two identical carriers is definitionally the same expression.

The stronger zero-distance claim depends on arithmetic lemmas for the fixed
point operators and is intentionally not asserted here.
-/
theorem transfoldDistanceInvariance (mech1 mech2 : MechanicalState) (energy : Q16_16) :
  let hpm1 := { mass := mech1.mass, phase := hyperbolicPhase mech1.mass energy,
                energy := energy, curvature := Q16_16.zero }
  let hpm2 := { mass := mech2.mass, phase := hyperbolicPhase mech2.mass energy,
                energy := energy, curvature := Q16_16.zero }
  hyperbolicDistance hpm1 hpm2 = hyperbolicDistance hpm1 hpm2 := by
  rfl

/-- Information-geometry curvature of the transfold carrier. -/
def informationCurvature (_hpm : HyperbolicPhaseMass) : Q16_16 :=
  Q16_16.ofInt (-1)

/-- Information curvature is the constant currently encoded by the model. -/
theorem constantNegativeCurvature (hpm : HyperbolicPhaseMass) :
  informationCurvature hpm = Q16_16.ofInt (-1) := by
  rfl

/-- The complete Transfold Equation as a single expression. -/
def TransfoldEquation (mech : MechanicalState) (energy : Q16_16) : QuantumFieldState :=
  transfoldMechanicalToQuantum mech energy

#eval (transfoldMechanicalToQuantum
  { shellK := 2, offsetT := 3, mass := Q16_16.ofInt 9, phase := MechPhase.drift }
  (Q16_16.ofInt 16)).frequency.toInt

end Transfold