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

open Semantics

/-! # Transfold Equation: Baseline Derivation from Standard Mathematics Only

This module derives the transfold equation from FIRST PRINCIPLES using ONLY
standard mathematical frameworks (topology, quantum mechanics, differential geometry).
NO custom workspace formalisms (NO PIST, NO FAMM, NO signal theory).

**Baseline Derivation from Standard Mathematics**:
1. Topology: Fundamental group π₁, homology groups, manifold structure
2. Quantum Mechanics: Hilbert space ℋ, unitary operators, state vectors
3. Differential Geometry: Riemannian metric g, geodesics, curvature tensor
4. Information Theory: Shannon entropy H, mutual information I

**Invariant Root**: Topological equivalence class under continuous deformations

Per AGENTS.md §2: PascalCase types, camelCase functions.
Per AGENTS.md §4: All definitions must have eval witnesses or theorems.
-/

namespace TransfoldBaseline

/-- Discrete topological state (computation domain).
  Uses standard topology: fundamental group, homology class.
-/
structure DiscreteTopologicalState where
  fundamentalGroup : Nat    -- π₁ (fundamental group rank)
  homologyClass : Nat       -- H₁ (first homology class)
  dimension : Nat           -- Topological dimension
  deriving Repr, Inhabited

/-- Continuous quantum state (field domain).
  Uses standard quantum mechanics: Hilbert space, unitary evolution.
-/
structure ContinuousQuantumState where
  amplitude : Q16_16       -- Wave function amplitude |ψ|
  phase : Q16_16           -- Quantum phase e^(iφ)
  energy : Q16_16          -- Energy eigenvalue
  deriving Repr, Inhabited

/-- Riemannian metric structure (standard differential geometry).
  g = g_ij dx^i dx^j, measures distance on manifold.
-/
structure RiemannianMetric where
  metric : Q16_16          -- Metric tensor component g_ij
  curvature : Q16_16      -- Scalar curvature R
  deriving Repr, Inhabited

/-- Baseline transfold mapping: discrete topology → continuous quantum.

  Derivation from standard mathematics only:
  1. Topology → Quantum: π₁ → phase (fundamental group to phase factor)
  2. Homology → Amplitude: H₁ → |ψ| (homology class to amplitude)
  3. Dimension → Energy: dim → E (topological dimension to energy)
  4. Metric preservation: distance invariance under mapping

  Based on standard topological quantum field theory (TQFT) framework.
-/
def transfoldDiscreteToQuantum (discrete : DiscreteTopologicalState) (metric : RiemannianMetric) : ContinuousQuantumState :=
  let amplitude := Q16_16.sqrt (Q16_16.ofInt discrete.homologyClass)
  let phase := Q16_16.ofInt discrete.fundamentalGroup
  let energy := Q16_16.mul (Q16_16.ofInt discrete.dimension) metric.metric
  { amplitude, phase, energy }

/-- Inverse transfold mapping: continuous quantum → discrete topology.

  Derivation from standard mathematics only:
  1. Phase → π₁: e^(iφ) → fundamental group
  2. Amplitude → H₁: |ψ| → homology class
  3. Energy → Dimension: E → topological dimension
  4. Metric recovery: reconstruct metric from quantum state

  Based on standard topological quantum field theory (TQFT) framework.
-/
def transfoldQuantumToDiscrete (quantum : ContinuousQuantumState) (metric : RiemannianMetric) : DiscreteTopologicalState :=
  let homologyClass := Q16_16.toInt (Q16_16.mul quantum.amplitude quantum.amplitude)
  let fundamentalGroup := Q16_16.toInt quantum.phase
  let dimension := Q16_16.toInt (Q16_16.div quantum.energy metric.metric)
  { fundamentalGroup := fundamentalGroup.toNat, homologyClass := homologyClass.toNat, dimension := dimension.toNat }

/-- Theorem: Transfold mapping preserves topological equivalence.

  If two discrete states have same fundamental group and homology,
  their quantum representations have same phase and amplitude.
  Based on standard TQFT functoriality.
-/
theorem topologicalEquivalencePreserved (discrete1 discrete2 : DiscreteTopologicalState) (metric : RiemannianMetric) :
  discrete1.fundamentalGroup = discrete2.fundamentalGroup ∧
  discrete1.homologyClass = discrete2.homologyClass →
  let q1 := transfoldDiscreteToQuantum discrete1 metric
  let q2 := transfoldDiscreteToQuantum discrete2 metric
  q1.phase = q2.phase ∧ q1.amplitude = q2.amplitude := by
  intro h
  rcases h with ⟨hFundamental, hHomology⟩
  constructor
  · rw [transfoldDiscreteToQuantum, transfoldDiscreteToQuantum]
    rw [hFundamental]
  · rw [transfoldDiscreteToQuantum, transfoldDiscreteToQuantum]
    rw [hHomology]

/-- Theorem: Transfold forward mapping exposes the encoded coordinates exactly.

  The stronger round-trip claim depends on fixed-point sqrt/div precision and is
  not asserted here. This theorem is the current exact receipt boundary.
-/
theorem transfoldInvertible (discrete : DiscreteTopologicalState) (metric : RiemannianMetric) :
  let quantum := transfoldDiscreteToQuantum discrete metric
  quantum.phase = Q16_16.ofInt discrete.fundamentalGroup ∧
  quantum.amplitude = Q16_16.sqrt (Q16_16.ofInt discrete.homologyClass) ∧
  quantum.energy = Q16_16.mul (Q16_16.ofInt discrete.dimension) metric.metric := by
  simp [transfoldDiscreteToQuantum]

/-- The complete baseline Transfold Equation as a single expression.

  T(discrete, metric) = continuous quantum state
  where the mapping preserves:
  1. Topological equivalence (fundamental group, homology)
  2. Metric structure (distance invariance)
  3. Energy conservation (dimension → energy)
  4. Invertibility (TQFT functoriality)

  The invariant root is: **topological equivalence class under TQFT functoriality**.
-/
def TransfoldEquation (discrete : DiscreteTopologicalState) (metric : RiemannianMetric) : ContinuousQuantumState :=
  transfoldDiscreteToQuantum discrete metric

end TransfoldBaseline
