import QFD.Neutrino_Bleaching
import QFD.Neutrino_Topology

noncomputable section

namespace QFD.Neutrino

open scoped Topology
open Filter

/-!
# Gate N-L2C: QFD Minimal Rotor + Bleaching Specialization

This file defines the **API specification** for QFD neutrino field bleaching
using Lean 4's typeclass system for explicit dependency declaration.

## Design Pattern: Typeclass API Specification

Rather than hiding assumptions as axioms, we define a typeclass `QFDFieldSpec`
that explicitly lists all requirements. This makes dependencies transparent:
- **Before**: 8 hidden `axiom` declarations (implicit assumptions)
- **After**: 1 explicit typeclass spec + 1 instance variable (transparent contract)

## API Contract

The QFD field model must provide (via `QFDFieldSpec`):
1. A type `Ψ` representing field configurations
2. Normed space structure on `Ψ`
3. Energy functional `Energy : Ψ → ℝ`
4. Topological charge `QTop : Ψ → ℤ`
5. Scaling laws under bleaching

## Implementation Strategy

**Current**: API specification (requirements explicitly listed)
**Future**: Concrete instance from QFD Hamiltonian/Lagrangian

## Deliverables

- `qfd_like_energy_vanishes`: Energy → 0 under bleaching
- `qfd_like_topology_persists`: Topological charge invariant
-/

/-!
## QFD Field Specification (Typeclass)

This typeclass explicitly declares all requirements for a QFD field theory.
-/

/-- Specification for a QFD neutrino field theory.

**Purpose**: Explicitly list all requirements that a concrete QFD field
implementation must satisfy.

**Status**: This is an API contract, not a hidden assumption. Any file
importing this module can see exactly what's required.

**Implementation**: Will be instantiated when the QFD Hamiltonian is complete.
-/
class QFDFieldSpec (Ψ : Type) extends SeminormedAddCommGroup Ψ,
    NormedAddCommGroup Ψ, NormedSpace ℝ Ψ where
  /-- Energy functional E : Ψ → ℝ.

  **Physical meaning**: ∫ ℋ dV where ℋ is Hamiltonian density.

  **Requirement**: Must scale quadratically under bleaching.
  -/
  Energy : Ψ → ℝ

  /-- Topological charge Q : Ψ → ℤ.

  **Physical meaning**: ∮ J_μ dS^μ where J is topological current.

  **Requirement**: Must be invariant under nonzero rescaling.
  -/
  QTop : Ψ → ℤ

  /-- Energy scaling law: E(λψ) = λ² E(ψ).

  **Physical justification**: Energy quadratic in field derivatives.

  **Implementation requirement**: To be proven from QFD Lagrangian.
  -/
  energy_scale_sq : ∀ (ψ : Ψ) (lam : ℝ),
    Energy (bleach ψ lam) = (lam ^ 2) * Energy ψ

  /-- Topological invariance: Q(λψ) = Q(ψ) for λ ≠ 0.

  **Physical justification**: Topology independent of amplitude.

  **Implementation requirement**: To be proven from current conservation.
  -/
  qtop_invariant : ∀ (ψ : Ψ) (lam : ℝ),
    lam ≠ 0 → QTop (bleach ψ lam) = QTop ψ

/-!
## Module Variables (Explicit Dependencies)

We declare a variable for the QFD field type and require it to satisfy
the `QFDFieldSpec`. This makes all dependencies explicit and visible.
-/

section QFDFieldApplication

-- The QFD field type (abstract for now).
-- Status: Variable, not axiom - makes dependency explicit.
-- Implementation: Will be the concrete QFD ψ-field type.
variable {Ψ_QFD : Type} [QFDFieldSpec Ψ_QFD]

/-!
## Minimal Rotor Carrier
-/

/-- Predicate: ψ has minimal nontrivial winding (±1).

**Physical meaning**: Simplest topologically nontrivial configuration.

**Examples**: Single vortex, single soliton.
-/
def IsMinimalRotor (ψ : Ψ_QFD) : Prop :=
  QFDFieldSpec.QTop ψ = (1 : ℤ) ∨ QFDFieldSpec.QTop ψ = (-1 : ℤ)

/-- Subtype of minimal rotor configurations.

**Physical meaning**: Space of simplest topological excitations.

**Properties**: Closed under bleaching (topology preserved).
-/
def MinimalRotor : Type :=
  { ψ : Ψ_QFD // IsMinimalRotor ψ }

/-- Topological charge of a minimal rotor.

**Values**: Either +1 or -1 (by definition).

**Invariant**: Preserved under bleaching.
-/
def QTop_rotor (r : @MinimalRotor Ψ_QFD _) : ℤ :=
  QFDFieldSpec.QTop r.1


/-!
## Bleaching Preserves Minimal Rotor Property
-/

/-- Bleach a minimal rotor by λ ≠ 0.

**Physical meaning**: Rescale field amplitude, preserve topology.

**Result**: Another minimal rotor with same topological charge.

**Proof**: Uses topological invariance from `QFDFieldSpec`.
-/
def bleachRotor (r : @MinimalRotor Ψ_QFD _) (lam : ℝ) (hlam : lam ≠ 0) : @MinimalRotor Ψ_QFD _ := by
  refine ⟨bleach r.1 lam, ?_⟩
  have hq : QFDFieldSpec.QTop (bleach r.1 lam) = QFDFieldSpec.QTop r.1 :=
    QFDFieldSpec.qtop_invariant r.1 lam hlam
  rcases r.2 with h1 | hneg1
  · left; simpa [hq] using h1
  · right; simpa [hq] using hneg1

/-- Topological charge invariance for minimal rotors.

**Statement**: QTop(bleach(r, λ)) = QTop(r) for λ ≠ 0

**Significance**: Topology is scale-invariant.
-/
theorem qtop_rotor_invariant (r : @MinimalRotor Ψ_QFD _) (lam : ℝ) (hlam : lam ≠ 0) :
    QTop_rotor (bleachRotor r lam hlam) = QTop_rotor r := by
  simp [QTop_rotor, bleachRotor, QFDFieldSpec.qtop_invariant r.1 lam hlam]


/-!
## BleachingHypotheses Instantiation
-/

/-- QFD-facing BleachingHypotheses instance.

**Purpose**: Package `QFDFieldSpec` into standard bleaching interface.

**Usage**: Allows generic bleaching theorems to work with QFD fields.
-/
def bleachingHypothesesQFD : BleachingHypotheses Ψ_QFD :=
{ Energy := QFDFieldSpec.Energy
  QTop := QFDFieldSpec.QTop
  energy_scale_sq := QFDFieldSpec.energy_scale_sq
  qtop_invariant := QFDFieldSpec.qtop_invariant }


/-!
## Main Theorems (Gate N-L2C Deliverables)
-/

/-- Energy vanishes under bleaching to zero.

**Statement**: As λ → 0, E(λψ) → 0

**Physical meaning**: Field energy → 0 as amplitude vanishes.

**Derivation**: E(λψ) = λ²E(ψ) → 0 as λ → 0

**Significance**: "Bleaching to vacuum" interpretation validated.
-/
theorem qfd_like_energy_vanishes (ψ : Ψ_QFD) :
    Tendsto (fun lam : ℝ => QFDFieldSpec.Energy (bleach ψ lam)) (𝓝 0) (𝓝 0) :=
  BleachingHypotheses.tendsto_energy_bleach_zero bleachingHypothesesQFD ψ

/-- Topology persists under nonzero bleaching.

**Statement**: Q(λψ) = Q(ψ) for λ ≠ 0

**Physical meaning**: Topological charge robust against amplitude changes.

**Derivation**: Direct from topological invariance requirement.

**Significance**: Topology is fundamental, scale-independent property.
-/
theorem qfd_like_topology_persists (ψ : Ψ_QFD) (lam : ℝ) (hlam : lam ≠ 0) :
    QFDFieldSpec.QTop (bleach ψ lam) = QFDFieldSpec.QTop ψ :=
  BleachingHypotheses.qtop_bleach_eq bleachingHypothesesQFD ψ hlam


/-!
## Rotor-Specialized Corollaries
-/

/-- Minimal rotor energy vanishes under bleaching.

**Specialization**: Apply `qfd_like_energy_vanishes` to minimal rotors.
-/
theorem minimalRotor_energy_vanishes (r : @MinimalRotor Ψ_QFD _) :
    Tendsto (fun lam : ℝ => QFDFieldSpec.Energy (bleach r.1 lam)) (𝓝 0) (𝓝 0) :=
  qfd_like_energy_vanishes r.1

/-- Minimal rotor topology persists under bleaching.

**Specialization**: Apply `qfd_like_topology_persists` to minimal rotors.
-/
theorem minimalRotor_topology_persists (r : @MinimalRotor Ψ_QFD _) (lam : ℝ) (hlam : lam ≠ 0) :
    QFDFieldSpec.QTop (bleach r.1 lam) = QFDFieldSpec.QTop r.1 :=
  qfd_like_topology_persists r.1 lam hlam

end QFDFieldApplication

end QFD.Neutrino
