import QFD.Neutrino_Bleaching
import Mathlib.Data.Real.Basic
import Mathlib.Order.Filter.Basic
import Mathlib.Topology.Basic

noncomputable section

open Filter Topology

namespace QFD.Neutrino

/-!
# Appendix N.2 — Toy Topology Instantiation (Gate N-L2B)

This file **instantiates** the abstract scaffold in `QFD.Neutrino_Bleaching` with a
concrete (toy) state space.

Purpose:
- Demonstrate that the `BleachingHypotheses` structure is satisfiable.
- Provide a working pattern for the *real* QFD instantiation (Energy functional + winding number).

Toy choices:
- State space Ψ := ℝ
- Energy(x) := x²  (quadratic by construction)
- QTop(x) := 0     (trivial topology; replaced later by a winding number)

This completes the Lean plumbing needed for Theorem N.1 at the "mechanism" level:
Energy collapses under bleaching, while QTop is invariant for λ ≠ 0.
-/

abbrev Ψtoy : Type := ℝ

def Energy_toy (x : Ψtoy) : ℝ := x ^ 2

def QTop_toy (_x : Ψtoy) : ℤ := 0

lemma energy_scale_sq_toy (x : Ψtoy) (lam : ℝ) :
    Energy_toy (bleach x lam) = (lam ^ 2) * Energy_toy x := by
  -- bleach x lam = lam * x for Ψtoy = ℝ
  -- (lam*x)^2 = lam^2 * x^2
  simp [bleach, Energy_toy, pow_two, mul_left_comm, mul_comm]

lemma qtop_invariant_toy (x : Ψtoy) (lam : ℝ) (_hlam : lam ≠ 0) :
    QTop_toy (bleach x lam) = QTop_toy x := by
  -- Trivial by definition.
  simp [QTop_toy]

/-- The toy instantiation of `BleachingHypotheses`. -/
def ToyBleachingHypotheses : BleachingHypotheses Ψtoy where
  Energy := Energy_toy
  QTop := QTop_toy
  energy_scale_sq := by
    intro x lam
    simpa using energy_scale_sq_toy x lam
  qtop_invariant := by
    intro x lam hlam
    simpa using qtop_invariant_toy x lam hlam

/-- Corollary: In the toy model, energy vanishes as lam → 0 under bleaching. -/
theorem tendsto_energy_bleach_zero_toy (x : Ψtoy) :
    Tendsto (fun lam : ℝ => (ToyBleachingHypotheses).Energy (bleach x lam)) (𝓝 0) (𝓝 0) :=
  (BleachingHypotheses.tendsto_energy_bleach_zero (H := ToyBleachingHypotheses) x)

/-- Corollary: In the toy model, QTop is invariant under any nonzero scaling. -/
theorem qtop_bleach_eq_toy (x : Ψtoy) {lam : ℝ} (hlam : lam ≠ 0) :
    (ToyBleachingHypotheses).QTop (bleach x lam) = (ToyBleachingHypotheses).QTop x :=
  (BleachingHypotheses.qtop_bleach_eq (H := ToyBleachingHypotheses) x hlam)

end QFD.Neutrino
