import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.Order.Filter.Basic
import Mathlib.Topology.Basic
import Mathlib.Data.Real.Basic

noncomputable section

open Filter Topology

namespace QFD.Neutrino

/-!
# Appendix N.2 — Bleaching Limit (Formal Scaffold)

This file formalizes the core mathematical content of Appendix N.2 in an **abstract**
and **reusable** form.

Appendix N.2 asserts two independent facts about the "bleaching" family ψ ↦ λ • ψ:

1. **Energy collapses** under amplitude scaling as λ → 0.
2. **Topological charge** (winding) is invariant under any **nonzero** scaling (λ ≠ 0).

We package only the hypotheses required for these claims in `BleachingHypotheses`.
The resulting theorems contain **no QFD-specific PDE** assumptions; they are meant to be
instantiated later with the concrete QFD energy functional and winding definition.
-/

/-- The bleaching family: uniform amplitude scaling. -/
def bleach {Ψ : Type*} [SMul ℝ Ψ] (ψ : Ψ) (lam : ℝ) : Ψ :=
  lam • ψ

/-- Hypotheses required to formalize the bleaching argument in Appendix N.2.

* `Energy` is a real-valued functional on states.
* `QTop` is an integer-valued topological invariant (winding number).
* `energy_scale_sq` encodes quadratic energy scaling under amplitude rescaling.
* `qtop_invariant` encodes invariance of winding under any nonzero rescaling. -/
structure BleachingHypotheses (Ψ : Type*) [SMul ℝ Ψ] where
  Energy : Ψ → ℝ
  QTop : Ψ → ℤ
  energy_scale_sq : ∀ (ψ : Ψ) (lam : ℝ), Energy (bleach ψ lam) = (lam ^ 2) * Energy ψ
  qtop_invariant : ∀ (ψ : Ψ) (lam : ℝ), lam ≠ 0 → QTop (bleach ψ lam) = QTop ψ

namespace BleachingHypotheses

variable {Ψ : Type*} [SMul ℝ Ψ] (H : BleachingHypotheses Ψ)

/-- Energy vanishes in the bleaching limit lam → 0, assuming quadratic energy scaling. -/
theorem tendsto_energy_bleach_zero (ψ : Ψ) :
    Tendsto (fun lam : ℝ => H.Energy (bleach ψ lam)) (𝓝 0) (𝓝 0) := by
  -- Reduce to a pure real limit: (λ^2) * const → 0.
  have hid : Tendsto (fun x : ℝ => x) (𝓝 (0 : ℝ)) (𝓝 (0 : ℝ)) := tendsto_id
  have hpow2 : Tendsto (fun x : ℝ => x ^ 2) (𝓝 (0 : ℝ)) (𝓝 (0 : ℝ)) := by
    -- x^2 = x*x and 0*0 = 0
    simpa [pow_two] using (hid.mul hid)
  have hmul : Tendsto (fun x : ℝ => (x ^ 2) * H.Energy ψ) (𝓝 (0 : ℝ)) (𝓝 (0 : ℝ)) := by
    -- Multiply by a constant.
    simpa using (hpow2.mul tendsto_const_nhds)
  -- Rewrite Energy(λ•ψ) using the scaling law.
  convert hmul using 1
  funext lam
  simp only [bleach]
  exact H.energy_scale_sq ψ lam

/-- Topological charge is invariant under bleaching for any lam ≠ 0. -/
theorem qtop_bleach_eq (ψ : Ψ) {lam : ℝ} (hlam : lam ≠ 0) :
    H.QTop (bleach ψ lam) = H.QTop ψ := by
  simpa [bleach] using H.qtop_invariant ψ lam hlam

end BleachingHypotheses

end QFD.Neutrino
