import Mathlib
import QFD.Hydrogen.PhotonSoliton

set_option autoImplicit false

namespace QFD

/-!
  # QFD Photon Sector: Topological Protection

  This module formalizes the "Topological Protection Hypothesis" (Issue #2 Resolution).

  It upgrades the `ShapeInvariant` predicate from a dynamical accident to a
  topological necessity.

  Physics Theory:
  - The Vacuum has degenerate ground states (phases).
  - A Soliton is a "domain wall" or "kink" connecting different vacua.
  - This connection is measured by an integer Topological Charge (Q).
  - Since Q is discrete, it cannot change continuously.
  - Therefore, the Soliton cannot spread (disperse) or decay.
-/

universe u
variable {Point : Type u}

/--
  Extension of the Kinematic Model to include Topology.
  We extend the base QFDModel to add topological invariants.
-/
structure QFDModelTopological (Point : Type u) extends QFDModel Point where

  /--
    The Topological Charge Operator (Q).
    Physically: The winding number ∫ ∇ψ · dS or kink charge ∫ dφ.
    Mathematically: Maps a configuration to an integer.
  -/
  Q : Config Point → ℤ

  /--
    Time Evolution Operator.
    (Required to state conservation laws).
  -/
  Evolve : ℝ → Config Point → Config Point

  -- === AXIOMS OF TOPOLOGY ===

  /--
    Axiom 1: Conservation of Topological Charge.
    The winding number cannot change under continuous time evolution.
    This is the definition of a topological invariant.
  -/
  conservation_of_Q :
    ∀ (c : Config Point) (t : ℝ), Q (Evolve t c) = Q c

  /--
    Axiom 2: Topological Protection.
    If a configuration has a non-zero topological charge, it is ShapeInvariant.

    Why? Because "spreading" or "dissipating" would require the configuration
    to continuously deform into the vacuum (Q=0). But Q cannot jump from
    ±1 to 0 continuously. Therefore, the shape is locked.
  -/
  protection_implies_invariant :
    ∀ (c : Config Point), (Q c ≠ 0) → ShapeInvariant c

  /--
    Axiom 3: The Photon is Topological.
    Photons are not just waves; they are defects with Q = ±1.
    (Corresponds to Right/Left Circular Polarization).
  -/
  photon_is_topological :
    ∀ (γ : Photon), ∃ (c : Config Point), (Q c = 1 ∨ Q c = -1)

namespace QFDModelTopological

variable {M : QFDModelTopological Point}

/-! ## The Zero Dispersion Proof -/

/--
  Definition: Zero Dispersion.
  A particle has zero dispersion if its shape remains invariant
  over time (no spreading width).
-/
def HasZeroDispersion (c : Config Point) : Prop :=
  M.ShapeInvariant c

/--
  Theorem: Topological Stability (The "Kink" Theorem).

  IF a configuration has non-zero topological charge (like a photon),
  THEN it has exactly zero dispersion.

  This replaces the "stiffness suppression" argument (ξ ~ e^-β)
  with an exact topological lock (ξ = 0).
-/
theorem zero_dispersion_of_topology
  (c : Config Point)
  (h_charged : M.Q c ≠ 0) :
  HasZeroDispersion c := by
  -- The proof follows directly from the Protection Axiom.
  -- In a full PDE model, this would involve showing the Soliton
  -- lies in a distinct homotopy class from the vacuum.
  apply M.protection_implies_invariant
  exact h_charged

/-! ## Application to Photons -/

/--
  Theorem: Photons are Non-Dispersive.

  Combines the physical postulate (Photons have Q=±1) with the
  mathematical theorem (Q≠0 → Stable).
-/
theorem photon_stability_theorem (γ : Photon) :
  ∃ (c : Config Point), (M.Q c ≠ 0) ∧ HasZeroDispersion c := by
  -- 1. Get the configuration corresponding to the photon
  obtain ⟨c, h_charge_val⟩ := M.photon_is_topological γ

  -- 2. Show that Q = ±1 implies Q ≠ 0
  have h_nonzero : M.Q c ≠ 0 := by
    cases h_charge_val with
    | inl h_pos => rw [h_pos]; decide -- 1 ≠ 0
    | inr h_neg => rw [h_neg]; decide -- -1 ≠ 0

  -- 3. Apply the Topological Stability Theorem
  have h_stable := zero_dispersion_of_topology c h_nonzero

  -- 4. Conclude existence
  use c
  exact ⟨h_nonzero, h_stable⟩

/-! ## Corollaries for Experimental Physics -/

/--
  Corollary: No "Half-Photons".
  Topological charge is an integer. You cannot have a photon with Q = 0.5.
  This explains why charge is quantized.
-/
theorem charge_quantization (c : Config Point) :
  ∃ (n : ℤ), M.Q c = n := by
  use (M.Q c) -- Trivial in this model, but foundational physically.

/--
  Corollary: Stability of Information.
  Because the shape is invariant, the spectral width (Δk) is constant.
  This ensures that atomic spectral lines observed from 13 billion light years
  away are as sharp as those in the lab.
-/
theorem spectral_sharpness_preserved
  (c : Config Point) (t : ℝ) (hQ : M.Q c ≠ 0) :
  M.ShapeInvariant (M.Evolve t c) := by
  -- 1. Evolve the particle
  let c_t := M.Evolve t c
  -- 2. Charge is conserved
  have hQ_t : M.Q c_t = M.Q c := M.conservation_of_Q c t
  -- 3. Since original Q ≠ 0, evolved Q ≠ 0
  rw [hQ_t] at *
  -- 4. Apply protection axiom
  exact M.protection_implies_invariant c_t hQ

end QFDModelTopological
end QFD