import Mathlib.Topology.Homotopy.Basic
import Mathlib.Geometry.Euclidean.Sphere.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
import QFD.GA.Cl33

/-!
# The Topological Protection Theorem (Why Matter is Stable)

**Status**: Core theorem proven (0 sorries)
**Axiom Status**: 3 axioms - standard algebraic topology results not yet in Mathlib4

## Physical Mechanism

Standard Model: Lepton number is conserved by fiat.

QFD: An electron is a topological defect (winding number 1) in the vacuum field.
The conservation law follows from π₃(S³) ≅ ℤ: winding numbers are homotopy invariants.
Continuous time evolution cannot change an integer winding number.

## Mathematical Foundation

The proof relies on three standard results from algebraic topology:
1. Maps S³ → S³ have an integer-valued degree (winding number)
2. Homotopic maps have equal degree (homotopy invariance)
3. Constant maps have degree 0 (vacuum state)

These are classical theorems but not yet formalized in Mathlib4.
See AXIOM_INVENTORY.md for elimination strategy (Mathlib singular homology).

-/

noncomputable section

namespace QFD.Lepton.Topology

open ContinuousMap

/-- The 3-sphere S³ ⊂ ℝ⁴ (local definition to avoid import cycle). -/
abbrev Sphere3 : Type := Metric.sphere (0 : EuclideanSpace ℝ (Fin 4)) 1

/-- The rotor group (SU(2) ≅ S³) for internal rotations (local definition). -/
abbrev RotorGroup : Type := Sphere3

/-- Minimal physics model structure for topological theorems. -/
structure TopologyModel where
  /-- Winding number / degree of field configuration. -/
  winding_number : C(Sphere3, RotorGroup) → ℤ
  /-- Homotopic maps have equal degree. -/
  homotopy_invariance : ∀ f g : C(Sphere3, RotorGroup),
    ContinuousMap.Homotopic f g → winding_number f = winding_number g
  /-- Constant map has degree 0. -/
  vacuum_degree : ∀ x : Sphere3, winding_number (ContinuousMap.const Sphere3 x) = 0

variable (P : TopologyModel)

/-!
## Algebraic Topology Axioms

The following three axioms encode standard results from algebraic topology
that are not yet formalized in Mathlib4:

1. **Degree map existence**: π₃(S³) ≅ ℤ (Hurewicz theorem)
2. **Homotopy invariance**: Degree is a homotopy invariant (fundamental in topology)
3. **Vacuum normalization**: Constant map has degree 0 (definition of degree)

**Mathlib Status**: Singular homology is formalized (Topaz, 2023), which provides
the mathematical foundation for degree theory. However, the explicit degree map
and homotopy invariance theorem are not yet available in Mathlib4.

**Elimination Path**: Once Mathlib4 includes degree theory for sphere maps,
these axioms can be replaced with `import Mathlib.AlgebraicTopology.DegreeTheory`.
-/

/-- The degree (winding number) of a map S³ → S³ is an integer.
    Standard result: This is the induced homomorphism on π₃(S³) ≅ ℤ. -/
def winding_number : C(Sphere3, RotorGroup) → ℤ :=
  P.winding_number

/-- Homotopic maps have equal degree (fundamental homotopy invariance).
    Standard result: Degree factors through homotopy classes [S³, S³] ≅ ℤ. -/
theorem degree_homotopy_invariant {f g : C(Sphere3, RotorGroup)} :
    ContinuousMap.Homotopic f g → winding_number P f = winding_number P g :=
  P.homotopy_invariance f g

-----------------------------------------------------------
-- Physical Definitions
-----------------------------------------------------------

/--
**Time Evolution is a Homotopy**
If a field evolves continuously from time t=0 to t=1 without tearing (amplitude singularity),
the function $F(x, t)$ defines a homotopy between State(0) and State(1).
-/
def ContinuousEvolution
  (initial_state final_state : C(Sphere3, RotorGroup)) : Prop :=
  ContinuousMap.Homotopic initial_state final_state

/--
**Stability Condition**
A state is stable if its winding number is non-zero.
(It is distinct from the vacuum, which has winding 0).
-/
def IsStableParticle (psi : C(Sphere3, RotorGroup)) : Prop :=
  winding_number P psi ≠ 0

/-- The trivial vacuum state has winding number 0.
    Standard result: Constant maps have degree 0 by definition. -/
theorem vacuum_winding :
    ∃ (vac : C(Sphere3, RotorGroup)), winding_number P vac = 0 := by
  -- The sphere S³ is nonempty (it contains the unit vector e₀)
  let e0 : EuclideanSpace ℝ (Fin 4) := EuclideanSpace.single 0 1
  have h_e0_mem : e0 ∈ Metric.sphere (0 : EuclideanSpace ℝ (Fin 4)) 1 := by
    rw [Metric.mem_sphere, dist_eq_norm, sub_zero, EuclideanSpace.norm_single, norm_one]
  let pt : Sphere3 := ⟨e0, h_e0_mem⟩
  exact ⟨ContinuousMap.const Sphere3 pt, P.vacuum_degree pt⟩

-----------------------------------------------------------
-- The Theorem
-----------------------------------------------------------

/--
Topological protection:
any configuration with nonzero winding number cannot continuously evolve into
the vacuum map supplied by the postulates. This is the Lean formalization of
the physical "matter cannot unwind" statement.
-/
theorem topological_protection
    {ψ vac : C(Sphere3, RotorGroup)}
    (h_stable : IsStableParticle P ψ)
    (h_vacuum : winding_number P vac = 0)
    (h_evolution : ContinuousEvolution ψ vac) :
    False := by
  have h_equal :
      winding_number P ψ = winding_number P vac :=
    degree_homotopy_invariant P
      (by simpa [ContinuousEvolution] using h_evolution)
  have hψ_zero : winding_number P ψ = 0 := h_equal.trans h_vacuum
  exact h_stable hψ_zero

/--
Convenient corollary: exhibit the canonical vacuum witness from the postulates
and record that no stable particle can homotope into it.
-/
theorem no_decay_into_vacuum
    {ψ : C(Sphere3, RotorGroup)}
    (h_stable : IsStableParticle P ψ) :
    ∀ {vac : C(Sphere3, RotorGroup)},
      winding_number P vac = 0 →
      ¬ ContinuousEvolution ψ vac := by
  intro vac hvac hpath
  exact topological_protection P h_stable hvac hpath

-----------------------------------------------------------
-- Significance Discussion
-----------------------------------------------------------

/-!
## Physical Implications

Lepton number conservation emerges from topology:

1. **Stability mechanism**: No continuous path connects winding-1 (electron) to winding-0 (vacuum)
   without passing through a field singularity.

2. **Energy barrier**: Creating a singularity (A → 0) requires concentrating energy density
   beyond typical interaction scales, providing kinetic stability over cosmological timescales.

3. **Discrete spectrum**: Winding numbers are integers, giving a discrete particle spectrum
   rather than a continuum.

This explains why matter is stable without invoking ad hoc conservation laws.
-/

end QFD.Lepton.Topology

end
