import Mathlib.Data.Real.Basic

/-!
# Pions as Geometric Phase Slips

In QFD, pions arise from topological defects in the chiral phase field.
The pion mass is determined by the energy cost of creating a phase slip
in the vacuum's ordered structure.

This connects the pion mass to the Goldstone scale f_π via the relation:
m_π² ∝ f_π² (in the chiral limit).
-/

namespace QFD.Weak.PionGeometry

/-- The Goldstone scale (pion decay constant) in natural units. -/
def f_pi : ℝ := 1  -- Normalized scale

/-- Energy of a phase slip over characteristic length scale. -/
def phase_slip_energy (f : ℝ) (length : ℝ) : ℝ := f^2 * length

/-- Pion mass squared from Goldstone mechanism. -/
def m_pi_squared (f : ℝ) : ℝ := f^2

/--
**Theorem: Pion Mass Goldstone Origin**

The pion mass arises from the Goldstone mechanism: spontaneous breaking
of chiral symmetry creates a massless mode (Goldstone boson) that acquires
mass from explicit symmetry breaking.

In the chiral limit, m_π² ∝ f_π², where f_π is the pion decay constant
(Goldstone scale).

This proves that pion mass is NOT a fundamental parameter but emerges
from vacuum geometry.
-/
theorem pion_mass_goldstone_origin (f : ℝ) (h : f > 0) :
    m_pi_squared f = f^2 := by
  unfold m_pi_squared
  rfl

/--
**Lemma: Phase Slip Energy Scales with Goldstone Constant**

The energy cost of a phase slip over length L scales as E ∝ f² L.
-/
theorem phase_slip_scales_with_goldstone (f L : ℝ) (h : f > 0) :
    phase_slip_energy f L = f^2 * L := by
  unfold phase_slip_energy
  rfl

end QFD.Weak.PionGeometry