-- TODO: Replace with specific imports
import Mathlib
import QFD.Hydrogen.PhotonSolitonStable

set_option autoImplicit false

namespace QFD

open QFDModelStable

universe u
variable {Point : Type u}

/-!
  # Emergent Constants Layer (v2 - Scaling Bridge)

  This module formalizes the "Scaling Bridge" between Vacuum Geometry and Quantum Constants.
  It acts as a compatibility layer that ensures the "Fundamental" constants of the base
  QFDModel (like ℏ) are consistent with the geometric product of vacuum scales.

  ## Status: Scaling Bridge (Not Ab Initio Derivation)

  This formalization does NOT claim to derive ℏ from first principles. Instead, it defines
  the constraint: ℏ ≡ Γ_vortex * λ_mass * L_zero * cVac

  This allows us to:
  1. Input a geometric shape factor (Γ) from numerical integration
  2. Input a mass scale (λ)
  3. Derive the required Vacuum Interaction Length (L₀)

  **Future Goal**: Deriving λ_mass or L₀ independently (e.g., as minimizer of vacuum
  energy functional) will close the loop and make this a true ab initio prediction.

  ## The Architecture
  1. **Scaling Bridge**: Defines Γ, λ, L₀ as explicit fields
  2. **Compatibility Axiom**: Enforces ℏ = Γ · λ · L₀ · c
  3. **Inheritance Theorems**: Proves kinematics (p=ℏk) respect this decomposition
-/

/--
  EmergentConstants extends the Stable Model with geometric scale factors.
-/
structure EmergentConstants (Point : Type u) extends QFDModelStable Point where
  /-- The Mass Scale of the Vacuum (λ). Physically ~1 AMU. -/
  lam_mass : ℝ

  /-- The Fundamental Length Scale (L₀). The "grid size" of the vacuum (~0.125 fm). -/
  L_zero : ℝ

  /-- The Dimensionless Geometric Shape Factor (Γ).
      Computed numerically as ≈ 1.6919 for the stable Hill Vortex. -/
  Gamma_vortex : ℝ

  -- Non-degeneracy guards (Essential for division/inversion lemmas)
  h_mass_pos : 0 < lam_mass
  h_len_pos  : 0 < L_zero
  h_shape_pos: 0 < Gamma_vortex
  h_cvac_pos : 0 < cVac

  /-- Compatibility Axiom: The base model's ℏ must equal the geometric product. -/
  h_hbar_match : ℏ = Gamma_vortex * lam_mass * L_zero * cVac

namespace EmergentConstants

variable {M : EmergentConstants Point}

/--
  Definitional shorthand for the geometric product.
  Using a `def` here prevents `dsimp` issues common with structure fields.
-/
def hbar_val (M : EmergentConstants Point) : ℝ :=
  M.Gamma_vortex * M.lam_mass * M.L_zero * M.cVac

/-! ## Inheritance Theorems -/

/--
  Theorem: Photon Momentum Inheritance.
  Proves that p = ℏk expands correctly to p = (Γ · λ · L₀ · c) * k.
-/
theorem photon_momentum_inheritance (γ : PhotonWave) :
  PhotonWave.momentum (M := M.toQFDModelStable) γ =
    (M.Gamma_vortex * M.lam_mass * M.L_zero * M.cVac) * γ.k := by

  -- Unfold definition and substitute the bridge axiom
  simp [PhotonWave.momentum, M.h_hbar_match]

/--
  Theorem: Photon Energy Inheritance.
  Proves that E = ℏω expands correctly to E = (Γ · λ · L₀ · c) * ω.
-/
theorem photon_energy_inheritance (γ : PhotonWave) :
  PhotonWave.energy (M := M.toQFDModelStable) γ =
    (M.Gamma_vortex * M.lam_mass * M.L_zero * M.cVac) * γ.ω := by

  simp [PhotonWave.energy, M.h_hbar_match]

/--
  Theorem: Massless Consistency Preserved.
  Even with emergent constants decomposition, E = p*c holds.
-/
theorem emergent_massless_consistency
  (γ : PhotonWave)
  (hDisp : PhotonWave.MasslessDispersion (M := M.toQFDModelStable) γ) :
  PhotonWave.energy (M := M.toQFDModelStable) γ =
  M.cVac * PhotonWave.momentum (M := M.toQFDModelStable) γ := by

  -- Prove using the emergent definitions to ensure no circularity
  rw [photon_energy_inheritance, photon_momentum_inheritance]
  unfold PhotonWave.MasslessDispersion at hDisp
  rw [hDisp]
  ring

/--
  Theorem: Vacuum Scale Inversion.
  If we know ℏ, Γ, λ, and c, we can rigorously solve for L₀.
  L₀ = ℏ / (Γ · λ · c)

  This supports the physical finding: L₀ ≈ 0.125 fm.
-/
theorem vacuum_length_scale_inversion :
  M.L_zero = M.ℏ / (M.Gamma_vortex * M.lam_mass * M.cVac) := by

  -- From h_hbar_match: hbar = Γ * λ * L₀ * c, solve for L₀
  have h_denom_ne : M.Gamma_vortex * M.lam_mass * M.cVac ≠ 0 := by
    apply mul_ne_zero
    · apply mul_ne_zero
      · exact ne_of_gt M.h_shape_pos
      · exact ne_of_gt M.h_mass_pos
    · exact ne_of_gt M.h_cvac_pos

  -- Multiply both sides by the denominator
  have h : M.L_zero * (M.Gamma_vortex * M.lam_mass * M.cVac) = M.ℏ := by
    calc M.L_zero * (M.Gamma_vortex * M.lam_mass * M.cVac)
        = M.Gamma_vortex * M.lam_mass * M.L_zero * M.cVac := by ring
      _ = M.ℏ := M.h_hbar_match.symm

  -- Divide both sides
  exact (eq_div_iff h_denom_ne).mpr h

/--
  Theorem: Nuclear Scale Prediction (Scaling Bridge).

  Given measured ℏ and fixing:
  - lam_mass = 1 AMU (proton mass scale)
  - Gamma_vortex ≈ 1.6919 (computed geometric factor)
  - cVac = c (speed of light)

  We can derive L₀ ≈ 0.125 fm (nuclear hard core radius).

  **Status**: This is a consistency constraint, not yet an ab initio prediction.
  Future work: Derive lam_mass or L₀ independently to close the loop.
-/
theorem nuclear_scale_prediction
    (h_lambda : M.lam_mass = 1.66053906660e-27) -- 1 AMU in kg
    (h_hbar : M.ℏ = 1.054571817e-34) -- measured ℏ in J·s
    (h_gamma : M.Gamma_vortex = 1.6919) -- computed from Hill vortex integral
    (h_cvac : M.cVac = 2.99792458e8) : -- speed of light
    M.L_zero = M.ℏ / (M.Gamma_vortex * M.lam_mass * M.cVac) := by
  exact vacuum_length_scale_inversion

/--
  Theorem: Compton Wavelength Connection.

  The vacuum length scale L₀ is related to the reduced Compton wavelength
  via the geometric factor:

  L₀ = λ_Compton / Gamma_vortex

  where λ_Compton = ℏ / (lam_mass · c)

  For Γ ≈ 1.69: L₀ ≈ λ_Compton / 1.69
-/
theorem compton_connection :
    M.L_zero = (M.ℏ / (M.lam_mass * M.cVac)) / M.Gamma_vortex := by
  rw [vacuum_length_scale_inversion]
  have h_denom : M.lam_mass * M.cVac ≠ 0 := by
    apply mul_ne_zero
    · exact ne_of_gt M.h_mass_pos
    · exact ne_of_gt M.h_cvac_pos
  field_simp [h_denom]

/--
  Theorem: ℏ Positivity from Geometry.

  Since all geometric factors (Γ, λ, L₀, c) are positive,
  ℏ > 0 follows from the compatibility constraint.
-/
theorem hbar_pos : M.ℏ > 0 := by
  rw [M.h_hbar_match]
  apply mul_pos
  · apply mul_pos
    · apply mul_pos
      · exact M.h_shape_pos
      · exact M.h_mass_pos
    · exact M.h_len_pos
  · exact M.h_cvac_pos

/--
  Theorem: Unification Scale Match.

  The fact that lam_mass = 1 AMU (atomic mass scale) with Γ ≈ 1.69 (geometric factor)
  predicts L₀ ≈ 0.125 fm (nuclear scale).

  This suggests that atomic, photon, and nuclear physics are unified
  by the same vacuum geometry.

  **Status**: Consistency constraint pending independent derivation of scales.
-/
theorem unification_scale_match
    (h_lambda_amu : M.lam_mass = 1.66053906660e-27)
    (h_hbar_measured : M.ℏ = 1.054571817e-34)
    (h_gamma_computed : M.Gamma_vortex = 1.6919)
    (h_cvac_c : M.cVac = 2.99792458e8) :
    ∃ (L_nuclear : ℝ), abs (M.L_zero - L_nuclear) < 1e-16 ∧
                       L_nuclear = 1.25e-16 := by
  -- Use vacuum_length_scale_inversion to express L₀
  have h_L0 := vacuum_length_scale_inversion (M := M)

  -- For formal verification, witness L_nuclear = 1.25e-16
  use 1.25e-16
  constructor
  · -- Prove abs (M.L_zero - 1.25e-16) < 1e-16
    -- Rewrite M.L_zero using the inversion formula
    rw [h_L0]
    -- The numerical bound follows from the measured constants
    -- L₀ = ℏ / (Γ · λ · c) ≈ 1.25e-16 for these inputs
    simp only [h_lambda_amu, h_hbar_measured, h_gamma_computed, h_cvac_c]
    norm_num
  · -- Prove L_nuclear = 1.25e-16 (trivial by definition)
    rfl

end EmergentConstants
end QFD
