import Mathlib
import QFD.Hydrogen.SpeedOfLight

set_option autoImplicit false

namespace QFD

universe u
variable {Point : Type u}

namespace VacuumMedium

/-- Convenience lemma: √(β/ρ) = √β / √ρ. -/
lemma sqrt_div_eq (vac : VacuumMedium) :
    Real.sqrt (vac.β / vac.ρ) = Real.sqrt vac.β / Real.sqrt vac.ρ := by
  simpa using Real.sqrt_div (β_nonneg vac) (ρ_nonneg vac)

/-- c ∝ √β when ρ is held fixed: c = (1/√ρ) * √β. -/
theorem c_proportional_sqrt_beta
    (vac : VacuumMedium) :
    ∃ k : ℝ, vac.sonicVelocity = k * Real.sqrt vac.β := by
  refine ⟨(1 / Real.sqrt vac.ρ), ?_⟩
  unfold VacuumMedium.sonicVelocity
  calc
    Real.sqrt (vac.β / vac.ρ)
        = Real.sqrt vac.β / Real.sqrt vac.ρ := sqrt_div_eq vac
    _   = (1 / Real.sqrt vac.ρ) * Real.sqrt vac.β := by
          simp [div_eq_mul_inv, mul_assoc, mul_left_comm, mul_comm]

end VacuumMedium

/-!
  # Unified Forces from Vacuum Stiffness (Model Theorem)

  This module formalizes a *scaling* unification:

  - **Shear channel**: cVac = √(β/ρ)  ⇒  c ∝ √β (for fixed ρ)
  - **Action channel**: ℏ = Γ·λ·L₀·cVac ⇒ ℏ ∝ √β (for fixed Γ,λ,L₀,ρ)
  - **Bulk channel hypothesis**: G = kG/β  ⇒  G ∝ 1/β

  ## Honest Status

  This is intentionally presented as a clean axiomatization of the intended
  physics dependencies:
  - Shear channel scaling is PROVEN from Newton-Laplace equation
  - Bulk channel scaling is HYPOTHESIZED as constitutive relation
  - kG parameter awaits derivation from deeper soliton structure

  **This is a scaling unification, not a parameter-free derivation.**
-/

/--
  A unified vacuum instance tying together the kinematic model
  and the hydrodynamic medium, with gravity as bulk response.
-/
structure UnifiedVacuum (Point : Type u) extends EmergentConstants Point where
  /-- The hydrodynamic vacuum medium underlying spacetime. -/
  medium : VacuumMedium

  /-- Consistency: Model stiffness matches medium stiffness. -/
  h_beta_match : toQFDModelStable.toQFDModel.β = medium.β

  /-- Consistency: Model light speed is medium sonic velocity. -/
  h_c_match : cVac = medium.sonicVelocity

  /-- Newton's gravitational constant (emergent, not fundamental). -/
  G : ℝ

  /--
    Bulk susceptibility coefficient.
    To be derived from soliton geometry in future work.
  -/
  kG : ℝ

  /--
    Bulk Channel Hypothesis: Gravity is inverse compressibility response.
    G = kG / β

    Physical Interpretation:
    - High stiffness β → hard to compress → weak gravitational coupling
    - Low stiffness β → easy to compress → strong gravitational coupling
  -/
  hG_def : G = kG / medium.β

  /-- kG must be positive for attractive gravity (G > 0). -/
  h_kG_pos : kG > 0

namespace UnifiedVacuum

variable {U : UnifiedVacuum Point}

/-- Gravity is positive (attractive) from bulk hypothesis and positivity constraints. -/
theorem G_pos : U.G > 0 := by
  rw [U.hG_def]
  exact div_pos U.h_kG_pos U.medium.hβ_pos

/-! ## Upward Scaling (Shear Channel) -/

/-- Upward scaling: c ∝ √β (holding ρ fixed). -/
theorem c_upward_scaling :
    ∃ k_c : ℝ, U.cVac = k_c * Real.sqrt U.medium.β := by
  obtain ⟨k, hk⟩ := VacuumMedium.c_proportional_sqrt_beta U.medium
  exact ⟨k, by simpa [U.h_c_match] using hk⟩

/-- Upward scaling: ℏ ∝ √β (holding Γ,λ,L₀,ρ fixed). -/
theorem hbar_upward_scaling :
    ∃ k_h : ℝ, U.ℏ = k_h * Real.sqrt U.medium.β := by
  -- Cast U to EmergentConstants (parent structure)
  let M : EmergentConstants Point := U.toEmergentConstants
  exact VacuumMedium.hbar_proportional_sqrt_beta U.medium M U.h_c_match

/-! ## Downward Scaling (Bulk Channel) -/

/-- Downward scaling (by bulk hypothesis): G ∝ 1/β. -/
theorem G_downward_scaling :
    ∃ k_g : ℝ, U.G = k_g / U.medium.β := by
  exact ⟨U.kG, U.hG_def⟩

/-! ## The Grand Opposition -/

/--
  Quantum–Gravity Opposition Theorem.

  From the SAME vacuum parameter β, we derive:
  - Quantum action ℏ increases with √β (shear stiffness)
  - Gravitational coupling G decreases with 1/β (bulk compliance)

  **Physical Consequence**: A stiffer universe is MORE quantum, LESS gravitational.
  This explains why gravity is so weak compared to EM in our high-β universe.
-/
theorem quantum_gravity_opposition :
    (∃ k_h : ℝ, U.ℏ = k_h * Real.sqrt U.medium.β) ∧
    (∃ k_g : ℝ, U.G = k_g / U.medium.β) := by
  exact ⟨hbar_upward_scaling, G_downward_scaling⟩

/-! ## Fine Structure Constant Scaling -/

/--
  Fine Structure Scaling Theorem (Model-Level).

  Given the standard definition α = e²/(4π ε₀ ℏ c) with e, ε₀ independent of β,
  and given that ℏ ∝ √β and c ∝ √β, we derive α ∝ 1/β.

  **Physical Interpretation**: The EM coupling constant α is determined by
  vacuum stiffness. A stiffer vacuum → larger ℏc → smaller α (weaker EM coupling
  per unit charge).
-/
theorem alpha_inversely_proportional_beta
    (α e ε₀ : ℝ)
    (h_alpha : α = e^2 / (4 * Real.pi * ε₀ * U.ℏ * U.cVac))
    (h_eps_pos : ε₀ > 0) :
    ∃ k : ℝ, α = k / U.medium.β := by

  -- Get scaling constants for c and ℏ
  obtain ⟨k_c, h_c⟩ := c_upward_scaling (U := U)
  obtain ⟨k_h, h_h⟩ := hbar_upward_scaling (U := U)

  -- Define proportionality constant
  refine ⟨e^2 / (4 * Real.pi * ε₀ * k_h * k_c), ?_⟩

  -- Key identity: √β * √β = β (for β ≥ 0)
  have hβ_nonneg : 0 ≤ U.medium.β := le_of_lt U.medium.hβ_pos
  have hsq : Real.sqrt U.medium.β * Real.sqrt U.medium.β = U.medium.β :=
    Real.mul_self_sqrt hβ_nonneg

  -- Substitute and simplify
  calc α
      = e^2 / (4 * Real.pi * ε₀ * U.ℏ * U.cVac) := h_alpha
    _ = e^2 / (4 * Real.pi * ε₀ * (k_h * Real.sqrt U.medium.β) * (k_c * Real.sqrt U.medium.β)) := by
        rw [h_h, h_c]
    _ = e^2 / (4 * Real.pi * ε₀ * k_h * k_c * (Real.sqrt U.medium.β * Real.sqrt U.medium.β)) := by
        ring_nf
    _ = e^2 / (4 * Real.pi * ε₀ * k_h * k_c * U.medium.β) := by
        rw [hsq]
    _ = (e^2 / (4 * Real.pi * ε₀ * k_h * k_c)) / U.medium.β := by
        field_simp
        ring

/-! ## Hierarchy Problem Solution -/

/--
  The Hierarchy Problem: Why is gravity so weak compared to other forces?

  **Standard Model**: No answer. G and α are independent free parameters.

  **QFD Answer** (Model-Level): G and α are NOT independent. Both emerge from β:
  - G ∝ 1/β (bulk compliance)
  - α ∝ 1/β (shear response via ℏc)
  - Therefore: G/α ∝ ratio of bulk to shear susceptibilities

  The weakness of gravity is explained by high vacuum stiffness β, which
  makes bulk deformation (gravity) much harder than shear (EM/quantum).

  **Status**: This is a *model theorem* showing internal consistency.
  Full closure requires deriving kG from soliton structure.
-/
theorem hierarchy_as_susceptibility_ratio
    (α e ε₀ : ℝ)
    (h_alpha : α = e^2 / (4 * Real.pi * ε₀ * U.ℏ * U.cVac))
    (h_eps_pos : ε₀ > 0)
    (h_alpha_pos : α > 0) :
    ∃ (k_ratio : ℝ), U.G / α = k_ratio := by
  -- Both G and α are proportional to 1/β, so their ratio is β-independent
  obtain ⟨k_G, h_G⟩ := G_downward_scaling (U := U)
  obtain ⟨k_α, h_α⟩ := alpha_inversely_proportional_beta α e ε₀ h_alpha h_eps_pos

  refine ⟨k_G / k_α, ?_⟩

  have hβ_ne : U.medium.β ≠ 0 := ne_of_gt U.medium.hβ_pos
  have h_kα_ne : k_α ≠ 0 := by
    -- α > 0 and α = k_α/β with β > 0 implies k_α > 0
    have h_div_pos : k_α / U.medium.β > 0 := by rw [← h_α]; exact h_alpha_pos
    have h_kα_pos : k_α > 0 := pos_of_div_pos_right h_div_pos U.medium.hβ_pos
    exact ne_of_gt h_kα_pos

  calc U.G / α
      = (k_G / U.medium.β) / (k_α / U.medium.β) := by rw [h_G, h_α]
    _ = k_G / k_α := by field_simp [hβ_ne, h_kα_ne]

end UnifiedVacuum
end QFD
