-- QFD/Cosmology/VacuumRefraction.lean
import QFD.Schema.Couplings
import QFD.Cosmology.RadiativeTransfer
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic

noncomputable section

namespace QFD.Cosmology

open QFD.Schema
open Real

/-!
# QFD Vacuum Refraction and CMB Power Spectrum Modulation

This module formalizes the mechanism by which photon-photon scattering mediated
by the ψ-field creates oscillatory modulation in the CMB angular power spectrum.

## Physical Framework

**Standard Cosmology**:
CMB acoustic peaks arise from sound waves in the primordial plasma at recombination
(z ~ 1100). Baryon-photon fluid oscillates, creating characteristic angular scale.

**QFD Alternative**:
CMB power spectrum modulation arises from vacuum refraction - photon-photon
scattering mediated by a scalar field ψ with characteristic correlation length r_ψ.
This creates interference patterns that mimic acoustic oscillations.

## Key Mechanism

The ψ-field correlation function:
  ⟨ψ(x)ψ(x')⟩ ~ exp(-|x-x'|/r_ψ)

imprints a characteristic scale on photon scattering. Photons traversing the
vacuum experience phase shifts that coherently add, creating oscillatory
modulation in angular power spectrum C_ℓ with period Δℓ ~ D_A/r_ψ where D_A
is angular diameter distance.

## Testable Predictions

1. **Oscillation amplitude**: A_osc bounded by unitarity (can't scatter > 100%)
2. **Correlation scale**: r_ψ determines oscillation frequency
3. **Phase coherence**: Oscillations persist across all ℓ (unlike acoustic damping)
4. **Polarization**: Different modulation for TT, TE, EE

References:
- QFD Appendix J (Time Refraction)
- Planck 2018 Power Spectra (arXiv:1807.06209)
-/

/-- Vacuum refraction parameters for CMB power spectrum modulation -/
structure VacuumRefractionParams where
  -- Correlation scale of ψ-field
  r_psi : ℝ           -- Correlation length (Mpc)

  -- Oscillation amplitude
  A_osc : Unitless    -- Modulation amplitude (dimensionless)

  -- Phase offset
  phi   : ℝ           -- Phase (radians)

  -- Angular diameter distance (cosmology-dependent)
  D_A   : ℝ           -- Mpc

  -- Base scattering parameters (from SNe fit)
  alpha : Unitless    -- From radiative transfer
  beta  : Unitless    -- From radiative transfer

/-- Physical constraints on vacuum refraction parameters -/
structure VacuumRefractionConstraints (p : VacuumRefractionParams) : Prop where
  -- Correlation length: must be positive and cosmologically reasonable
  r_psi_positive : p.r_psi > 0.0
  r_psi_bounded : p.r_psi < 1000.0  -- Mpc (smaller than Hubble radius)

  -- Oscillation amplitude: bounded by unitarity
  A_osc_nonneg : p.A_osc.val ≥ 0.0
  A_osc_unitary : p.A_osc.val < 1.0  -- Can't modulate by more than 100%

  -- Phase: periodic
  phi_range : -π ≤ p.phi ∧ p.phi ≤ π

  -- Angular diameter distance: positive
  D_A_positive : p.D_A > 0.0

  -- Scattering parameters (inherited from radiative transfer)
  alpha_bounds : 0.0 < p.alpha.val ∧ p.alpha.val < 2.0
  beta_bounds : 0.4 ≤ p.beta.val ∧ p.beta.val ≤ 1.0

/-! ## Oscillatory Modulation Function -/

/--
Characteristic angular scale for oscillations.

The ψ-field correlation length r_ψ subtends an angular scale:
  θ_scale = r_ψ / D_A

In multipole space:
  ℓ_scale = π / θ_scale = π D_A / r_ψ

This is the fundamental oscillation frequency in ℓ-space.
-/
def characteristic_ell_scale (r_psi : ℝ) (D_A : ℝ) : ℝ :=
  π * D_A / r_psi

/--
Oscillatory modulation function for CMB power spectrum.

The transfer function relates QFD power spectrum to standard (ΛCDM) prediction:
  C_ℓ^QFD = C_ℓ^ΛCDM × M(ℓ)

where the modulation function is:
  M(ℓ) = 1 + A_osc × cos(2π ℓ/ℓ_scale + φ)

Physical interpretation:
- Unity baseline: no modulation in limit A_osc → 0
- Cosine oscillation: periodic modulation from ψ-field interference
- Amplitude A_osc: strength of vacuum refraction effect
- Period Δℓ = ℓ_scale: determined by correlation length r_ψ
- Phase φ: depends on line-of-sight integration details
-/
def modulation_function (ell : ℝ) (p : VacuumRefractionParams) : ℝ :=
  let ell_scale := characteristic_ell_scale p.r_psi p.D_A
  1.0 + p.A_osc.val * cos (2 * π * ell / ell_scale + p.phi)

/--
**Theorem 1: Modulation function is bounded**

The modulation function M(ℓ) satisfies:
  1 - A_osc ≤ M(ℓ) ≤ 1 + A_osc

This ensures physical power spectrum: if C_ℓ^ΛCDM > 0 and A_osc < 1,
then C_ℓ^QFD = M(ℓ) × C_ℓ^ΛCDM > 0.
-/
theorem modulation_bounded (ell : ℝ) (p : VacuumRefractionParams)
    (h_A : p.A_osc.val < 1.0)
    (h_A_pos : p.A_osc.val ≥ 0.0) :
    1.0 - p.A_osc.val ≤ modulation_function ell p ∧
    modulation_function ell p ≤ 1.0 + p.A_osc.val := by
  unfold modulation_function
  let cos_val := cos (2 * π * ell / (characteristic_ell_scale p.r_psi p.D_A) + p.phi)
  have h_cos_bounds : -1 ≤ cos_val ∧ cos_val ≤ 1 := by
    constructor
    · exact neg_one_le_cos _
    · exact cos_le_one _

  have h_A_pos' : 0 ≤ p.A_osc.val := by linarith
  constructor
  · -- Lower bound: 1 - A_osc
    have h_mul_ge : p.A_osc.val * cos_val ≥ p.A_osc.val * (-1) := by
      exact mul_le_mul_of_nonneg_left h_cos_bounds.1 h_A_pos'
    linarith
  · -- Upper bound: 1 + A_osc
    have h_mul_le : p.A_osc.val * cos_val ≤ p.A_osc.val * 1 := by
      exact mul_le_mul_of_nonneg_left h_cos_bounds.2 h_A_pos'
    linarith

/--
**Theorem 2: Modulation preserves positivity**

If the baseline power spectrum is positive (C_ℓ > 0) and unitarity holds
(A_osc < 1), then the modulated spectrum remains positive.
-/
theorem modulation_preserves_positivity (ell : ℝ) (C_ell_base : ℝ)
    (p : VacuumRefractionParams)
    (h_C_pos : C_ell_base > 0)
    (h_A : p.A_osc.val < 1.0)
    (h_A_pos : p.A_osc.val ≥ 0.0) :
    C_ell_base * modulation_function ell p > 0 := by
  have h_mod_bounds := modulation_bounded ell p h_A h_A_pos
  have h_mod_pos : modulation_function ell p > 0 := by
    linarith [h_mod_bounds.1]
  exact mul_pos h_C_pos h_mod_pos

/--
**Theorem 3: Oscillation period determined by correlation scale**

The modulation function has period Δℓ = ℓ_scale in multipole space.
-/
theorem modulation_periodic (ell : ℝ) (p : VacuumRefractionParams) (h_p : VacuumRefractionConstraints p) :
    let ell_scale := characteristic_ell_scale p.r_psi p.D_A
    modulation_function (ell + ell_scale) p = modulation_function ell p := by
  unfold modulation_function characteristic_ell_scale
  simp only
  have h_r_psi_pos : 0 < p.r_psi := by linarith [h_p.r_psi_positive]
  have h_DA_pos : 0 < p.D_A := by linarith [h_p.D_A_positive]
  have h_r_psi_ne_zero : p.r_psi ≠ 0 := ne_of_gt h_r_psi_pos
  have h_DA_ne_zero : p.D_A ≠ 0 := ne_of_gt h_DA_pos
  have h_scale_ne_zero : π * p.D_A / p.r_psi ≠ 0 := by
    apply div_ne_zero
    · apply mul_ne_zero pi_ne_zero h_DA_ne_zero
    · exact h_r_psi_ne_zero
  -- Simplify: need to show cos(2π(ell + scale)/scale + φ) = cos(2πell/scale + φ)
  -- This follows from: (ell + scale)/scale = ell/scale + 1, so argument shifts by 2π
  congr 1
  have h_algebra : 2 * π * (ell + π * p.D_A / p.r_psi) / (π * p.D_A / p.r_psi) + p.phi =
                   2 * π * ell / (π * p.D_A / p.r_psi) + p.phi + 2 * π := by
    field_simp
    ring
  rw [h_algebra, cos_add_two_pi]

/--
**Definition: Unitarity bound**

The oscillation amplitude should not exceed what's physically allowed by
the survival fraction from scattering.
-/
def unitarity_bound (p : VacuumRefractionParams) (z : ℝ) : Prop :=
  p.A_osc.val ≤ 1.0 - exp (-(p.alpha.val * z ^ p.beta.val))

/--
**Theorem 4: Unitarity bound is physical**

If A_osc satisfies the unitarity bound, it is less than 1.
-/
theorem unitarity_implies_physical (p : VacuumRefractionParams) (z_CMB : ℝ)
    (h_z : z_CMB > 0)
    (h_alpha : p.alpha.val > 0)
    (h_beta : p.beta.val > 0)
    (h_unit : unitarity_bound p z_CMB) :
    p.A_osc.val < 1.0 := by
  unfold unitarity_bound at h_unit
  have h_tau_pos : p.alpha.val * z_CMB ^ p.beta.val > 0 := by
    apply mul_pos h_alpha
    apply rpow_pos_of_pos h_z
  have h_exp_pos : 0 < exp (-(p.alpha.val * z_CMB ^ p.beta.val)) := exp_pos _
  have h_exp_lt_one : exp (-(p.alpha.val * z_CMB ^ p.beta.val)) < 1 := by
    rw [exp_lt_one_iff]
    linarith
  linarith [h_exp_pos]

/-! ## CMB Power Spectrum Predictions -/

/--
QFD prediction for CMB TT power spectrum.

C_ℓ^TT,QFD = C_ℓ^TT,ΛCDM × M(ℓ, r_ψ, A_osc, φ)

where C_ℓ^TT,ΛCDM is the standard prediction from CAMB/CLASS.
-/
def C_ell_TT_QFD (ell : ℝ) (C_ell_LCDM : ℝ) (p : VacuumRefractionParams) : ℝ :=
  C_ell_LCDM * modulation_function ell p

/--
QFD prediction for CMB TE power spectrum.

The cross-correlation (temperature-E-mode polarization) may have a different
modulation pattern due to different scattering processes for temperature vs
polarization fluctuations.

For simplicity, we assume the same modulation (can be refined):
  C_ℓ^TE,QFD = C_ℓ^TE,ΛCDM × M(ℓ)
-/
def C_ell_TE_QFD (ell : ℝ) (C_ell_LCDM : ℝ) (p : VacuumRefractionParams) : ℝ :=
  C_ell_LCDM * modulation_function ell p

/--
QFD prediction for CMB EE power spectrum.

E-mode polarization power spectrum with same modulation assumption.
-/
def C_ell_EE_QFD (ell : ℝ) (C_ell_LCDM : ℝ) (p : VacuumRefractionParams) : ℝ :=
  C_ell_LCDM * modulation_function ell p

/-! ## Falsifiability -/

/--
**Distinguishing prediction: Phase coherence across all scales**

Standard acoustic oscillations have damping at high ℓ due to photon diffusion.
QFD vacuum refraction predicts persistent oscillations (constant A_osc) across
all ℓ where scattering is coherent.

Test: Fit A_osc separately in low-ℓ and high-ℓ bins.
- Standard: A_osc(high-ℓ) << A_osc(low-ℓ) (damping tail)
- QFD: A_osc(high-ℓ) ≈ A_osc(low-ℓ) (persistent modulation)
-/
def phase_coherence_test (A_osc_low : Unitless) (A_osc_high : Unitless) : Prop :=
  -- QFD predicts similar amplitudes
  abs (A_osc_low.val - A_osc_high.val) < 0.2 * A_osc_low.val

/--
**Falsification criterion 1: Unphysical oscillation amplitude**

If fitted A_osc > 1, the model violates unitarity and is falsified.
-/
def falsified_by_unitarity (A_osc : Unitless) : Prop :=
  A_osc.val ≥ 1.0

/--
**Falsification criterion 2: No oscillations**

If fitted A_osc is consistent with zero (A_osc < threshold), vacuum refraction
is undetectable and the model adds no explanatory power.
-/
def falsified_by_null_detection (A_osc : Unitless) (threshold : ℝ) : Prop :=
  A_osc.val < threshold

/--
**Theorem 5: Vacuum refraction model makes falsifiable predictions**

There exist parameter values that violate observational constraints.
-/
theorem vacuum_refraction_is_falsifiable :
    ∃ (p : VacuumRefractionParams),
      falsified_by_unitarity p.A_osc ∨
      falsified_by_null_detection p.A_osc 0.01 := by
  use { r_psi := 100.0, A_osc := ⟨1.5⟩, phi := 0.0, D_A := 1000.0,
        alpha := ⟨0.5⟩, beta := ⟨0.7⟩ }
  left
  unfold falsified_by_unitarity
  norm_num

/-! ## Integration with Grand Solver -/

/--
**Parameter space for Planck fit**

Free parameters to be fitted against Planck 2018 TT/TE/EE power spectra:
1. r_ψ: Correlation length (determines oscillation frequency)
2. A_osc: Oscillation amplitude (strength of modulation)
3. φ: Phase offset (line-of-sight integration detail)

Fixed/constrained:
- D_A: Angular diameter distance (from cosmology)
- α, β: Scattering parameters (from SNe + CMB spectrum fit)
-/
def planck_fit_parameters : List String :=
  ["r_psi", "A_osc", "phi"]

/--
**Success criterion for QFD hypothesis**

QFD successfully explains CMB acoustic peaks if:
1. Fitted parameters within physical bounds (Lean constraints satisfied)
2. χ² comparable to standard ΛCDM + primordial inflation
3. A_osc > detection threshold (oscillations present)
4. r_ψ consistent with independent QFD predictions (if any)
-/
def qfd_explains_acoustic_peaks (chi2_QFD : ℝ) (chi2_standard : ℝ)
    (p : VacuumRefractionParams)
    (constraints : VacuumRefractionConstraints p) : Prop :=
  -- Comparable fit quality
  chi2_QFD ≤ chi2_standard * 1.1 ∧
  -- Oscillations detected
  p.A_osc.val ≥ 0.05 ∧
  -- Physical parameters
  constraints.A_osc_unitary

end QFD.Cosmology
end