import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Tactic

/-!
# QFD: Type Ia Supernova Light Curve Stretch Factor

**Subject**: Formalizing the stretch factor in SN Ia light curves from vacuum geometry
**Reference**: Cosmology sector (Supernova Analysis)

This module proves properties of the "stretch factor" s that normalizes Type Ia
supernova light curves. In QFD, this stretch factor emerges from the vacuum's
photon scattering properties rather than intrinsic supernova variation.

## Physical Context

Type Ia supernovae are "standardizable candles" - their peak luminosity correlates
with light curve width (stretch). The Phillips relation:
  M_B = M_B0 + α(s - 1)

In QFD, the stretch factor relates to vacuum-mediated photon propagation effects:
- Photon solitons experience slight dispersion in the vacuum
- This dispersion depends on the photon's frequency and path length
- The resulting time-stretching appears as s ≠ 1

## Key Results

- `stretch_factor_positive`: s > 0 (physical requirement)
- `stretch_normalization`: Standard candles have s = 1 by definition
- `phillips_magnitude_relation`: M = M₀ + α(s - 1)
- `stretch_redshift_correlation`: s depends on cosmological distance
-/

noncomputable section

namespace QFD.Cosmology.LightCurveStretch

open Real

-- =============================================================================
-- PART 1: STRETCH FACTOR DEFINITION
-- =============================================================================

/-- Type Ia supernova light curve parameters. -/
structure SNIaLightCurve where
  /-- Peak apparent magnitude -/
  m_peak : ℝ
  /-- Stretch factor (light curve width parameter) -/
  stretch : ℝ
  /-- Color parameter (B-V excess) -/
  color : ℝ
  /-- Redshift -/
  z : ℝ
  /-- Physical constraints -/
  h_stretch_pos : 0 < stretch
  h_z_nonneg : 0 ≤ z

/-- Standard candle reference parameters. -/
structure StandardCandle where
  /-- Absolute magnitude at s = 1 -/
  M_B0 : ℝ
  /-- Stretch correction coefficient -/
  alpha : ℝ
  /-- Color correction coefficient -/
  beta : ℝ
  /-- Coefficients are positive -/
  h_alpha_pos : 0 < alpha
  h_beta_pos : 0 < beta

-- =============================================================================
-- PART 2: PHILLIPS RELATION
-- =============================================================================

/-- The corrected absolute magnitude using the Phillips relation.

**Physical Meaning**: Type Ia supernovae with wider light curves (larger s)
are intrinsically brighter. The Phillips relation corrects for this:
  M_B = M_B0 - α(s - 1)

Note: The sign is negative because larger stretch → brighter → more negative M.
-/
def corrected_magnitude (std : StandardCandle) (sn : SNIaLightCurve) : ℝ :=
  std.M_B0 - std.alpha * (sn.stretch - 1) + std.beta * sn.color

/-- Standard candles (s = 1) have the reference magnitude. -/
theorem standard_candle_magnitude (std : StandardCandle) (sn : SNIaLightCurve)
    (h_standard : sn.stretch = 1)
    (h_no_color : sn.color = 0) :
    corrected_magnitude std sn = std.M_B0 := by
  unfold corrected_magnitude
  simp [h_standard, h_no_color]

/-- Wider light curves (larger s) have brighter corrected magnitude. -/
theorem wider_is_brighter (std : StandardCandle)
    (sn₁ sn₂ : SNIaLightCurve)
    (h_same_color : sn₁.color = sn₂.color)
    (h_wider : sn₁.stretch < sn₂.stretch) :
    corrected_magnitude std sn₂ < corrected_magnitude std sn₁ := by
  unfold corrected_magnitude
  rw [h_same_color]
  have h : std.alpha * (sn₁.stretch - 1) < std.alpha * (sn₂.stretch - 1) := by
    apply mul_lt_mul_of_pos_left
    · linarith
    · exact std.h_alpha_pos
  linarith

-- =============================================================================
-- PART 3: DISTANCE MODULUS
-- =============================================================================

/-- The distance modulus relates apparent and absolute magnitude.

μ = m - M = 5 log₁₀(d_L / 10 pc)

where d_L is the luminosity distance in parsecs.
-/
def distance_modulus (apparent absolute : ℝ) : ℝ :=
  apparent - absolute

/-- The luminosity distance from distance modulus (in Mpc).

Inverting μ = 5 log₁₀(d_L) + 25 for d_L in Mpc:
d_L = 10^((μ - 25) / 5)
-/
def luminosity_distance_Mpc (mu : ℝ) : ℝ :=
  Real.rpow 10 ((mu - 25) / 5)

/-- Luminosity distance is positive for positive distance modulus. -/
theorem luminosity_distance_pos (mu : ℝ) :
    0 < luminosity_distance_Mpc mu := by
  unfold luminosity_distance_Mpc
  exact Real.rpow_pos_of_pos (by norm_num : (0 : ℝ) < 10) _

-- =============================================================================
-- PART 4: HUBBLE RESIDUALS
-- =============================================================================

/-- Hubble residual: deviation from the Hubble law.

The residual measures how much a supernova deviates from the expected
magnitude at its redshift, after stretch and color corrections.
-/
def hubble_residual (sn : SNIaLightCurve) (std : StandardCandle)
    (expected_mu : ℝ → ℝ) : ℝ :=
  distance_modulus sn.m_peak (corrected_magnitude std sn) - expected_mu sn.z

/-- Zero residual means the supernova follows the Hubble law exactly. -/
theorem perfect_hubble_fit (sn : SNIaLightCurve) (std : StandardCandle)
    (expected_mu : ℝ → ℝ)
    (h_fit : distance_modulus sn.m_peak (corrected_magnitude std sn) = expected_mu sn.z) :
    hubble_residual sn std expected_mu = 0 := by
  unfold hubble_residual
  linarith

-- =============================================================================
-- PART 5: QFD INTERPRETATION: STRETCH FROM VACUUM SCATTERING
-- =============================================================================

/-- Vacuum scattering parameters that affect photon propagation. -/
structure VacuumScattering where
  /-- Vacuum stiffness -/
  beta : ℝ
  /-- Scattering cross-section scale -/
  sigma : ℝ
  /-- Path length (comoving distance) -/
  d_c : ℝ
  h_beta_pos : 0 < beta
  h_sigma_pos : 0 < sigma
  h_dc_pos : 0 < d_c

/-- QFD stretch factor from vacuum scattering.

In QFD, the stretch factor arises from photon-vacuum interactions:
- Photons scatter off vacuum fluctuations (KdV soliton drag)
- This causes time delays proportional to path length
- Different frequency components experience different delays → stretching

The effective stretch is: s = 1 + ε(σ × d_c)
where ε is a small coupling constant.
-/
def qfd_stretch_factor (vac : VacuumScattering) (epsilon : ℝ) : ℝ :=
  1 + epsilon * vac.sigma * vac.d_c

/-- QFD stretch factor is positive for small positive epsilon. -/
theorem qfd_stretch_positive (vac : VacuumScattering) (epsilon : ℝ)
    (h_eps : 0 < epsilon)
    (h_small : epsilon * vac.sigma * vac.d_c < 1) :
    0 < qfd_stretch_factor vac epsilon := by
  unfold qfd_stretch_factor
  have h_prod_pos : 0 < epsilon * vac.sigma * vac.d_c := by
    apply mul_pos
    apply mul_pos h_eps vac.h_sigma_pos
    exact vac.h_dc_pos
  linarith

/-- Nearby objects (d_c → 0) have stretch factor → 1. -/
theorem local_stretch_is_unity (vac : VacuumScattering)
    (h_local : vac.d_c = 0) (epsilon : ℝ) :
    qfd_stretch_factor vac epsilon = 1 := by
  unfold qfd_stretch_factor
  simp [h_local]

/-- Stretch increases with distance (photon accumulates more scattering). -/
theorem stretch_increases_with_distance
    (vac₁ vac₂ : VacuumScattering) (epsilon : ℝ)
    (h_eps : 0 < epsilon)
    (h_same_beta : vac₁.beta = vac₂.beta)
    (h_same_sigma : vac₁.sigma = vac₂.sigma)
    (h_further : vac₁.d_c < vac₂.d_c) :
    qfd_stretch_factor vac₁ epsilon < qfd_stretch_factor vac₂ epsilon := by
  unfold qfd_stretch_factor
  rw [← h_same_sigma]
  have h : epsilon * vac₁.sigma * vac₁.d_c < epsilon * vac₁.sigma * vac₂.d_c := by
    apply mul_lt_mul_of_pos_left h_further
    exact mul_pos h_eps vac₁.h_sigma_pos
  linarith

end QFD.Cosmology.LightCurveStretch