import Mathlib.Data.Real.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic

/-!
# Real-Time Cosmological Drift (Sandage-Loeb Test)

**Priority**: 140 (Cluster 4)
**Goal**: Explicit prediction for frequency drift `dν/dt`.

## Physical Setup

The Sandage-Loeb test measures the redshift drift: how z(t) of a distant
source changes over observer time. For a source at rest in comoving
coordinates, the observed frequency ν evolves as:

  ν(t) = ν₀ / (1 + z(t))

where z(t) changes due to the difference between emission and observation
Hubble rates.

## QFD Prediction

In QFD's static vacuum model with tired-light redshift, z is constant
for a given source at fixed comoving distance, so dz/dt = 0. This gives:

  dν/dt = 0

This is a falsifiable prediction: if dz/dt ≠ 0 is ever measured (per
the Sandage-Loeb effect), QFD's static model would require modification.
-/

namespace QFD.Cosmology.RealTimeCosmology

/-- Redshift as a function of observer time (for a fixed source).
    In QFD static model, this is constant. -/
noncomputable def redshift_of_time (z₀ : ℝ) (_ : ℝ) : ℝ := z₀

/-- Observed frequency from emitted frequency and redshift -/
noncomputable def observed_frequency (ν₀ z : ℝ) : ℝ := ν₀ / (1 + z)

/--
**Theorem: Redshift Drift Vanishes in Static Model**

In QFD's static vacuum model, the redshift of a fixed source
does not change with observer time, i.e., dz/dt = 0.
-/
@[simp] theorem realtime_redshift_derivative (z₀ t : ℝ) :
    deriv (redshift_of_time z₀) t = 0 := by
  unfold redshift_of_time
  simp [deriv_const]

/--
**Corollary: Frequency Drift Vanishes**

Since z is constant, the observed frequency ν = ν₀/(1+z) is also constant.
-/
theorem frequency_drift_zero (ν₀ z₀ t : ℝ) :
    deriv (fun s => observed_frequency ν₀ (redshift_of_time z₀ s)) t = 0 := by
  unfold observed_frequency redshift_of_time
  simp [deriv_const]

end QFD.Cosmology.RealTimeCosmology