-- QLF_MercuryPerihelion.lean
-- Mercury perihelion advance Δφ = 6πGM/(c²a(1-e²)) from QLF substrate
-- (Mercury_Perihelion.md), composing Newton (Gravity_From_Delay) with
-- Schwarzschild weak-field metric (GR_Schwarzschild) for the famous
-- 43"/century test of General Relativity.
--
-- The substrate composition:
--
--   1. Newton's law F = GMm/r² from holographic event-counting
--      (QLF_GravityFromDelay.lean).
--
--   2. Schwarzschild metric weak-field components from substrate
--      Cross-Frequency Lorentz + radial event scaling
--      (GR_Schwarzschild.md, not yet Lean-anchored individually).
--
--   3. Effective orbital potential gains a 1/r³ correction from
--      g_rr ≠ 1: V_eff(r) = -GMm/r + L²/(2mr²) - GM L²/(mc² r³).
--
--   4. Perihelion advance per orbit (standard orbital perturbation
--      theory applied to the 1/r³ term):
--
--        Δφ = 6πGM/(c² a (1-e²)) = 3π R_s/(a (1-e²))
--
--      where R_s = 2GM/c² is the Schwarzschild radius.
--
-- Mercury numerical check:
--   M_Sun = 1.989 × 10³⁰ kg, a = 5.791 × 10¹⁰ m, e = 0.2056,
--   T = 87.969 days, R_s(Sun) = 2.953 km
--   Per-orbit Δφ = 3π × 2953 / (5.791e10 × 0.9577) = 5.02 × 10⁻⁷ rad
--   415 orbits/century × 5.02e-7 × 206265 arcsec/rad ≈ 42.99"/century
--   vs Park et al. (2017) measured 42.98 ± 0.04"/century  → 0.03% match.
--
-- This is the **eighth Lean-anchored fundamental-physics theorem** in
-- the QLF tree (after α, m_p/m_e, γ, Dirac, Lamb, g−2, Gravity).
--
-- Honest scope.  This module Lean-anchors:
--   • the Schwarzschild radius R_s = 2GM/c² definition
--   • the perihelion-advance formula Δφ = 3π R_s/(a(1-e²))
--   • the equivalent closed form Δφ = 6πGM/(c²a(1-e²))
--   • the dimensionless ratio (Δφ a (1-e²))/(3π) = R_s
--
-- It does NOT Lean-anchor:
--   • the orbital perturbation analysis itself (standard classical
--     mechanics; integration of 1/r³ correction gives the prefactor 6π).
--     Substrate-native derivation is Tier-3 open (Mercury_Perihelion.md §5).
--   • the substrate derivation of the Schwarzschild metric components
--     (sketched in GR_Schwarzschild.md, not yet Lean-anchored).
--   • the Mercury observables (M_Sun, a, e, T) as substrate-derived —
--     these are astronomical inputs.
--
-- Companion to:
--   • Mercury_Perihelion.md                                  — structural argument
--   • mercury_perihelion_demo.py                             — numerical demo
--   • GR_Schwarzschild.md                                   — substrate metric
--   • Gravity_From_Delay.md                                 — Newton's law
--   • Cross_Frequency_Lorentz.md                            — gravitational redshift
--   • Kitada_Local_Time_GR.md                               — Markov-blanket GR programme
--   • lean/QLF_GravityFromDelay.lean                        — Newton substrate Lean
--   • lean/QLF_SubstrateLightSpeed.lean                     — c substrate Lean
--   • lean/QLF_EinsteinGeometricFactor.lean                 — 8π = 4π · 2

import Mathlib.Analysis.SpecialFunctions.Pow.Real

namespace QLF

/-- **The Schwarzschild radius of a mass M**:
    `R_s = 2 G M / c²`.

    This is the substrate radius at which Schwarzschild metric component
    `g_tt = -(1 - R_s/r)` reaches singular behaviour.  For the Sun:
    `R_s ≈ 2.953 km`.  For an electron mass: `R_s ≈ 1.35 × 10⁻⁵⁷ m`
    (far below Planck length — quantum gravity regime, not classical
    Schwarzschild).

    `G` and `c` are substrate-derived in QLF:
    `G = L_Planck² c³ / ℏ` (substrate event quantum),
    `c = L_Planck / τ_Planck` (substrate event quantum).
    `M` is an empirical input for the specific astronomical body. -/
noncomputable def schwarzschild_radius (G M c : ℝ) : ℝ :=
  2 * G * M / c^2

/-- **Schwarzschild radius identity**: `R_s = 2GM/c²` (definitional). -/
theorem schwarzschild_radius_eq (G M c : ℝ) :
    schwarzschild_radius G M c = 2 * G * M / c^2 := rfl

/-- **Perihelion advance per orbit** for a test mass in Schwarzschild
    weak-field orbit:

      Δφ = 3π R_s / (a (1 - e²))   per orbit.

    Equivalently: `Δφ = 6π G M / (c² a (1 - e²))`.

    Derivation: in the Schwarzschild metric, the radial geodesic equation
    gives an effective potential `V_eff = -GMm/r + L²/(2mr²) - GM L²/(mc² r³)`.
    The 1/r³ correction (relative to Kepler) shifts the apsidal angle by
    `Δφ` per orbit, computed via standard perturbation theory as

      Δφ = (3 R_s) / (a (1 - e²)) × 2π × (1/2) = 3π R_s / (a (1 - e²)).

    The (1 - e²) factor enhances precession for non-circular orbits;
    for e = 0 (circular): `Δφ = 3π R_s / a`. -/
noncomputable def perihelion_advance_per_orbit
    (R_s a e : ℝ) : ℝ :=
  3 * Real.pi * R_s / (a * (1 - e^2))

/-- **Equivalent form**: `Δφ = 6πGM/(c²a(1-e²))`.

    Substituting `R_s = 2GM/c²` into the perihelion-advance formula
    above gives this standard textbook form. -/
noncomputable def perihelion_advance_from_GMc
    (G M c a e : ℝ) : ℝ :=
  6 * Real.pi * G * M / (c^2 * a * (1 - e^2))

/-- **The two perihelion formulas agree** when R_s = 2GM/c² (assuming
    c, a, (1 - e²) ≠ 0; physically Newton perihelion is only defined
    for bound non-degenerate orbits). -/
theorem perihelion_advance_form_equivalence
    (G M c a e : ℝ) (hc : c ≠ 0) (ha : a ≠ 0) (he : 1 - e^2 ≠ 0) :
    perihelion_advance_per_orbit (schwarzschild_radius G M c) a e
      = perihelion_advance_from_GMc G M c a e := by
  unfold perihelion_advance_per_orbit
         perihelion_advance_from_GMc
         schwarzschild_radius
  have hc2 : c^2 ≠ 0 := pow_ne_zero _ hc
  field_simp
  ring

/-- **Dimensionless ratio identity**: the perihelion advance per orbit
    relates linearly to the Schwarzschild radius via the orbital
    geometry.  Specifically:

      (Δφ × a × (1 - e²)) / (3π) = R_s.

    This is the perihelion advance expressed as a measurement of R_s. -/
theorem perihelion_advance_extracts_R_s
    (R_s a e : ℝ) (ha : a ≠ 0) (he : 1 - e^2 ≠ 0) :
    perihelion_advance_per_orbit R_s a e * a * (1 - e^2) / (3 * Real.pi)
      = R_s := by
  unfold perihelion_advance_per_orbit
  have hpi : Real.pi ≠ 0 := Real.pi_ne_zero
  field_simp

/-- **Per-century precession**: Δφ_century = N_orbits × Δφ_orbit
    where `N_orbits = T_century / T_orbit`.

    For Mercury: T_century = 100 × 365.25 days, T_Mercury = 87.969 days,
    N_orbits ≈ 415.2.  Per-century advance ≈ 5.02e-7 × 415.2 rad
    ≈ 2.08e-4 rad ≈ 42.99 arcsec. -/
noncomputable def perihelion_advance_per_century
    (delta_phi_orbit T_century T_orbit : ℝ) : ℝ :=
  delta_phi_orbit * (T_century / T_orbit)

/-- **Per-century expansion**. -/
theorem perihelion_advance_per_century_eq
    (delta_phi_orbit T_century T_orbit : ℝ) :
    perihelion_advance_per_century delta_phi_orbit T_century T_orbit
      = delta_phi_orbit * T_century / T_orbit := by
  unfold perihelion_advance_per_century
  ring

/-- **Composition of substrate primitives**: the Mercury perihelion
    advance assembles from the substrate primitives.

    Packaging:
    - Schwarzschild radius `R_s = 2GM/c²` is the substrate metric scale.
    - Perihelion advance `Δφ = 3π R_s / (a(1-e²))` per orbit follows
      from substrate Schwarzschild + orbital perturbation theory.
    - Equivalent form `Δφ = 6πGM/(c²a(1-e²))` substitutes R_s.
    - Per-century advance scales by N orbits per century. -/
theorem mercury_perihelion_substrate_summary
    (G M c a e : ℝ) (hc : c ≠ 0) (ha : a ≠ 0) (he : 1 - e^2 ≠ 0) :
    schwarzschild_radius G M c = 2 * G * M / c^2 ∧
    perihelion_advance_per_orbit (schwarzschild_radius G M c) a e
      = perihelion_advance_from_GMc G M c a e ∧
    (∀ R_s, perihelion_advance_per_orbit R_s a e
          = 3 * Real.pi * R_s / (a * (1 - e^2))) := by
  refine ⟨rfl, ?_, ?_⟩
  · exact perihelion_advance_form_equivalence G M c a e hc ha he
  · intro R_s
    rfl

/-- **What this module does NOT prove**.

    - It does NOT derive the orbital-perturbation prefactor `3π` from
      substrate first principles.  Standard classical mechanics gives
      the 3π via integration of the 1/r³ correction over one Kepler
      orbit; substrate-native derivation is Tier-3 open.

    - It does NOT individually Lean-anchor the Schwarzschild metric
      components `g_tt = -(1 - R_s/r)` and `g_rr = (1 - R_s/r)⁻¹`.
      These are sketched in `GR_Schwarzschild.md` via Cross-Frequency
      Lorentz + substrate event scaling; full Lean-anchoring is a
      sibling module to `QLF_GravityFromDelay`.

    - It does NOT derive Mercury observables (M_Sun, a, e, T) from
      substrate.  These are astronomical inputs.

    - It does NOT extend to other GR tests (light bending, Shapiro
      delay, frame-dragging).  Same substrate framework would extend
      but is open. -/
theorem mercury_perihelion_not_proved_here : True := trivial

end QLF
