-- QFD/Rift/SequentialEruptions.lean
import Mathlib.Data.Real.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Tactic
import QFD.Rift.ChargeEscape

/-!
# QFD Black Hole Rift Physics: Sequential Eruptions and Charge Accumulation

**Goal**: Prove that successive rift eruptions build up charge at the black
hole surface, creating conditions for subsequent eruptions.

**Physical Mechanism** (The Rift Cascade):
1. **Initial rift**: Plasma erupts, electrons escape first (lighter)
2. **Residual charge**: Net positive charge left on BH surface (ions remain)
3. **Charge accumulation**: Successive rifts deposit more charge
4. **Enhanced repulsion**: Accumulated charge repels new ions
5. **Feedback loop**: Each rift makes next rift easier

**Key Insight**: Earlier rifts were CLOSER to BH (higher charge density).
As charge builds up, eruption radius moves outward.

**Status**: DRAFT - Framework complete, proof details needed

## Reference
- Schema: `blackhole_rift_charge_rotation.json` (parameter: rift_history_depth)
- Python: `blackhole-dynamics/simulation.py` (track rift history)
- PHYSICS_REVIEW.md: Lines 33-35 (charge separation mechanism)
-/

noncomputable section

namespace QFD.Rift.SequentialEruptions

open Real QFD.Rift.ChargeEscape Nat

/-! ## 1. Rift Event Structure -/

/-- A single rift eruption event -/
structure RiftEvent where
  /-- Event index (0 = first rift, 1 = second, etc.) -/
  index : ℕ
  /-- Radius of eruption (modified Schwarzschild surface at this event) -/
  r_eruption : ℝ
  /-- Net charge deposited at surface after this event -/
  charge_deposited : ℝ
  /-- Number of particles ejected -/
  num_ejected : ℕ
  /-- Fraction that were electrons (vs ions) -/
  electron_fraction : ℝ
  /-- Constraints -/
  r_pos : 0 < r_eruption
  charge_sign : charge_deposited > 0  -- Net positive (ions left behind)
  elec_frac_bound : 0 ≤ electron_fraction ∧ electron_fraction ≤ 1
  electron_majority : electron_fraction > 0.5  -- Electrons escape first

/-- History of rift events -/
def RiftHistory := List RiftEvent

/-! ## 2. Charge Accumulation -/

/-- Total charge at BH surface after n rift events.

    **Formula**: Q_total(n) = Σᵢ₌₀ⁿ Q_deposited(i)

    **Monotonicity**: Q_total(n+1) > Q_total(n)
    Each rift adds positive charge (electrons preferentially escape).
-/
def total_surface_charge (history : RiftHistory) : ℝ :=
  history.foldl (fun acc event => acc + event.charge_deposited) 0

/-! ## 3. Main Theorem: Monotonic Charge Accumulation -/

/-- **Theorem**: Surface charge increases monotonically with each rift event.

    **Proof**:
    1. At rift event n, plasma erupts
    2. Electrons have higher v_thermal (lower mass) → escape preferentially
    3. Net positive charge left: Q_n = (N_ions - N_electrons) * e
    4. Since electron_fraction > 0.5, more electrons escape
    5. Q_n > 0 (positive charge deposited)
    6. Q_total(n+1) = Q_total(n) + Q_n > Q_total(n)
    7. Therefore: Q_total is strictly increasing
-/
theorem charge_accumulation_monotonic
    (history : RiftHistory)
    (h_nonempty : history.length > 0)
    -- Mathematical assumption: List fold accumulation property
    -- This states that when you add one more element to a list and fold with addition,
    -- the result equals the previous sum plus the new element's value.
    -- This is a standard property of list folding that could be proven from Mathlib.
    (foldl_take_succ_eq :
      ∀ (lst : List RiftEvent) (n : ℕ) (h_n_lt : n < lst.length),
        (lst.take (n+1)).foldl (fun acc e => acc + e.charge_deposited) 0 =
        (lst.take n).foldl (fun acc e => acc + e.charge_deposited) 0 +
        (lst.get ⟨n, h_n_lt⟩).charge_deposited) :
    ∀ (n : ℕ), n < history.length - 1 →
      total_surface_charge (history.take (n+1)) >
      total_surface_charge (history.take n) := by
  intro n hn
  unfold total_surface_charge
  have h_len : n < history.length := by omega
  -- The charge at position n is positive (electrons escape preferentially)
  have h_charge_pos : (history.get ⟨n, h_len⟩).charge_deposited > 0 :=
    (history.get ⟨n, h_len⟩).charge_sign
  -- Apply the fold accumulation hypothesis with explicit proof of n < history.length
  rw [foldl_take_succ_eq history n h_len]
  -- Now we have: sum(take n) + charge[n] > sum(take n)
  -- This is true because charge[n] > 0
  linarith

/-! ## 4. Eruption Radius Evolution -/

/-- **Theorem**: Eruption radius moves outward with each successive rift.

    **Physical reason**:
    1. More charge at surface → stronger Coulomb repulsion
    2. Repulsion reaches farther out
    3. Modified Schwarzschild surface (where E_total > E_binding) moves outward
    4. Therefore: r_eruption(n+1) > r_eruption(n)

    **Consequence**: Earlier rifts were closer to BH → higher charge density.

    **Physics Assumption** (now a hypothesis):
    Assumes radius increases with accumulated charge (Coulomb repulsion effect).
-/
theorem eruption_radius_increases
    (history : RiftHistory)
    (h_ordered : ∀ i j (hi : i < history.length) (hj : j < history.length),
                  i < j → (history.get ⟨i, hi⟩).index < (history.get ⟨j, hj⟩).index)
    -- Mathematical assumption: List fold accumulation (needed for charge monotonicity)
    (foldl_take_succ_eq :
      ∀ (lst : List RiftEvent) (n : ℕ) (h_n_lt : n < lst.length),
        (lst.take (n+1)).foldl (fun acc e => acc + e.charge_deposited) 0 =
        (lst.take n).foldl (fun acc e => acc + e.charge_deposited) 0 +
        (lst.get ⟨n, h_n_lt⟩).charge_deposited)
    -- Physics assumption: radius increases with charge
    (radius_increases_with_charge :
      ∀ {n : ℕ} (hn : n < history.length - 1),
        total_surface_charge (history.take (n+1)) > total_surface_charge (history.take n) →
        (history.get ⟨n+1, by omega⟩).r_eruption > (history.get ⟨n, by omega⟩).r_eruption) :
    ∀ (n : ℕ) (hn : n < history.length - 1),
      let hn1 : n + 1 < history.length := by omega
      (history.get ⟨n+1, hn1⟩).r_eruption > (history.get ⟨n, by omega⟩).r_eruption := by
  intro n hn
  have h_charge_inc : total_surface_charge (history.take (n + 1)) > total_surface_charge (history.take n) :=
    charge_accumulation_monotonic history (by omega) foldl_take_succ_eq n hn
  exact radius_increases_with_charge hn h_charge_inc

/-! ## 5. Charge Separation Fraction -/

/-- Fraction of ions left behind by previous rifts.

    **Definition**: f_sep = (N_ions_remaining) / (N_ions_total)

    **From schema**: f_sep ∈ [0.01, 0.5]
    - Low end (0.01): Weak separation, most ions escape
    - High end (0.5): Strong separation, half of ions remain

    **Dependence**: f_sep increases with rift history depth
    → More rifts → more accumulated charge → harder for ions to escape
-/
def charge_separation_fraction (history : RiftHistory) : ℝ :=
  if history.length = 0 then 0
  else
    let total_ions_deposited := history.foldl
      (fun acc event => acc + (event.num_ejected : ℝ) * (1 - event.electron_fraction))
      0
    let total_ions_erupted := history.foldl
      (fun acc event => acc + (event.num_ejected : ℝ))
      0
    total_ions_deposited / total_ions_erupted

/-- **Theorem**: Charge separation fraction increases with rift depth.

    More rift history → more accumulated charge → stronger ion retention.

    **Physics Assumption** (now a hypothesis):
    Assumes separation fraction increases with charge (Coulomb barrier effect).
-/
theorem separation_fraction_increases_with_depth
    (history : RiftHistory)
    (h_nonempty : history.length > 0)
    -- Mathematical assumption: List fold accumulation (needed for charge monotonicity)
    (foldl_take_succ_eq :
      ∀ (lst : List RiftEvent) (n : ℕ) (h_n_lt : n < lst.length),
        (lst.take (n+1)).foldl (fun acc e => acc + e.charge_deposited) 0 =
        (lst.take n).foldl (fun acc e => acc + e.charge_deposited) 0 +
        (lst.get ⟨n, h_n_lt⟩).charge_deposited)
    -- Physics assumption: separation fraction increases with charge
    (separation_fraction_increases_with_charge :
      ∀ {n : ℕ} (hn : n < history.length - 1),
        total_surface_charge (history.take (n+1)) > total_surface_charge (history.take n) →
        charge_separation_fraction (history.take (n+1)) > charge_separation_fraction (history.take n)) :
    ∀ (n : ℕ), n < history.length - 1 →
      charge_separation_fraction (history.take (n+1)) >
      charge_separation_fraction (history.take n) := by
  intro n hn
  have h_charge_inc : total_surface_charge (history.take (n + 1)) > total_surface_charge (history.take n) :=
    charge_accumulation_monotonic history h_nonempty foldl_take_succ_eq n hn
  exact separation_fraction_increases_with_charge hn h_charge_inc

/-! ## 6. Connection to Electron Thermal Advantage -/

/-- Use ChargeEscape theorem on electron thermal advantage.

    **Relevance**: Electrons escape first because of lighter mass.
    This is WHY charge accumulation happens (mass ratio m_p/m_e ≈ 1836).
-/
theorem electrons_escape_preferentially_in_rifts
    (T : ℝ) (m_e m_p : ℝ)
    (h_T : 0 < T) (h_me : 0 < m_e) (h_mp : 0 < m_p)
    (h_ratio : m_p > 1800 * m_e) :  -- Proton ~1836× heavier
    (2 * k_boltzmann * T / m_e) > (2 * k_boltzmann * T / m_p) := by
  exact electron_thermal_advantage T m_e m_p h_T h_me h_mp (by linarith)

end QFD.Rift.SequentialEruptions