/-
Copyright (c) 2026 SilverSight Contributors. All rights reserved.

MathlibConnect.lean — Connects our Eisenstein series to Mathlib's modular forms library.

Mathlib already defines:
  • Normalized Eisenstein series `E k : ModularForm Γ(1) k` for even k ≥ 3
    (EisensteinSeries/Basic.lean)
  • Their q-expansion coefficients: coeff₀ = 1, coeffₘ = -(2k/Bₖ)·σ_{k-1}(m)
    (EisensteinSeries/QExpansion.lean)

For k = 4: -(2·4 / B₄) = -(8 / (-1/30)) = 240 → E 4 = 1 + 240 Σ σ₃(n) qⁿ = our E4
For k = 8: -(2·8 / B₈) = -(16 / (-1/30)) = 480 → E 8 = 1 + 480 Σ σ₇(n) qⁿ = our E8

The missing piece is the dimension formula dim M₈(Γ(1)) = 1
(TODO in Mathlib/NumberTheory/ModularForms/LevelOne.lean).
Once that is available, E₄² = E₈ follows from:
  1. E₄ ∈ M₄, E₈ ∈ M₈ (via Eisenstein series)
  2. E₄² ∈ M₈ (ring structure)
  3. dim M₈ = 1 → E₄² = λ·E₈
  4. Constant term: 1 = λ·1 → λ = 1 → E₄² = E₈
  5. q-expansion coefficients: σ₇ = σ₃ + 120·(σ₃∗σ₃)
-/

import Mathlib
import CoreFormalism.Eisenstein

open SilverSight.Eisenstein

namespace SilverSight.MathlibConnect

set_option linter.unusedVariables false

-- ============================================================================
-- §1  Bernoulli normalization constants
-- ============================================================================

/-- -(2·4 / B₄) = 240 (in ℂ). -/
theorem E4_normalization : -(2 * (4 : ℂ) / ((bernoulli 4 : ℚ) : ℂ)) = (240 : ℂ) := by
  have hB4 : (bernoulli 4 : ℚ) = -1/30 := by native_decide
  rw [hB4]; norm_num

/-- -(2·8 / B₈) = 480 (in ℂ). -/
theorem E8_normalization : -(2 * (8 : ℂ) / ((bernoulli 8 : ℚ) : ℂ)) = (480 : ℂ) := by
  have hB8 : (bernoulli 8 : ℚ) = -1/30 := by native_decide
  rw [hB8]; norm_num

-- ============================================================================
-- §2  Connecting to Mathlib's normalized Eisenstein series
-- ============================================================================

/-- Mathlib's `E hk` (normalized Eisenstein series of weight k) has q-expansion
    coefficients matching our formal E4/E8 QExpansions.

    See EisensteinSeries.QExpansion.lean, lemma E_qExpansion_coeff:
      (qExpansion 1 (E hk)).coeff m = if m = 0 then 1 else -(2k/B_k) · σ_{k-1}(m)

    This is used below for k=4 and k=8. The proof uses native_decide for the
    Bernoulli constant and the divisor sum functions already defined in Mathlib. -/
theorem E4_qExpansion_matches (n : ℕ) : (ModularFormClass.qExpansion 1 (ModularForm.E (by decide : 3 ≤ 4))).coeff n = ((E4 n : ℚ) : ℂ) := by
  have hk4 : 3 ≤ (4 : ℕ) := by decide
  have hk4_even : Even (4 : ℕ) := by decide
  rcases n with (rfl | n)
  · simpa [E4] using EisensteinSeries.E_qExpansion_coeff_zero hk4 hk4_even
  · have hcoeff := EisensteinSeries.E_qExpansion_coeff hk4 hk4_even (n+1)
    simpa [E4, E4_normalization, ArithmeticFunction.sigma_apply, sigma, sigma3, Nat.succ_eq_add_one] using hcoeff

theorem E8_qExpansion_matches (n : ℕ) : (ModularFormClass.qExpansion 1 (ModularForm.E (by decide : 3 ≤ 8))).coeff n = ((E8 n : ℚ) : ℂ) := by
  have hk8 : 3 ≤ (8 : ℕ) := by decide
  have hk8_even : Even (8 : ℕ) := by decide
  rcases n with (rfl | n)
  · simpa [E8] using EisensteinSeries.E_qExpansion_coeff_zero hk8 hk8_even
  · have hcoeff := EisensteinSeries.E_qExpansion_coeff hk8 hk8_even (n+1)
    simpa [E8, E8_normalization, ArithmeticFunction.sigma_apply, sigma, sigma7, Nat.succ_eq_add_one] using hcoeff

end SilverSight.MathlibConnect
