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

Eisenstein.lean — The E₈ = E₄² divisor convolution identity.

Architecture (virtual proof):
  The analytic gap (E₄² = E₈ as formal q-series) is isolated into the single
  structure `EisensteinBridge`. From that bridge, the divisor convolution
  identity is deduced entirely by verified algebra — no sorries.

Proof status:
  ✓ Finite bounds (n ≤ 50) proven unconditionally via kernel computation.
  ✓ Algebraic deduction from bridge (0 sorries in deduction).
  ☐ Bridge lemma E₄² = E₈ — requires modular forms (dim M₈ = 1).
    Isolated in `EisensteinBridge`; provable once Mathlib has modular curves.
-/

import Mathlib

open scoped BigOperators

namespace SilverSight.Eisenstein

set_option linter.unusedVariables false

-- ============================================================================
-- §1  Divisor sums and arithmetic convolution
-- ============================================================================

/-- Divisor sum σₖ(n) = Σ_{d|n} dᵏ (in ℕ). -/
def sigma (k n : ℕ) : ℕ := ∑ d ∈ Nat.divisors n, d ^ k

def sigma3 (n : ℕ) : ℕ := sigma 3 n
def sigma7 (n : ℕ) : ℕ := sigma 7 n

/-- Self-convolution of σ₃: Σ_{j=1}^{n-1} σ₃(j)·σ₃(n-j). -/
def convolutionSum (n : ℕ) : ℕ :=
  ∑ j ∈ Finset.Ico 1 n, sigma3 j * sigma3 (n - j)

-- ============================================================================
-- §2  Finite bounds (unconditional, 0 sorries)
-- ============================================================================

/-- The identity holds for 1 ≤ n ≤ 50, proven by kernel computation. -/
theorem eisenstein_identity_finite (n : ℕ) (h1 : 1 ≤ n) (h2 : n ≤ 50) :
    sigma7 n = sigma3 n + 120 * convolutionSum n := by
  interval_cases n <;> decide

-- ============================================================================
-- §3  Formal q-expansions and the bridge
-- ============================================================================

/-- A q-expansion is a sequence of coefficients (formal power series ℕ → ℚ). -/
def QExpansion := ℕ → ℚ

/-- Cauchy product of two q-expansions: (f∗g)(n) = Σ_{j=0}^{n} f(j)·g(n-j). -/
def cauchyProduct (f g : QExpansion) (n : ℕ) : ℚ :=
  ∑ j ∈ Finset.range (n+1), f j * g (n - j)

/-- Normalized E₄: coefficient sequence 1, 240·σ₃(1), 240·σ₃(2), ... -/
noncomputable def E4 : QExpansion :=
  λ n => if n = 0 then 1 else 240 * (sigma 3 n : ℚ)

/-- Normalized E₈: coefficient sequence 1, 480·σ₇(1), 480·σ₇(2), ... -/
noncomputable def E8 : QExpansion :=
  λ n => if n = 0 then 1 else 480 * (sigma 7 n : ℚ)

/--
  THE BRIDGE — The single analytic gap.

  EisensteinBridge asserts the formal q-series identity E₄² = E₈.
  This is the coefficient-level equality corresponding to the modular form
  identity E₈ = E₄², which follows from dim M₈(SL₂(ℤ)) = 1 (Riemann-Roch).

  Once Mathlib's modular curves infrastructure is complete, this structure
  can be inhabited by a proof. The deduction below needs no other assumption.
-/
structure EisensteinBridge : Prop where
  square_eq : cauchyProduct E4 E4 = E8

-- ============================================================================
-- §4  Algebraic deduction from the bridge (0 sorries)
-- ============================================================================

/--
  The divisor convolution identity follows algebraically from the bridge.
  Given EisensteinBridge, deduces for all n > 0:
    σ₇(n) = σ₃(n) + 120 · Σ_{j=1}^{n-1} σ₃(j)·σ₃(n−j)
-/
theorem ramanujan_convolution_from_bridge (bridge : EisensteinBridge) (n : ℕ) (hn : 0 < n) :
    sigma7 n = sigma3 n + 120 * convolutionSum n := by
  -- Express (E₄²)ₙ in terms of sigma3 by expanding the Cauchy product
  have hE4sq : cauchyProduct E4 E4 n = (480 : ℚ) * (sigma 3 n : ℚ) + (57600 : ℚ) * ((convolutionSum n : ℕ) : ℚ) := by
    unfold cauchyProduct E4
    have hn0 : n ≠ 0 := by omega
    have hsplit : Finset.range (n+1) = ({0} : Finset ℕ) ∪ (Finset.Ico 1 n) ∪ ({n} : Finset ℕ) := by
      ext x; simp [Finset.mem_range, Finset.mem_Ico, Finset.mem_insert]; omega
    have h0mem : 0 ∉ Finset.Ico 1 n := by simp [Finset.mem_Ico]
    have hnmem : n ∉ insert 0 (Finset.Ico 1 n) := by
      simp [Finset.mem_insert, Finset.mem_Ico, hn0]
    have h0_union : ({0} : Finset ℕ) ∪ (Finset.Ico 1 n) = insert 0 (Finset.Ico 1 n) := by ext x; simp
    have h1_union : (insert 0 (Finset.Ico 1 n)) ∪ ({n} : Finset ℕ) = insert n (insert 0 (Finset.Ico 1 n)) := by
      ext x; simp [Finset.mem_insert, Finset.mem_Ico]; omega
    calc
      ∑ j ∈ Finset.range (n+1), (if j = 0 then (1 : ℚ) else 240 * (sigma 3 j : ℚ)) *
        (if n - j = 0 then (1 : ℚ) else 240 * (sigma 3 (n - j) : ℚ))
          = ∑ j ∈ ({0} : Finset ℕ) ∪ (Finset.Ico 1 n) ∪ ({n} : Finset ℕ),
              (if j = 0 then (1 : ℚ) else 240 * (sigma 3 j : ℚ)) *
              (if n - j = 0 then (1 : ℚ) else 240 * (sigma 3 (n - j) : ℚ)) := by rw [hsplit]
      _ = ∑ j ∈ insert n (insert 0 (Finset.Ico 1 n)),
          (if j = 0 then (1 : ℚ) else 240 * (sigma 3 j : ℚ)) *
          (if n - j = 0 then (1 : ℚ) else 240 * (sigma 3 (n - j) : ℚ)) := by
        rw [h0_union, h1_union]
      _ = (240 : ℚ) * (sigma 3 n : ℚ) + (57600 : ℚ) * ((convolutionSum n : ℕ) : ℚ) + (240 : ℚ) * (sigma 3 n : ℚ) := by
        rw [Finset.sum_insert hnmem, Finset.sum_insert h0mem]
        have hfn : (if n = 0 then (1 : ℚ) else 240 * (sigma 3 n : ℚ)) *
          (if n - n = 0 then (1 : ℚ) else 240 * (sigma 3 (n - n) : ℚ)) = (240 : ℚ) * (sigma 3 n : ℚ) := by
          simp [hn0]
        have hf0 : (if (0 : ℕ) = 0 then (1 : ℚ) else 240 * (sigma 3 0 : ℚ)) *
          (if n - 0 = 0 then (1 : ℚ) else 240 * (sigma 3 (n - 0) : ℚ)) = (240 : ℚ) * (sigma 3 n : ℚ) := by
          simp [hn0]
        have hIco : ∑ x ∈ Finset.Ico 1 n, (if x = 0 then (1 : ℚ) else 240 * (sigma 3 x : ℚ)) *
          (if n - x = 0 then (1 : ℚ) else 240 * (sigma 3 (n - x) : ℚ))
          = (57600 : ℚ) * ((convolutionSum n : ℕ) : ℚ) := by
          calc
            ∑ x ∈ Finset.Ico 1 n, (if x = 0 then (1 : ℚ) else 240 * (sigma 3 x : ℚ)) *
              (if n - x = 0 then (1 : ℚ) else 240 * (sigma 3 (n - x) : ℚ))
                = ∑ x ∈ Finset.Ico 1 n, (240 * (sigma 3 x : ℚ)) * (240 * (sigma 3 (n - x) : ℚ)) := by
              refine Finset.sum_congr rfl ?_
              intro x hx
              have hx0 : x ≠ 0 := by
                have hxmem := Finset.mem_Ico.1 hx; omega
              have hn_x0 : n - x ≠ 0 := by
                have hxmem := Finset.mem_Ico.1 hx; omega
              simp [hx0, hn_x0]
            _ = (57600 : ℚ) * ((convolutionSum n : ℕ) : ℚ) := by
              calc
                ∑ x ∈ Finset.Ico 1 n, (240 * (sigma 3 x : ℚ)) * (240 * (sigma 3 (n - x) : ℚ))
                    = ∑ x ∈ Finset.Ico 1 n, (57600 : ℚ) * ((sigma 3 x : ℚ) * (sigma 3 (n - x) : ℚ)) := by
                  refine Finset.sum_congr rfl (λ x hx => ?_)
                  ring
                _ = (57600 : ℚ) * (∑ x ∈ Finset.Ico 1 n, (sigma 3 x : ℚ) * (sigma 3 (n - x) : ℚ)) := by
                  simp [Finset.mul_sum]
                _ = (57600 : ℚ) * ((convolutionSum n : ℕ) : ℚ) := by
                  simp [convolutionSum, Nat.cast_sum, Nat.cast_mul, sigma3]
        rw [hfn, hf0, hIco]
        ring
      _ = (480 : ℚ) * (sigma 3 n : ℚ) + (57600 : ℚ) * ((convolutionSum n : ℕ) : ℚ) := by ring
  -- From the bridge: (E₄²)ₙ = (E₈)ₙ
  have hcoeff : cauchyProduct E4 E4 n = E8 n := by rw [bridge.square_eq]
  rw [hE4sq] at hcoeff
  have hn0 : n ≠ 0 := by omega
  have hE8val : E8 n = (480 : ℚ) * (sigma 7 n : ℚ) := by
    unfold E8; simp [hn0]
  have hcoeff' : (480 : ℚ) * (sigma 3 n : ℚ) + (57600 : ℚ) * ((convolutionSum n : ℕ) : ℚ) = (480 : ℚ) * (sigma 7 n : ℚ) :=
    calc
      (480 : ℚ) * (sigma 3 n : ℚ) + (57600 : ℚ) * ((convolutionSum n : ℕ) : ℚ) = E8 n := hcoeff
      _ = (480 : ℚ) * (sigma 7 n : ℚ) := by rw [hE8val]
  -- hcoeff: 480·σ₇(n) = 480·σ₃(n) + 57600·convolutionSum(n)  [in ℚ]
  have h_rat : (sigma 7 n : ℚ) = (sigma 3 n : ℚ) + (120 : ℚ) * ((convolutionSum n : ℕ) : ℚ) := by
    nlinarith
  exact_mod_cast h_rat

end SilverSight.Eisenstein
