/-
  EisensteinSeries.lean — Modular Forms Stub Infrastructure

  STATUS: SCAFFOLD — all theorems are HONESTY CLASS: CONJECTURE.
  Full formalization requires the Mathlib modular forms module (T2.1 in
  mathlib_missing_machinery.md). This stub provides the type signatures
  and honest CONJECTURE axioms so the codebase compiles honestly while
  waiting for Mathlib.

  References:
    Diamond-Shurman, "A First Course in Modular Forms", §4
    Serre, "A Course in Arithmetic", Ch. VII
    Koblitz, "Introduction to Elliptic Curves and Modular Forms", Ch. III

  Mathlib already has:
    - UpperHalfPlane (H := {z | im z > 0})
    - ModularGroup SL(2,Z) action on H
    - ArithmeticFunction.sigma (divisor sum) = σ_k(n) — PROVEN multiplicative
    - EisensteinSeries/QExpansion.lean (partial — q-expansion coefficients)

  What's MISSING (T2.1):
    - M_k(SL(2,Z)) as finite-dimensional C-vector space
    - Eisenstein series E_k for all even weights k ≥ 4
    - Proof that M_* ≅ C[E_4, E_6]
    - Rankin-Cohen brackets (for E_4^2 = E_8 identity)
    - Hecke operators
-/

import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion

namespace CoreFormalism.EisensteinSeries

open ArithmeticFunction

/-! ## §1  Divisor Sum Stubs

  σ_k(n) = Σ_{d|n} d^k. Already available in Mathlib (ArithmeticFunction.sigma).
  What's missing: the convolution identity that gives E_4^2 = E_8.
-/

/-- σ₃ convolution: Σ_{j=1}^{n-1} σ₃(j)·σ₃(n-j).
    This is the self-convolution that appears in E_4^2 = E_8. -/
def sigma3_conv (n : Nat) : Nat :=
  ∑ j ∈ Finset.Icc 1 (n - 1), sigma 3 j * sigma 3 (n - j)

/-- HONESTY CLASS: CONJECTURE
  The E_8 convolution identity:
  σ₇(n) = σ₃(n) + 120·Σ_{j=1}^{n-1} σ₃(j)·σ₃(n-j)

  Proven by computation for n ≤ 16 (E8Sidon.e8_conv_identity_16).
  Proven for n ≤ 200 by tab extension (E8Sidon.e8_conv_identity_200, T1.1).
  For all n: equivalent to E_4^2 = E_8 in the ring of modular forms.
  BLOCKED ON: Mathlib modular forms module (T2.1/T3.1). -/
axiom e8_convolution_identity (n : Nat) :
    sigma 7 n = sigma 3 n + 120 * sigma3_conv n

/-! ## §2  Eisenstein Series Type Stubs

  E_k : H → C is a holomorphic modular form of weight k for SL(2,Z).
  For even k ≥ 4, its Fourier expansion begins:
    E_k(z) = 1 + (-1)^{k/2}·(2k/B_k)·Σ_{n≥1} σ_{k-1}(n)·q^n
  where q = e^{2πiz}.
-/

/-- HONESTY CLASS: CONJECTURE
  The Eisenstein series of weight k (k ≥ 4 even) as a function H → C.
  Mathlib stub: only the type exists; the full definition needs T2.1. -/
opaque Eisenstein (k : Nat) : Type

/-- HONESTY CLASS: CONJECTURE
  The q-expansion of E_k gives divisor sum coefficients:
  E_k(z) = 1 + c_k·Σ_{n≥1} σ_{k-1}(n)·q^n
  where c_k = -2k/B_k is a rational constant. -/
axiom eisenstein_fourier (k n : Nat) (hk : 4 ≤ k) (hk_even : Even k) :
    True
  -- Placeholder: the actual statement needs q-expansion API

/-- HONESTY CLASS: CONJECTURE
  The weight-4 Eisenstein series squaring identity:
  E_4^2 = E_8 in M_8(SL(2,Z)).
  At the Fourier coefficient level, this yields the e8_convolution_identity.

  This is the single most important missing theorem for the number theory
  bridge between the E8 lattice and the Sidon set construction.
  Expected to be proven once Mathlib has Rankin-Cohen brackets. -/
axiom e4_squared_equals_e8 : True
  -- Placeholder: the actual statement needs modular forms ring API

end CoreFormalism.EisensteinSeries
