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- SidonAdapter.lean: updated Singer construction integration - UnitDistCandidateGen.lean: enhanced golden-angle perturbation - WeightCandidateGen.lean: cmix weight matrix -> ManifoldShortcut - ComplexProjectiveSpace.lean: Kaehler geometry scaffolding - EisensteinSeries.lean: modular forms stub infrastructure - test scripts: concurrency and Lean LLM integration tests
90 lines
3.4 KiB
Text
90 lines
3.4 KiB
Text
/-
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EisensteinSeries.lean — Modular Forms Stub Infrastructure
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STATUS: SCAFFOLD — all theorems are HONESTY CLASS: CONJECTURE.
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Full formalization requires the Mathlib modular forms module (T2.1 in
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mathlib_missing_machinery.md). This stub provides the type signatures
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and honest CONJECTURE axioms so the codebase compiles honestly while
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waiting for Mathlib.
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References:
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Diamond-Shurman, "A First Course in Modular Forms", §4
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Serre, "A Course in Arithmetic", Ch. VII
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Koblitz, "Introduction to Elliptic Curves and Modular Forms", Ch. III
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Mathlib already has:
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- UpperHalfPlane (H := {z | im z > 0})
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- ModularGroup SL(2,Z) action on H
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- ArithmeticFunction.sigma (divisor sum) = σ_k(n) — PROVEN multiplicative
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- EisensteinSeries/QExpansion.lean (partial — q-expansion coefficients)
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What's MISSING (T2.1):
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- M_k(SL(2,Z)) as finite-dimensional C-vector space
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- Eisenstein series E_k for all even weights k ≥ 4
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- Proof that M_* ≅ C[E_4, E_6]
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- Rankin-Cohen brackets (for E_4^2 = E_8 identity)
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- Hecke operators
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-/
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import Mathlib.NumberTheory.ArithmeticFunction
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import Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion
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namespace CoreFormalism.EisensteinSeries
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open ArithmeticFunction
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/-! ## §1 Divisor Sum Stubs
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σ_k(n) = Σ_{d|n} d^k. Already available in Mathlib (ArithmeticFunction.sigma).
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What's missing: the convolution identity that gives E_4^2 = E_8.
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-/
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/-- σ₃ convolution: Σ_{j=1}^{n-1} σ₃(j)·σ₃(n-j).
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This is the self-convolution that appears in E_4^2 = E_8. -/
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def sigma3_conv (n : Nat) : Nat :=
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∑ j ∈ Finset.Icc 1 (n - 1), sigma 3 j * sigma 3 (n - j)
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/-- HONESTY CLASS: CONJECTURE
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The E_8 convolution identity:
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σ₇(n) = σ₃(n) + 120·Σ_{j=1}^{n-1} σ₃(j)·σ₃(n-j)
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Proven by computation for n ≤ 16 (E8Sidon.e8_conv_identity_16).
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Proven for n ≤ 200 by tab extension (E8Sidon.e8_conv_identity_200, T1.1).
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For all n: equivalent to E_4^2 = E_8 in the ring of modular forms.
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BLOCKED ON: Mathlib modular forms module (T2.1/T3.1). -/
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axiom e8_convolution_identity (n : Nat) :
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sigma 7 n = sigma 3 n + 120 * sigma3_conv n
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/-! ## §2 Eisenstein Series Type Stubs
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E_k : H → C is a holomorphic modular form of weight k for SL(2,Z).
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For even k ≥ 4, its Fourier expansion begins:
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E_k(z) = 1 + (-1)^{k/2}·(2k/B_k)·Σ_{n≥1} σ_{k-1}(n)·q^n
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where q = e^{2πiz}.
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-/
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/-- HONESTY CLASS: CONJECTURE
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The Eisenstein series of weight k (k ≥ 4 even) as a function H → C.
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Mathlib stub: only the type exists; the full definition needs T2.1. -/
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opaque Eisenstein (k : Nat) : Type
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/-- HONESTY CLASS: CONJECTURE
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The q-expansion of E_k gives divisor sum coefficients:
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E_k(z) = 1 + c_k·Σ_{n≥1} σ_{k-1}(n)·q^n
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where c_k = -2k/B_k is a rational constant. -/
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axiom eisenstein_fourier (k n : Nat) (hk : 4 ≤ k) (hk_even : Even k) :
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True
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-- Placeholder: the actual statement needs q-expansion API
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/-- HONESTY CLASS: CONJECTURE
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The weight-4 Eisenstein series squaring identity:
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E_4^2 = E_8 in M_8(SL(2,Z)).
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At the Fourier coefficient level, this yields the e8_convolution_identity.
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This is the single most important missing theorem for the number theory
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bridge between the E8 lattice and the Sidon set construction.
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Expected to be proven once Mathlib has Rankin-Cohen brackets. -/
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axiom e4_squared_equals_e8 : True
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-- Placeholder: the actual statement needs modular forms ring API
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end CoreFormalism.EisensteinSeries
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