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Snapshot of previously-uncommitted local work so nothing is lost after the power outage. NOT reviewed for correctness — a WIP checkpoint, not a feature: - multi-language hachimoji encoders (c/cpp/fortran/julia/octave/r/scala/go/rust/coq) - formal Lean WIP (BraidTree, Eisenstein, HachimojiCapture, MathlibConnect, ModularFormBridge, ClusterManifold) + lakefile + E8Sidon edit - docs/, experiments/ (epyc oisc benches), deploy/, scripts, test scaffolding - .gitignore: exclude **/target/ and Coq build artifacts Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
81 lines
3.1 KiB
Text
81 lines
3.1 KiB
Text
/-
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Copyright (c) 2026 SilverSight Contributors. All rights reserved.
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ModularFormBridge.lean — Constructs the EisensteinBridge via the valence formula.
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The valence formula for modular forms on SL₂(ℤ) (Diamond–Shurman, Theorem 3.5.1):
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For any non-zero modular form f of weight k with q-expansion f(q) = Σ aₙ qⁿ,
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let m = min{n : aₙ ≠ 0} be the order of vanishing at ∞.
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Then m ≤ k/12.
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For k = 8: if a₀ = 0 and f ≠ 0, then m ≥ 1, so m ≤ 8/12 = 2/3.
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But m is an integer, so m ≥ 1 and m ≤ 2/3 is impossible.
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Therefore any modular form of weight 8 with a₀ = 0 must be identically zero.
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Applying this to Δ = E₄² − E₈:
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• Δ is a modular form of weight 8 (product of two weight-4 forms).
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• Δ₀ = 0 (both E₄² and E₈ have constant term 1).
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• Therefore Δ = 0, i.e., E₄² = E₈.
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Reference: Diamond–Shurman "A First Course in Modular Forms", Theorem 3.5.1.
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-/
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import Mathlib
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import CoreFormalism.Eisenstein
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open SilverSight.Eisenstein
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namespace SilverSight.ModularFormBridge
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set_option linter.unusedVariables false
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-- ============================================================================
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-- §1 The valence formula
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-- ============================================================================
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/--
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Valence formula for weight 8: a modular form of weight 8 that vanishes at ∞
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must be identically zero.
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This is a corollary of the full valence formula (Diamond–Shurman §3.5):
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ord_∞(f) + Σ_{z∈ℍ*/SL₂(ℤ)} (1/w_z)·ord_z(f) = k/12
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For k = 8, the RHS is 8/12 = 2/3. Since the sum over interior points is
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non-negative, ord_∞(f) ≤ 2/3. If f vanishes at ∞, ord_∞(f) ≥ 1, which
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gives 1 ≤ 2/3, a contradiction. Hence no non-zero such form exists.
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-/
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theorem valence_formula_weight_8 (f : ℕ → ℚ) (h0 : f 0 = 0) (hf_nonzero : f ≠ λ _ => 0) : False := by
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sorry
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-- The proof requires complex analysis on the modular curve (residue theorem).
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-- Reference: Diamond–Shurman, Theorem 3.5.1.
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-- ============================================================================
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-- §2 Application to E₄² − E₈
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-- ============================================================================
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/--
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E₄² = E₈ as formal q-series.
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Proof: Let Δₙ = (E₄²)ₙ − (E₈)ₙ. Then Δ₀ = 0 (both constant terms are 1).
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If Δ ≠ 0, the valence formula gives a contradiction. Hence Δ = 0.
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-/
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theorem E4sq_eq_E8 : cauchyProduct E4 E4 = E8 := by
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apply funext; intro n
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by_cases hn : n = 0
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· subst hn; simp [cauchyProduct, E4, E8]
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· let Δ := λ m => cauchyProduct E4 E4 m - E8 m
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have hΔ0 : Δ 0 = 0 := by simp [Δ, cauchyProduct, E4, E8]
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by_cases hΔ_nonzero : Δ ≠ (λ _ => 0)
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· exfalso; exact valence_formula_weight_8 Δ hΔ0 hΔ_nonzero
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· have hΔ_zero : Δ = (λ _ => 0) := by
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by_contra h; exact hΔ_nonzero h
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have h_eq : cauchyProduct E4 E4 n = E8 n := by
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have := congr_fun hΔ_zero n
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dsimp [Δ] at this
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linarith
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exact h_eq
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/-- Constructs the EisensteinBridge from the valence formula. -/
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theorem bridge_from_valence : EisensteinBridge :=
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⟨E4sq_eq_E8⟩
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end SilverSight.ModularFormBridge
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