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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>
76 lines
3.5 KiB
Text
76 lines
3.5 KiB
Text
/-
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Copyright (c) 2026 SilverSight Contributors. All rights reserved.
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MathlibConnect.lean — Connects our Eisenstein series to Mathlib's modular forms library.
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Mathlib already defines:
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• Normalized Eisenstein series `E k : ModularForm Γ(1) k` for even k ≥ 3
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(EisensteinSeries/Basic.lean)
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• Their q-expansion coefficients: coeff₀ = 1, coeffₘ = -(2k/Bₖ)·σ_{k-1}(m)
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(EisensteinSeries/QExpansion.lean)
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For k = 4: -(2·4 / B₄) = -(8 / (-1/30)) = 240 → E 4 = 1 + 240 Σ σ₃(n) qⁿ = our E4
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For k = 8: -(2·8 / B₈) = -(16 / (-1/30)) = 480 → E 8 = 1 + 480 Σ σ₇(n) qⁿ = our E8
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The missing piece is the dimension formula dim M₈(Γ(1)) = 1
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(TODO in Mathlib/NumberTheory/ModularForms/LevelOne.lean).
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Once that is available, E₄² = E₈ follows from:
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1. E₄ ∈ M₄, E₈ ∈ M₈ (via Eisenstein series)
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2. E₄² ∈ M₈ (ring structure)
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3. dim M₈ = 1 → E₄² = λ·E₈
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4. Constant term: 1 = λ·1 → λ = 1 → E₄² = E₈
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5. q-expansion coefficients: σ₇ = σ₃ + 120·(σ₃∗σ₃)
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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.MathlibConnect
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set_option linter.unusedVariables false
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-- ============================================================================
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-- §1 Bernoulli normalization constants
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-- ============================================================================
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/-- -(2·4 / B₄) = 240 (in ℂ). -/
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theorem E4_normalization : -(2 * (4 : ℂ) / ((bernoulli 4 : ℚ) : ℂ)) = (240 : ℂ) := by
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have hB4 : (bernoulli 4 : ℚ) = -1/30 := by native_decide
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rw [hB4]; norm_num
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/-- -(2·8 / B₈) = 480 (in ℂ). -/
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theorem E8_normalization : -(2 * (8 : ℂ) / ((bernoulli 8 : ℚ) : ℂ)) = (480 : ℂ) := by
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have hB8 : (bernoulli 8 : ℚ) = -1/30 := by native_decide
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rw [hB8]; norm_num
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-- ============================================================================
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-- §2 Connecting to Mathlib's normalized Eisenstein series
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-- ============================================================================
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/-- Mathlib's `E hk` (normalized Eisenstein series of weight k) has q-expansion
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coefficients matching our formal E4/E8 QExpansions.
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See EisensteinSeries.QExpansion.lean, lemma E_qExpansion_coeff:
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(qExpansion 1 (E hk)).coeff m = if m = 0 then 1 else -(2k/B_k) · σ_{k-1}(m)
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This is used below for k=4 and k=8. The proof uses native_decide for the
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Bernoulli constant and the divisor sum functions already defined in Mathlib. -/
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theorem E4_qExpansion_matches (n : ℕ) : (ModularFormClass.qExpansion 1 (ModularForm.E (by decide : 3 ≤ 4))).coeff n = ((E4 n : ℚ) : ℂ) := by
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have hk4 : 3 ≤ (4 : ℕ) := by decide
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have hk4_even : Even (4 : ℕ) := by decide
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rcases n with (rfl | n)
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· simpa [E4] using EisensteinSeries.E_qExpansion_coeff_zero hk4 hk4_even
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· have hcoeff := EisensteinSeries.E_qExpansion_coeff hk4 hk4_even (n+1)
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simpa [E4, E4_normalization, ArithmeticFunction.sigma_apply, sigma, sigma3, Nat.succ_eq_add_one] using hcoeff
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theorem E8_qExpansion_matches (n : ℕ) : (ModularFormClass.qExpansion 1 (ModularForm.E (by decide : 3 ≤ 8))).coeff n = ((E8 n : ℚ) : ℂ) := by
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have hk8 : 3 ≤ (8 : ℕ) := by decide
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have hk8_even : Even (8 : ℕ) := by decide
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rcases n with (rfl | n)
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· simpa [E8] using EisensteinSeries.E_qExpansion_coeff_zero hk8 hk8_even
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· have hcoeff := EisensteinSeries.E_qExpansion_coeff hk8 hk8_even (n+1)
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simpa [E8, E8_normalization, ArithmeticFunction.sigma_apply, sigma, sigma7, Nat.succ_eq_add_one] using hcoeff
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end SilverSight.MathlibConnect
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