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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>
164 lines
8.3 KiB
Text
164 lines
8.3 KiB
Text
/-
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Copyright (c) 2026 SilverSight Contributors. All rights reserved.
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Eisenstein.lean — The E₈ = E₄² divisor convolution identity.
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Architecture (virtual proof):
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The analytic gap (E₄² = E₈ as formal q-series) is isolated into the single
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structure `EisensteinBridge`. From that bridge, the divisor convolution
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identity is deduced entirely by verified algebra — no sorries.
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Proof status:
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✓ Finite bounds (n ≤ 50) proven unconditionally via kernel computation.
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✓ Algebraic deduction from bridge (0 sorries in deduction).
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☐ Bridge lemma E₄² = E₈ — requires modular forms (dim M₈ = 1).
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Isolated in `EisensteinBridge`; provable once Mathlib has modular curves.
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-/
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import Mathlib
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open scoped BigOperators
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namespace SilverSight.Eisenstein
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set_option linter.unusedVariables false
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-- ============================================================================
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-- §1 Divisor sums and arithmetic convolution
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-- ============================================================================
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/-- Divisor sum σₖ(n) = Σ_{d|n} dᵏ (in ℕ). -/
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def sigma (k n : ℕ) : ℕ := ∑ d ∈ Nat.divisors n, d ^ k
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def sigma3 (n : ℕ) : ℕ := sigma 3 n
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def sigma7 (n : ℕ) : ℕ := sigma 7 n
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/-- Self-convolution of σ₃: Σ_{j=1}^{n-1} σ₃(j)·σ₃(n-j). -/
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def convolutionSum (n : ℕ) : ℕ :=
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∑ j ∈ Finset.Ico 1 n, sigma3 j * sigma3 (n - j)
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-- ============================================================================
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-- §2 Finite bounds (unconditional, 0 sorries)
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-- ============================================================================
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/-- The identity holds for 1 ≤ n ≤ 50, proven by kernel computation. -/
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theorem eisenstein_identity_finite (n : ℕ) (h1 : 1 ≤ n) (h2 : n ≤ 50) :
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sigma7 n = sigma3 n + 120 * convolutionSum n := by
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interval_cases n <;> decide
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-- ============================================================================
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-- §3 Formal q-expansions and the bridge
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-- ============================================================================
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/-- A q-expansion is a sequence of coefficients (formal power series ℕ → ℚ). -/
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def QExpansion := ℕ → ℚ
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/-- Cauchy product of two q-expansions: (f∗g)(n) = Σ_{j=0}^{n} f(j)·g(n-j). -/
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def cauchyProduct (f g : QExpansion) (n : ℕ) : ℚ :=
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∑ j ∈ Finset.range (n+1), f j * g (n - j)
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/-- Normalized E₄: coefficient sequence 1, 240·σ₃(1), 240·σ₃(2), ... -/
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noncomputable def E4 : QExpansion :=
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λ n => if n = 0 then 1 else 240 * (sigma 3 n : ℚ)
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/-- Normalized E₈: coefficient sequence 1, 480·σ₇(1), 480·σ₇(2), ... -/
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noncomputable def E8 : QExpansion :=
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λ n => if n = 0 then 1 else 480 * (sigma 7 n : ℚ)
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/--
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THE BRIDGE — The single analytic gap.
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EisensteinBridge asserts the formal q-series identity E₄² = E₈.
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This is the coefficient-level equality corresponding to the modular form
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identity E₈ = E₄², which follows from dim M₈(SL₂(ℤ)) = 1 (Riemann-Roch).
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Once Mathlib's modular curves infrastructure is complete, this structure
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can be inhabited by a proof. The deduction below needs no other assumption.
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-/
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structure EisensteinBridge : Prop where
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square_eq : cauchyProduct E4 E4 = E8
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-- ============================================================================
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-- §4 Algebraic deduction from the bridge (0 sorries)
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-- ============================================================================
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/--
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The divisor convolution identity follows algebraically from the bridge.
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Given EisensteinBridge, deduces for all n > 0:
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σ₇(n) = σ₃(n) + 120 · Σ_{j=1}^{n-1} σ₃(j)·σ₃(n−j)
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-/
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theorem ramanujan_convolution_from_bridge (bridge : EisensteinBridge) (n : ℕ) (hn : 0 < n) :
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sigma7 n = sigma3 n + 120 * convolutionSum n := by
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-- Express (E₄²)ₙ in terms of sigma3 by expanding the Cauchy product
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have hE4sq : cauchyProduct E4 E4 n = (480 : ℚ) * (sigma 3 n : ℚ) + (57600 : ℚ) * ((convolutionSum n : ℕ) : ℚ) := by
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unfold cauchyProduct E4
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have hn0 : n ≠ 0 := by omega
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have hsplit : Finset.range (n+1) = ({0} : Finset ℕ) ∪ (Finset.Ico 1 n) ∪ ({n} : Finset ℕ) := by
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ext x; simp [Finset.mem_range, Finset.mem_Ico, Finset.mem_insert]; omega
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have h0mem : 0 ∉ Finset.Ico 1 n := by simp [Finset.mem_Ico]
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have hnmem : n ∉ insert 0 (Finset.Ico 1 n) := by
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simp [Finset.mem_insert, Finset.mem_Ico, hn0]
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have h0_union : ({0} : Finset ℕ) ∪ (Finset.Ico 1 n) = insert 0 (Finset.Ico 1 n) := by ext x; simp
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have h1_union : (insert 0 (Finset.Ico 1 n)) ∪ ({n} : Finset ℕ) = insert n (insert 0 (Finset.Ico 1 n)) := by
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ext x; simp [Finset.mem_insert, Finset.mem_Ico]; omega
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calc
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∑ j ∈ Finset.range (n+1), (if j = 0 then (1 : ℚ) else 240 * (sigma 3 j : ℚ)) *
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(if n - j = 0 then (1 : ℚ) else 240 * (sigma 3 (n - j) : ℚ))
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= ∑ j ∈ ({0} : Finset ℕ) ∪ (Finset.Ico 1 n) ∪ ({n} : Finset ℕ),
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(if j = 0 then (1 : ℚ) else 240 * (sigma 3 j : ℚ)) *
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(if n - j = 0 then (1 : ℚ) else 240 * (sigma 3 (n - j) : ℚ)) := by rw [hsplit]
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_ = ∑ j ∈ insert n (insert 0 (Finset.Ico 1 n)),
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(if j = 0 then (1 : ℚ) else 240 * (sigma 3 j : ℚ)) *
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(if n - j = 0 then (1 : ℚ) else 240 * (sigma 3 (n - j) : ℚ)) := by
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rw [h0_union, h1_union]
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_ = (240 : ℚ) * (sigma 3 n : ℚ) + (57600 : ℚ) * ((convolutionSum n : ℕ) : ℚ) + (240 : ℚ) * (sigma 3 n : ℚ) := by
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rw [Finset.sum_insert hnmem, Finset.sum_insert h0mem]
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have hfn : (if n = 0 then (1 : ℚ) else 240 * (sigma 3 n : ℚ)) *
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(if n - n = 0 then (1 : ℚ) else 240 * (sigma 3 (n - n) : ℚ)) = (240 : ℚ) * (sigma 3 n : ℚ) := by
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simp [hn0]
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have hf0 : (if (0 : ℕ) = 0 then (1 : ℚ) else 240 * (sigma 3 0 : ℚ)) *
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(if n - 0 = 0 then (1 : ℚ) else 240 * (sigma 3 (n - 0) : ℚ)) = (240 : ℚ) * (sigma 3 n : ℚ) := by
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simp [hn0]
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have hIco : ∑ x ∈ Finset.Ico 1 n, (if x = 0 then (1 : ℚ) else 240 * (sigma 3 x : ℚ)) *
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(if n - x = 0 then (1 : ℚ) else 240 * (sigma 3 (n - x) : ℚ))
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= (57600 : ℚ) * ((convolutionSum n : ℕ) : ℚ) := by
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calc
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∑ x ∈ Finset.Ico 1 n, (if x = 0 then (1 : ℚ) else 240 * (sigma 3 x : ℚ)) *
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(if n - x = 0 then (1 : ℚ) else 240 * (sigma 3 (n - x) : ℚ))
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= ∑ x ∈ Finset.Ico 1 n, (240 * (sigma 3 x : ℚ)) * (240 * (sigma 3 (n - x) : ℚ)) := by
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refine Finset.sum_congr rfl ?_
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intro x hx
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have hx0 : x ≠ 0 := by
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have hxmem := Finset.mem_Ico.1 hx; omega
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have hn_x0 : n - x ≠ 0 := by
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have hxmem := Finset.mem_Ico.1 hx; omega
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simp [hx0, hn_x0]
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_ = (57600 : ℚ) * ((convolutionSum n : ℕ) : ℚ) := by
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calc
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∑ x ∈ Finset.Ico 1 n, (240 * (sigma 3 x : ℚ)) * (240 * (sigma 3 (n - x) : ℚ))
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= ∑ x ∈ Finset.Ico 1 n, (57600 : ℚ) * ((sigma 3 x : ℚ) * (sigma 3 (n - x) : ℚ)) := by
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refine Finset.sum_congr rfl (λ x hx => ?_)
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ring
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_ = (57600 : ℚ) * (∑ x ∈ Finset.Ico 1 n, (sigma 3 x : ℚ) * (sigma 3 (n - x) : ℚ)) := by
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simp [Finset.mul_sum]
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_ = (57600 : ℚ) * ((convolutionSum n : ℕ) : ℚ) := by
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simp [convolutionSum, Nat.cast_sum, Nat.cast_mul, sigma3]
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rw [hfn, hf0, hIco]
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ring
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_ = (480 : ℚ) * (sigma 3 n : ℚ) + (57600 : ℚ) * ((convolutionSum n : ℕ) : ℚ) := by ring
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-- From the bridge: (E₄²)ₙ = (E₈)ₙ
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have hcoeff : cauchyProduct E4 E4 n = E8 n := by rw [bridge.square_eq]
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rw [hE4sq] at hcoeff
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have hn0 : n ≠ 0 := by omega
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have hE8val : E8 n = (480 : ℚ) * (sigma 7 n : ℚ) := by
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unfold E8; simp [hn0]
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have hcoeff' : (480 : ℚ) * (sigma 3 n : ℚ) + (57600 : ℚ) * ((convolutionSum n : ℕ) : ℚ) = (480 : ℚ) * (sigma 7 n : ℚ) :=
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calc
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(480 : ℚ) * (sigma 3 n : ℚ) + (57600 : ℚ) * ((convolutionSum n : ℕ) : ℚ) = E8 n := hcoeff
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_ = (480 : ℚ) * (sigma 7 n : ℚ) := by rw [hE8val]
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-- hcoeff: 480·σ₇(n) = 480·σ₃(n) + 57600·convolutionSum(n) [in ℚ]
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have h_rat : (sigma 7 n : ℚ) = (sigma 3 n : ℚ) + (120 : ℚ) * ((convolutionSum n : ℕ) : ℚ) := by
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nlinarith
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exact_mod_cast h_rat
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end SilverSight.Eisenstein
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