ZagierMZV.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

15 declarations, 2 contain sorry, 0 declared as axioms

Zagier's Conjecture on Multiple Zeta Values

References:

[Za94] Zagier, Don. "Values of zeta functions and their applications." First European Congress of Mathematics Paris, July 6–10, 1992: Vol. II: Invited Lectures (Part 2). Basel: Birkhäuser Basel, 1994.
[Co18] Combariza, Germán AG. "A few conjectures about the multiple zeta values." ACM Communications in Computer Algebra 52.1 (2018): 11-20.
[Te02] T. Terasoma. Mixed Tate motives and multiple zeta values. Invent. Math., 149(2):339–369, 2002.
[DG05] P. Deligne and A. Goncharov. Groupes fondamentaux motiviques de Tate mixte. Ann. Sci. Ecole Norm. Sup. (4), 38(1):1–56, 2005.
OEIS A000931

Imports (1)

FormalConjecturesUtil

def ZagierMZV.multiZeta The multiple zeta value line 50 source
Source documentation
The multiple zeta value $\zeta(s_1, s_2, \ldots, s_k)$, defined as $$\zeta(s_1, \ldots, s_k) = \sum_{n_1 > n_2 > \cdots > n_k > 0} \frac{1}{n_1^{s_1} n_2^{s_2} \cdots n_k^{s_k}}.$$ The argument is a list of positive natural numbers. The value is well-defined (i.e. the series converges) when the first entry is at least 2, but we define it for all inputs. For the empty list, multiZeta [] = 1 (the empty product convention).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: List, Nat, Real
Proof uses: ZagierMZV.multiZeta.match_1, Unit, OfNat.ofNat, tsum, HMul.hMul, HDiv.hDiv, HPow.hPow, HAdd.hAdd, ZagierMZV.multiZeta.aux, SummationFilter.unconditional
19 instances, coercions and auxiliary constants left out.
```
noncomputable def multiZeta : List ℕ → ℝ
  | [] => 1
  | s :: rest => ∑' n : ℕ, (1 / (n + 1 : ℝ) ^ s) * multiZeta.aux rest n
where
…
```
def ZagierMZV.weight The weight of an MZV index $(s_1, \ldots, s_k)$ is $s_1 + \cdots + s_k$ line 61 source
Source documentation
The weight of an MZV index $(s_1, \ldots, s_k)$ is $s_1 + \cdots + s_k$.

standard axioms: propext what this means
Statement uses: List, Nat
Proof uses: List.sum
3 instances, coercions and auxiliary constants left out.
```
def weight (s : List ℕ) : ℕ := s.sum
```
def ZagierMZV.AdmissibleIndex An MZV index is admissible if it is either empty or if the first entry is at least 2 and all entries are positive line 67 source
Source documentation
An MZV index is admissible if it is either empty or if the first entry is at least 2 and all entries are positive. The empty list convention ensures $\mathcal{Z}_0 = \mathbb{Q}$.

no axioms what this means
Statement uses: List, Nat
Proof uses: ZagierMZV.multiZeta.match_1, Unit, True, And, LT.lt, OfNat.ofNat, Membership.mem
3 instances, coercions and auxiliary constants left out.
```
def AdmissibleIndex : List ℕ → Prop
  | [] => True
  | s₁ :: rest => 1 < s₁ ∧ ∀ sᵢ ∈ rest, 0 < sᵢ

/-- The set of all MZV values of weight $n$. -/
```
Mentions: ZagierMZV.weight
def ZagierMZV.mzvSetOfWeight The set of all MZV values of weight $n$ line 72 source
Source documentation
The set of all MZV values of weight $n$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set, Real
Proof uses: Set.image, List, ZagierMZV.multiZeta, Set.ofPred, And, ZagierMZV.AdmissibleIndex, Eq, ZagierMZV.weight
```
def mzvSetOfWeight (n : ℕ) : Set ℝ :=
…
```
def ZagierMZV.mzvSpanOfWeight The $\mathbb{Q}$-submodule of $\mathbb{R}$ spanned by all MZVs of weight $n$ line 76 source
Source documentation
The $\mathbb{Q}$-submodule of $\mathbb{R}$ spanned by all MZVs of weight $n$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Submodule, Rat, Real
Proof uses: Submodule.span, ZagierMZV.mzvSetOfWeight
42 instances, coercions and auxiliary constants left out.
```
noncomputable def mzvSpanOfWeight (n : ℕ) : Submodule ℚ ℝ :=
…
```
def ZagierMZV.zagierDim The conjectured Zagier dimension sequence $d_n$, defined by $d_0 = 1$, $d_1 = 0$, $d_2 = 1$, and $d_n = d_{n-2} + d_{n-3}$ for $n \geq 3$ line 83 source
Source documentation
The conjectured Zagier dimension sequence $d_n$, defined by $d_0 = 1$, $d_1 = 0$, $d_2 = 1$, and $d_n = d_{n-2} + d_{n-3}$ for $n \geq 3$.

no axioms what this means
Statement uses: Nat
2 instances, coercions and auxiliary constants left out.
```
def zagierDim : ℕ → ℕ
  | 0 => 1
  | 1 => 0
  | 2 => 1
  | (n + 3) => zagierDim (n + 1) + zagierDim n

/--
**Zagier's conjecture**

The $\mathbb{Q}$-dimension of the vector space spanned by all multiple zeta values
of weight $n$ equals $d_n$, where $d_n$ is the Zagier dimension sequence
satisfying $d_0 = 1$, $d_1 = 0$, $d_2 = 1$, and $d_n = d_{n-2} + d_{n-3}$ for $n \geq 3$.
…
```
Mentions: ZagierMZV.weight
theorem ZagierMZV.zagier_conjecture Documented declaration: zagier_conjecture sorry line 97 source
Source documentation
Zagier's conjecture
The $\mathbb{Q}$-dimension of the vector space spanned by all multiple zeta values of weight $n$ equals $d_n$, where $d_n$ is the Zagier dimension sequence satisfying $d_0 = 1$, $d_1 = 0$, $d_2 = 1$, and $d_n = d_{n-2} + d_{n-3}$ for $n \geq 3$.

depends on sorry what this means
Statement uses: Iff, True, Nat, Eq, Module.finrank, Rat, Subtype, Real, Membership.mem, Submodule, ZagierMZV.mzvSpanOfWeight, ZagierMZV.zagierDim
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
59 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem zagier_conjecture :
    answer(sorry) ↔ ∀ n : ℕ, Module.finrank ℚ (mzvSpanOfWeight n) = zagierDim n := by
…
```
Mentions: ZagierMZV.mzvSpanOfWeight, ZagierMZV.zagierDim
theorem ZagierMZV.zagier_upper_bound Upper bound [Te02, DG05] The dimension of the $\mathbb{Q}$-vector space of MZVs of weight $n$ is at most $d_n$ sorry line 107 source
Source documentation
Upper bound [Te02, DG05]
The dimension of the $\mathbb{Q}$-vector space of MZVs of weight $n$ is at most $d_n$.

depends on sorry what this means
Statement uses: Nat, LE.le, Module.finrank, Rat, Subtype, Real, Membership.mem, Submodule, ZagierMZV.mzvSpanOfWeight, ZagierMZV.zagierDim
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
61 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem zagier_upper_bound :
    ∀ n : ℕ, Module.finrank ℚ (mzvSpanOfWeight n) ≤ zagierDim n := by
…
```
Mentions: ZagierMZV.mzvSpanOfWeight, ZagierMZV.zagierDim
theorem ZagierMZV.zagierDim_first_values The first few values of $d_n$ are $1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, \ldots$ line 117 source
Source documentation
The first few values of $d_n$ are $1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, \ldots$

no axioms what this means
Statement uses: Eq, List, Nat, List.cons, ZagierMZV.zagierDim, OfNat.ofNat, List.nil
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem zagierDim_first_values :
    [zagierDim 0, zagierDim 1, zagierDim 2, zagierDim 3, zagierDim 4,
     zagierDim 5, zagierDim 6, zagierDim 7, zagierDim 8, zagierDim 9] =
    [1, 0, 1, 1, 1, 2, 2, 3, 4, 5] := by
…
```
Mentions: ZagierMZV.zagierDim
theorem ZagierMZV.no_admissible_weight_one There is no admissible index of weight 1 (since $s_1 \geq 2$ is required) line 125 source
Source documentation
There is no admissible index of weight 1 (since $s_1 \geq 2$ is required).

standard axioms: Quot.sound, propext what this means
Statement uses: List, Nat, ZagierMZV.AdmissibleIndex, Ne, ZagierMZV.weight, OfNat.ofNat
Proof uses: Eq, List.nil, id, of_eq_true, Not, Eq.trans, False, True, congrArg, not_false_eq_true, Eq.symm, List.cons, List.sum, LT.lt, Membership.mem, ZagierMZV.multiZeta.match_1, Unit, And, HAdd.hAdd, Eq.refl
16 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
```
theorem no_admissible_weight_one (s : List ℕ) (h : AdmissibleIndex s) : weight s ≠ 1 := by
…
```
Mentions: ZagierMZV.AdmissibleIndex, ZagierMZV.weight
theorem ZagierMZV.dim_mzv_weight_zero $\mathcal{Z}_0 = \mathbb{Q}$, so $\dim_\mathbb{Q}(\mathcal{Z}_0) = 1$ line 139 source
Source documentation
$\mathcal{Z}_0 = \mathbb{Q}$, so $\dim_\mathbb{Q}(\mathcal{Z}_0) = 1$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Module.finrank, Rat, Subtype, Real, Membership.mem, Submodule, ZagierMZV.mzvSpanOfWeight, OfNat.ofNat
Proof uses: Set, ZagierMZV.mzvSetOfWeight, Singleton.singleton, Set.ext, Eq.mpr, Iff, Exists, List, And, ZagierMZV.AdmissibleIndex, ZagierMZV.weight, ZagierMZV.multiZeta, id, congr, congrArg, Set.ofPred, Iff.intro, List.nil, Eq.refl, Eq.symm and 19 more
85 instances, coercions and auxiliary constants left out.

Tactic trace: 22 steps
Open to load the checked before-and-after goal states.
```
theorem dim_mzv_weight_zero : Module.finrank ℚ (mzvSpanOfWeight 0) = 1 := by
…
```
Mentions: ZagierMZV.mzvSpanOfWeight
theorem ZagierMZV.dim_mzv_weight_one $\mathcal{Z}_1 = \emptyset$, so $\dim_\mathbb{Q}(\mathcal{Z}_1) = 0$ line 161 source
Source documentation
$\mathcal{Z}_1 = \emptyset$, so $\dim_\mathbb{Q}(\mathcal{Z}_1) = 0$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Module.finrank, Rat, Subtype, Real, Membership.mem, Submodule, ZagierMZV.mzvSpanOfWeight, OfNat.ofNat
Proof uses: Set, ZagierMZV.mzvSetOfWeight, EmptyCollection.emptyCollection, Set.ext, Eq.mpr, Iff, Exists, List, And, ZagierMZV.AdmissibleIndex, ZagierMZV.weight, ZagierMZV.multiZeta, False, id, congr, congrArg, Set.ofPred, Iff.intro, ZagierMZV.no_admissible_weight_one, False.elim and 6 more
69 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
```
theorem dim_mzv_weight_one : Module.finrank ℚ (mzvSpanOfWeight 1) = 0 := by
…
```
Mentions: ZagierMZV.mzvSpanOfWeight
theorem ZagierMZV.multiZeta_empty line 173 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Real, ZagierMZV.multiZeta, List.nil, Nat, OfNat.ofNat
Proof uses: of_eq_true, eq_self
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem ZagierMZV.multiZeta_two Euler's identity: $\zeta(2) = \pi^2/6$ line 178 source
Source documentation
Euler's identity: $\zeta(2) = \pi^2/6$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Real, ZagierMZV.multiZeta, List.cons, Nat, OfNat.ofNat, List.nil, HDiv.hDiv, HPow.hPow, Real.pi
Proof uses: tsum, HAdd.hAdd, SummationFilter.unconditional, id, of_eq_true, ZagierMZV.multiZeta.match_1, List, Unit, HMul.hMul, ZagierMZV.multiZeta.aux, Eq.trans, True, congrFun', congrArg, SummationFilter, optParam, funext, mul_one, eq_self, Eq.mpr and 20 more
80 instances, coercions and auxiliary constants left out.

Tactic trace: 16 steps
Open to load the checked before-and-after goal states.
```
theorem multiZeta_two : multiZeta [2] = Real.pi ^ 2 / 6 := by
…
```
Mentions: ZagierMZV.multiZeta
theorem ZagierMZV.multiZeta_four Euler's identity for $\zeta(4) = \pi^4/90$ line 198 source
Source documentation
Euler's identity for $\zeta(4) = \pi^4/90$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Real, ZagierMZV.multiZeta, List.cons, Nat, OfNat.ofNat, List.nil, HDiv.hDiv, HPow.hPow, Real.pi
Proof uses: tsum, HAdd.hAdd, SummationFilter.unconditional, id, of_eq_true, ZagierMZV.multiZeta.match_1, List, Unit, HMul.hMul, ZagierMZV.multiZeta.aux, Eq.trans, True, congrFun', congrArg, SummationFilter, optParam, funext, mul_one, eq_self, Eq.mpr and 24 more
81 instances, coercions and auxiliary constants left out.

Tactic trace: 18 steps
Open to load the checked before-and-after goal states.
```
theorem multiZeta_four : multiZeta [4] = Real.pi ^ 4 / 90 := by
…
```
Mentions: ZagierMZV.multiZeta