GraphConjecture103.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

7 declarations, 0 contain sorry, 0 declared as axioms

Written on the Wall II - Conjecture 103

Reference: E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc

Imports (3)

abbrev WrittenOnTheWallII.GraphConjecture103.wowii103Counterexample The 11-vertex counterexample: a triangle with four leaves on each of two vertices line 36 source
Source documentation
The 11-vertex counterexample: a triangle with four leaves on each of two vertices.

standard axioms: Quot.sound, propext what this means
Statement uses: SimpleGraph, Fin, OfNat.ofNat, Nat
Proof uses: SimpleGraph.fromEdgeSet, Insert.insert, Sym2, Set, Sym2.mk, Singleton.singleton
6 instances, coercions and auxiliary constants left out.
```
abbrev wowii103Counterexample : SimpleGraph (Fin 11) :=
…
```
theorem WrittenOnTheWallII.GraphConjecture103.wowii103Counterexample_connected The counterexample is connected line 45 source
Source documentation
The counterexample is connected.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph.Connected, Fin, OfNat.ofNat, Nat, WrittenOnTheWallII.GraphConjecture103.wowii103Counterexample
Proof uses: of_decide_eq_true, Insert.insert, Sym2, Set, Sym2.mk, Singleton.singleton, id, Eq, Bool, Decidable.decide, Bool.true, Eq.refl
13 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem wowii103Counterexample_connected : wowii103Counterexample.Connected := by
…
```
theorem WrittenOnTheWallII.GraphConjecture103.wowii103Counterexample_indepNum The counterexample has independence number nine line 50 source
Source documentation
The counterexample has independence number nine.

extra axioms: _private.0.WrittenOnTheWallII.GraphConjecture103.wowii103Counterexample_indepNum._native.decide.ax_1 what this means
Statement uses: Eq, Nat, SimpleGraph.indepNum, Fin, OfNat.ofNat, WrittenOnTheWallII.GraphConjecture103.wowii103Counterexample
Proof uses: Eq.mpr, SimpleGraph.computable_indep_num, Insert.insert, Sym2, Set, Sym2.mk, Singleton.singleton, id, congrArg, SimpleGraph.indep_num_eq_computable, of_decide_eq_true
14 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem wowii103Counterexample_indepNum : wowii103Counterexample.indepNum = 9 := by
…
```
theorem WrittenOnTheWallII.GraphConjecture103.wowii103Counterexample_bipartiteSize The largest induced bipartite subgraph of the counterexample has ten vertices line 56 source
Source documentation
The largest induced bipartite subgraph of the counterexample has ten vertices.

extra axioms: _private.0.WrittenOnTheWallII.GraphConjecture103.wowii103Counterexample_bipartiteSize._native.decide.ax_1 what this means
Statement uses: Eq, Nat, SimpleGraph.largestInducedBipartiteSubgraphSize, Fin, OfNat.ofNat, WrittenOnTheWallII.GraphConjecture103.wowii103Counterexample
Proof uses: Eq.mpr, SimpleGraph.computableLargestInducedBipartiteSubgraphSize, Insert.insert, Sym2, Set, Sym2.mk, Singleton.singleton, id, congrArg, SimpleGraph.largestInducedBipartiteSubgraphSize_eq_computable, of_decide_eq_true
14 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem wowii103Counterexample_bipartiteSize :
    wowii103Counterexample.largestInducedBipartiteSubgraphSize = 10 := by
…
```
theorem WrittenOnTheWallII.GraphConjecture103.wowii103Counterexample_averageEccentricity The counterexample has average eccentricity $30/11$ line 63 source
Source documentation
The counterexample has average eccentricity $30/11$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Real, SimpleGraph.averageEccentricity, Fin, OfNat.ofNat, Nat, WrittenOnTheWallII.GraphConjecture103.wowii103Counterexample, HDiv.hDiv
Proof uses: id, Finset.sum, Finset.univ, ENat.toNat, SimpleGraph.eccent, Eq.mpr, ENat, SimpleGraph.computable_eccent, Insert.insert, Sym2, Set, Sym2.mk, Singleton.singleton, congrFun', congrArg, Finset.sum_congr, Eq.refl, Finset, Membership.mem, SimpleGraph.eccent_eq_computable and 26 more
62 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
```
theorem wowii103Counterexample_averageEccentricity :
    averageEccentricity wowii103Counterexample = 30 / 11 := by
…
```
Mentions: WrittenOnTheWallII.GraphConjecture103.wowii103Counterexample
theorem WrittenOnTheWallII.GraphConjecture103.one_lt_log_thirty_eleven_and_lt_two The logarithm of the counterexample's average eccentricity lies strictly between one and two line 74 source
Source documentation
The logarithm of the counterexample's average eccentricity lies strictly between one and two.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: And, LT.lt, Real, OfNat.ofNat, Real.log, HDiv.hDiv, Nat
Proof uses: And.intro, Eq.mpr, Real.exp, id, Eq, congrArg, propext, Real.lt_log_iff_exp_lt, Mathlib.Meta.NormNum.isNNRat_lt_true, Mathlib.Meta.NormNum.IsNat.to_isNNRat, Mathlib.Meta.NormNum.isNat_ofNat, Nat.cast_zero, Mathlib.Meta.NormNum.isNNRat_div, Mathlib.Meta.NormNum.isNNRat_mul, HMul.hMul, Inv.inv, Eq.refl, Mathlib.Meta.NormNum.instAtLeastTwo, Mathlib.Meta.NormNum.isNNRat_inv_pos, Nat.mul and 29 more
70 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
```
theorem one_lt_log_thirty_eleven_and_lt_two :
    1 < Real.log (30 / 11 : ℝ) ∧ Real.log (30 / 11 : ℝ) < 2 := by
…
```
theorem WrittenOnTheWallII.GraphConjecture103.conjecture103 WOWII Conjecture 103 For a simple connected graph line 96 source
Source documentation
WOWII Conjecture 103
For a simple connected graph $G$, $\alpha(G) \le \lfloor b(G) - \ln(\mathrm{ecc\_avg}(G)) \rfloor$ where $\alpha(G) = G.\mathrm{indepNum}$ is the independence number, $b(G)$ is the largest induced bipartite subgraph size, and $\mathrm{ecc\_avg}(G) = G.\mathrm{averageEccentricity}$ is the average eccentricity of $G$. Uses Real.log (natural logarithm).
This conjecture is false. The graph wowii103Counterexample has independence number $9$, largest induced bipartite subgraph size $10$, and average eccentricity $30/11$. Since $1 < \ln(30/11) < 2$, the proposed upper bound is $8$.

L4DEL explanation
WOWII Conjecture 103 For a simple connected graph $G$, $\alpha(G) \le \lfloor b(G) - \ln(\mathrm{ecc\_avg}(G)) \rfloor$ where $\alpha(G) = G.\mathrm{indepNum}$ is the independence number, $b(G)$ is the largest induced bipartite subgraph size, and $\mathrm{ecc\_avg}(G) = G.\mathrm{averageEccentricity}$ is the average eccentricity of $G$. Uses Real.log (natural logarithm). This conjecture is false. The graph wowii103Counterexample has independence number $9$, largest induced bipartite subgraph size $10$, and average eccentricity $30/11$. Since $1 < \ln(30/11) < 2$, the proposed upper bound is $8$.
Sources and attribution: Pinned Lean source (1646ca16afd6)

extra axioms: _private.0.WrittenOnTheWallII.GraphConjecture103.wowii103Counterexample_bipartiteSize._native.decide.ax_1, _private.0.WrittenOnTheWallII.GraphConjecture103.wowii103Counterexample_indepNum._native.decide.ax_1 what this means
Statement uses: Iff, False, Fintype, DecidableEq, Nontrivial, SimpleGraph, SimpleGraph.Connected, LE.le, Real, SimpleGraph.indepNum, Int.floor, HSub.hSub, SimpleGraph.b, Real.log, SimpleGraph.averageEccentricity
Proof uses: id, Eq.mpr, Not, Eq, congrArg, false_iff, Fin, OfNat.ofNat, Nat, WrittenOnTheWallII.GraphConjecture103.wowii103Counterexample, WrittenOnTheWallII.GraphConjecture103.wowii103Counterexample_connected, Int, HDiv.hDiv, And, LT.lt, HAdd.hAdd, propext, Int.floor_eq_iff, And.intro, Mathlib.Meta.NormNum.instAtLeastTwo and 89 more
106 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture103 : answer(False) ↔
    ∀ (α : Type) [Fintype α] [DecidableEq α] [Nontrivial α]
      (G : SimpleGraph α) (_h : G.Connected),
      (G.indepNum : ℝ) ≤ ⌊b G - Real.log (averageEccentricity G)⌋ := by
…
```