Source labels openChecked July 26, 2026

Monochromatic quantum graphs (inherited vertex colorings)

For all even $N \geq 6$ and $D \geq 3$ , does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{Z}$ with weights in $\{-1, 0, 1\}$ ?

Combinatorics AMS 5 AMS 14 AMS 81

Mathematical statement

For all even $N \geq 6$ and $D \geq 3$ , does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{Z}$ with weights in $\{-1, 0, 1\}$ ?

Statement source: Papers statement material

Statement terms: Source-specific

Source-specific terms. Therefore does not assert reuse rights beyond attributed display.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

eqSystem_no_solution_ge6_ge3_trinary_int

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem eqSystem_no_solution_ge6_ge3_trinary_int :    answer(sorry) ↔      ∀ N D : Nat, N ≥ 6 → Even N → D ≥ 3 →        ¬ ∃ W : WeightsN N D ℤ,            (∀ e, W e = (-1 : ℤ) ∨ W e = 0 ∨ W e = 1) ∧              EqSystemN N D W := by  sorry

Statement source: Formal Conjectures
Lean version: v4.27.0
Placeholder: Present; no proof artifact
Source evidence: Pinned source index
Fidelity review: Community formulation

References

Pinned Formal Conjectures sourceformalization Vertex coloring inherited from perfect matchings (motivated by quantum physics)background Questions on the Structure of Perfect Matchings inspired by Quantum Physicsbackground Edge-coloured graphs with only monochromatic perfect matchings and their connection to quantum physicsbackground