The original Jaco addition law is false

Abstract

The distinct orders three and eleven refute the complete original two-branch Jaco addition assertion.

Definition 1.1 (A unique Jaconian vertex).

$$\forall n\in \mathrm{Nat},(\mathrm{UniqueJaconian}\left(n\right))\iff (\exists v\in \mathrm{Vertex}\left(n\right),(\mathrm{degree}\left(\mathrm{jaco}\left(n\right),v\right)=\mathrm{maxDegree}\left(\mathrm{jaco}\left(n\right)\right))\wedge (\forall w\in \mathrm{Vertex}\left(n\right),(\mathrm{degree}\left(\mathrm{jaco}\left(n\right),w\right)=\mathrm{maxDegree}\left(\mathrm{jaco}\left(n\right)\right))\Rightarrow (w=v)))$$

Formalization. D5/S3/ConceptDynamics/GraphColoring/JacoAdditionLawRefutation.UniqueJaconian (✓ std3).

Citation. Johan Kok; Paul Fisher; Bettina Wilkens; Mokhwetha Mabula; Vivian Mukungunugwa (2014). Characteristics of Finite Jaco Graphs, J_n(1), n in N. DOI: 10.48550/arXiv.1404.0484.

Commentary.

UniqueJaconian n means exactly one vertex of jaco n attains its ordinary maximum degree. The carrier consists precisely of positive labels 1 through n. This is uniqueness of a vertex within one graph, not uniqueness of a numerical degree across graph orders. The graph is the existing finite truncation of the recursive rule i<j<=2i-d^-(v_i), with orientation forgotten; every edge has exactly one forward arc, so simple degree counts incoming and retained outgoing arcs together.

Definition 1.2 (The whole original Theorem 2.10).

$$(claim)\iff (\forall n\in \mathrm{Nat},\forall m\in \mathrm{Nat},(3\le n)\Rightarrow ((3\le m)\Rightarrow ((n\ne m)\Rightarrow ((((\mathrm{UniqueJaconian}\left(n\right))\vee (\mathrm{UniqueJaconian}\left(m\right)))\Rightarrow (\mathrm{maxDegree}\left(\mathrm{jaco}\left(n+m\right)\right)=\mathrm{maxDegree}\left(\mathrm{jaco}\left(n\right)\right)+\mathrm{maxDegree}\left(\mathrm{jaco}\left(m\right)\right)))\wedge (((\neg \mathrm{UniqueJaconian}\left(n\right))\wedge (\neg \mathrm{UniqueJaconian}\left(m\right)))\Rightarrow (\mathrm{maxDegree}\left(\mathrm{jaco}\left(n+m\right)\right)=\mathrm{maxDegree}\left(\mathrm{jaco}\left(n\right)\right)+\mathrm{maxDegree}\left(\mathrm{jaco}\left(m\right)\right)+1))))))$$

Formalization. D5/S3/ConceptDynamics/GraphColoring/JacoAdditionLawRefutation.claim (✓ std3).

Citation. Johan Kok; Paul Fisher; Bettina Wilkens; Mokhwetha Mabula; Vivian Mukungunugwa (2014). Characteristics of Finite Jaco Graphs, J_n(1), n in N. DOI: 10.48550/arXiv.1404.0484.

Commentary.

Kok, Fisher, Wilkens, Mabula and Mukungunugwa, arXiv:1404.0484v1, printed page 7, Theorem 2.10 (Conjectured), assert both displayed branches for every ordered pair of distinct natural orders n,m at least three. The first branch uses inclusive OR: either graph having a unique Jaconian vertex suffices. Otherwise means neither has a unique Jaconian vertex. No cutoff or additional graph hypothesis is imposed. The explicit open-problem list on printed page 11 asks to prove this theorem.

Theorem 1.3 (Refutation of the complete addition assertion).

$$\neg claim$$

Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/GraphColoring/JacoAdditionLawRefutation.result (✓ std3). ∎

Resolves. Problems/kok-jaco-addition-law-refutation (refuted) by D5/S3/ConceptDynamics/GraphColoring/JacoAdditionLawRefutation.result.

Source. Repository-derived.

Acknowledgement. Johan Kok; Paul Fisher; Bettina Wilkens; Mokhwetha Mabula; Vivian Mukungunugwa (2014). Characteristics of Finite Jaco Graphs, J_n(1), n in N. DOI: 10.48550/arXiv.1404.0484.

Commentary.

Use the original graph and the existing JacoMaximumDegree.degree_formula and JacoMaximumDegree.result. Put phi=(1+sqrt(5))/2. Exact rational bounds 8/5<phi<13/8 give the needed floors. The three degrees of J_3 are 1,2,1, so its unique maximizing vertex is v_2 and its maximum is 2. The same maximum-degree formula gives maximum 7 for J_11 and maximum 8 for J_14. At the admissible pair (3,11), the first branch therefore requires 8=2+7, a contradiction. A failure at this pair negates the whole universal two-branch claim; it asserts neither failure at every pair nor a counterexample to the otherwise branch. These numerical values also occur in Kok’s prior published arXiv:2507.16500v1 Table 1; they are not new numerical data. Bounded source searches leave literature gaps and establish no worldwide priority.

References

• Truth anchor: D5/S3/ConceptDynamics/GraphColoring/JacoAdditionLawRefutation.UniqueJaconian
• Truth anchor: D5/S3/ConceptDynamics/GraphColoring/JacoAdditionLawRefutation.claim
• Truth anchor: D5/S3/ConceptDynamics/GraphColoring/JacoAdditionLawRefutation.result
• Dependency: D5/S3/ConceptDynamics/GraphColoring/JacoMaximumDegree