Graph Signature Arithmetic

A first-principles arithmetic of finite graphs

Blaize Rouyea · Corey Bourgeois

abstract

Graph signature arithmetic is an arithmetic on finite simple graphs built from two counts: vertices and edges. A graph G has signature σ(G) = (v(G),e(G)) and score s(G) = v(G) + e(G). Two operations make this arithmetic nontrivial.

The first is fusion of isolated numerals, which turns m and n into the complete bipartite graph K_m,n and gives the law s(m⊕n) = m+n+mn. The second is the lexicographic product O[R], read directionally: O is the organizer and R is the payload. Its exact score law is s(O[R]) = v(O)s(R) + e(O)v(R)².

For fixed payload R, product scores fall into arithmetic lanes. The payload is globally prime-dead exactly when gcd(v(R),e(R))>1; otherwise at least one lane is live, and Dirichlet's theorem gives infinitely many prime scores in that lane.

The score layer is deliberately blind: it depends on the payload only through (v,e). Therefore nonisomorphic payloads with the same signature cast the same arithmetic shadow, even when one is planar and another is forced nonplanar. Crossing number and quotient recovery are then introduced only as honest refinements, not as hidden assumptions.

1. The seed2
2. Objects and observables2
3. Isolated numerals and fusion3
4. The structural product3
5. Multiplicative primitiveness4
6. Arithmetic lanes5
7. The global death law6
8. The Dirichlet bridge7
9. Arithmetic shadows7
10. Topology as a second layer9
11. Crossing growth and bridge pressure9
12. Open problems10
13. Conclusion10

topics

graph theoryarithmeticprime numberscombinatoricsdirichlet theoremcrossing number

cite

@article{rouyea2026graphsignature,
  title={Graph Signature Arithmetic: A First-Principles Arithmetic of Finite Graphs},
  author={Rouyea, Blaize and Bourgeois, Corey},
  year={2026},
  note={Draft}
}

abstract

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