Factorizations in the Monoid of Integer Triangles

Integer triangles, namely triples (a,b,c) ∈ ℕ³ with a ≤ b ≤ c and a+b > c,form a commutative partial monoid under componentwise multiplication.We study the factorization theory of this monoid: atoms, atomic decompositions,length sets, and elasticity. The totalization of the triangle monoid is shownnot to be Krull, so the classical transfer machinery of factorization theorydoes not apply and is replaced here by direct structural arguments.The isosceles submonoid is shown to be a rich source of non-unique factorizationphenomena and is excluded in a strongly reduced class 𝒯₀, where we prove that themaximal factorization length is unbounded, establish a calculus of length sets,and formulate the isolated-gap conjecture together with the computational evidencefor it. On the analytic side, the set of integer triangles is a translate of aunimodular cone, which yields rational generating functions and a maximal-edgezeta function whose coefficients form a quadratic quasipolynomial.The multiplicative-norm zeta function, by contrast, is not a finite combinationof shifts of the Riemann zeta function; it admits a Hurwitz-type representationand a Pólya completion identity, and its counting function has the main termπ² x / 36.