# A broken stick and a Feynman diagram

Partial progress on an open problem. Vamshi Jandhyala.

> The chance that six random pieces of a stick make a tetrahedron is a three-loop Feynman diagram; it is 0.0125749944…, not 1/79.

Canonical: https://vamshij.com/research/broken-stick-feynman
Code and Lean proofs: https://github.com/jvvk/mathematics/tree/main/broken-stick-feynman
Paper (PDF): https://github.com/jvvk/mathematics/blob/main/broken-stick-feynman/paper/note.pdf

Break a stick at five random points and lay the six pieces, in a fixed order, on the edges of a tetrahedron. Benjamin Dickman asked on MathOverflow ([question 142983](https://mathoverflow.net/q/142983)) how often a tetrahedron with these edge lengths exists. No exact value is known, and the numbers so far were numerical estimates, one of which, `1/79`, turns out to be wrong.

The note shows that the probability is a Feynman integral. For `N = n(n+1)/2` pieces laid on the edges of an `n`-simplex,

`p_n = C_n ∫ ∏ β_e^((n−2)/2) e^(−Σβ) / U(β)^((n+1)/2) dβ`,   `C_n = 2^(N−n) π^(−N/2) Γ_n((n+1)/2)`,

where `U` is the first Symanzik polynomial of the complete graph `K_{n+1}` (a sum over spanning trees). So `p_n` is the vacuum diagram of `K_{n+1}` in dimension `n + 1` with unit masses and every propagator raised to the power `n/2`:

- `n = 2`: a one-loop triangle in three dimensions, which gives the classical `1/4`;
- `n = 3`: the three-loop tetrahedron in four dimensions with propagators `(q² + 1)^(−3/2)`, `p_3 = (π²/16) T`;
- `n = 4`: `p_4 = 1.0584 × 10^(−4)` (Monte Carlo).

The route is classical step by step (exponential spacings, Schoenberg's criterion, the Schwinger representation, Siegel's matrix gamma integral, Kirchhoff's matrix-tree theorem); the identity itself is new as far as a dated literature search could find. Two independent computations give `p_3 = 0.0125749944167…`, agreeing to within `10^(−12)`, which corrects the value `1/79` reported from simulation. For the question as asked, with the pieces placed in any order, the probability is `0.0652818 ± 0.0000004`, consistent with an earlier estimate. No closed form is known.

Preprint v1, 10 October 2026, not peer reviewed and not yet independently reviewed. The algebra of Theorem 1 for the triangle and the tetrahedron is formally verified in Lean 4 (`lean/`, standard axioms only); the analytic inputs are quoted. The author used AI tools (Claude, Anthropic) in this work, as described in the paper's acknowledgement, and is responsible for its content.