Shock, Swarm and Certificate: A Computational Anatomy of Machine-Discovered Mathematics in 2026

Abstract

In 2026 four widely reported episodes placed large language models at the point of mathematical discovery: Claude Opus 4.6 found the construction that settled Donald Knuth’s open Hamiltonian cycle decomposition problem for odd m, after which Knuth supplied the proof and titled his note Claude’s Cycles; Claude Fable 5 produced a one-line counterexample that disproved the Jacobian conjecture in dimension three and above; an unreleased research version of Claude raised the proven lower bound on zeta zeros lying on the critical line from 41.67 per cent to 67.25 per cent; and an internal OpenAI system directing roughly ten thousand concurrent agents produced a Lean-formalised proof that the three-dimensional Navier–Stokes equations admit a finite-time singularity. Parts I and II of this series treated such announcements as exogenous jumps that reprice equities. This paper opens the box. Treating the publicly disclosed budgets as data, we build a computational model of agentic discovery and derive five results. A diversity-limited parallel search gives speedup Nκ with κ = β/α < 1, so swarm width purchases latency and never efficiency: total cost rises as N1−κ, and halving the time to proof costs 2(1−κ)/κ times as much. The serial formalisation stage imposes an identification-free ceiling: the Navier–Stokes project could not have been compressed below 17 hours whatever the swarm width, a maximum further speedup of 6.18×. Credence factorises into proof soundness and statement faithfulness, and machine checking collapses only the first, which is why a Lean-verified claim can coexist with an unclaimed prize and an open Clay listing. The central empirical finding is a dichotomy. The two episodes in which the machine had to supply only a witness cost 1 and 4 agent-hours; the two in which it had to supply an argument cost 2,160 and 880,000. The separating variable is not fame or difficulty but whether the target admits a short independently checkable certificate, and the gap across the four episodes reaches 8.8×105. We conclude that the binding constraint on machine mathematics is certificate structure, not model capability, and that the cheapest available intervention is to reformulate open problems so that success leaves a witness.