Each mission turns a result from a paper or textbook into small Lean 4 statements anyone can tackle.
Campaigns (experimental)
Campaigns group missions around a shared mathematical goal. Each one tracks a quantity, such as an upper or lower bound. Have a good candidate in mind? Ping us on Slack, Zulip, or WeChat.
Is every odd number a sum of k primes? This campaign tracks formalized proofs of the smallest k that suffices.
Schnirelmann (1930) showed some finite k works. Vinogradov (1937) showed that three is enough for all sufficiently large odd numbers. Tao (2012) proved k=5 unconditionally. Helfgott (2013) proved that every odd number greater than 5 is a sum of three primes, though the proof is still unrefereed. Ideally, we can formalize this statement here. Note that three is optimal: 27 is neither prime nor 2 + prime.
Schoolbook matrix multiplication takes n3 operations. The exponent ω is the infimum of all τ such that two n×n matrices can be multiplied in O(nτ) arithmetic operations; trivially ω≥2, and ω=2 is conjectured but open.
Strassen gave the first nontrivial bound, ω<2.81, in 1969, and introduced the laser method in 1986 to reach ω<2.48. Coppersmith and Winograd's 1990 bound of 2.376 stood for two decades. Every subsequent improvement comes from analyzing higher tensor powers of their construction with refined laser-method variants. That line reached ω<2.371339 in 2025, and the current record is ω<2.371177, from August 2026. See Computational complexity of matrix multiplication for the full table. Can we formalize these results and even improve on them?
In a 1999 Vienna doctoral thesis, Gerhard Zauner conjectured that in every finite dimension d one can find d² unit vectors in complex d-space that are mutually as spread out as possible — any two sharing the same squared overlap 1/(d+1). Such a configuration, a symmetric informationally complete positive operator-valued measure (SIC-POVM), is the optimal minimal measurement for reconstructing an unknown quantum state, which is why the idea was rediscovered and named by Renes, Blume-Kohout, Scott, and Caves in 2004 and became central to quantum tomography, quantum cryptography, and the QBist reading of quantum mechanics. Geometrically these are maximal sets of complex equiangular lines; physically they are the most efficient quantum measurements; and, remarkably, they appear to be governed by deep number theory — recent work by Appleby, Flammia, Kopp, and others ties exact SICs to Stark units and Hilbert's twelfth problem on explicit class field theory. Exact solutions have been hand-built in scores of dimensions and numerical ones found in every dimension checked, yet a general existence proof remains out of reach. Formalizing Zauner's conjecture gives this problem — straddling quantum information, geometry, and algebraic number theory — a precise shared target.
A Hadamard matrix is a square array of +1s and −1s whose rows are mutually orthogonal — equivalently, one whose determinant attains the absolute maximum that Jacques Hadamard proved in 1893 any ±1 matrix can reach. The story opens earlier, with James Joseph Sylvester's 1867 doubling construction producing such matrices in every power-of-two order; Hadamard himself added orders 12 and 20. The conjecture bearing his name asserts that a Hadamard matrix exists for every order divisible by four. Raymond Paley's 1933 construction from finite fields settled vast new families, and computer searches filled stubborn gaps — beginning with order 92 at JPL in 1962 and reaching order 428 only in 2005, after which 668 became the smallest order whose existence is still unknown. Far from a curiosity, these matrices are workhorses of applied mathematics, underpinning error-correcting codes (the Reed–Muller code that sharpened Mariner spacecraft imagery), spread-spectrum and CDMA signal design, optimal statistical designs of experiments, and coded-aperture spectroscopy. Settling the conjecture would close a 130-year-old gap where combinatorics, number theory, and design theory meet.
In 1993 the Texas banker and self-taught number theorist Andrew Beal, tinkering on his own with generalizations of Fermat's Last Theorem, noticed a striking pattern: whenever A^x + B^y = C^z holds in positive integers with every exponent exceeding two, the bases A, B, C seem forced to share a common prime factor. Fermat's Last Theorem is exactly the slice x = y = z of this statement, so Beal's conjecture sweepingly generalizes one of history's most famous theorems. Beal backed his question with money, raising the prize from $5,000 in 1997 to $1,000,000, now held in trust by the American Mathematical Society. The conjecture is intimately tied to the Fermat–Catalan conjecture and the theory of the generalized Fermat equation, where 1/x + 1/y + 1/z < 1 forces only finitely many primitive solutions; individual exponent families such as (2,3,n) have been settled, often with the same Frey-curve and modularity machinery behind Wiles's proof, yet the full statement remains open. A clean formal statement turns this celebrated amateur's question into a shared, verifiable goal.
Formulated in 1985 by Joseph Oesterlé and David Masser as an arithmetic distillation of Szpiro's conjecture on elliptic curves, the abc conjecture makes a deceptively simple claim about coprime triples with a + b = c: the three numbers cannot all be built from many repeated small primes at once, so c can only rarely exceed rad(abc)^(1+ε). Dorian Goldfeld called it 'the most important unsolved problem in Diophantine analysis,' and for good reason — a single proof would cascade through number theory, delivering Fermat's Last Theorem for all large exponents almost for free, along with Roth's theorem, the Mordell–Faltings theorem, the Fermat–Catalan conjecture, infinitely many non-Wieferich primes, and all but finitely many counterexamples to Beal's conjecture. Since 2012 Shinichi Mochizuki has claimed a proof via inter-universal Teichmüller theory, published in 2021, but the community has not accepted it: in 2018 Peter Scholze and Jakob Stix identified a gap they regarded as fatal. A precise formal statement gives everyone a shared, machine-checkable target around which to organize verified progress.