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.

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.

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 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.

Which whole numbers are the area of a right triangle with rational sides? This is the congruent number problem, and it is astonishingly old — tabulated in tenth-century Arabic manuscripts (5 and 6 were among the first known cases), taken up by Fibonacci in the thirteenth century, and the subject of Fermat's celebrated infinite-descent proof that 1 is not congruent. The modern reformulation is a jewel of arithmetic geometry: n is congruent precisely when the elliptic curve y² = x³ − n²x has a rational point of infinite order, that is, positive rank. In 1983 Jerrold Tunnell, writing in Inventiones Mathematicae, turned this into a near-algorithm — counting integer representations of n by certain ternary quadratic forms (which arise as coefficients of weight-3/2 modular forms) yields a simple congruence criterion that settles the question by a finite computation. The catch, and the reason the problem remains officially open, is that the sufficiency of Tunnell's criterion rests on the Birch and Swinnerton-Dyer conjecture, itself a Millennium Prize Problem. This mission formalizes the converse of Tunnell's theorem in the odd case: for squarefree odd n, the representation-count identity 2|A_n| = |B_n| — where A_n and B_n count integer solutions of n = 2x² + y² + 32z² and n = 2x² + y² + 8z² — implies that n is a congruent number.

Which whole numbers are the area of a right triangle with rational sides? This is the congruent number problem, and it is astonishingly old — tabulated in tenth-century Arabic manuscripts (5 and 6 were among the first known cases), taken up by Fibonacci in the thirteenth century, and the subject of Fermat's celebrated infinite-descent proof that 1 is not congruent. The modern reformulation is a jewel of arithmetic geometry: n is congruent precisely when the elliptic curve y² = x³ − n²x has a rational point of infinite order, that is, positive rank. In 1983 Jerrold Tunnell, writing in Inventiones Mathematicae, turned this into a near-algorithm — counting integer representations of n by certain ternary quadratic forms (which arise as coefficients of weight-3/2 modular forms) yields a simple congruence criterion that settles the question by a finite computation. The catch, and the reason the problem remains officially open, is that the sufficiency of Tunnell's criterion rests on the Birch and Swinnerton-Dyer conjecture, itself a Millennium Prize Problem. This mission formalizes the converse of Tunnell's theorem in the even case: for squarefree even n, the representation-count identity 2|C_n| = |D_n| — where C_n and D_n count integer solutions of n = 8x² + 2y² + 64z² and n = 8x² + 2y² + 16z² — implies that n is a congruent number.

Among the most enduring mysteries in number theory is whether the primes keep producing twins — pairs like (11, 13) or (17, 19) that differ by exactly two — no matter how far out one looks. The general form was set down by Alphonse de Polignac in 1849, and the first deep theorem came from Viggo Brun in 1915, who proved that the reciprocals of the twin primes converge to a finite value, now called Brun's constant; in doing so he invented modern sieve theory and showed that twins must thin out even if there are infinitely many. Hardy and Littlewood went further, conjecturing a precise density of about 2C₂·x/(ln x)² for the count of twins below x. For nearly a century the infinitude itself stood untouched, until Yitang Zhang's stunning announcement on 17 April 2013 that some gap below 70 million recurs infinitely often — the first finite bound ever proved. A Polymath collaboration led by Terence Tao, together with James Maynard's independent multidimensional sieve, soon drove that bound down to 246, where it still stands. Closing the gap all the way to 2 — the twin prime conjecture itself — remains open. This mission states it cleanly: the set of primes p for which p + 2 is also prime is infinite.

No problem in mathematics carries more weight than the Riemann hypothesis. In his single eight-page paper of 1859, 'On the Number of Primes Less Than a Given Magnitude,' Bernhard Riemann linked the seemingly erratic distribution of the primes to the zeros of the analytic continuation of the zeta function ζ(s), and conjectured that every nontrivial zero lies exactly on the critical line where the real part equals 1/2. The truth of this statement would pin down the error term in the prime number theorem and tame the fluctuations of the primes around their expected count, and hundreds of theorems already stand proven only 'conditional on RH,' waiting for it to be settled. David Hilbert placed it in his eighth problem in 1900, alongside Goldbach and the twin primes; in 2000 the Clay Mathematics Institute named it one of the seven Millennium Prize Problems, with a million-dollar reward. G. H. Hardy proved in 1914 that infinitely many zeros lie on the critical line, and trillions more have since been verified by computation to do so — overwhelming evidence that is nonetheless not a proof. After more than 160 years it remains unresolved. This mission takes Mathlib's own definition of the hypothesis as its target.

Every even integer greater than $2$ is the sum of two primes. Christian Goldbach posed it in a 1742 letter to Euler, and it has resisted proof for nearly three centuries while being verified computationally up to $4\times10^{18}$ — making it one of the oldest and most famous open problems in all of mathematics. Its ternary sibling, the weak Goldbach conjecture, was settled by Helfgott in 2013, but the strong form stated here remains wide open: the circle method controls three-prime sums yet loses control at two. This headline mission hosts the conjecture as a machine-checked target for partial results, reductions between its variants, and any future attack.

Motivation

The traveling salesman problem — visit $n$ cities by the cheapest round trip — is the most widely known problem in combinatorial optimization, and its central open question concerns a linear program. The subtour-elimination relaxation (the Held–Karp bound) replaces tours by fractional edge weights, and both in theory and in practice (it powers the lower bounds inside the Concorde solver) it is remarkably close to the true optimum. How close, in the worst case, is the integrality gap of the relaxation: the supremum of $\mathrm{OPT}/\mathrm{LP}$ over metric instances. Explicit instance families push the gap up to $4/3$; the best proven upper bound sits just barely below $3/2$. The 4/3 conjecture — the gap is exactly $4/3$ — has been the benchmark question of approximation algorithms for four decades.

Timeline

• 1954. Dantzig, Fulkerson, and Johnson solve a 49-city instance by hand with the cutting planes that become the subtour-elimination LP.
• 1970–1971. Held and Karp introduce the 1-tree/Lagrangian bound and show it equals the subtour LP value — since then, "the Held–Karp bound".
• 1976/1978. Christofides, and independently Serdyukov, give the $3/2$-approximation: minimum spanning tree plus a matching on odd-degree vertices.
• 1980. Wolsey (Math. Prog. Study 13) shows Christofides' analysis goes through against the LP: $\mathrm{OPT} \le \frac{3}{2}\,\mathrm{LP}$, so the integrality gap is at most $3/2$. Shmoys and Williamson (IPL 1990) rediscover this via a monotonicity property.
• 1995. Goemans (Math. Programming 69) analyzes the worst-case ratios of TSP relaxations and states the $4/3$ conjecture explicitly; the $4/3$ lower-bound families (three parallel paths) are by then folklore.
• 2011–2014. For graph metrics (shortest-path metrics of unweighted graphs) the barrier breaks: Oveis Gharan–Saberi–Singh and Mömke–Svensson beat $3/2$, and Sebő–Vygen (Combinatorica 2014) reach $7/5$ — the conjectured-optimal shape of progress, but only for a special class.
• 2020–2022. Karlin, Klein, and Oveis Gharan prove a $3/2 - \varepsilon$ approximation for general metric TSP (STOC 2021) and then an integrality-gap bound $\gamma \le 3/2 - \varepsilon$ with $\varepsilon > 10^{-36}$ (FOCS 2022), via max-entropy sampling of spanning trees and strongly Rayleigh distributions — the first general improvement over Wolsey in forty years, by an astronomically small margin.
• Today. The gap between the $4/3$ lower bound and the $3/2 - 10^{-36}$ upper bound is the conjecture. For half-integral LP solutions — where the conjectured extremal instances live — the bound has been pushed to $1.4983$ (Gupta, Lee, Li, Mucha, Newman, and Sarkar, via matroid-based rounding).

Setting

An instance on $n \ge 3$ cities is a cost function $c$ assigning to each ordered pair of cities $u, v$ a real cost $c(u,v)$, required to be a metric cost: symmetric ($c(u,v) = c(v,u)$), zero on the diagonal ($c(v,v) = 0$), and satisfying the triangle inequality $c(u,w) \le c(u,v) + c(v,w)$. Nonnegativity follows; distinct cities at distance zero are allowed, as usual for metric TSP.

A tour visits every city exactly once and returns to its start. Formally a tour is given by an ordering: a permutation $\pi$ of the cities, traversed as $\pi(0), \pi(1), \dots, \pi(n-1)$ and back to $\pi(0)$; its cost $\mathrm{tourCost}(c, \pi)$ is the sum of the costs of consecutive steps, and $\mathrm{OPT}(c)$ — written tspOpt c — is the minimum over all orderings.

The subtour-elimination (Held–Karp) relaxation replaces the tour by a fractional edge weight $x(u,v)$ for each pair of cities. A weight vector $x$ is feasible (IsHeldKarp x) when it is symmetric with zero diagonal, has entries in $[0,1]$, gives every city fractional degree two ($\sum_u x(v,u) = 2$), and crosses every nontrivial cut at least twice: for every set $S$ of cities other than $\emptyset$ and all cities, $\sum_{u \in S} \sum_{v \notin S} x(u,v) \ge 2$. The Held–Karp bound hkValue c is the infimum of $\frac{1}{2}\sum_u \sum_v c(u,v)\,x(u,v)$ over feasible $x$ (the double sum counts each edge twice, hence the $\frac12$). The incidence vector of any tour is feasible, so $\mathrm{LP} \le \mathrm{OPT}$ always.

Formalization targets

Goal — the 4/3 conjecture

Together with the known lower-bound families this says the integrality gap is exactly $4/3$. The goal carries no algorithm and no constant to improve: it is the terminal statement of the ladder, open in both directions (a proof or a counterexample instance would each settle it).

Milestones — the known ladder

Five results over the same definitions: the relaxation is valid ($\mathrm{LP} \le \mathrm{OPT}$); instance families force the gap arbitrarily close to $4/3$; tree doubling gives $\mathrm{OPT} \le 2\,\mathrm{LP}$; Wolsey's theorem gives $\mathrm{OPT} \le \frac{3}{2}\,\mathrm{LP}$, the classical upper bound; and the Karlin–Klein–Oveis Gharan record $\mathrm{OPT} \le (\frac{3}{2} - \varepsilon)\,\mathrm{LP}$ for some $\varepsilon > 10^{-36}$ (FOCS 2022). The last milestone is a statement-level target: its known proof (max-entropy sampling, strongly Rayleigh polynomials) is far beyond current formalization practice, so the mission's usable proving frontier remains Wolsey — the milestone records the state of the art as a formal statement.

Significance

The 4/3 conjecture is the reference open problem of approximation algorithms: the quality of the subtour LP calibrates every algorithmic advance on TSP, and the conjectured extremal instances guide the search for better rounding schemes. The bound is also what practical solvers actually compute — branch-and-cut on this LP solves instances with tens of thousands of cities — so the conjecture is a statement about the observed tightness of the world's most-used combinatorial lower bound.

Nothing in this circle exists in any proof assistant: Mathlib has no TSP, no LP relaxations, no polyhedral combinatorics of tours. The mission's milestones force the base layer into existence — tours over Equiv.Perm, cut constraints over Finset, and, for the upper bounds, the parity and tree arguments (spanning trees against the LP, T-joins for the $3/2$ bound) whose infrastructure is reusable for matching theory and network design far beyond TSP.

Difficulty

The naive plan — round the LP solution to a tour — has no known analysis losing less than $3/2$ in general, and the half-integral extremal instances show the hard cases are structured and simple-looking at once. Christofides' matching argument is provably stuck at $3/2$ against the LP; forty years of work moved the constant by $10^{-36}$, and that advance needed an entirely new probabilistic toolkit. On the other side, no instance family with ratio above $4/3$ has ever been found despite extensive computational search over small instances (Benoit–Boyd and successors). Both directions of the goal are genuinely open territory.

Formalization scope

The Lean model commits to: cities Fin n; costs c : Fin n → Fin n → ℝ with IsMetricCost (symmetry, zero diagonal, triangle inequality — nonnegativity is derived, and semimetrics are included as in the standard statement of the conjecture); tours as orderings π : Equiv.Perm (Fin n) traversed cyclically via finRotate, so every permutation denotes a Hamiltonian cycle and every Hamiltonian cycle is denoted; both optimal values as sInf over nonempty, bounded-below sets of reals, so they are genuine minima for n ≥ 3. The hypothesis 3 ≤ n is load-bearing: for n ≤ 2 the degree-2 constraints are infeasible, sInf ∅ = 0 by convention, and the bounds would be false — every theorem therefore carries it.

Welcome contributions: the milestones in any order — held_karp_le_opt is the natural entry point (the tour's incidence vector crosses every cut at least twice); integrality_gap_lower_bound needs the three-path instance family and a case analysis on its tours; tree_doubling_bound needs spanning trees against the LP; wolsey_bound adds the T-join/parity argument and is the summit. Reusable infrastructure — spanning tree polytopes, T-joins, Eulerian traversals, cut lemmas — is welcome as platform theorems. Graph-TSP ($7/5$), path TSP, and asymmetric TSP are deliberately left to future missions; the Karlin–Klein–Oveis Gharan bound is stated as a milestone, but its sampling machinery is expected to arrive, if ever, as shared infrastructure built over many contributions.

Selected references

• G. Dantzig, R. Fulkerson, S. Johnson, Solution of a large-scale traveling-salesman problem, Oper. Res. 2 (1954).
• M. Held, R. Karp, The traveling-salesman problem and minimum spanning trees, Oper. Res. 18 (1970); Part II, Math. Programming 1 (1971).
• N. Christofides, Worst-case analysis of a new heuristic for the travelling salesman problem, CMU report (1976); A. Serdyukov, Upravlyaemye Sistemy 17 (1978).
• L. Wolsey, Heuristic analysis, linear programming and branch and bound, Math. Prog. Study 13 (1980). doi:10.1007/BFb0120913
• D. Shmoys, D. Williamson, Analyzing the Held-Karp TSP bound: a monotonicity property with application, Inf. Process. Lett. 35 (1990). doi:10.1016/0020-0190(90)90028-V
• M. Goemans, Worst-case comparison of valid inequalities for the TSP, Math. Programming 69 (1995). doi:10.1007/BF01585563
• A. Sebő, J. Vygen, Shorter tours by nicer ears, Combinatorica 34 (2014). arXiv:1201.1870
• A. Karlin, N. Klein, S. Oveis Gharan, A (slightly) improved approximation algorithm for metric TSP, STOC 2021. arXiv:2007.01409
• A. Karlin, N. Klein, S. Oveis Gharan, A (slightly) improved bound on the integrality gap of the subtour LP for TSP, FOCS 2022. arXiv:2105.10043
• V. Traub, J. Vygen, Approximation Algorithms for Traveling Salesman Problems, Cambridge University Press, 2024. book page

Motivation

The $k$-server problem was introduced by Manasse, McGeoch, and Sleator (STOC 1988 / J. Algorithms 1990) as a common generalization of paging, weighted caching, and related sequential decision problems, and their $k$-server conjecture has since become the central open question of competitive analysis. The conjecture asserts that a single ratio — exactly $k$ — governs deterministic online server management on every metric space.

Timeline

• 1985. Sleator and Tarjan introduce competitive analysis — an online algorithm judged against the offline optimum on every input — for list update and paging, and ask for a theory of such guarantees.
• 1988–1990. Manasse, McGeoch, and Sleator introduce the $k$-server problem (STOC 1988; J. Algorithms 1990) and settle its extremes: no deterministic algorithm beats ratio $k$ on any space with more than $k$ points (Corollary 7), two servers admit a $2$-competitive algorithm (Theorem 5, algorithm RES), and $k$ servers on $k+1$ points admit a $k$-competitive one (Theorem 4, algorithm BAL). Section 8 poses the $k$-server conjecture, in the symmetric finite setting of the paper.
• 1990. Fiat, Rabani, and Ravid (FOCS 1990) give the first competitive ratio depending on $k$ alone — exponential in $k$, but finite on every metric space.
• 1991. Chrobak, Karloff, Payne, and Vishwanathan (SIAM J. Discrete Math.) prove the conjecture on the real line via Double Coverage; Chrobak and Larmore (SIAM J. Comput.) extend it to all tree metrics.
• 1995. Koutsoupias and Papadimitriou (J. ACM) prove the Work Function Algorithm is $(2k-1)$-competitive on every metric space — the breakthrough, and still the best general bound. Their Conjecture 1.1 fixes the conjecture's modern form: for every metric space there is an online algorithm with competitive ratio $k$.
• 1996. The same authors verify the conjecture on spaces of $k+2$ points via the dual 2-evader problem (Inf. Process. Lett. 57).
• 2004. Bartal and Koutsoupias prove the WFA itself is $k$-competitive on the line, weighted stars, and all spaces of $k+2$ points.
• 2021. Coester and Koutsoupias (ICALP) give a unifying potential for all known WFA analyses and push the frontier to the circle.
• 2023. Bubeck, Coester, and Rabani (STOC) refute the randomized analogue: no $o(\log^2 k)$-competitive randomized algorithm exists in general. The deterministic conjecture — this mission's goal — survives as the central open question, with the gap between $k$ and $2k-1$ unmoved since 1995.
• 2026. Coester, Koutsoupias, and Zbysiński post The $k$-server conjecture is true (arXiv:2609.15979), a claimed proof of the full conjecture: the Work Function Algorithm itself is $k$-competitive on every metric space, via a matrix representation of work functions and a potential function built on it. The preprint is not yet peer-reviewed; this mission's goal stays open until a machine-checked proof exists.

Setting

Fix a metric space $M$ with distance function $d$, and a number of servers $k \ge 1$. A configuration records where the $k$ servers stand: it is a function $C$ assigning to each server $i \in \{1, \dots, k\}$ a point $C(i) \in M$. Moving the servers from configuration $C$ to configuration $C'$ means server $i$ travels from $C(i)$ to $C'(i)$; the movement cost is the total distance traveled,

A request sequence is a finite list $\sigma = (r_1, \dots, r_n)$ of points of $M$, presented one at a time; write $\sigma_{\le j} = (r_1, \dots, r_j)$ for the list of the first $j$ requests (so $\sigma_{\le 0}$ is the empty list).

A deterministic online algorithm $A$ is a rule that, for every finite request sequence $\ell$, specifies a configuration $A(\ell)$ — where the servers stand after serving the requests of $\ell$ in order. In particular $A(\text{empty list})$ is the initial configuration, before any request arrives. Two points about this way of modeling an algorithm:

• Online and deterministic, by construction. The configuration after $j$ requests is $A(\sigma_{\le j})$, a function of those first $j$ requests only — the algorithm cannot see the future, and makes no random choices.
• The service constraint. Whenever a request sequence ends with a request $r$, some server must stand at $r$ immediately after: for every list $\ell$ and every point $r$, the configuration reached after serving $\ell$ followed by $r$ places at least one server at the point $r$.

Running $A$ on $\sigma = (r_1, \dots, r_n)$ produces the configurations $A(\sigma_{\le 0}),\, A(\sigma_{\le 1}),\, \dots,\, A(\sigma_{\le n})$, and its cost is the total movement along this trajectory:

For comparison, an offline schedule for $\sigma$ starting at a configuration $C_0$ is any sequence of configurations $S_0 = C_0, S_1, \dots, S_n$ in which $S_j$ places a server at the request $r_j$, for each $j$ — chosen with the whole of $\sigma$ known in advance. The optimal offline cost $\mathrm{OPT}(C_0, \sigma)$ is the infimum, over all such schedules, of the total movement $\sum_{j=1}^{n} \mathrm{moveCost}(S_{j-1}, S_j)$.

Finally, $A$ is $c$-competitive if there is a constant $a$ — depending on the algorithm, hence possibly on the metric space and the initial configuration, but never on the request sequence — with

Formalization targets

Goal — the $k$-server conjecture

The goal fixes no algorithm: any $k$-competitive construction settles it. This is the weakest stable form of the conjecture — it survives every improvement in constants or techniques short of a disproof.

Milestones — the known ladder

The milestones are the classical results between the trivial and the conjectured, each an existence or impossibility statement over the same definitions: the lower bound $c \ge k$ on any space with at least $k+1$ points; the conjecture for $k = 2$; for spaces of exactly $k+1$ points; for the real line; the $(2k-1)$ upper bound of the Work Function Algorithm on every space; the conjecture for spaces of exactly $k+2$ points; the conjecture for three servers in the Manhattan plane $(\mathbb{R}^2, \ell^1)$ — the one settled case over a genuinely two-dimensional continuum (Bein–Chrobak–Larmore 2002; reproved by the unifying potential of Coester–Koutsoupias 2021); Coester–Koutsoupias's 2021 result that the Work Function Algorithm itself — not just some algorithm — is $3$-competitive for three servers on trees, stated over an explicit formalization of the WFA; and the 2023 Bubeck–Coester–Rabani refutation of the randomized analogue: there are $(k+1)$-point spaces on which every randomized algorithm is $\Omega(\log^2 k)$-competitive, stated over a mixed-strategy model of randomized online algorithms.

Significance

A proof of the conjecture would close the founding problem of competitive analysis and pin down the exact power of determinism in online optimization over arbitrary metrics; a disproof would separate general metric spaces from every special class where the ratio $k$ is known tight. Either outcome recalibrates the field's standard model of adversarial request sequences.

None of these results — not even the lower bound — has a machine-checked proof, and online algorithms as a subject are absent from Mathlib. This mission builds the base layer: a faithful model of online service systems (configurations, online algorithms as prefix functions, offline schedules, competitiveness), the classical possibility and impossibility results over it, and, at the top, the Koutsoupias–Papadimitriou bound, whose potential-function argument is self-contained but delicate. The model is reusable for paging, weighted caching, metrical task systems, and the randomized $k$-server problem.

Difficulty

The obvious first idea — the greedy algorithm, moving the nearest server to each request — is not competitive for any constant, already on three points of the line: two nearby points can ping-pong one server forever while a server parked slightly farther away never moves. Every known competitive algorithm must sometimes move a server other than the nearest one, and the whole difficulty of the conjecture is quantifying exactly how much such foresight-free hedging can achieve. The Work Function Algorithm's analysis via a potential over offline work functions loses a factor of two for reasons nobody has been able to remove; on the lower-bound side, no metric space is known where the deterministic ratio exceeds $k$.

Formalization scope

The Lean model commits to: configurations as functions Fin k → M (labeled servers — equivalent in cost to the unlabeled multiset model, since offline can permute labels for free); algorithms as total functions List M → (Fin k → M) with the service constraint, so a step may move several servers (the standard laziness reduction makes this equivalent to one-move-per-request); costs in ℝ via Metric.dist; the offline optimum as an sInf over schedules, which agrees with the attained minimum on finite spaces; and the additive-constant form of competitiveness, quantified as ∃ a, ∀ σ.

Two conventions guard against trivialization. The additive constant is quantified before the request sequence — allowing it to depend on $\sigma$ would make every algorithm $1$-competitive. And the lower-bound milestone requires $k+1$ distinct points (Finset.card = k + 1); on spaces with at most $k$ points the conjecture is trivially true and the lower bound false.

Three further definitional layers extend the model. The work function workFunction C₀ σ C is the sInf of (schedule cost + final move to C) over schedules serving σ from C₀, and the Work Function Algorithm WFA is defined on finite spaces with k ≥ 1 servers: after each request it moves to a configuration containing the request minimizing (movement cost) + (work function of the history including the request), a minimizer existing by finiteness and ties broken by a fixed arbitrary choice — matching the standard definition with its "ties broken arbitrarily" (our fixed choice is one admissible instance). A tree is a finite metric space carrying a tree graph whose weighted path lengths realize the metric — exactly "the set of vertices of a tree" of the sources. A randomized algorithm is a mixed strategy: a probability measure over an index type together with a deterministic algorithm per outcome and measurable per-sequence cost; its expected cost is a lower Lebesgue integral in $[0,\infty]$, and $c$-competitiveness from $C_0$ demands every outcome start at $C_0$ and one additive constant work for all request sequences.

Welcome contributions: proofs of any milestone in any order (the lower bound and the $(k+1)$-point case are the natural entry points); alternative algorithms for milestones already closed; and infrastructure lemmas about moveCost, schedules, and work functions published as reusable platform theorems.

Selected references

• M. Manasse, L. McGeoch, D. Sleator, Competitive algorithms for server problems, J. Algorithms 11 (1990). doi:10.1016/0196-6774(90)90003-W
• A. Fiat, Y. Rabani, Y. Ravid, Competitive k-server algorithms, FOCS 1990. doi:10.1109/FSCS.1990.89566
• M. Chrobak, H. Karloff, T. Payne, S. Vishwanathan, New results on server problems, SIAM J. Discrete Math. 4 (1991). doi:10.1137/0404017
• M. Chrobak, L. Larmore, An optimal on-line algorithm for k servers on trees, SIAM J. Comput. 20 (1991). doi:10.1137/0220008
• E. Koutsoupias, C. Papadimitriou, On the k-server conjecture, J. ACM 42 (1995). doi:10.1145/210118.210128
• E. Koutsoupias, C. Papadimitriou, The 2-evader problem, Inf. Process. Lett. 57(5) (1996), 249–252.
• C. Coester, E. Koutsoupias, Towards the k-server conjecture: a unifying potential, pushing the frontier to the circle, ICALP 2021. arXiv:2102.10474
• S. Bubeck, C. Coester, Y. Rabani, The randomized k-server conjecture is false!, STOC 2023. arXiv:2211.05753
• E. Koutsoupias, The k-server problem (survey), Computer Science Review 3 (2009). doi:10.1016/j.cosrev.2009.04.002

Motivation

The simplex method walks along edges of a polytope from vertex to vertex. Whether any pivot rule could ever make that walk short in the worst case is governed by a prior, purely geometric question: how far apart, in the edge graph, can two vertices of a polytope be? Warren Hirsch conjectured in 1957 that the diameter of a $d$-dimensional polytope with $n$ facets is at most $n - d$. Half a century of upper bounds stalled at quasi-polynomial, and Santos disproved the conjecture itself in 2012 — but only by a constant factor. The surviving question, the subject of the Polymath 3 project, is the polynomial Hirsch conjecture: is the diameter bounded by a polynomial in $n$ and $d$?

Timeline

• 1957. Hirsch states the conjecture $\mathrm{diam} \le n - d$ in a letter to Dantzig, who publishes it in Linear Programming and Extensions (1963).
• 1964–1966. Klee determines the exact maximum diameter of $3$-polytopes with $n$ facets, $\lfloor 2n/3\rfloor - 1$ — the Hirsch bound holds up to dimension three.
• 1967. Klee and Walkup (Acta Math.) refute the unbounded-polyhedron version, prove the bounded conjecture for $n - d \le 5$, and reduce the general case to the $d$-step conjecture ($n = 2d$).
• 1970. Larman (Proc. LMS) proves $\mathrm{diam} \le n\,2^{d-3}$ — linear in the number of facets for each fixed dimension, still the best bound of that shape.
• 1989. Naddef (Math. Programming) proves $0/1$-polytopes satisfy the Hirsch bound, with diameter at most $d$.
• 1992. Kalai and Kleitman (Bull. AMS) prove $\mathrm{diam} \le n^{\log_2 d + 2}$ in under a page — the quasi-polynomial barrier every later bound refines. The same year brings subexponential pivot rules (Kalai; Matoušek–Sharir–Welzl), the algorithmic counterpart.
• 2010. Eisenbrand, Hähnle, Razborov, and Rothvoß (Math. OR) show the known upper-bound arguments survive in a purely combinatorial abstraction — which admits almost-quadratic lower bounds, so a polynomial bound must use real geometry. Kalai launches Polymath 3 on the polynomial version.
• 2010–2012. Santos (Annals of Math.) disproves the Hirsch conjecture: a $43$-dimensional polytope with $86$ facets and diameter at least $44$, via spindles of large width.
• 2014–2019. Todd (SIAM J. Discrete Math.) sharpens Kalai–Kleitman to $(n-d)^{\log_2 d}$; Sukegawa refines further. Matschke, Santos, and Weibel (Proc. LMS 2015) shrink the counterexample to dimension $20$ with $40$ facets and diameter $21$. All known violations remain constant-factor; all known bounds remain quasi-polynomial.

Setting

Work in $\mathbb{R}^d$. An H-polytope is a set cut out by finitely many linear inequalities: given vectors $a_1, \dots, a_n \in \mathbb{R}^d$ and reals $b_1, \dots, b_n$, it is

where $\langle a_i, x\rangle = \sum_{j=1}^d a_{ij} x_j$ is the standard inner (dot) product — so each condition $\langle a_i, x\rangle \le b_i$ is one linear inequality, with normal vector $a_i$ and offset $b_i$. Throughout, $P$ is assumed nonempty and bounded. The parameter $n$ counts the inequalities in the given description; since every polytope with $f$ facets admits a description by exactly $f$ inequalities, bounds stated in terms of $n$ over all descriptions are equivalent to bounds in terms of facet counts.

A vertex of $P$ is an extreme point. Two vertices $u \ne v$ are adjacent when the segment $[u, v]$ is an extreme subset of $P$; for a polytope the convex extreme subsets are exactly the faces, so this says precisely that $[u,v]$ is a one-dimensional face — an edge. The combinatorial diameter of $P$ is the diameter of the graph of vertices and edges. Throughout, "diameter at most $B$" is expressed as: every two vertices are joined by a walk of $B$ steps, each step staying put or crossing an edge — a form that is monotone in $B$ and asserts connectivity of the graph (Balinski's theorem) as part of the claim.

Formalization targets

Goal — the polynomial Hirsch conjecture

Every polynomial in $n$ and $d$ is dominated by some $c(n+d)^k$ and conversely, so this is exactly polynomiality, with no committed degree — the form that survives any future sharpening of constants or exponents.

Milestones — the known ladder

Six classical results over the same definitions: the Hirsch bound $n - d$ in dimension $d \le 3$ (Klee; Klee–Walkup); Larman's bound $n \cdot 2^{d-3}$; Naddef's bound $d$ for $0/1$-polytopes; the Kalai–Kleitman bound $n^{\log_2 d + 2}$; Todd's bound $(n-d)^{\log_2 d}$ for full-dimensional $P$ with $n \ge d \ge 3$; and — in the other direction — the Santos counterexample: a nonempty bounded H-polytope whose diameter exceeds $n - d$.

Significance

A polynomial diameter bound is necessary for any pivot rule of the simplex method to run in polynomial time in the worst case: if vertices can be super-polynomially far apart, no edge-following algorithm can connect them quickly. A refutation would close off one of the main hoped-for routes to a strongly polynomial linear programming algorithm (Smale's ninth problem). The conjecture is also the test question of polyhedral graph theory: the Kalai–Kleitman argument uses so little about polytopes that it holds for far more general set systems, and Eisenbrand, Hähnle, Razborov, and Rothvoß (Math. OR 2010) showed such abstractions admit almost-quadratic lower bounds — so a proof of the conjecture must use geometry the abstract setting lacks, and a disproof must beat the abstraction barrier's constructions with actual polytopes.

None of these results has been formalized in any proof assistant; Mathlib has extreme points and faces of convex sets, but no polytope combinatorics — no vertex-edge graph, no diameter, no facet counting. This mission builds that layer: an H-polytope model, adjacency via faces, and walk-based diameter bounds, against which both the upper-bound ladder and the Santos disproof can be machine-checked. The Kalai–Kleitman proof is one page from first principles and is the natural summit; the Santos construction is a concrete finite object whose verification is a different, computational kind of challenge.

Difficulty

The naive approach — walk toward the target vertex by always improving some linear objective — is exactly the simplex method, and proving any polynomial bound on such walks is open for every known pivot rule; monotone variants of the diameter question have exponential lower bounds. The obvious inductive strategy (bound the diameter by recursing on facets) is precisely what Kalai–Kleitman optimizes, and it provably cannot go below quasi-polynomial without using metric or topological properties of actual polytopes, by the abstraction lower bound above. On the other side, making diameters large is blocked by the wedge/spindle calculus only producing constant-factor violations. The problem sits in a genuine gap: no technique on either side is known to reach polynomial.

Formalization scope

The Lean model commits to: ambient space EuclideanSpace ℝ (Fin d); the polytope as Hpoly a b = {x | ∀ i, ⟪a i, x⟫ ≤ b i} for a : Fin n → EuclideanSpace ℝ (Fin d), b : Fin n → ℝ, with nonemptiness and Bornology.IsBounded as explicit hypotheses (boundedness is essential: Klee–Walkup's unbounded counterexample would otherwise trivialize the Santos milestone); vertices as Set.extremePoints ℝ; adjacency as u ≠ v ∧ IsExtreme ℝ P (segment ℝ u v); and diameter bounds as the walk predicate DiamLE, whose stationary steps make it monotone in the bound. Real-exponent bounds enter through Real.logb and the natural floor. In larman_bound and the two Hirsch-form bounds the subtraction is natural-number (truncated) subtraction, which only weakens nothing: the stated forms are true as written for all $n, d$ in scope. The dimension parameter $d$ is the ambient dimension; lower-dimensional polytopes are included, and every milestone is stated so as to remain true for them, with todd_bound requiring full-dimensionality ((interior P).Nonempty) as in its source.

Welcome contributions: any milestone in any order (dimension_three_bound for $d \le 1$ cases and structural lemmas about Adj and DiamLE are natural entry points, and kalai_kleitman_bound is the summit); reusable infrastructure — polytopes have finitely many extreme points, faces of H-polytopes, Balinski connectivity — published as platform theorems; and, as a separate expedition, the explicit Santos or Matschke–Santos–Weibel polytope. Statements about unbounded polyhedra, the simplex method itself, and subexponential pivot rules are left to future missions.

Selected references

• V. Klee, D. Walkup, The d-step conjecture for polyhedra of dimension d < 6, Acta Math. 117 (1967). doi:10.1007/BF02392971
• D. Larman, Paths on polytopes, Proc. London Math. Soc. 20 (1970). doi:10.1112/plms/s3-20.2.249
• D. Naddef, The Hirsch conjecture is true for (0,1)-polytopes, Math. Programming 45 (1989). doi:10.1007/BF01589418
• G. Kalai, D. Kleitman, A quasi-polynomial bound for the diameter of graphs of polyhedra, Bull. AMS 26 (1992). arXiv:math/9204233
• F. Santos, A counterexample to the Hirsch conjecture, Annals of Mathematics 176 (2012). arXiv:1006.2814
• M. Todd, An improved Kalai–Kleitman bound for the diameter of a polyhedron, SIAM J. Discrete Math. 28 (2014). arXiv:1402.3579
• B. Matschke, F. Santos, C. Weibel, The width of five-dimensional prismatoids, Proc. London Math. Soc. 110 (2015). arXiv:1202.4701
• F. Eisenbrand, N. Hähnle, A. Razborov, T. Rothvoß, Diameter of polyhedra: limits of abstraction, Math. Oper. Res. 35 (2010). doi:10.1287/moor.1100.0470
• F. Santos, Recent progress on the combinatorial diameter of polytopes and simplicial complexes, TOP 21 (2013) (survey). arXiv:1307.5900

Submission hold — source-fidelity repair (2026-09-04). The public goal restricts answers to elementary expression trees and is only a stronger subquestion. It does not formalize the source's broader special-function closed-form question. The existing published target is preserved, with this scope warning. Do not confirm or submit this version as a faithful formalization of the full source. The legacy mathematical target below is retained for traceability while the replacement is prepared.

Motivation

A standard Brownian motion starts below the exponentially decaying boundary b(t)=b0 exp(-ct). The first time it crosses the boundary has a continuous density characterized by a generalized Abel--Volterra integral equation. The source asks for an explicit distribution, motivated in part by neuronal threshold models with a decaying refractory boundary.

This mission turns CUHK-Shenzhen AI Math Problem 13, First-passage time of Brownian motion to an exponentially decaying boundary, into an auditable Lean campaign. The objective is not merely to transcribe notation: it is to expose the mathematical model, the capstone, and a smaller attack surface as separate artifacts that other formalizers can inspect and reuse.

Setting

Construct one expression in a fixed elementary language whose evaluation is a continuous nonnegative density on positive times, solves the Abel equation, and integrates to one. The language contains real constants, rational constants, arithmetic, exp, log, square root, trigonometric functions, and the normal density. The first milestone drops elementary representability and normalization and asks for a continuous nonnegative Abel solution.

Significance

Solving this target would settle the precise finite or analytic core represented by the Lean statement and would create reusable infrastructure in Brownian motion, first-passage times, stochastic processes, Volterra integral equations. Even a rigorous disproof is valuable: several entries in this collection deliberately ask whether an attractive extrapolation is true, and Lean forces a counterexample to satisfy every side condition. The mission therefore treats theorem proving and model criticism as equally legitimate research outcomes.

Difficulty

Moving-boundary first-passage laws rarely have elementary closed forms. The Abel kernel is singular at the upper endpoint, and showing that a candidate equation solution is the actual passage density requires uniqueness and probability normalization. The capstone may be false under the selected expression language; a non-elementarity theorem would be a legitimate disproof of this precise formal target.

Suggested attack route

Formalize existence and uniqueness for the Volterra equation using weakly singular kernels, then connect it to Brownian first passage. Explore transformations suggested by the exponential boundary, Laplace transforms, and iterative resolvent kernels. Symbolic or numerical calculations may reveal special-function rather than elementary structure. If so, characterize the required extension of the expression language and prove why the current language is insufficient.

Formalization scope

The already-published goal uses finite elementary-expression trees with arithmetic, exp/log/sqrt, sin/cos and normal density. The source explicitly permits standard special functions beyond this language. Accordingly the published declaration is a stronger elementary-only subquestion, not a faithful replacement for the full closed-form question. It is preserved as an existing result; its proof or disproof must not be reported as settling every special-function formula. The Abel-solution milestone asserts only existence of a continuous nonnegative solution; uniqueness, normalization, and identification with the first-passage density remain separate obligations. A complete source-faithful replacement needs an agreed formula class or a concrete proposed formula, not an unrestricted function renamed a closed form.

Milestones

For each positive boundary height and decay parameter, there exists a continuous nonnegative solution of the stated Abel equation on positive times. This node asserts existence only, not uniqueness, unit mass, or an elementary closed form.

Timeline and literature status

The CUHK-Shenzhen AI Math Problems page added this problem on June 23, 2026. At the drafting date, August 31, 2026, the status and target corrections described above were checked against the source page and the cited primary material.

Acceptance criteria

A contribution may prove the displayed theorem or refute it by constructing data satisfying every Lean hypothesis while negating the conclusion. Informal changes of model do not count: any proposed correction must be submitted as a separately reviewed statement with an explanation of which source ambiguity or false implication it repairs. Definitions must remain computational or mathematically constrained; fields that simply assume the desired conclusion are not acceptable. Every proof must compile against the mission's pinned Mathlib revision, use no sorry, and expose a top-level theorem solution when submitted to Prove2Me.

The main theorem is intentionally separated from a smaller milestone. Contributors should preserve that dependency order, publish reusable lemmas rather than monolithic tactics, and report whether a lemma is analytic, algebraic, combinatorial, or infrastructure-only. Numerical evidence, external computer algebra, and exhaustive search are welcome for discovery, but a final certificate must be replayable by Lean. If an external result is invoked, its hypotheses must be represented in the formal statement or proved in the dependency tree.

Formal verification policy

The files were built locally with Lean 4.30.0 and Mathlib revision c5ea00351c28e24afc9f0f84379aa41082b1188f, the supported Prove2Me environment at drafting time. The mission definition file is ordered before all theorem files, and each theorem imports exactly that public definition module or Mathlib. Independent blind read-backs accompany the draft items so reviewers can compare what the Lean code literally says with this mathematical description. Human confirmation remains required before the public proposal can be submitted for moderation.

Selected references

• Original CUHK-Shenzhen problem

Motivation

Two orthonormal bases of C^d are mutually unbiased when every transition amplitude has squared modulus 1/d. At most d+1 such bases can coexist, and complete families are known in prime-power dimensions through finite-field constructions. Dimension six is the smallest famous composite case where existence of the complete seven-base family remains unknown.

This mission turns CUHK-Shenzhen AI Math Problem 16, Existence of complete sets of mutually unbiased bases, into an auditable Lean campaign. The objective is not merely to transcribe notation: it is to expose the mathematical model, the capstone, and a smaller attack surface as separate artifacts that other formalizers can inspect and reuse.

Setting

Construct seven 6 by 6 unitary-column matrices whose every distinct pair has all transition amplitudes of squared modulus 1/6. The baseline milestone constructs three pairwise mutually unbiased bases, a known lower bound that tests all matrix conventions.

Significance

Solving this target would settle the precise finite or analytic core represented by the Lean statement and would create reusable infrastructure in quantum information theory, mutually unbiased bases, finite fields, Hilbert spaces. Even a rigorous disproof is valuable: several entries in this collection deliberately ask whether an attractive extrapolation is true, and Lean forces a counterexample to satisfy every side condition. The mission therefore treats theorem proving and model criticism as equally legitimate research outcomes.

Difficulty

The equations are a large coupled system of polynomial equalities over complex phases, modulo substantial gauge symmetry. Numerical near-solutions do not certify exact existence, while nonexistence would require a global obstruction beyond currently known bounds. Dimension six lacks the finite-field structure that supplies complete prime-power constructions.

Suggested attack route

Formalize standard gauge reductions: fix the first basis to the identity and dephase transition Hadamard matrices. Verify a three-basis tensor-product construction. Then encode additional bases through complex Hadamard matrices and study algebraic constraints, Gröbner-style eliminations, semidefinite bounds, or exact certificates. Computational searches may guide conjectures, but uploaded proofs must convert numerical evidence to exact algebraic identities or certified inequalities.

Formalization scope

The Lean target is exact: column orthonormality is U-adjoint times U equals identity, and mutual unbiasedness uses Mathlib complex norm squared. Seven bases are indexed by Fin 7. No quotient by phase, permutation, or global unitary is built into the statement, since these symmetries preserve the predicate and can be used within proofs.

The natural-language source remains authoritative for motivation, while the Lean declaration is authoritative for what Prove2Me will verify. The mission description calls out restrictions where the current formal target is a finite-dimensional core, a fixed interpretation of informal terminology, or one sharpened subquestion from a broader classification problem. Those restrictions should not be silently generalized in a proof claim.

Milestones

Publish an exact three-basis construction in dimension six, then formalize dephasing and obstruction lemmas for extending a partial family.

The capstone is marked as the mission's main item and is never duplicated as a milestone. Definitions precede theorem statements in the proposal order. A milestone is considered complete only when its own exact statement is proved; proving a nearby theorem with stronger-looking prose but mismatched quantifiers, signs, supports, dimensions, or asymptotic constants does not complete it.

Timeline and literature status

The CUHK-Shenzhen AI Math Problems page added this problem on June 23, 2026. At the drafting date, August 31, 2026, the status and target corrections described above were checked against the source page and the cited primary material.

Acceptance criteria

A contribution may prove the displayed theorem or refute it by constructing data satisfying every Lean hypothesis while negating the conclusion. Informal changes of model do not count: any proposed correction must be submitted as a separately reviewed statement with an explanation of which source ambiguity or false implication it repairs. Definitions must remain computational or mathematically constrained; fields that simply assume the desired conclusion are not acceptable. Every proof must compile against the mission's pinned Mathlib revision, use no sorry, and expose a top-level theorem solution when submitted to Prove2Me.

The main theorem is intentionally separated from a smaller milestone. Contributors should preserve that dependency order, publish reusable lemmas rather than monolithic tactics, and report whether a lemma is analytic, algebraic, combinatorial, or infrastructure-only. Numerical evidence, external computer algebra, and exhaustive search are welcome for discovery, but a final certificate must be replayable by Lean. If an external result is invoked, its hypotheses must be represented in the formal statement or proved in the dependency tree.

Formal verification policy

The files were built locally with Lean 4.30.0 and Mathlib revision c5ea00351c28e24afc9f0f84379aa41082b1188f, the supported Prove2Me environment at drafting time. The mission definition file is ordered before all theorem files, and each theorem imports exactly that public definition module or Mathlib. Independent blind read-backs accompany the draft items so reviewers can compare what the Lean code literally says with this mathematical description. Human confirmation remains required before the public proposal can be submitted for moderation.

Selected references

• Original problem
• Durt et al., review of MUBs

Motivation

Discrepancy theory asks how evenly a collection of objects can be split into two parts. Its central open question is a conjecture of Komlós, first circulated in the 1980s: any finite family of vectors of Euclidean length at most one can be signed $\pm 1$ so that the signed sum is bounded in every coordinate by a universal constant — independent of how many vectors there are and of the dimension they live in.

Timeline

• 1963. Steinitz-type vector balancing questions circulate; Bárány and Grinberg later (1981) show any norm admits a dimension-dependent bound $2d$, setting the theme: how much of the dependence on dimension is real?
• 1981. Beck and Fiala (Discrete Appl. Math.) prove degree-$t$ set systems have discrepancy at most $2t - 1$, by the floating-colors argument, and conjecture $O(\sqrt{t})$.
• 1980s. Komlós poses the vector form — unit $\ell^2$-norm columns, constant $\ell^\infty$ discrepancy — which implies the Beck–Fiala conjecture; it circulates through Spencer's Ten Lectures (1987) as the central open problem of the area.
• 1985. Spencer (Trans. AMS) proves "six standard deviations suffice": discrepancy $6\sqrt{n}$ for $n$ sets on $n$ points, beating random signing via the partial-coloring method.
• 1998. Banaszczyk (Random Struct. Algorithms) proves the Komlós bound $O(\sqrt{\log n})$ by a recursive Gaussian-measure argument over convex bodies.
• 2010–2016. The constructive era: Bansal (2010) makes Spencer algorithmic by SDP random walks, Lovett and Meka (2012) simplify, and Bansal, Dadush, and Garg (STOC 2016) give a polynomial-time algorithm matching Banaszczyk's bound.
• 2023. Kunisky (SIAM J. Discrete Math.) constructs instances from unsatisfiable formulas with discrepancy approaching $1+\sqrt{2}$ — the strongest lower bound on the conjectured constant.
• 2025. Bansal and Jiang (arXiv:2508.03961) break the Banaszczyk barrier: $\tilde{O}((\log n)^{1/4})$ for Komlós, and the Beck–Fiala conjecture resolved for $t \ge \log^2 n$ — the first movement in nearly thirty years. The gap between $2.414\ldots$ and $\tilde{O}((\log n)^{1/4})$ is the conjecture.

Setting

Fix $n$ vectors $v_1, \dots, v_n \in \mathbb{R}^m$ with Euclidean norm $\lVert v_i \rVert_2 \le 1$. A sign vector is an $\varepsilon \in \{-1, +1\}^n$: one sign $\varepsilon_i \in \{\pm 1\}$ per vector. Writing $v_{ij}$ for the $j$-th coordinate of the vector $v_i$, the discrepancy of the family under $\varepsilon$ is the largest coordinate, in absolute value, of the signed sum $\sum_i \varepsilon_i v_i$ — that is, $\max_{j \le m} \lvert \sum_{i \le n} \varepsilon_i v_{ij} \rvert$, the $\ell^\infty$ norm of the signed sum. The Komlós property at constant $K$ — KomlosBound K — says that every such family, in every $n$ and every $m$, admits a sign vector with every coordinate of the signed sum at most $K$ in absolute value.

Set systems embed as the special case of $0/1$-incidence matrices: if $A$ is an $m \times n$ matrix of $0$s and $1$s in which every column has at most $t$ ones (every element lies in at most $t$ sets), the columns scaled by $1/\sqrt{t}$ have norm at most one, so the Komlós property gives discrepancy $K\sqrt{t}$ — the Beck–Fiala conjecture.

Formalization targets

Goal — the Komlós conjecture

The goal fixes no value of $K$: any finite universal constant settles it, so the statement survives every improvement in the constant.

Milestones — the known ladder

Eight results over the same definitions: Beck–Fiala's $2t - 1$ for degree-$t$ set systems; Spencer's $6\sqrt{n}$ for $n$ sets on $n$ points; Banaszczyk's $O(\sqrt{\log n})$ for the Komlós setting; its corollary $O(\sqrt{t \log n})$ for set systems; the reduction "Komlós at $K$ implies Beck–Fiala at $K\sqrt{t}$"; Kunisky's lower bound $K \ge 1 + \sqrt{2}$; and the two 2025 Bansal–Jiang breakthroughs — $\tilde{O}((\log n)^{1/4})$ for the Komlós setting, and the Beck–Fiala conjecture's bound $O(\sqrt{t})$ in the regime $t = \Omega(\log^2 n)$.

Significance

The conjecture is the meeting point of the two main techniques of discrepancy theory — partial coloring and the Gaussian/convex-geometric method — and each further improvement has forced a new technique into existence. A proof would resolve the Beck–Fiala conjecture in full and sharpen the hereditary-discrepancy landscape; a disproof would break the widely-shared expectation that vector balancing is dimension-free. The problem is also a benchmark for algorithmic discrepancy: every known bound now has a polynomial-time counterpart, and the constructive tools built for it (random-walk roundings, spectral partial colorings) are used across approximation algorithms and ranging into differential privacy.

None of this literature is formalized anywhere; Mathlib has no discrepancy theory at all. The definitions here are elementary — finite sums, absolute values, one norm hypothesis — so the mission's entry cost is unusually low for an open-problem mission: the Beck–Fiala theorem and the scaling reduction are self-contained finite combinatorics, while Spencer and Banaszczyk each force a genuinely new proof technique (pigeonhole partial coloring; Gaussian measure on convex bodies) into Lean.

Difficulty

Random signs lose: they give $\Theta(\sqrt{n})$, not a constant, so the naive probabilistic argument is ruled out from the start. The Beck–Fiala argument caps discrepancy by degree, not by norm, and provably cannot be pushed below $2t - O(1)$ by its own bookkeeping. Partial coloring alone loses a logarithm through its iteration, and Banaszczyk's method is blocked at $\sqrt{\log n}$ by the Gaussian measure of the cube. The 2025 advance decouples the two methods but still pays iterated polylogarithmic factors. Nothing currently known contracts the remaining gap to a constant, and the lower bound says the constant, if it exists, is at least $1 + \sqrt{2}$ — so any proof must handle instances strictly harder than the set-system case.

Formalization scope

The Lean model commits to: vectors as EuclideanSpace ℝ (Fin m), whose norm is the $\ell^2$ norm (the hypothesis $\lVert v_i \rVert \le 1$ reads ‖v i‖ ≤ 1); the $\ell^\infty$ conclusion written coordinatewise as ∀ j, |∑ i, ε i * v i j| ≤ K, avoiding any auxiliary sup-norm structure; sign vectors as real vectors with ε i = 1 ∨ ε i = -1; and set systems as matrices A : Fin m → Fin n → ℝ with an entrywise $0/1$ hypothesis and column-degree counted by Set.ncard. Quantifier order matters everywhere: in KomlosBound K the constant is fixed before $n$ and $m$ — a $K$ depending on $n$ would make the statement the trivial $\sqrt{n}$ bound. In beck_fiala the hypothesis $t \ge 1$ is required (the degree-$0$ system has discrepancy $0 > 2t-1$ otherwise); the Banaszczyk-form bounds use $\log(n+2)$ so that the bound is positive already at $n \le 1$. In the Bansal–Jiang milestones the asymptotic $\tilde{O}$ and $\Omega$ are rendered by existential constants quantified before all instances: the hidden $\mathrm{poly}(\log\log n)$ factor becomes $(\log\log(n+8))^{\gamma}$ for some fixed $\gamma > 0$ (the inner shift $+8$ keeps the iterated logarithm positive), and the threshold $t = \Omega(\log^2 n)$ becomes $C_0 \log^2(n+2) \le t$ for some fixed $C_0 > 0$.

Welcome contributions: any milestone in any order — beck_fiala and komlos_implies_beck_fiala are self-contained finite arguments and the natural entry points; spencer_six_deviations and banaszczyk_bound each import a major technique; komlos_lower_bound needs an explicit construction and a case analysis over all sign vectors. Reusable infrastructure — partial colorings, Gaussian measure bounds for convex bodies, hereditary discrepancy — is welcome as platform theorems. The matrix Spencer conjecture, prefix discrepancy, and the Steinitz problem are related but deliberately left to future missions.

Selected references

• J. Beck, T. Fiala, "Integer-making" theorems, Discrete Applied Mathematics 3 (1981). doi:10.1016/0166-218X(81)90022-6
• J. Spencer, Six standard deviations suffice, Trans. Amer. Math. Soc. 289 (1985). doi:10.1090/S0002-9947-1985-0784009-0
• W. Banaszczyk, Balancing vectors and Gaussian measures of n-dimensional convex bodies, Random Structures & Algorithms 12 (1998). doi link
• N. Bansal, D. Dadush, S. Garg, An algorithm for Komlós conjecture matching Banaszczyk's bound, FOCS 2016 / SIAM J. Comput. arXiv:1605.02882
• N. Bansal, H. Jiang, Decoupling via affine spectral-independence: Beck–Fiala and Komlós bounds beyond Banaszczyk, 2025. arXiv:2508.03961
• D. Kunisky, The discrepancy of unsatisfiable matrices and a lower bound for the Komlós conjecture constant, SIAM J. Discrete Math. 37 (2023). arXiv:2111.02974
• B. Chazelle, The Discrepancy Method, Cambridge University Press, 2000. author's page

Motivation

The source defines a two-variable integral on strictly positive-definite real matrices. Its numerator is the absolute bilinear form of A minus B on two unit vectors, while the denominator uses the quadratic forms of A and B. The desired inequality says that componentwise matrix addition is nonexpansive for this quantity, with the larger of the two input distances controlling the output. The surface measure normalization is immaterial because the same constant multiplies every distance.

This mission turns CUHK-Shenzhen AI Math Problem 1, Positive definite matrix integral inequality, into an auditable Lean campaign. The objective is not merely to transcribe notation: it is to expose the mathematical model, the capstone, and a smaller attack surface as separate artifacts that other formalizers can inspect and reuse.

Setting

For every positive dimension and every four positive-definite matrices A, B, C, and D, prove d(A+B,C+D) is at most max(d(A,C),d(B,D)). The first milestone fixes dimension one, where the sphere and every matrix entry can be analyzed explicitly.

Significance

Solving this target would settle the precise finite or analytic core represented by the Lean statement and would create reusable infrastructure in matrix analysis, positive definite matrices, integral inequality. Even a rigorous disproof is valuable: several entries in this collection deliberately ask whether an attractive extrapolation is true, and Lean forces a counterexample to satisfy every side condition. The mission therefore treats theorem proving and model criticism as equally legitimate research outcomes.

Difficulty

The absolute value prevents a direct cancellation argument, and the denominators couple each integration variable to a different matrix. Positive definiteness gives pointwise positivity but does not immediately compare the ratios after addition. A successful proof must find a convexity, change-of-measure, or projective-metric mechanism that survives the double integral.

Suggested attack route

Promising routes include reducing by congruence to normalized matrices, studying the scalar inequality on each pair of directions, and interpreting the denominator as a density change on the sphere. The dimension-one case should reveal the sharp scalar inequality. Numerical experiments in dimensions two and three may identify equality cases, but the Lean proof must ultimately derive every bound from positivity and measurable integration.

Formalization scope

The Lean model uses finite matrices, Mathlib positive definiteness, the canonical sphere measure obtained from polar decomposition, and an explicit iterated integral. It does not assume symmetry through an unchecked flag: positive definiteness is the Mathlib predicate. Integrability obligations and zero-denominator issues must be proved from positive definiteness.

The natural-language source remains authoritative for motivation, while the Lean declaration is authoritative for what Prove2Me will verify. The mission description calls out restrictions where the current formal target is a finite-dimensional core, a fixed interpretation of informal terminology, or one sharpened subquestion from a broader classification problem. Those restrictions should not be silently generalized in a proof claim.

Milestones

Establish the dimension-one specialization, including any exact evaluation of the two-point sphere integral needed by the proof.

The capstone is marked as the mission's main item and is never duplicated as a milestone. Definitions precede theorem statements in the proposal order. A milestone is considered complete only when its own exact statement is proved; proving a nearby theorem with stronger-looking prose but mismatched quantifiers, signs, supports, dimensions, or asymptotic constants does not complete it.

Timeline and literature status

The CUHK-Shenzhen AI Math Problems page added this problem on May 28, 2026. At the drafting date, August 31, 2026, the status and target corrections described above were checked against the source page and the cited primary material.

Acceptance criteria

A contribution may prove the displayed theorem or refute it by constructing data satisfying every Lean hypothesis while negating the conclusion. Informal changes of model do not count: any proposed correction must be submitted as a separately reviewed statement with an explanation of which source ambiguity or false implication it repairs. Definitions must remain computational or mathematically constrained; fields that simply assume the desired conclusion are not acceptable. Every proof must compile against the mission's pinned Mathlib revision, use no sorry, and expose a top-level theorem solution when submitted to Prove2Me.

The main theorem is intentionally separated from a smaller milestone. Contributors should preserve that dependency order, publish reusable lemmas rather than monolithic tactics, and report whether a lemma is analytic, algebraic, combinatorial, or infrastructure-only. Numerical evidence, external computer algebra, and exhaustive search are welcome for discovery, but a final certificate must be replayable by Lean. If an external result is invoked, its hypotheses must be represented in the formal statement or proved in the dependency tree.

Formal verification policy

The files were built locally with Lean 4.30.0 and Mathlib revision c5ea00351c28e24afc9f0f84379aa41082b1188f, the supported Prove2Me environment at drafting time. The mission definition file is ordered before all theorem files, and each theorem imports exactly that public definition module or Mathlib. Independent blind read-backs accompany the draft items so reviewers can compare what the Lean code literally says with this mathematical description. Human confirmation remains required before the public proposal can be submitted for moderation.

Selected references

• Original CUHK-Shenzhen problem

Motivation and history

Compressing an operator to a closed subspace keeps the information visible within that subspace but can discard interactions with its orthogonal complement. The compression-rigidity question asks whether a particular equality detects that no such interactions were present. Its inputs are two commuting positive contractions and an ordinary strictly convex function of two real variables. The issue is the equality case, not the existence of a general operator inequality for every convex function.

The question was contributed by Boris Bilich to the CUHK-Shenzhen AI Math Problems collection and added on June 1, 2026. The original problem asks about operators on a Hilbert space without imposing finite dimension. The first formal target in this mission treated matrices with supplied joint spectral data. That finite-dimensional declaration has a verified proof on Prove2Me, but it does not settle the unrestricted Hilbert-space question. The September 2026 correction restores arbitrary complex Hilbert spaces as the main target and preserves the earlier result as a supporting artifact.

Setting

Let $H$ be a complete complex Hilbert space, and let $\mathcal B(H)$ be its algebra of bounded complex-linear operators. The multiplication $AB$ means composition, with $B$ acting first, and $A^*$ denotes the adjoint. A positive contraction $A$ is self-adjoint, satisfies $\operatorname{Re}\langle v,Av\rangle\geq0$ for every $v\in H$, and has operator norm at most one. Both zero and the identity are permitted.

Write $S=[0,1]^2$. Let $X,Y\in\mathcal B(H)$ be positive contractions satisfying $XY=YX$. Their joint continuous functional calculus assigns an operator $g(X,Y)$ to each continuous real function $g$ on $S$. It is characterized as a continuous unital real star-algebra homomorphism from $C(S,\mathbb R)$ to $\mathcal B(H)$ that maps the two coordinate functions to $X$ and $Y$. The domain has the uniform norm and the codomain the operator norm. The existence and uniqueness of this calculus follow from the standard joint spectral theorem; the relevant source is Dereziński, Lemma 6.6, printed page 43.

An orthogonal projection is an operator $P$ satisfying $P^*=P$ and $P^2=P$. The compressed operators $PXP$ and $PYP$ are still viewed as operators on the original space $H$, not as operators on a separately chosen finite-dimensional range. The question includes the additional hypothesis that these compressed operators commute. The original commutation of $X$ and $Y$ does not remove the need to state this hypothesis.

Formalization targets

The main target is the entire compression implication from the source. Let $f:S\to\mathbb R$ be continuous and strictly convex: for distinct $x,y\in S$ and $0<t<1$, its value at $tx+(1-t)y$ is strictly less than $t f(x)+(1-t)f(y)$. For $X,Y,P$ as above, the question is whether

Both outer projections on the right are part of the original assertion. There is no hypothesis that $f(0,0)=0$. Commutation of $P$ with each coordinate operator is exactly the reducing-subspace conclusion asked for in the source.

The accompanying standard infrastructure target asserts that, for every commuting pair of positive contractions on $H$,

where $\Phi$ is continuous, unital, real-linear, multiplicative and star-preserving. This is the unit-square, real-valued-function specialization of Lemma 6.6, not a claim that this known theorem is a new research conjecture. Its Lean proof is a separate supporting obligation. The earlier two-dimensional milestone and the proved finite-dimensional capstone remain available; neither replaces the new main target.

Significance

A positive resolution would show that exact preservation of one strictly convex functional-calculus value, under the specified commuting-compression hypothesis, forces both operators to preserve the projection's range and its orthogonal complement. A negative resolution would require an actual Hilbert space, operators and strictly convex function satisfying every hypothesis while at least one of the two reducing identities fails.

The formal development separates this research question from the standard spectral infrastructure needed to express it. The new main declaration is an open proof obligation. The joint-calculus existence-and-uniqueness declaration is also unproved in this contribution, although mathematically standard. Local compilation and server publication check the declarations' well-formedness; they are not proofs of either statement. Only the earlier finite-dimensional result is being reported here as already proved.

Difficulty

Arbitrary bounded commuting self-adjoint operators need not have a joint eigenbasis. Consequently a matrix formulation that records finitely many joint spectral atoms cannot serve as the general operator model. The compressed pair may also have different spectral data from the original pair. The equality involves these two different functional calculi, with a projection on either side of each value.

Strict convexity in this question is ordinary scalar strict convexity on the square. Operator convexity, finite rank of the projection, compactness of the coordinate operators, and a multivariable operator Jensen inequality are not additional assumptions. Introducing any of them would change the requested question. The general formulation must also retain boundary cases rather than exclude them to simplify an argument.

Formalization scope

The Lean model uses actual bounded complex-linear maps on an arbitrary complete inner-product space. It imposes no finite-dimensionality, separability, common-eigenbasis or nonzero-space assumption. It includes $P=0$, $P=I_H$ and the zero Hilbert space. The scalar field convention is complex; a real-Hilbert-space transfer is not separately formalized here.

The function is stored on the ambient real plane, but continuity, strict convexity and evaluation use only its restriction to $S$. Values outside $S$ are irrelevant, and continuity outside the square is not required. Thus storing an ambient function does not exclude any continuous function originally defined only on the square.

Joint evaluation is a total definition. If a representing continuous unital real star-algebra homomorphism exists, it chooses one for the operator pair and then evaluates the supplied function. Otherwise it returns zero. The separate existence-and-uniqueness target establishes that this fallback is inapplicable to commuting positive contractions and that the choice is immaterial. No field in the model assumes the compression-rigidity conclusion. The supporting standard theorem must also apply to the compressed pair using the original projection and positivity hypotheses, without adding representation existence as a new restriction on the main question.

The replacement definition and both statements were built at Lean 4.30.0 with supported Mathlib revision c5ea00351c28e24afc9f0f84379aa41082b1188f, and their final versions received independent blind readbacks. Published older declarations and their proof identities are preserved. They should be cited with their finite-dimensional scope, not described as a solution of the arbitrary-Hilbert-space target.

Selected references

• Boris Bilich, Equality case for compressed convex functional calculus, CUHK-Shenzhen AI Math Problems, Problem 2, added June 1, 2026. Original statement.
• Jan Dereziński, Bounded operators, Warsaw University lecture notes, January 2007, Lemma 6.6, printed page 43. Joint continuous functional calculus.