Hefferon Linear Algebra II: Dimension and RankTextbook
Chapter Two of Jim Hefferon's *Linear Algebra* builds the vector space vocabulary — spanning, independence, basis — and turns it into a theory of dimension. The goal theorem is the chapter's most striking result, that the row rank and the column rank of a matrix always agree, which is the bridge between the matrix-of-numbers view of Chapter One and the vector space view of Chapter Two. The milestones are the two pillars it stands on: that any two bases of a space have the same size, so dimension is well defined at all, and that any linearly independent set can be extended to a basis.
Introduction to Linear Optimization III: Fourier–Motzkin Elimination and Projections of PolyhedraTextbook
Is the shadow of a polyhedron again a polyhedron? §2.8 of Bertsimas–Tsitsiklis answers this with perhaps the oldest method for solving linear programming problems: Fourier–Motzkin elimination. Given $P = \{x \in \mathbb{R}^n \mid \sum_{j=1}^n a_{ij}x_j \ge b_i,\ i = 1, \dots, m\}$, one sorts the constraints by the sign of the coefficient of $x_n$ — rewriting them as $x_n \ge d_i + \mathbf{f}_i'\bar{x}$, $d_j + \mathbf{f}_j'\bar{x} \ge x_n$, or $0 \ge d_k + \mathbf{f}_k'\bar{x}$ — and forms the polyhedron $Q \subset \mathbb{R}^{n-1}$ whose constraints are all pairwise combinations $d_j + \mathbf{f}_j'\bar{x} \ge d_i + \mathbf{f}_i'\bar{x}$ together with the constraints not involving $x_n$. The capstone, Theorem 2.10, states that $Q$ is exactly the projection $\Pi_{n-1}(P)$ of $P$ onto its first $n-1$ coordinates: a value of $x_n$ can be interpolated if and only if every lower bound is below every upper bound. Though hopeless as an algorithm (the number of constraints can grow exponentially), elimination has powerful theoretical corollaries, all formalized here: projections $\Pi_k(P)$ of polyhedra are polyhedra (Corollary 2.4), the image of a polyhedron under any linear mapping is a polyhedron (Corollary 2.5), and the convex hull of finitely many vectors is a polyhedron (Corollary 2.6) — the first half of the finite-basis picture completed by the resolution theorem of Mission VII.
Hefferon Linear Algebra I: Gauss's Method and the Solution SetTextbook
Chapter One of Jim Hefferon's *Linear Algebra* develops Gauss's method and asks what row reduction actually preserves. The answer arrives as the Linear Combination Lemma: row operations change the rows of a matrix but never the subspace those rows span, and that invariant is complete. The goal theorem is that completeness — two matrices are row equivalent exactly when they have the same row space — which is what makes reduced echelon form a genuine canonical form. The milestones are the two results the chapter builds on the way: that row operations leave a system's solution set alone, and that a solution set is always one particular solution translated by the solutions of the associated homogeneous system.
Introduction to Linear Optimization II: Existence and Optimality of Extreme PointsTextbook
Where should one look for the optimum of a linear programming problem? Chapter 1 of Bertsimas–Tsitsiklis suggests that optima "tend to occur at corners" of the feasible polyhedron; §§2.5–2.6 turn this intuition into theorems. Not every polyhedron has a corner — a halfspace in $\mathbb{R}^n$ ($n > 1$) has none — and the exact dividing line is the presence of an infinite line: a nonempty polyhedron $$P = \{x \mid a_i'x \ge b_i,\ i = 1, \dots, m\}$$ has an extreme point if and only if it does not contain a line, if and only if $n$ of the vectors $a_1, \dots, a_m$ are linearly independent (Theorem 2.6). In particular every nonempty bounded polyhedron and every nonempty standard-form polyhedron has a basic feasible solution (Corollary 2.2). The capstone, Theorem 2.8, is the sharpest form of the corner principle: if $P$ has at least one extreme point, then for any cost vector $c$ either the optimal cost is $-\infty$, or there is an extreme point of $P$ that is optimal — existence of an optimal solution comes for free once the cost is bounded below. Its companion Theorem 2.7 places an optimal extreme point under the weaker assumption that an optimal solution exists, and Corollary 2.3 — the fundamental theorem of linear programming — concludes that every feasible LP either has optimal cost $-\infty$ or attains an optimal solution, in stark contrast with nonlinear problems such as minimizing $1/x$ over $x \ge 1$. These results license the extreme-point search that the simplex method (Mission IV) performs.
Erdős Problem 390: Exact Second-Order AsymptoticResearch Paper
Determine the exact second-order term in the least possible largest factor in a factorization of $n!$ into distinct integers exceeding $n$, with the proposed rational constant $4029639598/25970038185$.
Bandit feedback is only one point on a spectrum: a learner might see more than its own loss (full information) or less (a spam filter never learns what happened to mail it deleted). Chapter 37 of Lattimore–Szepesvári studies finite adversarial games $G = (\mathcal{L}, \Phi)$ where the loss matrix and the feedback matrix are decoupled. The goal theorem is the celebrated classification theorem: every finite partial-monitoring game has minimax regret exactly $0$, $\Theta(\sqrt{n})$, $\Theta(n^{2/3})$ or $\Omega(n)$ — determined by two purely combinatorial conditions, global and local observability, on the game's neighbourhood structure. A single geometric dichotomy thus governs the price of information in every online decision problem with finite actions and feedback.
Bandit Algorithms XII: Follow-the-Regularised-Leader and Mirror DescentTextbook
Beneath Exp3, Exp4 and their relatives lies one algorithm: minimize past losses plus a convex regularizer. Chapters 26–28 of Lattimore–Szepesvári develop this unifying view. For a Legendre potential $F$ with Bregman divergence $D_F$, both mirror descent and follow-the-regularised-leader satisfy the master bound $R_n(a) \le \frac{F(a) - F(a_1)}{\eta} + \frac{1}{\eta}\sum_t D_F(a_t, \tilde a_{t+1})$; the negentropy potential on the simplex recovers Exp3 exactly. The goal theorem is the payoff for adversarial *linear* bandits: FTRL on the unit ball with the self-concordant-flavoured potential $F(a) = -\log(1-\|a\|) - \|a\|$ achieves $R_n \le 2\sqrt{3nd\log n}$ — improving the $\sqrt{d}$ factor over the Exp3-style approach of Chapter 27 and matching the $\Omega(d\sqrt{n})$ lower bound of Mission XI up to logarithms.
Bandit Algorithms XI: Lower Bounds for Stochastic Linear BanditsTextbook
Is the $d\sqrt{n}$ regret of LinUCB (Mission X) an artifact of the algorithm or a law of nature? Chapters 24–25 of Lattimore–Szepesvári prove it is essentially unimprovable. On the unit ball there is a parameter $\theta$ with $\|\theta\|_2^2 = d^2/(48n)$ forcing $R_n \ge \frac{d\sqrt{n}}{16\sqrt{3}}$ — the goal theorem — and the hypercube gives the same $\Omega(d\sqrt{n})$ rate. The asymptotic chapter is more striking still: for fixed finite action sets, the instance-optimal constant $c(\mathcal{A},\theta)$ is characterized by an allocation program, and *optimism itself is provably suboptimal* — LinUCB and Thompson sampling cannot achieve it, because exploration must sometimes deliberately play actions optimism would never touch. These lower bounds define the targets for the entire linear-bandit literature.
Bandit Algorithms X: Stochastic Linear Bandits and LinUCBTextbook
When actions are feature vectors and the mean reward is linear — $X_t = \langle \theta_*, A_t\rangle + \eta_t$ — a bandit can generalize across arms: pulling one arm reveals information about all of them. Chapter 19 of Lattimore–Szepesvári carries the optimism principle into this setting: LinUCB (a.k.a. OFUL) plays the action maximizing $\max_{\theta\in\mathcal{C}_t}\langle\theta, a\rangle$ over the confidence ellipsoid $\mathcal{C}_t$ of Mission IX. The goal theorem: with probability $1-\delta$, $\hat R_n \le \sqrt{8n\beta_n \log\frac{\det V_n}{\det V_0}} \le \sqrt{8dn\beta_n\log\frac{d\lambda + nL^2}{d\lambda}}$ — regret $\tilde O(d\sqrt{n})$ independent of the number of actions. The combinatorial engine is the elliptical potential lemma, bounding how many times adaptively chosen directions can be surprising. Chapter 22's phased elimination with G-optimal design (Mission IX) sharpens this to $\tilde O(\sqrt{dn\log k})$ for finite action sets.
Bandit Algorithms IX: Self-Normalized Concentration and Optimal DesignTextbook
Least-squares estimation from *adaptively* collected data is the statistical heart of linear bandits: the actions $A_t$ depend on past noise, so classical fixed-design theory does not apply. Chapter 20 of Lattimore–Szepesvári resolves this with the method of mixtures: the process $M_t(x) = \exp(\langle x, S_t\rangle - \frac{1}{2}\|x\|^2_{V_t(\lambda)})$ is a supermartingale, and integrating over a Gaussian mixture yields the self-normalized bound — the goal theorem — $\mathbb{P}\big(\exists t : \|S_t\|^2_{V_t(\lambda)^{-1}} \ge 2\log\frac{1}{\delta} + \log\frac{\det V_t(\lambda)}{\lambda^d}\big) \le \delta$, valid uniformly over all times. The resulting confidence ellipsoids for the regularized least-squares estimator (Abbasi-Yadkori et al.) calibrate every algorithm of Mission X. The mission also formalizes the Kiefer–Wolfowitz theorem of Chapter 21: G-optimal and D-optimal experimental designs coincide, with optimal value exactly $d$ — the classical equivalence theorem of optimal design theory.
Bandit Algorithms VIII: Contextual Bandits and Exp4Textbook
Real decisions come with context: a news site chooses an article *for a particular user*. Competing with the single best arm is then meaningless; the right benchmark is the best mapping from contexts to arms, or more generally the best of $M$ expert policies. Chapter 18 of Lattimore–Szepesvári formalizes this via Exp4 — exponential weighting over experts, fed by the importance-weighted estimator of Mission V. The goal theorem: with learning rate $\eta = \sqrt{2\log(M)/(nk)}$, Exp4 satisfies $R_n \le \sqrt{2nk\log M}$ against the best of $M$ experts. Since $M$ enters only logarithmically, the learner can compete with exponentially large policy classes — the conceptual gateway from bandits to reinforcement learning with function approximation.
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.
Single-Server Queueing Convergence via Forward CouplingTextbook
Formalize sample-path stability for a continuous-time, unit-rate, infinite-buffer single-server queue. Starting from cumulative arriving service work, define the reflected transient workload, the workload constructed from the infinite past, long-run offered load, and two-time stationarity. Prove that subcritical load forces finite-time coupling and consequently that every finite initial workload converges in its two-time finite-dimensional distributions to the stationary workload law.
Formalize sample-path Little's Law for deterministic continuous-time queueing trajectories, decomposed into area, sojourn, arrival-rate, boundary, and squeeze lemmas.
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.
First raised for two variables by Ludwig Kraus in 1884 and stated in full generality by Ott-Heinrich Keller in 1939, the Jacobian conjecture asks something that sounds almost like freshman calculus: if a polynomial map from complex n-space to itself has a Jacobian determinant equal to a nonzero constant, must it be invertible by another polynomial map? That constant-Jacobian condition is precisely the algebraic shadow of the inverse function theorem, yet producing a polynomial — not merely analytic — inverse has resisted every attack for over eighty years. Shreeram Abhyankar championed the problem because it can be stated 'using little beyond a knowledge of calculus,' and Stephen Smale placed it sixteenth on his 1998 list of problems for the new century. Its notoriety is sharpened by a graveyard of published 'proofs' that later collapsed. Deep reductions exist — Bass, Connell, and Wright showed in 1982 that the general case reduces to maps of degree three — and the problem is equivalent, through work of Tsuchimoto, Belov-Kanel, and Kontsevich, to the Dixmier conjecture on the Weyl algebra. A formal statement anchors this famously slippery problem so that progress can be verified rather than merely believed.
The Hardy-Littlewood Method I: Weyl's InequalityTextbook
## Motivation
The Hardy--Littlewood circle method is the principal analytic tool for counting solutions to additive equations in integers. Introduced by Hardy and Ramanujan for the partition function and developed by Hardy and Littlewood in their *Partitio Numerorum* series (1920--1928), it produces asymptotic formulae for the number of representations of a large integer $n$ as a sum of $s$ terms drawn from a prescribed set — $k$-th powers, primes, values of a polynomial.
Its engine is an estimate for **exponential sums**. If a sum $\sum_{x<N} e(\alpha x^k)$, where $e(\theta)=\exp(2\pi i\theta)$, exhibits cancellation for every $\alpha$ not well approximable by a rational with small denominator, the method delivers an asymptotic formula; if it does not, the method stalls. Weyl's inequality (Weyl 1916) was the first such estimate and remains the standard one for moderate $k$.
A short timeline of the estimate this mission targets. Weyl (1916) proved the inequality below with the exponent $2^{1-k}$, in the course of his work on uniform distribution. Hardy and Littlewood (1920--1928) built the circle method on it, obtaining $G(k)\le (k-2)2^{k-1}+5$ for Waring's problem. Vinogradov (1935) replaced Weyl differencing by his mean value theorem, superior for large $k$, reducing the bound to $O(k\log k)$; Wooley's efficient congruencing (2012) and the Bourgain--Demeter--Guth decoupling theorem (2016) settled the main conjecture of Vinogradov's mean value theorem. For small $k$ — and as the entry point to the subject — Weyl's inequality is still the right tool, and it is the natural first capstone for a formalization of the method.
## Setting
For a real number $\theta$ write
$$e(\theta) \;=\; \exp(2\pi i \theta),$$
the standard additive character of $\mathbb{R}/\mathbb{Z}$: it satisfies $e(x+y)=e(x)e(y)$, $|e(x)|=1$, and $e(x)=1$ exactly when $x\in\mathbb{Z}$.
For a real number $\theta$ write $\|\theta\|$ for the distance from $\theta$ to the nearest integer. It is periodic with period $1$, vanishes exactly on $\mathbb{Z}$, satisfies the triangle inequality, and is at most $\tfrac12$.
Given a finite set $A\subseteq\mathbb{Z}$, its generating function is $f_A(\theta)=\sum_{a\in A}e(a\theta)$. The basic identity of the subject is
$$\int_0^1 f_A(\theta)^s\,e(-n\theta)\,d\theta \;=\; \#\{(a_1,\dots,a_s)\in A^s : a_1+\cdots+a_s=n\},$$
a consequence of the orthogonality relation $\int_0^1 e(m\theta)\,d\theta=[\,m=0\,]$.
A **Weyl sum** of degree $k$ is $\sum_{0\le x<N} e(\alpha x^k)$. The whole difficulty is to bound it for $\alpha$ in the *minor arcs* — those $\alpha$ admitting no rational approximation $a/q$ with $q$ small.
## Target
Fix $k\ge 2$. For every $\varepsilon>0$ there is a constant $C=C(k,\varepsilon)$ such that whenever $(a,q)=1$, $q\ge 1$, and $\left|\alpha-\frac{a}{q}\right|\le \frac{1}{q^2}$,
$$\left|\sum_{0\le x<N} e(\alpha x^{k})\right| \;\le\; C\,N^{1+\varepsilon}\left(\frac{1}{q}+\frac{1}{N}+\frac{q}{N^{k}}\right)^{2^{1-k}}.$$
The intermediate targets, weakest first, are the milestone list: the counting identity, the Weyl differencing (squaring) step, the Farey covering with coprime numerator, the divisor bound $d(n)\ll_\varepsilon n^\varepsilon$, Hua's fourth-moment inequality for $k=2$, and the degree-two case of the inequality itself.
## Significance
*The result itself.* Weyl's inequality is what makes the minor arcs negligible. Applied with $q$ in the range $N^{\delta}\le q\le N^{k-\delta}$ it gives a power saving over the trivial bound $N$, and integrating that saving over the minor arcs shows their contribution is smaller than the main term produced by the major arcs. Every classical application of the circle method — the asymptotic formula in Waring's problem, Vinogradov's three primes theorem, the Birch--Davenport theory of forms in many variables — passes through an estimate of this shape. Without it the method produces an identity, not a theorem.
*Formalizing it.* Mathlib currently contains the analytic prerequisites — Fourier characters on `AddCircle`, Dirichlet's approximation theorem, Abel summation, Gauss sums — but no circle-method apparatus whatsoever: no Weyl sums, no arc dissection, no singular series, no mean value estimates. This mission supplies the first layer. The foundational tier is already machine-checked: **53 theorems** covering the character $e$, the norm $\|\cdot\|$, the geometric sum bound $\left|\sum_{x<N}e(x\theta)\right|\le\min\!\left(N,\frac{1}{2\|\theta\|}\right)$, both orthogonality relations, both forms of Dirichlet's theorem, and the basic theory of $f_A$, are published on the platform with verified proofs and may be imported freely. What remains open is the combinatorial and analytic core listed in the milestones. None of the milestone statements is currently formalized anywhere, to the best of our knowledge.
## Difficulty
The obvious approach fails immediately. One would like to sum $\left|\sum_{x<N}e(\alpha x^k)\right|$ by comparing it to the linear case, where the geometric series gives $\min(N,\frac{1}{2\|\alpha\|})$ outright. But for $k\ge2$ the summand is not a geometric progression and there is no closed form.
Weyl's device is to square and difference: $\left|\sum_x e(\phi(x))\right|^2=\sum_{x,y}e(\phi(x)-\phi(y))$, and the substitution $y=x+h$ turns the inner polynomial into one of degree $k-1$ in $x$. Iterating $k-1$ times reduces to a linear sum, at the cost of raising the estimate to the power $2^{1-k}$ — which is why the saving is so weak for large $k$, and why Vinogradov's method eventually supersedes it.
The genuine obstacles in a formalization are: (i) bookkeeping the shifted ranges produced by each differencing step, which are not $[0,N)$ and must be handled uniformly; (ii) the divisor bound $d(n)\ll_\varepsilon n^\varepsilon$, needed to count the $h$ for which the resulting linear coefficient is close to an integer, and which is not currently in Mathlib in this form; (iii) tracking the $\varepsilon$-dependent constants through $k-1$ iterations without the informal $\ll$ notation.
## Formalization scope
Statements are given over the Prove2Me default environment (Lean `v4.30.0`, Mathlib `c5ea003`), in the shared namespace `CircleMethod`, and build on two published definitions: `CircleMethod_char` (the character `e` and the norm `nrm`) and `CircleMethod_genfun` (the generating function `f`).
Conventions this mission commits to:
- $\|\theta\|$ is `nrm θ = |θ - round θ|`. Mathlib's `round` breaks ties upwards, so `round` is *not* an odd function; the characterisation to use is minimality, `nrm θ ≤ |θ - n|` for every integer `n`, which is published as `CircleMethod.nrm_le`.
- Sums run over `Finset.range N`, that is $0\le x<N$, and $N$ is a natural number. Hypotheses `0 < N` and `0 < q` are stated explicitly rather than left implicit.
- Asymptotic notation is eliminated in favour of explicit existential constants: $X\ll_\varepsilon Y$ is rendered as `∀ ε > 0, ∃ C > 0, ∀ …, X ≤ C * Y`, with the constant quantified *outside* the parameters it may depend on and *inside* nothing else. Solvers should not weaken this by allowing $C$ to depend on $N$, $q$ or $\alpha$.
- Exponents such as $N^{1+\varepsilon}$ and $2^{1-k}$ are real powers (`Real.rpow`), not natural powers.
- Coprimality is `Nat.Coprime a.natAbs q`, which is the correct notion for a possibly negative numerator.
One trivialising formalization to rule out: the goal must not be read with $C$ permitted to depend on $N$, since then $C=N$ makes it vacuous. The quantifier order in the Lean statement already forbids this, and solvers should preserve it exactly.
Contributions welcome on any milestone independently; the divisor bound and the Farey covering are self-contained and need no other milestone. Both are reusable well beyond this mission.
## Selected references
- H. Weyl, *Über die Gleichverteilung von Zahlen mod. Eins*, Mathematische Annalen 77 (1916), 313--352. [DOI:10.1007/BF01475864](https://doi.org/10.1007/BF01475864)
- G. H. Hardy and J. E. Littlewood, *Some problems of 'Partitio Numerorum' I--VI*, 1920--1928.
- R. C. Vaughan, *The Hardy--Littlewood Method*, 2nd ed., Cambridge Tracts in Mathematics 125, Cambridge University Press, 1997. (Weyl's inequality is Lemma 2.4; the geometric sum bound is Lemma 2.1.)
- I. M. Vinogradov, *New estimates for Weyl sums*, Doklady Akademii Nauk SSSR 8 (1935), 195--198.
- T. D. Wooley, *Vinogradov's mean value theorem via efficient congruencing*, Annals of Mathematics 175 (2012), 1575--1627. [DOI:10.4007/annals.2012.175.3.12](https://doi.org/10.4007/annals.2012.175.3.12)
- J. Bourgain, C. Demeter and L. Guth, *Proof of the main conjecture in Vinogradov's mean value theorem for degrees higher than three*, Annals of Mathematics 184 (2016), 633--682. [DOI:10.4007/annals.2016.184.2.7](https://doi.org/10.4007/annals.2016.184.2.7)
Existence of complete sets of mutually unbiased basesOpen 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](https://rybindmitry.github.io/problems/16.html)
- [Durt et al., review of MUBs](https://arxiv.org/abs/1004.3348)
First-passage time of Brownian motion to an exponentially decaying boundaryOpen Problem
# 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 Lean capstone focuses on the analytic density characterization and a deliberately auditable syntax of elementary functions. It does not yet construct Brownian motion or prove Durbin's derivation inside Lean. Real constants are allowed as expression leaves, so the substantive restriction is functional form, not the arithmetic nature of parameters. This interpretation must be considered when evaluating any impossibility claim.
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
Build a continuous nonnegative solution of the Abel equation, prove uniqueness and unit mass, and only then address elementary closed form.
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 CUHK-Shenzhen problem](https://rybindmitry.github.io/problems/13.html)
Davie–Stothers Fourth-Power Bound: omega < 2.3737Research Paper
## Motivation
The **matrix-multiplication exponent** $\omega$ measures the asymptotic arithmetic cost of multiplying square matrices. An upper bound $\omega<c$ means that, over the field under consideration, two $n\times n$ matrices can be multiplied using $O(n^{c+\varepsilon})$ field operations for every $\varepsilon>0$. It is a central benchmark in algebraic complexity and controls the exponent of many algorithms that use matrix multiplication as a subroutine.
Coppersmith and Winograd's 1990 analysis of the square of their tensor established $\omega<2.375477$. That number remained the record for roughly two decades. Stothers' 2010 thesis first obtained a smaller exponent by analyzing the fourth tensor power, and Davie and Stothers later supplied a self-contained journal treatment. Their Theorem 5.3 and numerical parameters give $\omega<2.373689703$; see [Davie--Stothers](https://www.maths.ed.ac.uk/~sandy/a11164.pdf), printed pp. 367--368. The result is the first historical step below the classical tensor-square barrier and is the natural next capstone after a formal proof of the $2.375477$ bound.
This mission formalizes the Davie--Stothers fourth-power argument at the exact rational endpoint $2.3737$. It concentrates on the new mathematical layer introduced by the fourth power: five non-matrix constituents, their recursive value estimates, and the two-dimensional same-marginal ambiguity in the final distribution count.
## Setting
For a field $K$, an order-three tensor represents a bilinear map. The **matrix-multiplication tensor**
$$
\langle a,b,c\rangle_K
=\sum_{i<a}\sum_{j<b}\sum_{k<c}
x_{ij}\otimes y_{jk}\otimes z_{ki}
$$
encodes multiplication of an $a\times b$ matrix by a $b\times c$ matrix. Restrictions apply linear maps to the three tensor legs; degenerations permit polynomial families of maps. A direct sum of matrix-multiplication tensors has disjoint variable blocks and can be converted into an exponent inequality by Schönhage's asymptotic sum inequality.
The **Coppersmith--Winograd tensor** $CW_q$ has border rank at most $q+2$ and a three-class coordinate partition. Its square decomposes into fifteen coarse constituents $\varphi_{ijk}$ with $i+j+k=4$. Davie--Stothers square this decomposition again. The fourth power has forty-five constituents with indices summing to eight, grouped into ten symmetry classes represented by
$$
\varphi_{008},\ \varphi_{017},\ \varphi_{026},\ \varphi_{035},\ \varphi_{044},
\ \varphi_{116},\ \varphi_{125},\ \varphi_{134},\ \varphi_{224},\ \varphi_{233}.
$$
The first five classes are rectangular matrix-multiplication tensors. The last five require recursive value bounds. With $\rho\in[2,3]$, the paper writes
$$
E=(2q)^\rho,\qquad H=(q^2+2)^\rho,\qquad
L=4q^\rho(q^\rho+2),
$$
and states the five lower bounds in Lemma 5.1. The final fourth-power extraction assigns frequencies to the ten symmetry classes. Their coordinate marginals are encoded by the $9\times10$ matrix $Q$ in Equation (5.2); its kernel is the two-dimensional space $Y$ displayed immediately after that equation.
The paper's bounds are limiting exponential rates and may carry subexponential losses in their finite Salem--Spencer extractions. Prove2Me's `HasTauValueAtLeast` predicate instead records a constant-relative finite witness. The source-faithful formal statements therefore assert attainment of every fixed nonnegative base strictly below each displayed limiting rate, rather than unjustified attainment of the limiting endpoint itself. This downward-closed form retains the complete asymptotic conclusion and is exactly what the final strict numerical surplus needs.
## Formalization targets
### Goal: the Davie--Stothers fourth-power bound
For every field $K$,
$$
\operatorname{matMulExp}(K)<\frac{23737}{10000}=2.3737.
$$
The source's computed endpoint $2.373689703$ is strictly smaller, giving slack for an exact rational certificate. The Lean goal has exactly the same field quantification and `matMulExp` definition as the existing Coppersmith--Winograd mission; only the theorem identifier and endpoint change.
### Source-level milestones
The mission records the canonical nine-grading of $CW_6^{\otimes4}$ and the ten symmetry classes of Table 1. It formalizes all five clauses of Lemma 5.1 for $\varphi_{116}$, $\varphi_{125}$, $\varphi_{134}$, $\varphi_{224}$, and $\varphi_{233}$ in every-strict-lower-base form; Equation (5.2) and the stated basis of $\ker Q$; Lemma 5.2's entropy minimization along that kernel; Theorem 5.3's downward-closed fourth-power value inequality; and the Table 2 numerical specialization. The final milestones connect the resulting tau-value surplus to the border-rank budget and transfer the Strassen-preorder exponent bound to `matMulExp`.
## Significance
Mathematically, this theorem is the first improvement obtained by passing from the square to the fourth power of the Coppersmith--Winograd tensor. It establishes the recursive constituent pattern used by the later eighth-, sixteenth-, and higher-power analyses. In particular, the five formulas in Lemma 5.1 are the first complete catalogue of genuinely recursive fourth-power constituents.
For formalization, the mission creates a reusable representation of higher-power CW gradings and their symmetry orbits. It also forces a distinction between a locally chosen joint type and all other types with the same marginals. Lemma 5.2 is the exact finite-dimensional entropy correction needed when the marginal map has nontrivial kernel. That infrastructure can be reused by later refined-laser and complete-split missions.
The result is known mathematically. The open task is a machine-checked reconstruction. Prove2Me already contains the CW tensor, its characteristic-free border-rank degeneration, its canonical square grading and constituent restrictions, the Salem--Spencer layer, direct-sum tau-value witnesses, the asymptotic sum inequality, and the exponent bridge. The exact optimizer identity for the $\varphi_{116}$ profile is also proved. The remaining frontier is to connect the literal fourth-power constituents to finite direct-sum extractions, then assemble all five value estimates and the final kernel-corrected distribution count.
## Difficulty
The fourth power contains 225 ordered products before symmetry grouping. A formal proof must show that each claimed constituent is the literal block of $CW_q^{\otimes4}$ and that its recursive decomposition uses the correct variable spaces. Replacing a sum of overlapping blocks by an external direct sum would make the value bound artificially strong.
The five non-matrix classes have different feasible frequency polytopes. Their optimizer formulas are valid only after the corresponding nonnegativity and normalization conditions are checked. The $\varphi_{233}$ class already has a nontrivial same-marginal family. At the global level the map $Q$ has a two-dimensional kernel, so marginal counts alone do not determine a unique joint distribution. Ignoring that kernel removes the entropy penalty and invalidates Theorem 5.3.
Finally, Table 2 contains decimal witnesses obtained numerically. A formal proof must replace floating-point evaluation by exact rational parameters and certified bounds for logarithms and real powers, while retaining strict slack at $23737/10000$.
## Formalization scope
The development uses environment `777aaa61dcd2a1258d2b4962dbe983ede4d23b2e` and the existing `TensorObj`, `MMObj`, restriction, degeneration, `HasTauValueAtLeast`, `tensorAsymptoticRank`, `matMulExp_strassen`, and `matMulExp` declarations. Top-level theorems quantify over an arbitrary field. Finite block indices and symmetry classes use finite types; frequency vectors and entropy inequalities use real numbers; exact finite profiles use natural numbers before passing to cofinal asymptotics.
The capstone specializes to $q=6$ and the fourth tensor power. Generic grading, orbit, multinomial, entropy, and optimizer lemmas are welcome when they shorten later missions. Every value theorem must ultimately be backed by restrictions or degenerations to direct sums of concrete matrix-multiplication tensors. An opaque value functional, a constituent definition that is an external sum rather than the source block, or a numerical hypothesis that assumes the desired endpoint is outside scope.
Contributions are welcome for the literal nine-grading, symmetry-orbit classification, the five constituent extractions, exact address factorizations, optimizer feasibility, the kernel calculation and Lemma 5.2, exact Table 2 arithmetic, and the final exponent assembly.
## Selected references
- A. M. Davie and A. J. Stothers, *Improved Bound for Complexity of Matrix Multiplication*, Proceedings of the Royal Society of Edinburgh Section A: Mathematics 143(2), 2013, pp. 351--369. [Author PDF](https://www.maths.ed.ac.uk/~sandy/a11164.pdf) and [DOI 10.1017/S0308210511001646](https://doi.org/10.1017/S0308210511001646).
- A. J. Stothers, *On the Complexity of Matrix Multiplication*, PhD thesis, University of Edinburgh, 2010. [Edinburgh Research Archive](https://era.ed.ac.uk/handle/1842/4734).
- Don Coppersmith and Shmuel Winograd, *Matrix Multiplication via Arithmetic Progressions*, Journal of Symbolic Computation 9, 1990, pp. 251--280. [DOI 10.1016/S0747-7171(08)80013-2](https://doi.org/10.1016/S0747-7171(08)80013-2).
- Arnold Schönhage, *Partial and Total Matrix Multiplication*, SIAM Journal on Computing 10(3), 1981, pp. 434--455. [DOI 10.1137/0210032](https://doi.org/10.1137/0210032).
Algorithmic Game Theory I: Existence of Nash EquilibriumTextbook
## Motivation
The strategic-form game is the basic object of noncooperative game theory, and the **Nash equilibrium** — a profile of randomized strategies from which no player benefits by deviating unilaterally — is its central solution concept. Nash proved in 1951 that every game with finitely many players and finite strategy sets has such an equilibrium ([Nash, *Non-cooperative games*, Ann. Math. 54 (1951)](https://doi.org/10.2307/1969529)); this single existence theorem is the reason the concept organizes the rest of the field, from the computational complexity of finding equilibria to the price of anarchy. The theorem is stated as Theorem 1.8 of Nisan–Roughgarden–Tardos–Vazirani (eds.), *Algorithmic Game Theory* (Cambridge, 2007), the source text of this mission series, whose first chapter (Tardos–Vazirani) also treats the two special cases that admit direct algorithmic proofs: two-person zero-sum games, where equilibria are exactly the optimal solutions of a dual pair of linear programs (von Neumann 1928; Theorem 1.11), and a simple linear market, where equilibrium prices are computed by an ascending tight-set algorithm (Theorem 1.17).
A timeline of the existence theorem: von Neumann (1928) proved the minimax theorem for two-person zero-sum games; Nash (1950, 1951) extended existence to arbitrary finite games, first via Kakutani's fixed-point theorem and then via Brouwer's. All known proofs of the general theorem pass through a fixed-point principle, and this is not an artifact: computing a Nash equilibrium is PPAD-complete (Daskalakis–Goldberg–Papadimitriou 2009; Chen–Deng–Teng 2009), and PPAD is precisely the complexity class of the fixed-point arguments.
## Setting
A **finite strategic-form game** consists of a finite set $\iota$ of players, for each player $i$ a finite nonempty set $S_i$ of **pure strategies**, and for each player a **payoff function** $u_i : \prod_j S_j \to \mathbb{R}$; all players are utility maximizers. A **mixed strategy** for player $i$ is a probability distribution on $S_i$, represented as a weight function $\sigma_i : S_i \to \mathbb{R}$ with $\sigma_i \ge 0$ and $\sum_{s} \sigma_i(s) = 1$ (a **lottery**). Players randomize independently, so a **mixed profile** $\sigma = (\sigma_i)_{i}$ induces the product distribution on pure strategy vectors, and player $i$'s **expected payoff** is
$$U_i(\sigma) \;=\; \sum_{s \in \prod_j S_j} \Big(\prod_j \sigma_j(s_j)\Big)\, u_i(s).$$
A mixed profile $\sigma$ is a **(mixed) Nash equilibrium** if for every player $i$ and every lottery $\tau$ on $S_i$, replacing $\sigma_i$ by $\tau$ does not increase $U_i$.
A **two-person zero-sum game** is given by a matrix $A \in \mathbb{R}^{m \times n}$: the row player picks a row distribution $p$, the column player a column distribution $q$, and the column player pays the row player $p^{\mathsf T} A q$ in expectation.
The market of §1.8.1 of the source has finitely many divisible **goods**, good $a$ in $s_a$ units, and finitely many **buyers**, buyer $j$ bringing budget $m_j > 0$ and interested in a nonempty set of goods; utilities are linear 0/1, so a buyer wants any goods from her interest set and none other. **Market-clearing prices** are positive prices under which each buyer can spend her whole budget on cheapest goods in her interest set while every good sells out exactly.
## Formalization targets
### Goal (capstone) — Theorem 1.8
$$\text{Every finite strategic-form game has a mixed Nash equilibrium.}$$
Stated for an arbitrary finite family of finite nonempty strategy types; no bound on the number of players, no genericity assumptions.
### Supporting — Brouwer fixed-point theorem
$$K \subseteq E \text{ nonempty compact convex},\ E \text{ finite-dimensional},\ f : K \to K \text{ continuous} \implies \exists x,\ f(x) = x.$$
Mathlib currently has no form of Brouwer's theorem; every known proof of Theorem 1.8 needs it (or an equivalent), so it enters the mission as an explicit milestone rather than an assumed library fact.
### Theorem 1.11 — zero-sum games
$$\exists\, p^\ast, q^\ast:\quad \forall p,\ p^{\mathsf T} A q^\ast \le {p^\ast}^{\mathsf T} A q^\ast, \quad \forall q,\ {p^\ast}^{\mathsf T} A q^\ast \le {p^\ast}^{\mathsf T} A q, \quad\text{and}\quad (p^\ast, q^\ast) \text{ is a mixed Nash equilibrium}$$
of the explicit two-player game with payoffs $A_{xy}$ to the row player and $-A_{xy}$ to the column player. The source states the result as: optimal solutions of a dual pair of LPs form a Nash equilibrium of the zero-sum game; the first two conjuncts are the saddle point that LP optimality amounts to, and the third states the Nash-equilibrium clause against the mission's own game vocabulary, so "zero-sum" is formal (the two payoffs sum to zero) rather than implicit in the shape of the statement.
### Theorem 1.17 (existence form)
$$\text{The 0/1-utilities linear market admits market-clearing prices and allocations.}$$
The source proves this by an ascending-price algorithm and also bounds its running time; the complexity half has no formal counterpart in this mission.
## Significance
The capstone is the foundation of the whole mission series: correlated equilibria, price-of-anarchy bounds, and mechanism-design characterizations in later missions all quantify over or compare against Nash equilibria, and the series inherits its game vocabulary (`IsLottery`, `IsMixedProfile`, `expectedPayoff`, `IsMixedNash`) from this mission.
Formalizing it produces the first Brouwer fixed-point theorem in this environment — a well-known gap in mathlib with reuse value far beyond game theory (every degree-theoretic and equilibrium-existence argument needs it). The zero-sum milestone yields the minimax theorem, reusable for the learning-dynamics mission that follows. All results here are classical and proved on paper; the work requested is machine-checked proof, not new mathematics.
## Difficulty
The central difficulty is Brouwer. The standard routes are (i) Sperner's lemma plus a limit argument, which needs a formal theory of simplicial subdivisions that does not exist in mathlib; (ii) algebraic topology (no retraction of the ball onto the sphere), for which mathlib has singular homology but not yet the homology of spheres in usable form; (iii) analytic proofs (Milnor–Rogers). None is short; the milestone is deliberately stated for a general nonempty compact convex set in a finite-dimensional normed space so that any route serves, and so the lemma lands in reusable generality.
Given Brouwer, Theorem 1.8 still requires Nash's gain-function construction on the product of simplices and the verification that fixed points are equilibria — bookkeeping-heavy but standard. Theorem 1.11 does not need Brouwer: mathlib's Sion minimax theorem (`Mathlib.Topology.Sion`) applies to the bilinear payoff on the product of standard simplices, or one can argue by LP duality directly. Theorem 1.17 needs the tight-set/max-flow argument of Lemmas 1.15–1.16 or any direct construction of the equilibrium.
## Formalization scope
Games are presented concretely: players form a finite index type, strategies a finite type per player, payoffs are functions into $\mathbb{R}$; mixed strategies are weight functions with a `IsLottery` predicate, not measure-theoretic distributions. Deviations in the equilibrium definition range over all lotteries (not only pure strategies): the pure-deviation reduction is a lemma a solver may prove, not part of the definition. Strategy sets are assumed nonempty in the capstone; the player set need not be. In the zero-sum milestone both dimensions are positive (`Fin (m+1)`, `Fin (n+1)`), payoffs flow from the column player to the row player, `stdSimplex` plays the role of the mixed-strategy space, and the Nash-equilibrium conjunct is stated for the Boolean-indexed two-player game built by `matrixGameStrat`/`zeroSumPayoff`/`matrixGameProfile` from the definitions bundle. In the market milestone all supplies and budgets are positive, every buyer's interest set is nonempty, and every good has an interested buyer, matching the standing assumptions of §1.8.1; allocations are recorded as money spent, so the clearing condition is $\sum_j x_{ja} = p_a s_a$ with no division anywhere.
Trivializing readings are ruled out: the empty simplex has no lotteries, so nonemptiness hypotheses appear exactly where their absence would make an existence claim false (Brouwer on the empty set, games with an empty strategy set, zero-dimensional matrix games).
## Selected references
- J. F. Nash, *Non-cooperative games*, Annals of Mathematics 54 (1951), 286–295. [DOI](https://doi.org/10.2307/1969529)
- J. von Neumann, *Zur Theorie der Gesellschaftsspiele*, Mathematische Annalen 100 (1928), 295–320. [DOI](https://doi.org/10.1007/BF01448847)
- N. Nisan, T. Roughgarden, É. Tardos, V. V. Vazirani (eds.), *Algorithmic Game Theory*, Cambridge University Press, 2007, Chapter 1. [DOI](https://doi.org/10.1017/CBO9780511800481)
- C. Daskalakis, P. W. Goldberg, C. H. Papadimitriou, *The complexity of computing a Nash equilibrium*, SIAM J. Computing 39 (2009), 195–259. [DOI](https://doi.org/10.1137/070699652)
Zhang–Si–Si: Corrected Resonant Almost-Periodic Poincaré–Treshchev PersistenceResearch Paper
## Motivation
The classical persistence problem asks which invariant tori of an integrable Hamiltonian survive a small perturbation. A resonance makes some internal angular frequencies vanish, and an **almost-periodic perturbation** introduces infinitely many additional forcing frequencies. Zhang, Si, and Si state a Poincaré–Treshchev persistence theorem for this combined regime: most lower-dimensional resonant tori associated with nondegenerate relative equilibria persist as infinite-dimensional invariant tori [in Theorem 2.7](https://doi.org/10.3934/dcds.2026043). The formal mission targets that headline result in corrected form. It retains the paper’s nonresonance, spatial-shell, and averaged-critical-point hypotheses, while making explicit the full frequency twist and reduced-frequency parameter chart used later in the paper but absent from the printed theorem statement.
## Setting
Let $n,m>0$ and $d=n+m$. The internal variables are $x\in\mathbb T^d$ and $y\in G\subset\mathbb R^d$, where $G$ is bounded and closed. The external forcing angles and their conjugate actions are $\theta\in\mathbb T^{\mathbb Z}$ and $J\in\ell^1(\mathbb Z;\mathbb R)$. The **suspended Hamiltonian** is
$$
\mathcal H_\epsilon(\theta,J,x,y)=
\sum_{j\in\mathbb Z}\omega_jJ_j+N(y)+\epsilon P(\theta,x,y,\epsilon).
$$
Here $N$ is real analytic near $G$. A primitive rank-$m$ resonance lattice is represented by an integer matrix $K_0=(K_1,K_2)$ with $\det K_0=1$, where the last $m$ columns generate the lattice. The adapted angles are $(\psi,\phi)=K_0^{\mathsf T}x$, with $\psi\in\mathbb T^n$ and $\phi\in\mathbb T^m$. Writing $\nabla N$ for the actual derivative of $N$, define
$$
O(g,G)=\{y\in G:K_2^{\mathsf T}\nabla N(y)=0\},
\qquad \Omega(y)=K_1^{\mathsf T}\nabla N(y).
$$
The **averaged potential** $h_0(\phi,y)$ is the zero external and zero $\psi$ Fourier coefficient of $P$ at $\epsilon=0$. A pair $(\phi,y)$ is associated and nondegenerate when
$$
\nabla_\phi h_0(\phi,y)=0,
\qquad \det D_\phi^2h_0(\phi,y)\ne0.
$$
Let $O_0$ be the subset of $O(g,G)$ admitting such a $\phi$, let $\Omega_0=\Omega(O_0)$, and trim in reduced-frequency coordinates:
$$
\Omega_\xi=\{\eta\in\Omega_0:\operatorname{dist}(\eta,\partial\Omega_0)\ge\xi\},
\qquad O_\xi=O_0\cap\Omega^{-1}(\Omega_\xi).
$$
The perturbation has the paper’s **spatial-shell Fourier expansion** over finite shells $A$ in a spatial structure $\mathcal S$. With
$[A]=1+\sum_{j\in A}\log^{\varrho}(1+|j|)$ for $\varrho>2$, each shell has a uniform analytic Fourier bound $B_A$, and $\sum_AB_Ae^{s[A]}<\infty$ for some $s>0$. Every nonzero admissible finite external mode $k$ satisfies
$$
|\langle k,\omega\rangle|\ge
\frac{\gamma}{\Delta([[k]])\Delta(|k|_1)},
$$
where $\gamma>0$, $[[k]]$ is the minimum shell weight containing its support, and $\Delta$ is a nondecreasing approximation function with $\Delta(0)=1$, $\log\Delta(t)/t\downarrow0$, and $\int_0^\infty\log\Delta(t)t^{-2}\,dt<\infty$.
## Target
Assume that $O_0$ is nonempty. On every sufficiently small trim $O_\xi$, require compactness and positive $n$-dimensional measure in the $\Omega$ chart, injectivity of $D(\nabla N)(y)$, an analytic lower-Lipschitz diffeomorphism $\Omega:O_\xi\to\Omega_\xi$, and nondegeneracy of every averaged critical point retained by the conclusion. Then, for every $0<\xi\le\xi_*$, there are $0<\epsilon_0\le1$, a rate $c(\epsilon)\to0$ as $\epsilon\downarrow0$, and closed measurable nonempty sets $\Lambda_\epsilon\subset O_\xi$ for $0<\epsilon\le\epsilon_0$ such that
$$
\operatorname{vol}_n\bigl(\Omega(O_\xi\setminus\Lambda_\epsilon)\bigr)
\longrightarrow0\qquad(\epsilon\downarrow0).
$$
For every $y\in\Lambda_\epsilon$ and every associated nondegenerate $\phi$, construct a topological embedding
$$
\iota_{\epsilon,y,\phi}:\mathbb T^{\mathbb Z}\times\mathbb T^n
\longrightarrow
(\mathbb T^{\mathbb Z}\times\ell^1)\times
(\mathbb T^d\times\mathbb R^d)
$$
that is analytic almost periodic with the same shell structure, is the image of the standard resonant torus under a local canonical transformation, is $c(\epsilon)$-close to that torus, and is invariant under $\mathcal H_\epsilon$ with rotation vector $(\omega,\Omega(y))$.
## Significance
The result combines three features that are usually separated: a positive-rank internal resonance, an infinite external frequency vector, and persistence for an asymptotically full-measure parameter set. It identifies the surviving object in the full suspended phase, including the external actions, rather than only in the finite internal fibre. The paper contains a proof of its printed theorem; the mission’s open work is a Lean proof of the corrected target. The correction exposes the parameter twist and chart assumptions on which the reduction and measure assertion depend, so a completed formalization would distinguish the theorem’s stated content from hypotheses needed to make that content mathematically controlled.
## Difficulty
Finite-dimensional Diophantine notation does not control infinitely many external modes: admissibility, support weights, and the decay of $\Delta$ must interact without leaving an empty mode class. Resonance also removes $m$ internal frequencies and replaces the original parameter set by a lower-dimensional surface. Its ambient Euclidean boundary is therefore unsuitable for a positive trim, and ordinary ambient volume is unsuitable for the majority statement. Finally, persistence must be expressed simultaneously as an invariant solution of the actual Hamilton equations, an analytic almost-periodic embedding, and a canonical equivalence in the suspended phase; any one of these conditions alone admits objects that do not express the theorem.
## Formalization scope
The Lean representation uses finitely supported integer external modes, a covering spatial structure, the literal shell-indexed norm, complex-neighborhood coefficient analyticity, and a separate absolute-summability guard for every infinite Fourier sum. Unit modes are provably admissible. The averaged potential, its gradient and Hessian, the internal frequency $\nabla N$, the resonant set, and the reduced-frequency map are transparent definitions. Nonemptiness of $O_0$, positive volume of every allowed $\Omega_\xi$, and nonempty $\Lambda_\epsilon$ rule out empty-set and zero-measure trivializations.
The formal target strengthens the printed assumptions only where the source’s reduction requires missing control: $D(\nabla N)$ is injective on the retained trim, and $\Omega$ is an analytic diffeomorphism there with a uniform lower Lipschitz bound. The boundary distance and excluded volume are both taken in the $n$-dimensional reduced-frequency chart. The word “Cantor” is represented by closedness, measurability, nonemptiness, and asymptotically full measure; perfectness and total disconnectedness are not additional targets.
The torus lives in a genuine $\ell^1$ external-action space. Its action component has one weighted-$\ell^1$-valued shell expansion, and the Hamiltonian pairing and external action velocity carry convergence guards. A local conjugacy is a homeomorphism between open neighborhoods, fixes $\theta$, is differentiable along all $\ell^1$ action directions and finitely supported external-angle directions, and preserves $\sum_jd\theta_j\wedge dJ_j+\sum_i dx_i\wedge dy_i$ on those cylinder directions. Reusable contributions include spatial-shell Fourier classes, approximation functions, reduced-manifold measure interfaces, weighted infinite canonical forms, and coordinatewise Hamiltonian invariance.
## Selected references
- Yuan Zhang, Wen Si, and Jianguo Si, *Poincaré–Treshchev Mechanism in Integrable Hamiltonian Systems Under Almost-Periodic Perturbations*, Discrete and Continuous Dynamical Systems **52** (2026), 32–69. [DOI: 10.3934/dcds.2026043](https://doi.org/10.3934/dcds.2026043). Main result: Theorem 2.7, journal p. 39 (PDF p. 8); spatial and nonresonance definitions: Definitions 2.2–2.4 and equations (5)–(7), journal pp. 35–38 (PDF pp. 4–7); reduction exposing the additional twist and parameter-change requirements: Lemma 3.2, journal pp. 41–43 (PDF pp. 10–12).
Cubic Congruence for the q-Secant Inversion EnumeratorResearch Paper
## Motivation
Alternating permutations are a classical meeting point of enumerative combinatorics, permutation statistics, and special functions. An up--down permutation alternates between rises and falls, and their ordinary counts are the Euler secant and tangent numbers. Refining this count by the inversion statistic produces the $q$-secant polynomial $E_{2n}(q)$. Its values and congruences retain information that disappears after setting $q=1$: they distinguish how the alternating permutations are distributed by inversion number and reveal cancellation at roots such as $q=-1$. Ji-Cai Liu's article isolates the next nontrivial term in the $(1+q)$-adic expansion of this polynomial, strengthening an earlier Andrews--Foata congruence. The mission formalizes the article's main result, Theorem 1.1, as an exact polynomial-divisibility statement.
## Setting
For $n\ge 0$, let $A(2n)$ be the set of permutations $\sigma=(\sigma_1,\ldots,\sigma_{2n})$ of $\{1,\ldots,2n\}$ satisfying
$$
\sigma_1<\sigma_2>\sigma_3<\sigma_4>\cdots<\sigma_{2n}.
$$
The empty permutation is the unique member of $A(0)$. The inversion number is
$$
\operatorname{inv}(\sigma)
=\#\{(i,j):1\le i<j\le 2n,\ \sigma_i>\sigma_j\}.
$$
The $q$-secant inversion enumerator is the integer polynomial
$$
E_{2n}(q)=\sum_{\sigma\in A(2n)}q^{\operatorname{inv}(\sigma)}\in\mathbb Z[q].
$$
Congruence modulo $(1+q)^3$ means divisibility in $\mathbb Z[q]$: two polynomials $F$ and $G$ are congruent precisely when $(1+q)^3$ divides $F-G$. This formulation avoids evaluation at a single number and records the first three orders of behavior at $q=-1$.
In Lean, a permutation is represented as an equivalence of `Fin (2*n)`. The alternating inequalities and inversion number are finite predicates and counts on this zero-based type. The polynomial variable is the canonical indeterminate in `Polynomial ℤ`.
## Formalization targets
### Cubic congruence
For every integer $n\ge0$, prove
$$
E_{2n}(q)\equiv
q^{2n(n-1)}-\binom n2(1+q)^2
\pmod{(1+q)^3}.
$$
Equivalently,
$$
(1+q)^3\mid
E_{2n}(q)-
\left(q^{2n(n-1)}-\binom n2(1+q)^2\right)
\quad\text{in }\mathbb Z[q].
$$
The boundary value $n=0$ is included. With the empty-permutation convention and natural-number truncated subtraction in the exponent, both sides reduce correctly, so the formal target does not hide a separate exceptional case.
## Significance
The theorem identifies the exact quadratic correction to the highest-inversion monomial near $q=-1$. It therefore explains why the prior congruence modulo $(1+q)^2$ does not generally lift unchanged to the cubic modulus. Specializing at $q=1$ also yields the corresponding refinement modulo $8$ for the ordinary secant numbers. More broadly, the statement is a compact test case for formal reasoning that combines finite permutations, order predicates, inversion statistics, generating polynomials, binomial coefficients, and divisibility in a polynomial ring.
A machine-checked proof would contribute reusable infrastructure for permutation enumerators and polynomial congruences. The published article supplies a human proof; the Prove2me goal is the formal reconstruction of its theorem in Lean. The mission does not encode a proof certificate, an orbit count, or the desired divisibility inside a definition. A successful submission must derive the divisibility from the concrete finite definitions.
The result also gives a useful interface between two styles of formal combinatorics. On one side, alternating permutations are finite objects that can be enumerated, mapped, and partitioned. On the other, their aggregate is an algebraic object in $\mathbb Z[q]$ whose divisibility can be studied without referring to individual permutations. Infrastructure connecting these levels can be reused for other $q$-Euler numbers, descent and major-index enumerators, and congruences obtained from finite weighted actions. The mission keeps that infrastructure general-purpose by making the final target an equality in a quotient of the polynomial ring rather than a specialized computational procedure.
## Difficulty
Direct expansion of $E_{2n}(q)$ is factorial in $n$ and gives no uniform explanation of divisibility by a third power. Divisibility by $(1+q)^3$ is stronger than merely checking the value at $q=-1$: it simultaneously constrains the value and the first two formal orders there. A formal solution must control the entire finite family of alternating permutations while preserving exact inversion exponents and polynomial coefficients. Index conventions are also delicate, because the paper numbers positions and values from $1$, whereas Lean uses `Fin` indices from $0$.
The source argument introduces combinatorial structure beyond the bare statement. Formalizers may contribute reusable lemmas about switching consecutive values, invariance of alternation under permitted switches, inversion-number changes, finite group actions, and divisibility of orbit enumerators. Those are natural milestones, but the present root goal deliberately remains the stable polynomial congruence rather than committing to one decomposition.
## Formalization scope
The mission fixes the coefficient ring to $\mathbb Z$ and uses exact polynomial divisibility. It does not replace congruence by coefficientwise arithmetic modulo $8$, evaluation at $q=-1$, or a numerical check for bounded $n$. `UpDown` is defined directly on permutations of `Fin (2*n)`, `invNumber` counts ordered index pairs with the required inequality, and `qSecant` is the finite sum of monomials $q^{\operatorname{inv}(\sigma)}$.
The formal statement quantifies over every natural number. The conventions at $n=0$ and $n=1$ are part of the same theorem and have been audited explicitly. The uploaded definition bundle is transparent and sorry-free; the only admitted declaration is the mission theorem itself. Useful contributions include general lemmas about polynomial divisibility, finite involutions and orbit sums, or bridges between one-based paper notation and Lean's finite types.
## Selected references
- Ji-Cai Liu, *A Combinatorial Proof of a Cubic Congruence for the $q$-Secant Inversion Enumerator*, Electronic Journal of Combinatorics 33(3), P3.10, 2026. [DOI](https://doi.org/10.37236/15666)