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Each mission turns a result from a paper or textbook into small Lean 4 statements anyone can tackle.

Campaigns (experimental)

Campaigns group missions around a shared mathematical goal. Each one tracks a quantity, such as an upper or lower bound. Have a good candidate in mind? Ping us on Slack, Zulip, or WeChat.

Block Sensitivity vs. Sensitivity

How much larger can block sensitivity be than sensitivity for a total Boolean function?

For a total Boolean function $f:\{0,1\}^n\to\{0,1\}$ , a block of coordinates is sensitive at $x$ if flipping every bit in that block changes $f(x)$ . Let $s(f,x)$ count sensitive singletons and $bs(f,x)$ be the maximum number of pairwise disjoint sensitive blocks. Define $s(f)=\max_x s(f,x)$ and $bs(f)=\max_x bs(f,x)$ .

This campaign seeks larger exponents $\alpha$ for which there exists an infinite family of total Boolean functions $f_k:\{0,1\}^{n_k}\to\{0,1\}$ , indexed by $k=1,2,\ldots$ , satisfying

s(f_k)\to\infty,\qquad bs(f_k)\ge s(f_k)^{\alpha-o(1)}\quad\text{as }k\to\infty.

Here $o(1)$ denotes a quantity tending to zero as $k\to\infty$ . Higher is better. The classical exponent is $2$ , and Huang's bound gives a ceiling of $4$ .

OpenAI's construction achieves an exponent strictly greater than $2$ .

One sufficient way to establish an exponent $\alpha$ is to find a single total Boolean function $f$ with $s(f)>1$ and $bs(f)\ge s(f)^\alpha$ . Input and output normalization followed by self-composition then yields the required asymptotic family.

≥ 2.13981Formalized record≥ 2.15Open frontier

11 provers on it22 of 23 missions formalized

Integer Multiplication Below n log n

Turn proposed improvements to integer multiplication into complete Lean proofs, and push the exponent saving further.

Harvey and van der Hoeven established an $O(n\log n)$ algorithm in 2021. This campaign builds on that foundation, the OpenAI manuscript, and subsequent community constructions to pursue a strict asymptotic improvement.

For two $n$ -bit integers, the target is

T(n)=O\!\left(n\,L(n)^{1-\kappa}\right),\qquad L(n)=\max(\lceil\log_2 n\rceil,1).

A positive $\kappa$ beats $n\log n$ asymptotically; larger $\kappa$ is better. Every entry must exhibit one deterministic multitape Turing machine, with a fixed finite alphabet and tape count, that computes the exact product at every positive input length and meets the eventual worst-case time bound. The tracked number measures an asymptotic exponent saving.

≥ 3.6 × 10⁻⁵Formalized record≥ 5.1 × 10⁻⁵Open frontier

8 provers on it4 of 7 missions formalized

3SUM Exponent

Classical algorithms solve 3SUM in $O(n^2)$ time. In a 2026 breakthrough, Alman and Vassilevska Williams gave a deterministic $O(n^{1.9992})$ algorithm, refuting the integer 3SUM hypothesis. How low can the exponent go?

Building on existing Lean formalizations, this campaign tracks upper bounds for 3SUM on polynomially bounded integers, using a word RAM with $O(\log n)$ -bit words, and pursues smaller exponents.

≤ 1.960614Formalized record≤ 1.9202796Open frontier

4 provers on it7 of 8 missions formalized

All-Pairs Shortest Paths (APSP) Exponent

Classical algorithms solve all-pairs shortest paths in $O(n^3)$ time. In a 2026 breakthrough, Alman and Vassilevska Williams refuted the APSP conjecture with a deterministic $O(n^{2.99942})$ algorithm. How low can the exponent go?

Building on existing Lean formalizations, this campaign tracks upper bounds for exact APSP and pursues smaller exponents.

≤ 2.995561Formalized record

3 provers on it5 of 5 missions formalized

The irrationality measure of π

The irrationality measure of π quantifies how closely rational numbers can approximate it. This campaign seeks formal proofs of sharper upper bounds, starting with Mahler’s bound of 42.

≤ 2Formalized record

19 provers on it9 of 9 missions formalized

Sharp diagonal Hlawka constant

The sharp Hlawka inequality for Schatten $p$ -norms is a cousin of the triangle inequality: it relates the norms of three matrices to the norms of their pairwise sums and their total sum. For complex diagonal matrices, an exact formula for the best possible comparison constant has been proved in Lean for every real $p\ge256$ . We conjecture that the same formula holds for all $p\ge2$ .

What is the smallest cutoff $p'$ for which this formula holds for every real $p\ge p'$ ?

References:

Wolfram MathWorld, Hlawka's Inequality.
Audenaert and Kittaneh, Problems and Conjectures in Matrix and Operator Inequalities, §8.2 (2017).
Marinescu and Niculescu, A New Look at the Hornich–Hlawka Inequality (2025).
Analytic argument for $p\ge90$ , awaiting formalization in Lean.

≤ 4Formalized record≤ 2Open frontier

8 provers on it17 of 18 missions formalized

Odd numbers as sums of primes

Is every odd number a sum of $k$ primes? This campaign tracks formalized proofs of the smallest $k$ that suffices.

Schnirelmann (1930) showed some finite $k$ works. Vinogradov (1937) showed that three is enough for all sufficiently large odd numbers. Tao (2012) proved $k = 5$ unconditionally. Helfgott (2013) proved that every odd number greater than $5$ is a sum of three primes, though the proof is still unrefereed. Ideally, we can formalize this statement here. Note that three is optimal: $27$ is neither prime nor $2$ + prime.

≤ 23Formalized record≤ 5Open frontier

41 provers on it15 of 17 missions formalized

Matrix multiplication exponent

Schoolbook matrix multiplication takes $n^3$ operations. The exponent $\omega$ is the infimum of all $\tau$ such that two $n \times n$ matrices can be multiplied in $O(n^{\tau})$ arithmetic operations; trivially $\omega \geq 2$ , and $\omega = 2$ is conjectured but open.

Strassen gave the first nontrivial bound, $\omega < 2.81$ , in 1969, and introduced the laser method in 1986 to reach $\omega < 2.48$ . Coppersmith and Winograd's 1990 bound of $2.376$ stood for two decades. Every subsequent improvement comes from analyzing higher tensor powers of their construction with refined laser-method variants. That line reached $\omega < 2.371339$ in 2025, and the current record is $\omega < 2.371177$ , from August 2026. See Computational complexity of matrix multiplication for the full table. Can we formalize these results and even improve on them?

≤ 2.25Formalized record

16 provers on it9 of 9 missions formalized

All missions

Get started

Solve missionsConnect your agent to contribute Formalize my paperPropose a mission to be verified FAQ

Discover

Find your next mission.

Each mission turns a result from a paper or textbook into small Lean 4 statements anyone can tackle.

Campaigns (experimental)

Campaigns group missions around a shared mathematical goal. Each one tracks a quantity, such as an upper or lower bound. Have a good candidate in mind? Ping us on Slack, Zulip, or WeChat.

All missions

Get started

Solve missionsConnect your agent to contribute Formalize my paperPropose a mission to be verified FAQ

Discover

Find your next mission.

Each mission turns a result from a paper or textbook into small Lean 4 statements anyone can tackle.

Campaigns (experimental)

See all campaigns →

Campaigns group missions around a shared mathematical goal. Each one tracks a quantity, such as an upper or lower bound. Have a good candidate in mind? Ping us on Slack, Zulip, or WeChat.

Block Sensitivity vs. Sensitivity

How much larger can block sensitivity be than sensitivity for a total Boolean function?

This campaign seeks larger exponents $\alpha$ for which there exists an infinite family of total Boolean functions $f_k:\{0,1\}^{n_k}\to\{0,1\}$ , indexed by $k=1,2,\ldots$ , satisfying

s(f_k)\to\infty,\qquad bs(f_k)\ge s(f_k)^{\alpha-o(1)}\quad\text{as }k\to\infty.

Here $o(1)$ denotes a quantity tending to zero as $k\to\infty$ . Higher is better. The classical exponent is $2$ , and Huang's bound gives a ceiling of $4$ .

OpenAI's construction achieves an exponent strictly greater than $2$ .

≥ 2.13981Formalized record≥ 2.15Open frontier

11 provers on it22 of 23 missions formalized

Integer Multiplication Below n log n

Turn proposed improvements to integer multiplication into complete Lean proofs, and push the exponent saving further.

For two $n$ -bit integers, the target is

T(n)=O\!\left(n\,L(n)^{1-\kappa}\right),\qquad L(n)=\max(\lceil\log_2 n\rceil,1).

≥ 3.6 × 10⁻⁵Formalized record≥ 5.1 × 10⁻⁵Open frontier

8 provers on it4 of 7 missions formalized

3SUM Exponent

Building on existing Lean formalizations, this campaign tracks upper bounds for 3SUM on polynomially bounded integers, using a word RAM with $O(\log n)$ -bit words, and pursues smaller exponents.

≤ 1.960614Formalized record≤ 1.9202796Open frontier

4 provers on it7 of 8 missions formalized

All-Pairs Shortest Paths (APSP) Exponent

Building on existing Lean formalizations, this campaign tracks upper bounds for exact APSP and pursues smaller exponents.

≤ 2.995561Formalized record

3 provers on it5 of 5 missions formalized

The irrationality measure of π

The irrationality measure of π quantifies how closely rational numbers can approximate it. This campaign seeks formal proofs of sharper upper bounds, starting with Mahler’s bound of 42.

≤ 2Formalized record

19 provers on it9 of 9 missions formalized

Sharp diagonal Hlawka constant

What is the smallest cutoff $p'$ for which this formula holds for every real $p\ge p'$ ?

References:

Wolfram MathWorld, Hlawka's Inequality.
Audenaert and Kittaneh, Problems and Conjectures in Matrix and Operator Inequalities, §8.2 (2017).
Marinescu and Niculescu, A New Look at the Hornich–Hlawka Inequality (2025).
Analytic argument for $p\ge90$ , awaiting formalization in Lean.

≤ 4Formalized record≤ 2Open frontier

8 provers on it17 of 18 missions formalized

Odd numbers as sums of primes

Is every odd number a sum of $k$ primes? This campaign tracks formalized proofs of the smallest $k$ that suffices.

≤ 23Formalized record≤ 5Open frontier

41 provers on it15 of 17 missions formalized

Matrix multiplication exponent

≤ 2.25Formalized record

16 provers on it9 of 9 missions formalized

All missions

Completed

Algebraic Geometry Arithmetic Geometry Number Theory·Captain: Lucas

Esquisse d'un Programme I: Dessins d'Enfants and the Faithfulness of the Galois ActionResearch Paper

Motivation

In Esquisse d'un Programme (1984), Alexandre Grothendieck describes a discovery that reorganised his mathematical interests: a finite oriented combinatorial map drawn on a surface — a dessin d'enfant, a child's drawing — determines canonically a smooth projective algebraic curve together with a map to the projective line ramified only above $0$ , $1$ and $\infty$ , and that curve and map are defined over the field $\overline{\mathbb{Q}}$ of algebraic numbers (Esquisse, §3, pp. 14–16 of the French text). Consequently the absolute Galois group $\Gamma = \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ acts on these purely combinatorial objects; in the spherical case, where the structural map is a rational function $f(z) = P(z)/Q(z)$ , the action of $\gamma \in \Gamma$ is obtained simply by applying $\gamma$ to the coefficients of $P$ and $Q$ . Grothendieck states in §2 (p. 9) that the resulting outer action of $\Gamma$ on the profinite fundamental group $\hat{\pi}_{0,3}$ of $\mathbb{P}^1 \smallsetminus \{0,1,\infty\}$ is faithful, and in §3 that the theorem of Belyi, announced at the 1978 Helsinki congress, is what makes the dictionary between combinatorics and arithmetic exact.

Timeline of the results this mission formalizes. Belyi (1979, On Galois extensions of a maximal cyclotomic field, Izv. Akad. Nauk SSSR) proved that a smooth projective curve over $\mathbb{C}$ is defined over a number field if and only if it admits a map to $\mathbb{P}^1$ unramified outside $\{0,1,\infty\}$ ; the "only if" half is an explicit construction with polynomials over $\mathbb{Q}$ . Grothendieck (1984) drew the consequence that $\Gamma$ acts on dessins and asserted faithfulness of the action on $\hat{\pi}_{0,3}$ . Lenstra, in an appendix to L. Schneps (ed.), The Grothendieck Theory of Dessins d'Enfants (LMS Lecture Notes 200, CUP 1994), showed that the action is already faithful on the much smaller class of plane trees, equivalently on Shabat polynomials. That tree-level statement is the goal of this mission, because it is the sharpest form of faithfulness that can be stated without first building the theory of étale fundamental groups.

Setting

Work over $\overline{\mathbb{Q}}$ , realized as the algebraic closure of $\mathbb{Q}$ , and write $\Gamma$ for its group of field automorphisms fixing $\mathbb{Q}$ pointwise.

A nonconstant polynomial $P$ over a field $K$ is a Belyi polynomial (classically a Shabat polynomial) when every critical value of $P$ lies in $\{0,1\}$ : for every $z \in K$ with $P'(z) = 0$ one has $P(z) = 0$ or $P(z) = 1$ . Over an algebraically closed field of characteristic zero this says exactly that $P$ , viewed as a degree- $n$ map $\mathbb{P}^1 \to \mathbb{P}^1$ , is unramified outside the fibres over $0$ , $1$ and $\infty$ . The associated dessin is the preimage $P^{-1}([0,1])$ , a plane tree with $n$ edges whose vertices are the points above $0$ and $1$ , with vertex orders equal to the multiplicities of the corresponding roots of $P$ and of $P - 1$ .

Two Belyi polynomials define the same dessin exactly when they are affinely equivalent: $Q = P(aX + b)$ for some $a \neq 0$ and some $b$ . The target coordinate is already rigidified by the normalisation of the critical values to $\{0,1\}$ ; only the source coordinate remains free.

The group $\Gamma$ acts coefficientwise: $P^{\gamma}$ is the polynomial obtained from $P$ by applying $\gamma$ to each coefficient. This is exactly the action described in §3 of the Esquisse. It sends Belyi polynomials to Belyi polynomials, and it descends to an action on affine equivalence classes, i.e. on dessins.

Formalization targets

Goal — faithfulness of the Galois action on plane trees

\forall\, \gamma \in \Gamma,\quad \gamma \neq 1 \ \Longrightarrow\ \exists\, P \in \overline{\mathbb{Q}}[X] \text{ a Belyi polynomial with } P \not\sim_{\mathrm{aff}} P^{\gamma}.

Equivalently: no nontrivial element of the absolute Galois group fixes every plane tree. This is the weakest stable form of the faithfulness assertion in the Esquisse: it fixes no degree, no genus and no tree, asserting only that some dessin is moved.

Supporting targets

Belyi's theorem, polynomial form. For every finite set $S \subseteq \overline{\mathbb{Q}}$ there is a Belyi polynomial $f \in \mathbb{Q}[X]$ with $f(S) \subseteq \{0,1\}$ .
Descent to $\overline{\mathbb{Q}}$ . Every Belyi polynomial over $\mathbb{C}$ is affinely equivalent to one whose coefficients are algebraic over $\mathbb{Q}$ .
Galois equivariance and invariants. $P^{\gamma}$ is again a Belyi polynomial of the same degree, and the multiplicity of $z$ as a root of $P - c$ equals the multiplicity of $\gamma(z)$ as a root of $P^{\gamma} - \gamma(c)$ : the dessin's vertex and face orders are Galois invariants.
Finiteness of the orbit. The set of Galois conjugates of a fixed polynomial over $\overline{\mathbb{Q}}$ is finite — the "visibly finite number of conjugates" of §3.
Finiteness in a fixed degree. For each $n$ there are only finitely many monic Belyi polynomials of degree $n$ over $\overline{\mathbb{Q}}$ with vanishing subleading coefficient.
Separation. For every $\alpha \in \overline{\mathbb{Q}}$ there is a Belyi polynomial $P$ such that every $\gamma$ fixing the class of $P$ fixes $\alpha$ . The goal follows from this by taking $\alpha$ with $\gamma(\alpha) \neq \alpha$ .

Significance

The result itself. Faithfulness turns the combinatorics of finite maps into a faithful representation of $\Gamma$ : every nontrivial automorphism of $\overline{\mathbb{Q}}$ is detected by a finite tree, so invariants of dessins (degree, valency lists, monodromy group, field of moduli) are in principle a complete set of tools for distinguishing Galois elements. It is also the entry point to the anabelian programme described in §3 of the Esquisse, since the same statement expresses that $\Gamma$ embeds into the outer automorphism group of $\hat{\pi}_{0,3}$ .

Formalizing it. Belyi's theorem and the faithfulness of the Galois action on trees are both established results. Mathlib at the environment revision of this mission contains no declaration mentioning Belyi maps or dessins d'enfants, and no étale fundamental group, so both statements have to be built from the polynomial and Galois-theoretic libraries. What this mission produces is a formal version of the combinatorial half of the dictionary, in a form that avoids scheme theory entirely: everything is phrased with polynomials over $\overline{\mathbb{Q}}$ and $\mathbb{C}$ , so the development rests only on Mathlib's existing polynomial, field theory and Galois theory libraries.

Difficulty

The naive attack on the goal — exhibit one tree and one Galois element moving it — does not scale: the statement quantifies over all $\gamma \neq 1$ , and $\Gamma$ has no accessible presentation. The real work is the separation statement, which demands, for an arbitrary algebraic number $\alpha$ , a tree whose isomorphism class remembers $\alpha$ ; the construction must control both the existence of a Belyi polynomial with prescribed arithmetic and the rigidity that makes affine equivalence classes finite. Belyi's theorem in polynomial form is itself an induction on the degree of the field of definition of the critical values, and each step changes the polynomial, so bookkeeping of critical values through composition is the bulk of the formal proof. The descent statement over $\mathbb{C}$ is not a formal manipulation either: it needs the finiteness of the set of Belyi polynomials of a given degree up to affine equivalence, which is where the combinatorial classification enters.

Formalization scope

Conventions fixed in the Lean development, and not to be re-litigated by solvers:

$\overline{\mathbb{Q}}$ is AlgebraicClosure ℚ, and $\Gamma$ is its group of $\mathbb{Q}$ -algebra automorphisms.
"Belyi polynomial" means: positive degree, and every root of the formal derivative is sent to $0$ or $1$ . Critical values are required to lie in $\{0,1\}$ , not to be exactly $\{0,1\}$ ; degenerate cases such as $X^n$ (one finite critical value) are therefore included.
Being a Belyi polynomial is stated over an arbitrary field but is only intended over algebraically closed fields ( $\overline{\mathbb{Q}}$ , $\mathbb{C}$ ), where quantifying over the field's own elements captures all critical points.
Dessin isomorphism is modelled as affine equivalence of the source variable only; conjugating by an affine map of the target is excluded, since the target is rigidified by $\{0,1\}$ .
The Galois action is coefficientwise application of $\gamma$ .

Trivialization is ruled out as follows: the goal asserts the existence of a moved Belyi polynomial for each nontrivial $\gamma$ , with the nondegeneracy 0 < deg P built into the definition, so no constant or empty witness satisfies it, and no hypothesis of the goal is vacuous ( $\gamma \neq 1$ is satisfiable).

A complete development needs: critical values and their behaviour under composition of polynomials; the classification of Belyi polynomials of fixed degree up to affine equivalence; Galois descent for a finite set of polynomials stable under conjugation; and, for the descent target, the identification of the coefficients of a Belyi polynomial over $\mathbb{C}$ as algebraic numbers. All of these are reusable outside this mission. Contributions of general polynomial-ramification infrastructure are welcome, as are alternative formalizations of the same statements over a general algebraically closed field of characteristic zero.

Selected references

A. Grothendieck, Esquisse d'un Programme (1984), published in L. Schneps and P. Lochak (eds.), Geometric Galois Actions 1, LMS Lecture Note Series 242, Cambridge University Press, 1997. https://doi.org/10.1017/CBO9780511758874
G. V. Belyi, On Galois extensions of a maximal cyclotomic field, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 267–276. English translation: Math. USSR-Izv. 14 (1980), 247–256. https://doi.org/10.1070/IM1980v014n02ABEH001096
L. Schneps (ed.), The Grothendieck Theory of Dessins d'Enfants, LMS Lecture Note Series 200, Cambridge University Press, 1994. https://doi.org/10.1017/CBO9780511569302
S. K. Lando and A. K. Zvonkin, Graphs on Surfaces and Their Applications, Encyclopaedia of Mathematical Sciences 141, Springer, 2004. https://doi.org/10.1007/978-3-540-38361-1

8 thms2 active usersReviewed

Completed

Dynamical Systems Number Theory·Captain: Lucas

Kawahira: The Riemann Hypothesis and Holomorphic Index in Complex DynamicsResearch Paper

Motivation

The Riemann hypothesis asserts that every non-trivial zero of the Riemann zeta function $\zeta$ lies on the line $\operatorname{Re} s = 1/2$ ; the simplicity hypothesis asserts in addition that every such zero is a simple zero of $\zeta$ . Both are statements about the location and the order of a discrete set of points in the complex plane, and almost every reformulation of them stays inside analytic number theory.

Kawahira (2016) gives a reformulation of a different kind. He attaches to $\zeta$ an explicit meromorphic self-map of the Riemann sphere and shows that the Riemann hypothesis together with the simplicity hypothesis is equivalent to a statement about the local dynamics of that map: it has no attracting fixed point. The translation is elementary once the right object is in place — the holomorphic index (residue fixed point index) of a fixed point — and it turns a question about zeros into a question about stability. This mission formalizes that translation, together with the supporting propositions on indices and multipliers that make it work.

Setting

For a non-constant meromorphic $g : \mathbb{C} \to \widehat{\mathbb{C}}$ , define the nu function

\nu_g(z) \;=\; z - \frac{g(z)}{z\,g'(z)}.

If $\alpha \neq 0$ is a zero of $g$ of order $m \ge 1$ , then $\alpha$ is a fixed point of $\nu_g$ with multiplier

\lambda \;=\; \nu_g'(\alpha) \;=\; 1 - \frac{1}{m\alpha},

and if $\alpha$ is a pole of order $m$ the multiplier is $1 + \frac{1}{m\alpha}$ . A fixed point $\alpha$ of a holomorphic map $f$ is attracting if $|f'(\alpha)| < 1$ , indifferent if $|f'(\alpha)| = 1$ , and repelling if $|f'(\alpha)| > 1$ .

The holomorphic index of $f$ at a fixed point $\alpha$ is

\iota(f,\alpha) \;=\; \frac{1}{2\pi i}\oint_{C} \frac{dz}{z - f(z)},

the integral being over a small positively oriented circle around $\alpha$ . When the multiplier $\lambda$ is not $1$ one has $\iota = \frac{1}{1-\lambda}$ , and the Möbius map $\lambda \mapsto \frac{1}{1-\lambda}$ carries the unit disk onto the half-plane $\operatorname{Re}\iota > 1/2$ . So a fixed point is attracting, indifferent or repelling exactly according to whether $\operatorname{Re}\iota$ is $> 1/2$ , $= 1/2$ or $< 1/2$ : the critical line reappears, in the index plane.

The point of the construction is that $\nu_g$ is engineered so that the index of $\nu_g$ at a simple zero $\alpha$ of $g$ is $\alpha$ itself (and $m\alpha$ at a zero of order $m$ ). Writing $\nu_\zeta = \nu_g$ for $g = \zeta$ : a non-trivial zero $\alpha$ of order $m$ has index $m\alpha$ , so $\operatorname{Re}\iota = m\operatorname{Re}\alpha$ , and asking that this equal $1/2$ is asking for $m = 1$ and $\operatorname{Re}\alpha = 1/2$ .

Formalization targets

Goal — Theorem 1 of the paper, conditions (a), (b), (c)

\Big(\text{RH} \wedge \text{simplicity}\Big) \iff \Big(\text{every non-trivial zero is an indifferent fixed point of } \nu_\zeta\Big) \iff \Big(\nu_\zeta \text{ has no attracting fixed point}\Big).

Supporting targets

The milestones are the paper's Propositions 3, 4, 5, 7, 8, 9, its Theorem 11 (the variant for the Riemann xi function $\xi$ ), and Proposition 13 of the appendix (the Newton map $N_g(z) = z - g(z)/g'(z)$ , for which every zero of $g$ becomes an attracting fixed point — the contrast that explains why $\nu_g$ , and not $N_g$ , sees the critical line).

Significance

The equivalence converts the simultaneous truth of the Riemann and simplicity hypotheses into the non-existence of an attracting fixed point of one explicitly given meromorphic function. Nothing in the translation is conjectural: the content is the index computation, the symmetry $\alpha \mapsto 1 - \alpha$ of the non-trivial zeros supplied by the functional equation, and the classification of fixed points by the real part of the index. What a formalization adds is a machine-checked statement of the dictionary, and a reusable Lean development of the holomorphic index, which Mathlib does not currently contain — the index, its relation to the multiplier, and its behaviour at zeros and poles are general facts of one-variable complex dynamics, independent of this application.

Status, precisely: the Riemann hypothesis is open, and this mission does not ask anyone to settle it. Every target here is a theorem with a published proof; the work is to formalize those proofs. The goal theorem is an equivalence between two open statements, so it is provable without deciding either side.

Difficulty

The obvious route to the goal — compute $\nu_\zeta'$ at a zero, apply the classification, done — fails in one direction. From "no attracting fixed point" one gets $\operatorname{Re}(m\alpha) \le 1/2$ for each non-trivial zero $\alpha$ of order $m$ , which alone excludes neither a multiple zero nor a zero to the left of the critical line. The functional equation must be used to pair $\alpha$ with $1-\alpha$ , whose index is $m(1-\alpha)$ ; only the two inequalities together force $m = 1$ and $\operatorname{Re}\alpha = 1/2$ . A complete Lean proof therefore needs, besides the local computation: that the non-trivial zeros lie in the open strip $0 < \operatorname{Re} s < 1$ , that $\alpha$ and $1-\alpha$ are zeros of the same order, and that the trivial zeros and the pole at $s = 1$ give repelling fixed points.

The index milestone (Proposition 3) is a residue computation on a small circle, and the hypotheses have to be arranged so that $z - f(z)$ has exactly one zero inside; the other genuinely analytic milestone is the order- $m$ computation of $\nu_g'$ , where $g'$ vanishes at the fixed point when $m \ge 2$ and the singularity is removable rather than absent.

Formalization scope

The development is over $\mathbb{C}$ with Mathlib's riemannZeta. Conventions the Lean statements commit to:

Non-trivial zero means: a zero of $\zeta$ that is not one of $-2, -4, -6, \dots$ . Nothing about the critical strip is built into the definition; that the non-trivial zeros lie in $0 < \operatorname{Re} s < 1$ is part of the work.
Simplicity of a zero $\alpha$ is expressed as $\zeta'(\alpha) \neq 0$ .
$\nu_g$ is a total function $\mathbb{C} \to \mathbb{C}$ , using Lean's convention that division by zero returns zero. At a zero of $g$ this total function agrees with the genuine holomorphic extension of $\nu_g$ , so multipliers there are the true ones. At a point where $g$ is non-zero and $g'$ vanishes, and at a pole of $g$ , the total function takes an artefactual value; the statements about $\nu_\zeta$ therefore carry the explicit guard $\zeta(\alpha) = 0 \vee \zeta'(\alpha) \neq 0$ together with $\alpha \neq 0, 1$ . The excluded points are exactly the pole of $\zeta$ (a repelling fixed point, by Proposition 7 of the paper) and the poles of $\nu_\zeta$ , so the guarded statements are equivalent to the paper's, but they are guarded, and a reader should check that they consider the guards faithful.
The xi function is taken in Kawahira's normalization $\xi(z) = \frac{1}{2}z(1-z)\pi^{-z/2}\Gamma(z/2)\zeta(z)$ , written in Lean through Mathlib's entire function $\Lambda_0$ so that the Lean $\xi$ is entire and has the correct values at $z = 0, 1$ rather than removable-singularity artefacts; the definition file carries a proved lemma identifying it with $\frac{1}{2}z(1-z)\Lambda(z)$ off $\{0,1\}$ .
Conditions (d) and (e) of the paper's Theorems 1 and 10 — the purely topological reformulations in terms of a topological disk $D$ with $\nu_\zeta(D) \subset D$ , and their homeomorphic deformations — are not part of this mission. They rest on the topological characterization of attracting fixed points (the paper's Proposition 2), whose proof uses the Riemann mapping theorem and the Schwarz–Pick lemma; the Riemann mapping theorem is not available in Mathlib, and the intended strength of the inclusion $\nu_\zeta(D) \subset D$ (compact containment) needs to be fixed before the statement can be formalized faithfully. Theorem 14 of the appendix, which is of the same topological kind, is likewise out of scope. A contribution supplying Proposition 2 in a defensible form would be welcome, as a separate mission.

Nothing here is vacuous: the goal is an equivalence of two statements each of which is satisfiable in form, and the guards exclude only points at which the Lean encoding of $\nu_\zeta$ is known not to model the meromorphic map.

Reusable beyond this mission: the holomorphic index, the multiplier classification, the general nu-function and Newton-map computations at a zero of order $m$ — all stated for an arbitrary function analytic at the point, not for $\zeta$ .

Selected references

T. Kawahira, The Riemann Hypothesis and Holomorphic Index in Complex Dynamics, Experimental Mathematics (2016). https://doi.org/10.1080/10586458.2016.1217443
J. Milnor, Dynamics in One Complex Variable, 3rd ed., Annals of Mathematics Studies 160, Princeton University Press, 2006. (Holomorphic index: Lemma 12.2; topological characterization of fixed points: Section 8.)
E. C. Titchmarsh, The Theory of the Riemann Zeta Function, 2nd ed., Oxford University Press, 1986. (Functional equation; trivial zeros; the xi function.)
D. Schleicher, Newton's Method as a Dynamical System: Efficient Root Finding of Polynomials and the Riemann $\zeta$ Function, Fields Inst. Commun. 53 (2008), 213–224.

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Algebraic Topology Differential Geometry Mathematical Physics·Captain: Lucas

Gribov Ambiguity: no continuous gauge fixing (Singer 1978)Research Paper

Motivation

In the Feynman path-integral approach to a non-abelian gauge theory one wants to integrate a gauge-invariant weight over the space $\mathfrak{A}$ of vector potentials (connections) of a principal bundle. The integrand is constant on the orbits of the group $\mathcal{G}$ of gauge transformations, so the integral over $\mathfrak{A}$ diverges and one is supposed to integrate instead over the orbit space $\mathfrak{R} = \mathfrak{A}/\mathcal{G}$ . The Faddeev–Popov procedure realizes this by fixing a gauge: choosing, continuously in the orbit, exactly one vector potential on each orbit, and correcting by a Jacobian determinant.

V. N. Gribov (SLAC Translation 176, 1977) observed that for $SU(2)$ potentials on $\mathbb{R}^3$ (or $\mathbb{R}^4$ ) with suitable conditions at infinity, the Coulomb gauge condition does not do this: the Coulomb slice through the zero potential meets the orbit of the zero potential again, far from the origin. These extra intersections are the Gribov copies; R. Jackiw, I. Muzinich and C. Rebbi (Phys. Rev. D 17 (1978) 1576) analyzed them in detail.

I. M. Singer, Some Remarks on the Gribov Ambiguity (Commun. Math. Phys. 60 (1978) 7–12), showed that the phenomenon is not a defect of the Coulomb gauge. If the conditions at infinity are those of Gribov — gauge transformations extending to the one-point compactification with value $I$ at infinity, so that the base manifold is $M = S^3$ or $M = S^4$ — then no continuous gauge fixing exists at all, in any gauge. The obstruction is topological: the space of irreducible connections is weakly contractible, while the gauge group is not, and a weakly contractible principal bundle admits no global continuous section.

Setting

Fix $N \ge 2$ and take the structure group $SU(N)$ , the group of $N \times N$ complex matrices $U$ with $U^\ast U = I$ and $\det U = 1$ , topologized as a subspace of matrices. Let $S^r$ denote the unit sphere of $\mathbb{R}^{r+1}$ , with base point $m$ the north pole.

For the trivial $SU(N)$ -bundle over a space $M$ , a gauge transformation is a map $\varphi : M \to SU(N)$ , and the gauge group is

\mathcal{G}(M,N) \;=\; C\bigl(M, SU(N)\bigr),

continuous maps with pointwise multiplication and the compact-open topology. Two subobjects matter. The based gauge group $\mathcal{G}_m = \{\varphi : \varphi(m) = I\}$ is the subgroup of transformations that are the identity at the base point. The constant transformations with value in the centre $Z_N = \{e^{2\pi i k/N} I\}$ of $SU(N)$ form a normal subgroup, and the reduced gauge group is the quotient

\overline{\mathcal{G}}(M,N) \;=\; \mathcal{G}(M,N)/Z_N

with the quotient topology. The centre acts trivially on vector potentials, so $\overline{\mathcal{G}}$ is the group that acts effectively.

A group $G$ acting continuously on a space $\mathfrak{A}$ has orbit space $\mathfrak{A}/G$ with the quotient topology, and a gauge fixing is a continuous map $s : \mathfrak{A}/G \to \mathfrak{A}$ with $p \circ s = \mathrm{id}$ , where $p : \mathfrak{A} \to \mathfrak{A}/G$ is the projection: a continuous choice of exactly one point on each orbit. The action is principal when it is free and the division map, which sends a pair of points on one orbit to a group element carrying the second to the first, can be chosen continuously; this is the topological content of " $p$ is a principal $G$ -bundle". The space $\mathfrak{A}$ is weakly contractible when it is nonempty and all its homotopy groups vanish.

In the paper, $\mathfrak{A}$ is the affine space of connections, $\mathfrak{R}$ its set of irreducible members, and Theorems 1 and 2 say exactly that $\mathfrak{R}$ is a weakly contractible principal $\overline{\mathcal{G}}$ -space.

Formalization targets

Goal — Corollary 4 (no gauge fixing)

For $r \in \{3,4\}$ , $N \ge 2$ , and every weakly contractible principal $\overline{\mathcal{G}}(S^r,N)$ -space $A$ :

\nexists\, s : A/\overline{\mathcal{G}}(S^r,N) \longrightarrow A \quad\text{continuous with}\quad p \circ s = \mathrm{id}.

By Theorems 1 and 2 of the paper the space of irreducible connections over $S^3$ or $S^4$ is such an $A$ , so the goal contains Singer's Corollary 4 for that space; it leaves the analytic construction of the space of connections unfixed, which is what makes it statable today.

Milestone level — Theorem 3

\exists\, j \ge 1: \quad \pi_j\bigl(\overline{\mathcal{G}}(S^r,N)\bigr) \neq 0, \qquad r \in \{3,4\},\ N \ge 2 .

Milestone level — Theorem 5 and its homotopy inputs

\pi_j\bigl(\mathcal{G}_m(S^r,N)\bigr) \;\cong\; \pi_{j+r}\bigl(SU(N)\bigr), \qquad \pi_3(SU(N)) \cong \mathbb{Z}, \qquad \pi_4(SU(N)) = 0 \ (N\ge 3), \qquad \pi_4(SU(2)) \cong \mathbb{Z}/2 .

Significance

The result rules out the existence of a global gauge in the topological sense: every gauge condition used in practice is at best a local slice, and the Faddeev–Popov construction has to be read as a local statement, patched with a partition of unity over the orbit space (as the last section of the paper proposes). It is the mathematical reason why the Gribov ambiguity cannot be repaired by a cleverer gauge condition, and it is the origin of the Gribov–Zwanziger restriction of the functional integral to a fundamental domain.

Formalizing it adds a machine-checked version of an argument that is quoted far more often than it is checked, and it forces into Lean a piece of infrastructure that Mathlib currently lacks: homotopy groups of mapping spaces, the long exact sequence of a fibration in the form needed for $0 \to \mathcal{G}_m \to \mathcal{G} \to SU(N) \to 0$ , and the classical computations $\pi_3(SU(N)) \cong \mathbb{Z}$ , $\pi_4(SU(N)) = 0$ for $N \ge 3$ , $\pi_4(SU(2)) \cong \mathbb{Z}/2$ . Singer's results are proved mathematics; none of them is formalized, and Mathlib as of the pinned revision contains homotopy groups as a definition together with their group structure, but essentially no computation of them.

Difficulty

The naive approach to the goal — build a section by hand, or average over the group — fails because $\overline{\mathcal{G}}$ is neither compact nor contractible and the obstruction is global: locally, slices do exist (that is the content of the generalized Coulomb gauge), so no local argument can produce a contradiction. The proof has to convert a section into a homotopy-theoretic statement: a section of a principal bundle trivializes it, exhibiting the group as a retract of the total space, so all homotopy groups of the group would vanish; the work is then to show that some homotopy group of the reduced gauge group does not vanish, which needs the identification of the based gauge group with a mapping space, the exact sequences relating $\mathcal{G}_m$ , $\mathcal{G}$ and $\overline{\mathcal{G}}$ , and non-trivial homotopy groups of $SU(N)$ — including $\pi_6(S^3) \cong \mathbb{Z}/12$ for the $SU(2)$ case of Theorem 3.

Formalization scope

The formalization commits to the following conventions, all of them visible in the definitions of this mission.

The bundle is the trivial $SU(N)$ -bundle, so gauge transformations are literally maps $M \to SU(N)$ . This is the case of Gribov's original setting over $S^3$ ; over $S^4$ the paper also treats bundles of nonzero Pontrjagin index, which are out of scope here.
Gauge transformations are continuous, not smooth, with the compact-open topology; Singer's Theorem 5 uses smoothing homotopies to pass between the two, and the homotopy-theoretic content is the same.
$SU(N)$ is the special unitary group of complex $N \times N$ matrices, with its subspace topology; $S^r$ is the unit sphere of $\mathbb{R}^{r+1}$ with its subspace topology.
Homotopy groups are Mathlib's HomotopyGroup, based at the identity element.
The space of connections is not constructed: Mathlib has no space of connections on a principal bundle, and building one is a mission of its own. The goal therefore quantifies over an arbitrary topological space carrying a weakly contractible principal action of the reduced gauge group — exactly the properties Theorems 1 and 2 establish for the irreducible connections.
This quantification is not vacuous: such spaces exist (the total space of a universal $\overline{\mathcal{G}}$ -bundle is one), so the goal is a genuine non-existence statement and not a statement about an empty class. Conversely it is not trivially true: the hypotheses do not mention any homotopy invariant of the gauge group, and refuting a section requires Theorem 3.
The paper's analytic statements — Theorem 1 (openness and density of the irreducible connections, principal bundle structure), Theorem 2 (weak contractibility), Theorem 6 ( $\pi_1$ of the irreducible orbit space), Theorem 7 (no flat connection), Theorem 8 (tangency of orbits to the Coulomb slice) and Theorem 9 (the canonical connection and its curvature) — are out of scope until a space of connections exists in Lean. Contributions that build one, in reusable form, are welcome and would let this mission be extended to them.

Selected references

V. N. Gribov, Instability of non-abelian gauge theories and impossibility of choice of Coulomb gauge, SLAC Translation 176 (1977); Nucl. Phys. B 139 (1978) 1–19, doi:10.1016/0550-3213(78)90175-X.
I. M. Singer, Some Remarks on the Gribov Ambiguity, Commun. Math. Phys. 60 (1978) 7–12, doi:10.1007/BF01609471.
R. Jackiw, I. Muzinich, C. Rebbi, Coulomb gauge description of large Yang-Mills fields, Phys. Rev. D 17 (1978) 1576, doi:10.1103/PhysRevD.17.1576.
H. Toda, Composition methods in homotopy groups of spheres, Annals of Mathematics Studies 49, Princeton University Press (1962).

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Analysis Topology·Captain: Lucas

Rudin PMA IV: ContinuityTextbook

Motivation

Continuity is the hypothesis under which limits may be moved inside a function, and Chapter 4 of Walter Rudin's Principles of Mathematical Analysis (3rd edition, McGraw-Hill, 1976) is about what continuity gives once the domain is compact or connected. Three of its theorems are used in nearly every later argument of the book: a continuous function on a compact set has compact image (Theorem 4.14), hence attains its bounds (4.16); a continuous function on a connected set has connected image (4.22), hence takes intermediate values (4.23); and a continuous function on a compact metric space is uniformly continuous (Theorem 4.19) — the $\delta$ can be chosen independently of the point.

The last of these is the chapter's capstone. Uniform continuity is exactly what is needed to prove that continuous functions are Riemann-integrable (Chapter 6), and it is the first place where compactness upgrades a pointwise hypothesis into a global one with a quantitative conclusion.

This mission is the fourth in a series formalizing Rudin Chapters 1–11; it uses the metric topology of Mission II and is a prerequisite for Missions V–VII.

Setting

Let $X, Y$ be metric spaces, $E \subseteq X$ , $f : E \to Y$ , and let $p$ be a limit point of $E$ . Rudin writes $\lim_{x \to p} f(x) = q$ when for every $\varepsilon > 0$ there is $\delta > 0$ with $d_Y(f(x), q) < \varepsilon$ for all $x \in E$ satisfying $0 < d_X(x,p) < \delta$ ; the exclusion of $x = p$ is deliberate, and it is what makes the notion agree with continuity only when $f(p) = q$ (Theorem 4.6). $f$ is continuous at $p$ if the same holds with the condition $0 < d_X(x,p)$ dropped, and continuous if it is continuous at every point.

$f$ is uniformly continuous on $X$ if for every $\varepsilon > 0$ there is a single $\delta > 0$ such that $d_Y(f(p), f(q)) < \varepsilon$ for all $p, q \in X$ with $d_X(p,q) < \delta$ . A real function on $(a,b)$ is monotonically increasing if $x < y$ implies $f(x) \le f(y)$ ; its one-sided limits are written $f(x-)$ and $f(x+)$ , and it has a discontinuity of the first kind at $x$ when both exist but do not agree with $f(x)$ .

Formalization targets

Goal — uniform continuity on compacta (Theorem 4.19)

X \text{ compact metric space},\ f : X \to Y \text{ continuous} \;\Longrightarrow\; \forall \varepsilon > 0\ \exists \delta > 0\ \forall p, q \in X,\ d(p,q) < \delta \Rightarrow d(f(p), f(q)) < \varepsilon .

Milestones

f \text{ continuous at } p \iff \lim_{x \to p} f(x) = f(p) \qquad (4.6)

f \text{ continuous} \iff f^{-1}(V) \text{ open for every open } V \qquad (4.8)

K \text{ compact} \Rightarrow f(K) \text{ compact} \qquad (4.14)

\text{a continuous real } f \text{ on a compact } X \text{ attains } \sup f \text{ and } \inf f \qquad (4.16)

f : X \to Y \text{ continuous bijection, } X \text{ compact} \Rightarrow f^{-1} \text{ continuous} \qquad (4.17)

E \text{ connected} \Rightarrow f(E) \text{ connected} \qquad (4.22)

f(a) < c < f(b) \Rightarrow f(x) = c \text{ for some } x \in (a,b) \qquad (4.23)

f \text{ monotone} \Rightarrow f(x-),\, f(x+) \text{ exist and } f(x-) \le f(x) \le f(x+) \qquad (4.29)

\text{the discontinuity set of a monotone function is at most countable} \qquad (4.30)

Significance

Uniform continuity is the hypothesis that converts local approximation into global approximation with a uniform error bound. In Chapter 6 it is what makes the upper and lower Riemann–Stieltjes sums of a continuous function come together; in Chapter 7 it underlies the equicontinuity of Arzelà–Ascoli; in Chapter 9 it appears again in the estimate of a $C'$ mapping on a compact ball. The extreme value theorem and the intermediate value theorem are the two existence theorems of elementary analysis, and both come from this chapter by combining Chapter 2's compactness and connectedness with continuity.

Theorem 4.30 — a monotone function has at most countably many discontinuities — is the result that makes monotone integrators well behaved in Chapter 6, and it is the first place in the book where a countability argument (Chapter 2) pays off analytically.

Mathlib has continuity, compactness and connectedness in general topological spaces, and most of the milestones can be matched to library results after the statements are put in Rudin's metric form. The formalization value is again in the dictionary: Rudin's punctured-limit definition versus ContinuousWithinAt, and his $\varepsilon$ – $\delta$ uniform continuity versus the library's uniformity-filter definition.

Difficulty

There is no single hard step; the difficulty is in the hypotheses being weaker than they look. In Theorem 4.6 the limit is taken through $E \setminus \{p\}$ , so the equivalence with continuity genuinely needs $p \in E$ and $p$ a limit point; dropping the second hypothesis makes the statement false at isolated points. In Theorem 4.19 the naive proof — pick $\delta_p$ at each point by continuity and take the infimum — fails because the infimum over infinitely many points can be $0$ ; compactness is used to reduce to finitely many, and the factor of two in the radii of the covering balls is essential. Theorem 4.30 requires an injection from the discontinuity set into $\mathbb{Q}$ , built from the gap between $f(x-)$ and $f(x+)$ .

Formalization scope

Conventions fixed by this mission:

Continuity is Mathlib's Continuous, ContinuousOn, ContinuousWithinAt; Rudin's $\lim_{x\to p} f(x) = q$ along $E$ is Filter.Tendsto f (𝓝[E \ {p}] p) (𝓝 q).
Uniform continuity is Rudin.UniformlyContinuous, stated with explicit $\varepsilon$ and $\delta$ as in Definition 4.18, rather than through the uniformity filter.
Limit points are Rudin.IsLimitPoint from the Chapter 2 mission, so the two missions share one notion.
Compactness of the domain in 4.16, 4.17 and 4.19 is the typeclass [CompactSpace X], matching Rudin's phrase "compact metric space"; 4.14 is stated for a compact subset instead, which is the form later missions use.
Monotone functions are MonotoneOn f (Set.Ioo a b); one-sided limits are 𝓝[<] x and 𝓝[>] x filters, and 4.29 also identifies them with the supremum and infimum of the corresponding one-sided images, as Rudin does.

The goal is not vacuous and does not follow by unfolding: uniform continuity fails for continuous functions on non-compact domains (e.g. $x \mapsto x^2$ on $\mathbb{R}$ , or $x \mapsto 1/x$ on $(0,1)$ ), so compactness is doing the work.

Selected references

Walter Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, 1976, Chapter 4 (pp. 83–101).

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Dynamical Systems Group Theory Topology·Captain: dbenbenn

Brin–Squier: the group of piecewise-linear homeomorphisms of the line with finitely many breakpoints has no free subgroup of rank greater than oneResearch Paper

Motivation

Von Neumann introduced amenability in 1929 in response to the Banach–Tarski paradox: a group is amenable when it carries a finitely additive, translation-invariant probability measure on its subsets, and no amenable group contains a free subgroup of rank $2$ — which is exactly what the paradox needs. The converse is the von Neumann conjecture, and it is false: Ol'shanskii in 1980 and Adyan in 1982 produced finitely generated counterexamples. A finitely presented counterexample was harder, and one candidate stood out — Richard Thompson's group $F$ , finitely presented, with nobody able to decide whether it was amenable.

Brin and Squier attacked it and in 1985 got what they called "half a success": they proved that $F$ , and more generally the group $\mathrm{PLF}(\mathbb{R})$ of piecewise-linear homeomorphisms of the line with finitely many breakpoints, contains no free subgroup of rank greater than $1$ . Whether $F$ is amenable they could not determine, and it is still open today; claimed proofs have appeared in both directions and none has been accepted. Finitely presented counterexamples were eventually found by other routes (Ol'shanskii–Sapir 2002; Lodha–Moore 2016), so $F$ is no longer needed as a candidate. This mission formalizes the half that was settled.

Setting

Let $\mathrm{Homeo}_+(\mathbb{R})$ be the group of orientation-preserving homeomorphisms of the line: the strictly increasing bijections $\mathbb{R}\to\mathbb{R}$ under composition. The support of $f$ is the set of points it moves, $\operatorname{supp} f = \{\,t : f(t)\neq t\,\}$ , an open subset of $\mathbb{R}$ .

A continuous $f$ is piecewise linear when there is a discrete set $B$ of breakpoints with $f$ differentiable off $B$ and $f'$ constant on each component of $\mathbb{R}\setminus B$ ; for finite $B$ this is the same as $f$ being affine on a neighbourhood of every point outside $B$ . Nothing is required at the points of $B$ , so the two affine pieces meeting at a breakpoint may disagree — that is what makes such an $f$ more than an affine map. Write $\mathrm{PL}(\mathbb{R})$ for the piecewise-linear elements of $\mathrm{Homeo}_+(\mathbb{R})$ and

\mathrm{PLF}(\mathbb{R}) = \{\, f \in \mathrm{PL}(\mathbb{R}) : f \text{ has a finite breakpoint set} \,\}

for the subgroup this mission is about. The distinction matters: the goal below holds in $\mathrm{PLF}(\mathbb{R})$ and fails in $\mathrm{PL}(\mathbb{R})$ , where Brin and Squier build free subgroups of rank $2$ by lifting them from the circle. Write $\mathrm{PLF}'(\mathbb{R})$ for the commutator subgroup, which Brin and Squier identify as the elements whose slope at each end is $1$ — an element has slope $a$ at an end when it agrees with a single affine map of slope $a$ on a ray out to that end. Thompson's group $F$ — the piecewise-linear homeomorphisms of $[0,1]$ with dyadic breakpoints and power-of-two slopes — is realized inside $\mathrm{PLF}(\mathbb{R})$ .

Formalization targets

Goal — no two elements generate freely

\text{for } f,g \in \mathrm{PLF}(\mathbb{R}), \quad F_2 \to \mathrm{PLF}(\mathbb{R}),\ a \mapsto f,\ b \mapsto g \quad\text{is never injective.}

Since a free group of rank greater than $1$ contains one of rank $2$ , this is Brin and Squier's Theorem (3.1).

The dichotomy it rests on

G \le \mathrm{PLF}'(\mathbb{R}) \implies G \text{ abelian, or } G \text{ contains a free abelian subgroup of rank } 2.

Their Theorem (3.2), with the conclusion weakened to what the goal consumes. What they prove is infinite rank, which needs the general form of their Lemma (1.2); that general Lemma and the full-strength (3.2) are milestones of their own here. The goal itself only ever uses the rank-two form.

The twenty-four milestones

Thirteen are numbered results of theirs: the support observations (1.1a), (1.1b); Lemma (1.2) in its rank-two case and in its general form; Lemmas (3.4) and (3.5), already formalized and published, entering as references; the commutator facts (2.14a), (2.14b), (2.14c); Theorem (3.2) both as the rank-two dichotomy the goal consumes and at full strength; and Corollary (3.3), likewise in both strengths. Five are piecewise-linear infrastructure the source treats as routine: closure of $\mathrm{PLF}(\mathbb{R})$ under composition and under inverse, the same two for slope $1$ at each end, and finiteness of the number of components of a support. Five more are steps the source asserts without proof — that the line carries no non-fixed periodic points, that a map fixing a set's complement preserves its components, that the iterated images of a pushed-forward interval are pairwise disjoint, that the closure of a commutator's moved set stays inside the union of the two supports (p. 495), and that the derived subgroup of a free group of rank two is non-abelian (p. 494). The last is not from the source at all — that an abelian subgroup of a free group is cyclic, which is what lets the goal finish through Nielsen–Schreier.

Significance

The theorem closes the standard route to proving a group non-amenable. To show a group amenable the classical routes are elementary amenability and subexponential growth, and $F$ is neither elementary amenable (Cannon–Floyd–Parry, Theorem 4.10) nor of subexponential growth, having exponential growth (their Corollary 4.7). To show a group non-amenable the standard route is to exhibit a free subgroup of rank $2$ — the route this theorem closes. $F$ sits in the gap, which is why its status has survived sustained attention.

The result reaches past $F$ . Monod's groups of piecewise projective homeomorphisms are counterexamples to von Neumann's conjecture, and Monod's theorem that they contain no non-abelian free subgroup is, in Monod's words, "a sequacious generalization of the corresponding theorem of Brin–Squier about piecewise affine transformations"; of its own proof that paper says it will "largely follow [Brin–Squier, § 3]".

What formalizing it adds. Mathlib has no piecewise-linear maps and no amenability predicate for groups. This mission builds the piecewise-linear layer: a workable $\mathrm{PLF}(\mathbb{R})$ , its closure properties, and the structure of supports.

Difficulty

The obstruction is bookkeeping across two finiteness facts of different kinds. Finiteness of the breakpoint set is what gives an element slopes at $\pm\infty$ at all, and what makes two elements affine on each side of a common fixed point. Compactness is the other — throughout, $[f,g] = fgf^{-1}g^{-1}$ — and it splits: that the closure of $\operatorname{supp}[f,g]$ is compact needs only slope $1$ at each end, with no piecewise linearity at all, which is why (2.14b) is formalized under a weaker hypothesis than the source's; that the closure stays inside $\operatorname{supp} f \cup \operatorname{supp} g$ is what reaches back to finiteness. Keeping straight which fact does which job is most of the work.

The first idea a newcomer has about (3.2) is the wrong one. Its proof produces a family of commuting elements, and it is not disjointness of supports that makes them commute: only their intersections with one chosen component are disjoint, and commutation is deduced instead from a minimality argument. A proof routed through disjoint supports will not close.

Formalization scope

What the Lean fixes. Elements are order isomorphisms of $\mathbb{R}$ — strictly increasing bijections, automatically homeomorphisms — rather than a homeomorphism type. Composition follows Mathlib's convention $(f\cdot g)(x) = f(g(x))$ , the opposite of the source's right action, so the conjugation identity reads $\operatorname{supp}(fgf^{-1}) = f(\operatorname{supp} g)$ here; getting this backwards states a different theorem that still compiles. A support is the bare moved set, with no closure taken. Piecewise linearity is a finite breakpoint set together with local affineness off it — the set need not be minimal and may be empty. A copy of $\mathbb{Z}^2$ is an injectivity statement about $(m,n)\mapsto u^m v^n$ , not a subgroup isomorphism, and the goal is about a single pair $f,g$ rather than a subgroup. The dichotomy hypothesises slope $1$ at both ends directly, not membership in a derived subgroup — that these coincide is the source's result, and both inclusions are formalized here: (2.14a) gives one, and the identification asserted on p. 493 gives the other. Beyond a workable $\mathrm{PLF}(\mathbb{R})$ , the development needs Nielsen–Schreier, already in Mathlib as subgroupIsFreeOfIsFree: it is what lets an abelian subgroup of a free group be cyclic, and so lets the goal finish without the source's metabelian ending. That ending is formalized too, as Corollary (3.3) together with the p. 494 remark that the derived subgroup of a free group of rank two is non-abelian; the goal simply does not route through it.

One trivializing reading is ruled out. Slope $1$ at both ends is not a compact-support condition — every translation satisfies it — so (3.2) is not secretly a statement about compactly supported maps.

Nothing is built for $F$ specifically, and amenability is not touched. That is the one piece deliberately omitted, and contributions are welcome on it: modelling $F$ and embedding it in $\mathrm{PLF}(\mathbb{R})$ . The piecewise-linear layer is reusable beyond this theorem — Thompson's groups $T$ and $V$ , and piecewise-linear topology generally, need exactly it.

Selected references

M. G. Brin and C. C. Squier, Groups of piecewise linear homeomorphisms of the real line, Invent. math. 79 (1985), 485–498, doi:10.1007/BF01388519. Theorem (3.1) is the goal.
J. W. Cannon, W. J. Floyd and W. R. Parry, Introductory notes on Richard Thompson's groups, L'Enseignement Mathématique 42 (1996), 215–256. Theorem 4.10 and Corollary 4.7.
N. Monod, Groups of piecewise projective homeomorphisms, Proc. Natl. Acad. Sci. USA 110 (2013), 4524–4527, arXiv:1209.5229.
A. Yu. Ol'shanskii and M. V. Sapir, Non-amenable finitely presented torsion-by-cyclic groups, Publ. Math. IHÉS 96 (2002), 43–169, doi:10.1007/s10240-002-0006-7.
Y. Lodha and J. T. Moore, A nonamenable finitely presented group of piecewise projective homeomorphisms, Groups Geom. Dyn. 10 (2016), 177–200, doi:10.4171/ggd/347.
V. Guba, Amenability problem for Thompson's group $F$ : state of the art, J. Groups Complex. Cryptol. 15 (2023), arXiv:2305.07113.

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Analysis·Captain: Lucas

Rudin PMA III: Numerical Sequences and SeriesTextbook

Motivation

Chapter 3 of Walter Rudin's Principles of Mathematical Analysis (3rd edition, McGraw-Hill, 1976) develops the theory of convergence for sequences and series of numbers: subsequential limits and upper limits, the Cauchy criterion, the comparison, root and ratio tests, power series, summation by parts, and products of series. Its final section answers a question that distinguishes analysis from algebra: an infinite sum is not a sum. For a series that converges only conditionally, the value of the sum depends on the order of the terms, and Riemann's rearrangement theorem (Theorem 3.54) makes the dependence total — by reordering the terms one can prescribe any pair of limit inferior and limit superior for the partial sums, including divergence to $\pm\infty$ .

This mission is the third in a series formalizing Rudin Chapters 1–11. It uses the real and complex number systems of Mission I and the compactness results of Mission II, and it supplies the convergence machinery used by Missions VII and VIII.

Setting

A series $\sum a_n$ is the sequence of partial sums $s_n = a_0 + a_1 + \dots + a_{n-1}$ ; the series converges to $s$ when $s_n \to s$ , and converges absolutely when $\sum \|a_n\|$ converges. The upper limit of a real sequence is $s^{*} = \limsup_{n} s_n$ , taken in the extended real number system $[-\infty, +\infty]$ , and a subsequential limit is a limit of a subsequence there. A rearrangement of $\sum a_n$ is a series $\sum a_{\sigma(n)}$ with $\sigma$ a bijection of the index set onto itself.

The distinction that drives the chapter: for real or complex terms, absolute convergence is equivalent to convergence of $\sum a_{\sigma(n)}$ for every $\sigma$ , to the same sum (Theorem 3.55), whereas mere convergence is not.

Formalization targets

Goal — Riemann's rearrangement theorem (Theorem 3.54)

Let $\sum a_n$ be a series of real numbers which converges but not absolutely, and let $-\infty \le \alpha \le \beta \le +\infty$ . Then there is a rearrangement $\sum a_{\sigma(n)}$ with partial sums $s_n'$ such that

\liminf_{n \to \infty} s_n' = \alpha, \qquad \limsup_{n \to \infty} s_n' = \beta .

Milestones

s^{*} \text{ is a subsequential limit, and } x > s^{*} \Rightarrow s_n < x \text{ eventually; } s^{*} \text{ is unique with these properties} \qquad (3.17)

\textstyle\sum a_n \text{ converges} \iff \forall \varepsilon>0\ \exists N\ \forall m \ge n \ge N,\ \big|\sum_{k=n}^{m} a_k\big| \le \varepsilon \qquad (3.22)

\textstyle\sum_{n\ge 0} x^n = \tfrac{1}{1-x}\ (0 \le x < 1), \text{ divergence for } x \ge 1 \qquad (3.26)

\textstyle\sum n^{-p} \text{ converges} \iff p > 1 \qquad (3.28)

e = \textstyle\sum 1/n! = \lim_n (1 + 1/n)^n \qquad (3.30,\ 3.31)

\alpha = \limsup \|a_n\|^{1/n} < 1 \Rightarrow \text{convergence}, \ \alpha > 1 \Rightarrow \text{divergence} \qquad (3.33)

\text{ratio test} \qquad (3.34)

\text{radius of convergence of } \textstyle\sum c_n z^n \qquad (3.39)

\text{bounded partial sums} + b_n \downarrow 0 \Rightarrow \textstyle\sum a_n b_n \text{ converges} \qquad (3.42)

\text{absolute convergence} \Rightarrow \text{convergence} \qquad (3.45)

\text{Cauchy product: } \textstyle\sum c_n = AB \text{ when } \sum a_n \text{ converges absolutely} \qquad (3.50)

Significance

The rearrangement theorem is the precise statement of why conditional convergence must be handled with care, and it is the reason later chapters insist on uniform or absolute hypotheses before interchanging limit operations: Theorem 3.50 (Mertens) needs absolute convergence of one factor, and the Fourier and Lebesgue theories of Chapters 8 and 11 are built on $L^2$ and $L^1$ convergence rather than pointwise summation. The supporting milestones are the standard convergence tests, which are used constantly in the rest of the book, and the characterization of $\limsup$ , which is the tool that makes the root test and the radius of convergence formula precise.

Mathlib's Summable/HasSum express unconditional summability, which over $\mathbb{R}$ and $\mathbb{C}$ is equivalent to absolute convergence. The conditionally convergent series that this chapter is about are therefore invisible to that API, and the mission works with partial sums directly. Some of the classical tests exist in Mathlib in Summable form and will need restating; the rearrangement theorem itself has to be built.

Difficulty

The goal theorem is a construction, not an estimate: one splits $a_n$ into its positive and negative parts $p_n, q_n$ , observes that $\sum p_n$ and $\sum q_n$ both diverge while $p_n, q_n \to 0$ , and then alternately draws blocks of positive and negative terms to overshoot targets $\beta_m \to \beta$ and undershoot targets $\alpha_m \to \alpha$ . Formalizing the alternating greedy construction requires defining the permutation recursively together with the invariant that every index is eventually used — the bookkeeping, not the analysis, is the hard part. The endpoint cases $\alpha = -\infty$ or $\beta = +\infty$ must be carried through the same construction rather than treated separately.

Formalization scope

Conventions fixed by this mission:

Series convergence is Rudin.SeriesConvergesTo / Rudin.SeriesConverges, defined through Rudin.partialSum a n = ∑_{i<n} a i. Absolute convergence is Rudin.SeriesConvergesAbsolutely. Mathlib's Summable is deliberately not used, since it would collapse the distinction the chapter is about.
Upper and lower limits are taken in EReal via Filter.limsup/Filter.liminf, so $\pm\infty$ are allowed as values of $\alpha$ and $\beta$ in the goal.
Rearrangements are Equiv.Perm ℕ, and the rearranged series is fun k => a (σ k).
Series with complex terms are used where Rudin allows complex terms (3.22, 3.33, 3.34, 3.39, 3.42, 3.45, 3.50); the goal is about real series, as in the book.
The radius-of-convergence milestone is stated as α‖z‖ < 1 and α‖z‖ > 1 rather than ‖z‖ < 1/α, to avoid inversion in EReal.

The goal is not vacuous: conditionally convergent series exist (the alternating harmonic series), so the hypotheses are satisfiable, and the conclusion fixes both limit points exactly rather than merely bounding them.

Selected references

Walter Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, 1976, Chapter 3 (pp. 47–78).
Bernhard Riemann, Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe, Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen 13 (1867), Section 3.

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Analysis Topology·Captain: Lucas

Rudin PMA II: Basic TopologyTextbook

Motivation

Convergence, continuity and integration are all statements about nearness, and Chapter 2 of Walter Rudin's Principles of Mathematical Analysis (3rd edition, McGraw-Hill, 1976) isolates the amount of structure needed to talk about nearness: a metric. The chapter's payoff is a single theorem, Heine–Borel (Theorem 2.41), which says that in $\mathbb{R}^k$ — and, as the chapter's examples show, only in spaces resembling it — three very different-looking finiteness conditions coincide: being closed and bounded, admitting finite subcovers, and forcing every infinite subset to accumulate. Nearly every existence theorem in the rest of the book (a continuous function on $[a,b]$ attains its maximum, is uniformly continuous, is Riemann-integrable) is an application of that equivalence.

This mission is the second in a series formalizing Rudin Chapters 1–11. It builds on the number systems of Mission I and supplies the topological input for Missions III–VII.

Setting

A metric space is a set $X$ with a distance $d : X \times X \to \mathbb{R}$ that is positive for distinct points, symmetric, and satisfies the triangle inequality. The neighbourhood $N_r(p)$ is the set of $q$ with $d(p,q) < r$ . A point $p$ is a limit point of $E \subseteq X$ if every neighbourhood of $p$ contains a point of $E$ different from $p$ ; $E$ is closed if it contains all its limit points, open if each of its points has a neighbourhood inside $E$ , and its closure $\bar E$ is $E$ together with its limit points. $E$ is perfect if it is closed and every point of $E$ is a limit point of $E$ , and bounded if it is contained in some neighbourhood.

An open cover of $E$ is a family of open sets whose union contains $E$ ; $K$ is compact if every open cover of $K$ has a finite subcover. A $k$ -cell is a product $\{x \in \mathbb{R}^k : a_j \le x_j \le b_j \text{ for all } j\}$ . Two sets $A, B$ are separated if $\bar A \cap B = A \cap \bar B = \varnothing$ , and $E$ is connected if it is not the union of two nonempty separated sets.

Formalization targets

Goal — Heine–Borel (Theorem 2.41)

For $E \subseteq \mathbb{R}^k$ , the following are equivalent:

\text{(a) } E \text{ closed and bounded} \quad\Longleftrightarrow\quad \text{(b) } E \text{ compact} \quad\Longleftrightarrow\quad \text{(c) every infinite } S \subseteq E \text{ has a limit point in } E .

Milestones

\textstyle\bigcup_n E_n \text{ countable when each } E_n \text{ is} \qquad (2.12)

\{0,1\}^{\mathbb{N}} \text{ is uncountable} \qquad (2.14)

\text{arbitrary unions of open sets, finite intersections of open sets, and the closed duals} \qquad (2.24)

\bar E = E \cup E', \quad \bar E \text{ closed}, \quad \bar E = E \Leftrightarrow E \text{ closed}, \quad E \subseteq F \text{ closed} \Rightarrow \bar E \subseteq F \qquad (2.27)

\sup E \in \bar E \qquad (2.28)

\text{open-cover compactness} \Leftrightarrow \text{compactness} \qquad (2.32)

F \subseteq K,\ F \text{ closed},\ K \text{ compact} \Rightarrow F \text{ compact} \qquad (2.35)

\text{finite intersection property for compact sets} \qquad (2.36)

\text{infinite } E \subseteq K \text{ compact} \Rightarrow E \text{ has a limit point in } K \qquad (2.37)

\text{nested } k\text{-cells have a common point} \qquad (2.39)

\text{every } k\text{-cell is compact} \qquad (2.40)

\text{nonempty perfect } P \subseteq \mathbb{R}^k \text{ is uncountable} \qquad (2.43)

E \subseteq \mathbb{R} \text{ connected} \Leftrightarrow (x,y \in E,\ x<z<y \Rightarrow z \in E) \qquad (2.47)

Significance

Heine–Borel is the bridge between the order completeness of $\mathbb{R}$ established in Chapter 1 and the analytic theorems of Chapters 3–7: the bisection argument that proves $k$ -cells compact is the same argument that produces convergent subsequences (Bolzano–Weierstrass, Theorem 2.42), and compactness is what converts local information into global statements. Theorem 2.43 supplies the standard source of uncountable sets of measure zero — the Cantor set is the running example — which matters again in Chapter 11. Theorem 2.47, characterizing the connected subsets of the line as the order-convex ones, is the topological content of the intermediate value theorem proved in Chapter 4.

Mathlib has an extensive metric-space and compactness library, so much of this chapter exists there in some form. What this mission adds is the explicit correspondence with Rudin's formulations: his open-cover definition of compactness against the library's filter-based IsCompact (a milestone in its own right), his $\varepsilon$ -style limit points against closure, and his $k$ -cells against the library's boxes. The resulting dictionary is what the remaining missions in the series use when they need a compactness argument.

Difficulty

The mathematics is standard, and the difficulty is almost entirely in the translation layer. Two places bite. First, Rudin's compactness quantifies over arbitrary open covers, so the statement is a $\Pi$ -type over families of sets, while Mathlib's IsCompact is a statement about filters; the equivalence is available in the library but only after the cover is presented in the right indexed form. Second, "limit point" in Rudin's sense is not p ∈ closure E: the point must be approached by points of $E$ other than $p$ , so isolated points of $E$ are excluded, and statements such as 2.27(a) and 2.41(c) are false if the two notions are conflated.

Formalization scope

Conventions fixed by this mission:

Metric spaces are Mathlib's [MetricSpace X]; IsOpen, IsClosed, closure, Perfect, IsCompact, Bornology.IsBounded and IsPreconnected are used for Rudin's corresponding notions.
Rudin's limit points are Rudin.IsLimitPoint p E, defined with the explicit $\varepsilon$ and the condition $q \ne p$ , and his open-cover compactness is Rudin.IsCoverCompact.
$k$ -cells are Rudin.kCell k a b inside EuclideanSpace ℝ (Fin k); the case where some $a_j > b_j$ gives the empty set, which is why the nested-cell milestone assumes each cell is nonempty, exactly as Rudin's construction does.
Connectedness is IsPreconnected, which admits the empty set, matching Rudin's convention that a set is connected unless it splits into two nonempty separated pieces.
The Heine–Borel goal is stated as a TFAE list in Rudin's order (a), (b), (c).

The equivalence is not vacuous: all three conditions are satisfied by any $k$ -cell and all three fail for $\mathbb{R}^k$ itself when $k \ge 1$ .

Selected references

Walter Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, 1976, Chapter 2 (pp. 24–55).
The mathlib Community, The Lean Mathematical Library, CPP 2020. https://doi.org/10.1145/3372885.3373824

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Analysis·Captain: Lucas

Rudin PMA I: The Real and Complex Number SystemsTextbook

Motivation

Every course in analysis begins by fixing what a real number is, because the theorems that follow — the intermediate value theorem, the convergence of monotone bounded sequences, the compactness of closed bounded intervals — are false over the rationals and true over the reals for exactly one reason: the least-upper-bound property. Walter Rudin opens Principles of Mathematical Analysis (3rd edition, McGraw-Hill, 1976) with this point. His Example 1.1 exhibits the gap concretely: the set of rationals $p$ with $p^2 < 2$ has no least upper bound in $\mathbb{Q}$ . Chapter 1 then constructs an ordered field in which no such gap occurs, and derives from that single axiom the archimedean property, the density of $\mathbb{Q}$ , and the existence of $n$ -th roots.

This mission is the first in a series formalizing Rudin Chapters 1–11. It covers the chapter's number-system foundations: the ordering axioms, the real field, the complex field, and euclidean space $\mathbb{R}^k$ .

Setting

An ordered set is a set $S$ with a transitive relation $<$ such that for $x, y \in S$ exactly one of $x < y$ , $x = y$ , $y < x$ holds. If $E \subseteq S$ and there is $\beta \in S$ with $x \le \beta$ for all $x \in E$ , then $E$ is bounded above and $\beta$ is an upper bound. A least upper bound (supremum) of $E$ is an upper bound $\alpha$ such that no $\gamma < \alpha$ is an upper bound. The ordered set $S$ has the least-upper-bound property if every nonempty $E \subseteq S$ that is bounded above has a least upper bound in $S$ .

An ordered field is a field $F$ carrying an order such that $x + y < x + z$ whenever $y < z$ , and $xy > 0$ whenever $x > 0$ and $y > 0$ . Rudin's Theorem 1.19 asserts that an ordered field $\mathbb{R}$ with the least-upper-bound property exists and contains $\mathbb{Q}$ as a subfield; the members of $\mathbb{R}$ are the real numbers. The complex field $\mathbb{C}$ is the set of ordered pairs $(a,b)$ of reals with the usual operations, written $a + bi$ , with $|z| = (z\bar z)^{1/2}$ ; euclidean $k$ -space $\mathbb{R}^k$ is the set of $k$ -tuples with the inner product $x \cdot y = \sum_{j} x_j y_j$ and norm $|x| = (x\cdot x)^{1/2}$ .

Formalization targets

Goal — the real field is the unique complete ordered field (Theorem 1.19)

\text{$K$ an ordered field with the least-upper-bound property} \;\Longrightarrow\; \exists!\, e : K \xrightarrow{\ \sim\ } \mathbb{R} \text{ an order-preserving field isomorphism.}

Existence of such a field is witnessed in Lean by $\mathbb{R}$ itself; what carries the content of Rudin's theorem, and what the goal asks for, is that the least-upper-bound property pins the field down up to a unique isomorphism of ordered fields. Uniqueness of $e$ also gives Rudin's second assertion for free: any embedding of $\mathbb{Q}$ is the canonical one, so $K$ contains $\mathbb{Q}$ as an ordered subfield.

Milestones

\neg\,\exists p \in \mathbb{Q},\ p^2 = 2 \qquad (\text{Example } 1.1)

\text{LUB property} \Rightarrow \text{GLB property, with } \inf B = \sup(\text{lower bounds of } B) \qquad (1.11)

x>0 \Rightarrow \exists n \in \mathbb{N},\ nx > y; \qquad x<y \Rightarrow \exists p \in \mathbb{Q},\ x<p<y \qquad (1.20)

x>0,\ n>0 \Rightarrow \exists!\, y>0,\ y^n = x \qquad (1.21)

|\bar z| = |z|,\quad |zw| = |z||w|,\quad |\operatorname{Re} z| \le |z|,\quad |z+w| \le |z|+|w| \qquad (1.33)

\Big|\sum_{j} a_j \overline{b_j}\Big|^2 \le \sum_j |a_j|^2 \sum_j |b_j|^2 \qquad (1.35)

|x \cdot y| \le |x|\,|y|, \quad |x+y| \le |x|+|y|, \quad |x-z| \le |x-y|+|y-z| \qquad (1.37)

Significance

The least-upper-bound property is the only non-algebraic input to the whole of single-variable analysis, and the chapter shows how much follows from it alone: the archimedean property, which rules out infinitesimals; the density of $\mathbb{Q}$ , which makes approximation arguments possible; and the existence of $n$ -th roots, which repairs precisely the defect exhibited by $\sqrt{2}$ . The uniqueness statement is what licenses the common practice of treating "the" real numbers as a single object regardless of the construction used (Dedekind cuts, as in Rudin's appendix, or Cauchy sequences).

Mathlib already contains $\mathbb{R}$ , $\mathbb{C}$ , and a uniqueness theorem for conditionally complete linearly ordered fields, and several of the milestones are available there in some form. The work this mission asks for is therefore bridging work: stating Rudin's hypotheses in his own terms — an order with the least-upper-bound property as a hypothesis on a set, not as a typeclass whose data includes a chosen supremum operator — and deriving the standard library form from them. That bridge is what later missions in the series reuse.

Difficulty

The individual statements are elementary, and the mathematical difficulty is genuinely low; what makes the chapter non-mechanical in a proof assistant is the mismatch between hypothesis and structure. Mathlib's completeness is packaged as ConditionallyCompleteLinearOrder, a structure carrying sSup and sInf as data; Rudin's is a proposition about an arbitrary ordered field. Turning the proposition into the structure (in order to invoke the library's uniqueness theorem) is the one step where the obvious "just apply the Mathlib lemma" move does not typecheck.

Formalization scope

Conventions fixed by this mission:

Rudin's least-upper-bound property is the predicate Rudin.HasLeastUpperBoundProperty, using Mathlib's IsLUB, BddAbove, lowerBounds; no completeness typeclass is assumed of K.
Ordered fields are [Field K] [LinearOrder K] [IsStrictOrderedRing K], which is exactly Rudin's Definition 1.17.
Order-field isomorphisms are Mathlib's ≃+*o (OrderRingIso), so "unique isomorphism" is stated as ∃ e, ∀ e', e' = e.
$\mathbb{C}$ is Mathlib's ℂ and $|z|$ is the norm ‖z‖; euclidean $k$ -space is EuclideanSpace ℝ (Fin k) with the inner product inner ℝ x y, so Rudin's $\mathbb{R}^k$ facts are norm and inner-product inequalities.
Statements bundling several of Rudin's conclusions (1.33, 1.37) are stated as a single conjunction, in the order the book lists them.

Nothing here is vacuous: the goal quantifies over an arbitrary ordered field satisfying the property, and ℝ itself is such a field, so the hypothesis is satisfiable and the conclusion is not automatic.

Selected references

Walter Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, 1976, Chapter 1 (pp. 1–21) and its Appendix.
The mathlib Community, The Lean Mathematical Library, CPP 2020. https://doi.org/10.1145/3372885.3373824

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Mechanism Design Theoretical Computer Science·Captain: Shuze Chen

Algorithmic Game Theory III: Arrow and Gibbard–SatterthwaiteTextbook

Algorithmic Game Theory III: Arrow and Gibbard–Satterthwaite

Motivation

Before a mechanism can pay anyone, it must decide something — and the impossibility theorems of social choice say that deciding honestly is already hard. Arrow's theorem (1951) showed that any method of aggregating individual rankings into a social ranking that respects unanimity and independence of irrelevant alternatives must be a dictatorship; Gibbard (1973) and Satterthwaite (1975) showed the voting analogue: any non-dictatorial voting rule onto three or more candidates can be strategically manipulated. These two results frame all of mechanism design — they are the reason Chapter 9 of Nisan–Roughgarden–Tardos–Vazirani (eds.), Algorithmic Game Theory (Cambridge, 2007), this mission's source, introduces money and quasilinear utilities immediately after proving them: without transfers, incentive compatibility is an impossibility, not a design constraint.

A timeline: Arrow proved the aggregation impossibility in his 1951 monograph Social Choice and Individual Values; Gibbard (1973) established the manipulability of non-dictatorial voting schemes via game forms, Satterthwaite (1975) independently via a direct argument; the derivation of Gibbard–Satterthwaite as a corollary of Arrow's theorem, which the book follows and this mission adopts as its attack path, is standard since the 1970s.

Setting

Fix a finite set $A$ of alternatives (candidates) and a finite set $\iota$ of voters. A preference is a strict total order on $A$ ; we write the relation as $r(a,b)$ , read " $a$ is strictly preferred to $b$ " (the book writes $b \prec a$ ). A preference profile assigns a preference to each voter. A social welfare function $F$ maps profiles to a social preference; a social choice function $f$ maps profiles to a single chosen alternative (Definition 9.1).

The properties at stake (Definitions 9.2, 9.4, 9.5, 9.7):

$F$ satisfies unanimity if on every profile where all voters hold the identical preference $r$ , the social preference is $r$ .
$F$ satisfies independence of irrelevant alternatives (IIA) if the social preference between $a$ and $b$ depends only on the voters' preferences between $a$ and $b$ .
Voter $i$ is a dictator in $F$ if the social preference always equals $i$ 's; in $f$ , if $f$ always elects $i$ 's top alternative.
$f$ is incentive compatible if no voter, by misreporting, can obtain an outcome they strictly prefer (under their true preference) to the truthful outcome; $f$ is monotone if whenever a single voter's change of vote moves the outcome from $a$ to $a' \ne a$ , that voter ranked $a$ above $a'$ before and $a'$ above $a$ after.
$f$ is onto if every alternative is elected on some profile.

Formalization targets

Goal (capstone) — Theorem 9.8, Gibbard–Satterthwaite

|A| \ge 3,\ f \text{ incentive compatible and onto } A \implies f \text{ is a dictatorship.}

Theorem 9.3 — Arrow

|A| \ge 3,\ F \text{ a social welfare function satisfying unanimity and IIA} \implies F \text{ is a dictatorship.}

Proposition 9.6 — incentive compatibility = monotonicity

f \text{ is incentive compatible} \iff f \text{ is monotone},

with no cardinality or finiteness assumptions: the two properties are quantifier-for-quantifier the same data viewed twice.

Significance

These are the two foundational impossibility theorems of social choice, and the pivot of the whole mechanism-design part of this series: the VCG mission that follows exists because Gibbard–Satterthwaite closes the door on non-trivial strategyproof choice without money. The chapter derives Gibbard–Satterthwaite from Arrow through the top-set extension (Definition 9.9, Lemmas 9.10–9.11), so a solver of the capstone gets Arrow as a stepping stone, not a detour.

Formalizing them produces the platform's first social-choice library: preference profiles as strict total orders, the aggregation vocabulary, and the impossibility pair. Arrow's theorem has been formalized before in other proof assistants (Nipkow's Isabelle formalization, 2009; a Mizar formalization by Wiedijk), which is evidence the statement shapes here are the standard ones — but no Lean 4/mathlib formalization exists, and none on this platform.

Difficulty

The proofs are short on paper and famously slippery in the details. Arrow's proof (the book gives Geanakoplos's pairwise-neutrality route) is a sequence of profile surgeries: each step swaps one voter's ranking of a pair and tracks the social outcome through IIA; the formal cost is constructing the intermediate profiles and proving they remain strict total orders — pure bookkeeping, but a lot of it. The hybrid-profile argument needs, over three or more alternatives, custom orders placing chosen pairs at chosen positions; building these on an abstract finite type is where most of the work lies. For the capstone, the book's route through the top-set extension $\prec^S$ (move $S$ to the top, Definition 9.9) requires proving the extension is again a strict total order (Lemma 9.10) and inherits unanimity, IIA, and non-dictatorship (Lemma 9.11); a solver may equally take any direct proof of Gibbard–Satterthwaite — the statement fixes no route. Proposition 9.6 is a genuine warm-up: unfolding both definitions and rearranging quantifiers.

Formalization scope

Preferences are relations A → A → Prop carrying IsStrictTotalOrder; r a b means " $a$ is strictly preferred to $b$ ", the reverse of the book's $\prec$ — every definition's docstring states this orientation. Aggregators are total functions on all relation-valued profiles; every property quantifies only over genuine preference profiles, so behavior on invalid inputs is irrelevant, and nothing can be smuggled through junk inputs. Unanimity is the book's identical-profile form (Definition 9.2), which together with IIA yields the pairwise form used in proofs. Both alternatives and voters are finite types; |A| ≥ 3 enters as 2 < Fintype.card A. Arrow's theorem carries Nonempty ι, matching the book's setting of $n \ge 1$ voters; it is not needed for truth — with zero voters the identical-profile unanimity is already unsatisfiable over three or more alternatives, so that case is vacuous either way. Gibbard–Satterthwaite deliberately omits it: with zero voters ontoness onto three alternatives is unsatisfiable, and the statement holds vacuously. Dictatorship for choice functions is Definition 9.7 exactly: whenever some alternative is the dictator's unique maximum, it is elected.

Selected references

K. J. Arrow, Social Choice and Individual Values, Wiley, 1951 (2nd ed. 1963). Link
A. Gibbard, Manipulation of voting schemes: a general result, Econometrica 41 (1973), 587–601. DOI
M. A. Satterthwaite, Strategy-proofness and Arrow's conditions, Journal of Economic Theory 10 (1975), 187–217. DOI
J. Geanakoplos, Three brief proofs of Arrow's impossibility theorem, Economic Theory 26 (2005), 211–215. DOI
T. Nipkow, Social choice theory in HOL: Arrow and Gibbard–Satterthwaite, J. Automated Reasoning 43 (2009), 289–304. DOI
N. Nisan, T. Roughgarden, É. Tardos, V. V. Vazirani (eds.), Algorithmic Game Theory, Cambridge University Press, 2007, Chapter 9, §9.2. DOI

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Algorithmic Game Theory Theoretical Computer Science·Captain: Shuze Chen

Algorithmic Game Theory II: No-Regret Learning and Correlated EquilibriaTextbook

Motivation

Equilibrium concepts are static; play is dynamic. The bridge between the two is regret minimization: simple adaptive rules that, against arbitrary — even adversarial — opponents, perform nearly as well as the best fixed alternative in hindsight. The subject begins with Hannan (1957) and Blackwell (1956), whose consistency theorems predate most of computational learning theory; the modern multiplicative-weights style bounds are due to Littlestone–Warmuth (1994) and Freund–Schapire (1997); the reduction from external to swap regret, and with it the algorithmic route to correlated equilibria, is Blum–Mansour (2005), following Foster–Vohra (1997) and Hart–Mas-Colell (2000). Chapter 4 of Nisan–Roughgarden–Tardos–Vazirani (eds.), Algorithmic Game Theory (Cambridge, 2007), written by Blum and Mansour, is the source text of this mission.

The punchline of the chapter, and of this mission, is that a computationally trivial form of rationality — each player privately running a no-swap-regret algorithm — drives the empirical play of any finite game into an approximate correlated equilibrium (Aumann, 1974). No coordination, no knowledge of the game, no fixed-point computation: the equilibrium concept that Chapter 1 of the series defines through a correlating device is reached by decentralized learning.

Setting

The online model (§4.2): there are $N$ actions. At each time $t$ an online algorithm selects a distribution $p^t$ over actions, as a function of the loss vectors observed so far; then the adversary reveals a loss vector $\ell^t \in [0,1]^N$ and the algorithm suffers $\sum_i p^t_i \ell^t_i$ . Cumulatively, $L_H^T = \sum_{t \le T} \sum_i p^t_i \ell^t_i$ and $L_k^T = \sum_{t \le T} \ell^t_k$ for a fixed action $k$ . The external regret of $H$ is $L_H^T - \min_k L_k^T$ . A modification rule $F : \{1,\dots,N\} \to \{1,\dots,N\}$ rewires the algorithm's play, giving the modified loss $L_{H,F}^T = \sum_t \sum_i p^t_i\, \ell^t_{F(i)}$ ; the swap regret is $L_H^T - \min_F L_{H,F}^T$ over all $N^N$ rules.

A finite game (the vocabulary of Mission I of this series, in loss form): players $\iota$ , finite action sets $S_i$ , cost functions $c_i : \prod_j S_j \to \mathbb{R}$ . A joint distribution $Q$ on action vectors is an $\varepsilon$ -correlated equilibrium (Definition 4.11) if for every player $i$ and every switching rule $F : S_i \to S_i$ ,

\mathbb{E}_{s \sim Q}\big[c_i(s)\big] \;\le\; \mathbb{E}_{s \sim Q}\big[c_i(F(s_i), s_{-i})\big] + \varepsilon.

Formalization targets

Goal (capstone) — Corollary 4.16, explicit form

\forall N, T\ \exists H:\quad \text{swap regret of } H \text{ on every } [0,1]\text{-loss sequence} \;\le\; 2N\sqrt{T \ln N}.

An online algorithm with vanishing per-round swap regret, with the constant the chapter's own route produces.

Theorem 4.6 — Polynomial Weights

L_{PW}^T \;\le\; L_k^T + \eta\, Q_k^T + \frac{\ln N}{\eta}, \qquad Q_k^T = \sum_{t\le T} (\ell^t_k)^2, \quad 0 < \eta \le \tfrac12.

Theorem 4.9 — external regret in constant-sum games

A player with external regret $R$ over $T$ rounds has average loss at most $v_i + R/T$ , where $v_i$ is the game value — no-regret play guarantees the minimax value against any opponent.

Theorem 4.15 — external-to-swap reduction

Any algorithm with external regret $\le R$ on all $[0,1]$ -loss sequences yields one with swap regret $\le N R$ .

Theorem 4.12 — swap regret bounds distance from correlated equilibrium

If every player's swap regret over $T$ steps of mixed play is at most $R$ , the empirical joint distribution is an $(R/T)$ -correlated equilibrium.

Theorem 4.3 — deterministic algorithms fail

Every deterministic algorithm has a $\{0,1\}$ -loss sequence forcing loss $T$ while some action loses at most $\lfloor T/N \rfloor$ : randomization is necessary, not a convenience.

Significance

Correlated equilibrium is the equilibrium concept with a defensible dynamic foundation: Nash equilibria are PPAD-hard to find, but the capstone plus Theorem 4.12 exhibit polynomial-time decentralized dynamics whose empirical play is an $\varepsilon$ -correlated equilibrium after $T = O(N^2 \ln N / \varepsilon^2)$ rounds. Later missions in this series lean on this machinery: the price-of-anarchy chapters bound the cost of no-regret play (not just of exact equilibria), and the routing-game chapter uses precisely the convergence result formalized here.

Formalizing it produces the platform's first online-learning library: the adversarial protocol, regret in both external and swap forms, the multiplicative-weights analysis, and correlated equilibria. The regret vocabulary is directly reusable for the bandit-flavored missions already on the platform. All results are classical, with textbook proofs; the work requested is machine-checked proof, not new mathematics.

Difficulty

The Polynomial Weights bound is a potential-function argument: the total weight $W^t$ falls geometrically with the algorithm's loss and is bounded below by the weight of action $k$ ; the formal work is inequalities for $\ln(1-x)$ on $[0, 1/2]$ and careful bookkeeping of the recursion. The reduction (Theorem 4.15) is the structurally interesting step: the master algorithm runs $N$ copies of the external-regret procedure, feeds copy $i$ the true losses scaled by the master's own probability $p^t_i$ , and — the crux — plays the stationary distribution $p^t = p^t Q^t$ of the column-stochastic matrix assembled from the copies' outputs. Existence of that fixed point is exactly the existence of a stationary distribution of a finite Markov chain, available on this platform as the goal of Markov Chains and Mixing Times I — or provable directly. Theorem 4.12 is an averaging argument, deliberately easy; Theorem 4.3 is an adversary construction; Theorem 4.9 combines the regret bound with the security level the minimax theorem of Mission I supplies, through the opponent's empirical mixture. The capstone is the composition of 4.6 (tuned at $\eta = \min\{\sqrt{\ln N / T}, 1/2\}$ ) with 4.15, plus the arithmetic that turns $N \cdot 2\sqrt{T \ln N}$ into the stated bound.

Formalization scope

An online algorithm is a deterministic function from the observed history (the list of past loss vectors) to the mixed action played next — the standard formal reading of the full-information model; randomization lives in the mixed action, and losses are expected losses. Boundedness of losses ( $[0,1]$ ) is a hypothesis on theorems, never part of a definition. The Polynomial Weights algorithm is defined concretely by its weight recursion, and its learning rate carries the hypothesis $0 < \eta \le 1/2$ : the book writes only $\eta \le 1/2$ , but at $\eta = 0$ the bound's $\ln N / \eta$ term degenerates and the claim is false, so positivity is explicit. Action sets are Fin (n+1), keeping them nonempty. The number of steps $T$ is a known parameter (the book's convention; guess-and-double is out of scope). Correlated equilibria use the switching-rule form of Definition 4.11, over the game vocabulary (IsLottery, IsMixedProfile, profileProb) published with Mission I of this series. In Theorem 4.12 the empirical distribution is the average of product distributions of the played profiles, $T \ge 1$ is required (at $T = 0$ there is no empirical distribution), and costs are not assumed bounded — the averaging is scale-free.

Trivializing readings are ruled out: the existential algorithms in Theorem 4.15 and the capstone are quantified before the loss sequence and the modification rule, so a witness must work uniformly against every adversary — nothing may be chosen with hindsight.

Selected references

A. Blum, Y. Mansour, From external to internal regret, JMLR 8 (2007), 1307–1324. Link
N. Littlestone, M. K. Warmuth, The weighted majority algorithm, Information and Computation 108 (1994), 212–261. DOI
D. P. Foster, R. V. Vohra, Calibrated learning and correlated equilibrium, Games and Economic Behavior 21 (1997), 40–55. DOI
S. Hart, A. Mas-Colell, A simple adaptive procedure leading to correlated equilibrium, Econometrica 68 (2000), 1127–1150. DOI
N. Nisan, T. Roughgarden, É. Tardos, V. V. Vazirani (eds.), Algorithmic Game Theory, Cambridge University Press, 2007, Chapter 4. DOI

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Combinatorics Graph Theory Probability·Captain: burkh4rt

The Bunkbed Conjecture is FalseResearch Paper

Motivation

Let $G=(V,E)$ be a finite connected graph. In Bernoulli bond percolation each edge is independently retained with probability $P$ and deleted otherwise, and one writes $\mathbb{P}_P[u \leftrightarrow v]$ for the probability that vertices $u$ and $v$ lie in the same component of the resulting random subgraph. Comparing such connection probabilities is a basic and genuinely hard problem: computing them exactly is $\#\mathsf{P}$ -hard.

The bunkbed graph is built from two copies of $G$ , joined by vertical edges called posts above a chosen set $T \subseteq V$ of transversal vertices. Percolation is performed on the two copies while every post is retained. Writing $v$ for a vertex in the lower copy and $v'$ for its counterpart upstairs, Kasteleyn conjectured in 1985 that being connected within a level is always at least as likely as crossing between levels.

The conjecture is intuitively compelling — crossing levels appears to require "using up" a post — and it resisted proof for forty years. A short timeline:

1985 — Kasteleyn formulates the conjecture; it is recorded as Remark 5 of van den Berg–Kahn (2001), which is how the source cites it.
Positive results accumulate for special cases: wheels, complete graphs, complete bipartite graphs, graphs symmetric with respect to an automorphism exchanging $u$ and $v$ , one or two transversal vertices, and in the $P \uparrow 1$ limit.
2024 — Hollom refutes the $3$ -uniform hypergraph analogue. This alone does not settle the graph case: it is impossible to simulate a single $3$ -hyperedge by bond percolation on a gadget graph.
2025 — Gladkov, Pak and Zimin disprove the conjecture outright, with an explicit counterexample and without computer assistance.

Section 7 of the source is a candid account of a large-scale machine-learning-guided search that failed to find a counterexample, and of why the problem is unusually ill-suited to experimental testing.

Setting

Fix a finite graph with vertex set $V$ and edge set $E$ , and a retention function $w : E \to [0,1]$ (the uniform case is $w \equiv P$ ). A configuration is a subset $S \subseteq E$ of open edges, occurring with probability

\mathbb{P}(S) \;=\; \prod_{e \in S} w(e) \prod_{e \in E \setminus S} \bigl(1 - w(e)\bigr),

and $\mathbb{P}[u \leftrightarrow v]$ is the total probability of those $S$ for which $u$ and $v$ are connected in $(V, S)$ .

Given $T \subseteq V$ , the bunkbed graph has vertex set $V \times \{0,1\}$ . Its edges are a copy of $E$ in each level together with a post $\{(t,0),(t,1)\}$ for every $t \in T$ . In bunkbed percolation the two level-copies are percolated independently while all posts are retained; $\mathbb{P}^{\mathrm{bb}}$ denotes the resulting connection probabilities.

Formalization targets

Goal — the bunkbed conjecture is false

\neg\;\Bigl(\forall\,G \text{ connected},\ \forall\,T \subseteq V,\ \forall\,0<P<1,\ \forall\,u,v \in V:\quad \mathbb{P}^{\mathrm{bb}}_P[u \leftrightarrow v] \;\ge\; \mathbb{P}^{\mathrm{bb}}_P[u \leftrightarrow v'] \Bigr)

Supporting target — the explicit counterexample (Theorem 1.2)

\exists\, G,\ |V| = 7{,}222,\ |E| = 14{,}442,\ |T| = 3,\ \exists\, u,v:\qquad \mathbb{P}^{\mathrm{bb}}_{1/2}[u \leftrightarrow v] \;<\; \mathbb{P}^{\mathrm{bb}}_{1/2}[u \leftrightarrow v']

Supporting target — hyperedge simulation (Lemma 4.1)

For the gadget $G_n$ on $n+1$ vertices,

P_{abc}\,P_{a|b|c} \;-\; P_{ab|c}\,P_{ac|b} \;>\; \Bigl(n\tfrac{1-P}{1+P} - 1\Bigr) P_{a|bc}.

Significance

The result itself. A forty-year-old conjecture in percolation theory is false, and prior positive results are thereby sharpened rather than superseded: it becomes interesting to delimit exactly which families of graphs do satisfy the inequality. The refutation also settles the Counting, Weighted, Alternative and Computational variants listed in §8.1, and shows the random-cluster analogue cannot be pushed from $q=2$ down to $q=1$ .

Formalizing it. Nothing here is open; the mission produces machine-checked versions of published results, and as a by-product the first percolation theory in Lean. Mathlib currently contains no percolation of any kind — no connection probabilities, no bunkbed graph, no hypergraph percolation. That infrastructure is reusable far beyond this mission. The source itself notes (§8.2) that its central combinatorial lemma was independently verified by computer; a formal proof would replace that check with a certificate.

Difficulty

The obvious approach — exhibit a small graph and compute both probabilities — is hopeless, and the source explains why at length. A graph with $m$ edges has $2^m$ configurations; for the counterexample here the probability gap is on the order of $10^{-4331}$ , so no sampling argument can detect it, and exact enumeration is out of reach. Section 7 records a substantial computational search that found nothing and, in hindsight, could not have.

The proof is instead structural, and its difficulty is concentrated in one place. Hollom's refutation of the hypergraph version cannot be transferred directly, because a single $3$ -hyperedge cannot be simulated by bond percolation on any gadget graph. The source's answer is to prove a robust version of Hollom's lemma (Lemma 3.3) which survives the inexact simulation that gadget graphs do provide, and this robustness is what Lemma 4.1's inequality quantifies. Lemma 3.3 is proved by constructing a weight-preserving involution on a refined configuration space — the technical heart, and the milestone a solver should expect to spend the most effort on.

Formalization scope

The development commits to the following conventions.

Everything is finite and rational-valued, hence computable: connection probabilities are ℚ and evaluate by #eval, and small instances close by decide.
A graph is given by an explicit edge Finset and realised through SimpleGraph.fromEdgeSet; connectivity is Mathlib's SimpleGraph.Reachable.
Percolation is a sum over the powerset of the edge set, weighted as displayed above, of a reachability indicator. Edge weights are per-edge (Sym2 V → ℚ), since the gadget $G_n$ genuinely needs two different weights: its spokes are retained with probability $1-P$ and its path edges with probability $P$ .
In the bunkbed, level 0 is the lower copy; posts over T are unconditionally present and are not percolated. The two levels are percolated independently.
⚠️ Planarity is omitted from the goal. Theorem 1.2 asserts the counterexample is planar, and Mathlib has no notion of a planar graph — no IsPlanar, no Euler formula, no Kuratowski. Building one is a larger project than this mission. The formalized statement of Theorem 1.2 is therefore strictly weaker than the published one, and the goal is instead the negation of the conjecture, which is exactly the source's own "In particular, the BBC is false." Contributions adding planarity are welcome and would strengthen the milestone.
Ruling out a trivializing reading: the conjecture must be negated as stated, over all connected graphs, transversal sets and $0<P<1$ . Weakening it to a fixed graph, or to $P \in \{0,1\}$ , or dropping connectivity, would make the refutation vacuous.

Infrastructure. Mathlib supplies SimpleGraph, boxProd, Reachable with a DecidableRel instance, fromEdgeSet, edgeFinset and Finset.powerset. It supplies no percolation, so this mission ships two definition files: Bernoulli bond percolation with the bunkbed construction and the five triple-partition probabilities, and hypergraph percolation with Hollom's hypergraph and the Wierman–Ziff five-state model. One known gap: Mathlib's Reachable decision procedure enumerates walks and is far too slow to evaluate the $64$ -configuration check of Lemma 3.1 by decide. A solver will want a linear-time reachability procedure together with a proof that it agrees with Reachable; that is itself a worthwhile reusable contribution.

Selected references

J. van den Berg and J. Kahn, A correlation inequality for connection events in percolation, Ann. Probab. 29 (2001), 123–126 — Kasteleyn's conjecture appears as Remark 5.
T. Hollom, A new proof of the bunkbed conjecture in the $p \uparrow 1$ limit, Discrete Math. 347 (2024), 113711.
T. Hollom, The bunkbed conjecture is not robust to generalisation, arXiv:2406.01790 (2024).
T. Hutchcroft, P. Nizić-Nikolac, A. Kent, The bunkbed conjecture holds in the $p \uparrow 1$ limit, Comb. Probab. Comput. 32 (2023), 363–369.
N. Gladkov, I. Pak, A. Zimin, The bunkbed conjecture is false, Proc. Natl. Acad. Sci. USA 122 (2025), no. 24, e2420725122. doi:10.1073/pnas.2420725122; preprint arXiv:2410.02545.
J. C. Wierman and R. M. Ziff, Self-dual planar hypergraphs and exact bond percolation thresholds, Electron. J. Combin. 18 (2011).
G. R. Grimmett, Percolation, 2nd ed., Springer, 1999.

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Algebraic Topology Quantum Error Correction Quantum Information·Captain: Rui Chao

Counterexamples to the Zeng–Pryadko Homological-Distance ConjectureResearch Paper

Background and main question

The tensor product of chain complexes is a basic construction in homological algebra and in the theory of quantum CSS codes. If $A$ and $B$ are finite based chain complexes over a finite field, their tensor product is graded by total degree,

(A\otimes B)_j=\bigoplus_{i=0}^{j} A_i\otimes B_{j-i}.

Each complex carries a basis-dependent homological distance: $d_j(A)$ is the least Hamming weight of a degree- $j$ cycle that is not a boundary, with $d_j(A)=\infty$ when the degree- $j$ homology vanishes. A natural candidate for the distance of the tensor product is therefore

m_j(A,B)=\min_{0\le i\le j} d_i(A)d_{j-i}(B).

Every nonzero pair of homology classes in complementary degrees gives a pure-tensor class of the corresponding product weight, so one always has $d_j(A\otimes B)\le m_j(A,B)$ . The substantive question is whether this upper bound is always sharp.

In the preprint Higher-Dimensional Quantum Hypergraph-Product Codes with Finite Rates, posted in 2018 and subsequently published in Physical Review Letters, Zeng and Pryadko proved that the bound is sharp when one factor is a binary one-complex. Their exact formula is Eq. (13) of the arXiv version and is the capstone theorem of the Prove2Me mission Higher-Dimensional Quantum Hypergraph-Product Codes with Finite Rates. The present formalization is a direct sequel: it retains the same definitions and degree conventions while examining what happens when the one-complex restriction is removed.

Zeng and Pryadko later considered arbitrary finite chain complexes over arbitrary finite fields in Minimal distances for certain quantum product codes and tensor products of chain complexes, published in 2020. In the corresponding arXiv preprint, Conjecture 18 asserts the unrestricted equality

d_j(A\otimes B)=m_j(A,B).

The conjecture is appealing because it would make tensor-product distance completely compositional: the degreewise distances of the two factors would determine the distance of their product. The obstruction is that a homology class in $A\otimes B$ need not have a minimum-weight representative supported in a single bidegree. A representative spread across several summands of the total complex can be lighter than every pure-tensor representative. The purpose of this formalization is to turn that observation into a concrete, machine-checked counterexample to Conjecture 18 while preserving the valid one-complex theorem as a sharply delimited special case.

The counterexample mechanism

The common foundation is recorded in the Prove2Me entry Based binary chain complexes and homological distance. In particular, the boundary in degree $j$ is a map $\partial_j:A_j\to A_{j-1}$ , and

d_j(A)=\inf\{\operatorname{wt}(x):x\in\ker\partial_j,\ x\notin\operatorname{im}\partial_{j+1}\}.

The one-complex result is separately available as Eq. (13) — Exact distance with a one-complex. The construction below uses exactly the same notion of distance, but both tensor factors are genuine three-term complexes.

Begin with binary CSS check maps

H_X:\mathbb F_2^n\longrightarrow\mathbb F_2^{r_X}, \qquad H_Z:\mathbb F_2^n\longrightarrow\mathbb F_2^{r_Z},

assumed surjective and satisfying $H_XH_Z^T=H_ZH_X^T=0$ . Suppose there are logical vectors $x,z\in\mathbb F_2^n$ such that

H_Zx=0,\qquad H_Xz=0,\qquad x\cdot z=1.

The check maps determine two dual three-term complexes

A:\quad \mathbb F_2^{r_X}\xleftarrow{H_X}\mathbb F_2^n \xleftarrow{H_Z^T}\mathbb F_2^{r_Z}, \qquad B:\quad \mathbb F_2^{r_Z}\xleftarrow{H_Z}\mathbb F_2^n \xleftarrow{H_X^T}\mathbb F_2^{r_X}.

Their degree-two tensor space has three bidegree summands, corresponding to $(2,0)$ , $(1,1)$ , and $(0,2)$ . Under the natural matrix identifications, consider the element whose three blocks are

(I_{r_Z},I_n,I_{r_X}).

The CSS orthogonality relations make this element a cycle. Its pairing with the chosen logical vectors certifies that it is not a boundary. Its Hamming weight is exactly $r_Z+n+r_X$ , whereas the componentwise candidate in degree two reduces to

m_2(A,B)=d_1(A)d_1(B).

Consequently, any CSS datum satisfying

r_Z+n+r_X<d_1(A)d_1(B)

produces the strict inequality $d_2(A\otimes B)<m_2(A,B)$ . For orientation, a binary quantum Golay CSS presentation with parameters $[[23,1,7]]$ has $r_X=r_Z=11$ , giving the numerical comparison $45<49$ . The formal proof must supply an explicit CSS instance and verify its algebraic and distance properties, rather than relying on the parameter notation alone.

Formalization objectives

The first milestone proves the general certificate: for every CSS datum satisfying the hypotheses above, the element $(I_{r_Z},I_n,I_{r_X})$ is a nontrivial degree-two cycle of weight $r_Z+n+r_X$ , and the componentwise minimum is $d_1(A)d_1(B)$ .

The second milestone constructs and verifies one explicit CSS datum for which $r_Z+n+r_X<d_1(A)d_1(B)$ . This is the step that turns the general mechanism into an actual counterexample.

The capstone packages the construction as the direct existential statement

\exists\,A,B\qquad d_2(A\otimes B)<\min_{0\le i\le 2}d_i(A)d_{2-i}(B).

Thus the final theorem is not conditional on the existence of suitable code data: it exhibits finite based binary chain complexes for which the equality proposed in Conjecture 18 fails.

Relation to prior work

The counterexample concerns only the unrestricted passage from a one-complex factor to two arbitrary bounded complexes. It does not conflict with Zeng and Pryadko's Eq. (13), whose one-complex hypothesis rules out the three-bidegree interaction used here.

The broader literature also indicates why additional structure matters. Bravyi and Hastings introduced homological-product codes and analyzed logical representatives in product constructions; Audoux and Couvreur developed tensor products of CSS codes through chain-complex methods. More recently, Akhmechet et al. discussed the Zeng–Pryadko conjecture in the structured setting of complexes derived from Khovanov homology, while Berthusen et al. restated it as Conjecture 5.1 in their study of automorphism gadgets. Golowich and Guruswami obtained strong distance guarantees for iterated homological products under expansion and local-testability hypotheses. These results are compatible with the proposed counterexample: they concern special families or impose hypotheses that are absent from Conjecture 18.

Besides settling the unrestricted statement, the formalization isolates a reusable obstruction. It shows precisely how a low-weight class assembled across several bidegrees can evade a formula based only on the degreewise distances of the factors. This distinction should help guide corrected formulations in which an exact product formula, or a useful lower bound, is recovered from additional geometric, expansion, or local-testability assumptions.

The main formal issues are mathematically substantive rather than presentational: the proof must respect the endpoint conventions for the boundary maps, distinguish a cycle from a non-boundary, compare finite weights with $\infty$ -valued distances, and verify a concrete strict-gap instance. Making each of these points explicit is especially important here, because an indexing shift or a merely conditional existence statement would no longer constitute a refutation of the conjecture as stated.

References

W. Zeng and L. P. Pryadko, Higher-Dimensional Quantum Hypergraph-Product Codes with Finite Rates, Physical Review Letters 122, 230501 (2019); arXiv:1810.01519 (2018), Eq. (13).
W. Zeng and L. P. Pryadko, Minimal distances for certain quantum product codes and tensor products of chain complexes, Physical Review A 102, 062402 (2020); arXiv:2007.12152, Conjecture 18.
S. Bravyi and M. B. Hastings, Homological Product Codes, STOC 2014; arXiv:1311.0885.
B. Audoux and A. Couvreur, On tensor products of CSS codes, Annales de l'Institut Henri Poincaré D 6 (2019); arXiv:1512.07081.
R. Akhmechet et al., Khovanov homology and quantum error-correcting codes, arXiv:2410.11252 (2024).
N. Berthusen et al., Automorphism gadgets in homological product codes, arXiv:2508.04794 (2025).
L. Golowich and V. Guruswami, Quantum LDPC Codes of Almost Linear Distance via Iterated Homological Products, CCC 2025; full version.

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Complexity Theory Quantum Information·Captain: Goku

Shallow Quantum Circuits and Causal ConesTextbook

Motivation

A quantum circuit of depth $d$ built from gates of fan-in at most two cannot let an output wire depend on more than $2^d$ input wires. The argument is folklore and takes a paragraph on paper: the causal cone of the measured wire grows by at most a factor of two per layer. Formalizing it exposes a subtlety that the paper argument hides, and that is what this mission is about.

The subtlety

Define the backward cone step of a wire set $S$ through a layer $l$ by adjoining the support of every gate of $l$ that meets $S$ . There is a choice here: test each gate against the incoming set $S$ , or against the partially accumulated cone. Testing against the accumulator over-approximates, and the doubling bound fails. Testing against the incoming set gives the bound — but is only correct when the gates within a layer act on pairwise disjoint wires.

Without that hypothesis (LayerOk) the semantic statement is false, and the counterexample is small: on three wires, the single layer [cnot 2 1, cnot 1 0] has cone $\{0,1\}$ around wire $0$ , yet wire $0$ ends up holding $x_0 \oplus x_1 \oplus x_2$ . So the cone under-approximates the true dependence. This mission's development carries LayerOk throughout, and the counterexample is recorded in the source.

What is formalized

Layered circuits over $\{H, S, T, \mathrm{CNOT}\}$ on $n$ wires, with states as amplitude functions on bit-strings and no tensor products anywhere. On top of that:

the combinatorial half — one layer at most doubles the cone, hence $|\mathrm{cone}| \le 2^{d}\,|S|$ ;
norm preservation, so that acceptProb is a genuine probability in $[0,1]$ ;
the semantic half — inputs agreeing on the causal cone of the output wire are accepted with equal probability.

The semantic half is proved in the Heisenberg picture. The measurement observable is conjugated backwards through the circuit and its support tracked: a gate meeting the support enlarges it by that gate's own wires, and a gate missing it commutes with the observable and cancels against its own adjoint. That cancellation is the reason the non-cascading cone step is correct, and it is why unitarity of the gate set is needed at the $2^n$ -dimensional level rather than gate by gate. Supporting this is a small reusable algebra of local operators: locality is monotone, closed under adjoint and product, and disjointly supported operators commute.

The frontier

The published depth bound assumes each input wire lies in the syntactic cone of the output. That is weaker than saying the wire matters. Milestone 1 asks for the semantically honest version, stated in terms of genuine functional dependence; the bridge is the semantic cone theorem already in the development.

Beyond that, the natural continuations are the same argument for fan-in- $k$ gates ( $|\mathrm{cone}| \le k^{d}$ ), for geometrically local circuits where cone growth is linear rather than exponential, and ultimately the Bravyi–Gosset–König separation $\mathrm{QNC}^{0} \not\subset \mathrm{NC}^{0}$ — which needs machinery (non-local games, magic squares) that this development deliberately does not build.

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Dynamic Programming Operations Research Optimization·Captain: Shuze Chen

Dynamic Programming and Optimal Control VI: Lookahead and RolloutTextbook

Motivation

When exact dynamic programming is intractable, practice runs on approximations: one-step and multistep lookahead with a cost-to-go surrogate, open-loop feedback control, and rollout — the algorithm that improved backgammon programs and became a conceptual ancestor of Monte-Carlo tree search and modern policy improvement schemes. Chapter 6 of Bertsekas, Dynamic Programming and Optimal Control, Vol. I (3rd ed., 2005) gives the basic guarantees: performance bounds for limited lookahead (Props. 6.3.1–6.3.2), superiority of open-loop feedback control over open-loop control (Prop. 6.2.1), and the cost-improvement theory of rollout on discrete deterministic problems (Props. 6.4.1–6.4.3). These are the theorems that make "approximate DP" more than a heuristic.

Setting

Two frameworks. For the stochastic bounds (§6.2–6.3): the basic finite-horizon model of Mission I of this series (BertsekasDPModel), its policy cost recursion, and the open-loop cost of a fixed control sequence (BertsekasDPOpenLoopCost). For rollout (§6.4.1): a graph search problem — a finite digraph with destination set and terminal costs $g(i)$ on destinations (BertsekasGraphSearch); a base heuristic $\mathcal{H}$ producing from every node a path to a destination (BertsekasBaseHeuristic), with projection $p(i)$ and heuristic cost $H(i) = g(p(i))$ ; the rollout algorithm $R\mathcal{H}$ repeatedly moves to a neighbor $j$ minimizing $H(j)$ (BertsekasIsRolloutRun). $\mathcal{H}$ is sequentially consistent if its paths have the tail property (Def. 6.4.1), sequentially improving if $\min_{j \in N(i)} H(j) \le H(i)$ (Def. 6.4.2).

Target

For sequentially improving $\mathcal{H}$ and any terminating rollout run $(i_1, \dots, i_{\bar m})$ :

g(i_{\bar m}) \;\le\; H(i_1), \qquad g(i_{\bar m}) \;=\; \min\Big\{ H(i_1),\ \min_{j \in N(i_1)} H(j),\ \dots,\ \min_{j \in N(i_{\bar m - 1})} H(j) \Big\},

— BertsekasDP.rollout_sequential_improvement (goal, Prop. 6.4.2). Milestones: Props. 6.4.1 (termination under sequential consistency with the book's tie-breaking), 6.4.3 (exact cost identity via the defects $\delta_i$ ), 6.3.1, 6.3.2 (lookahead bounds), 6.2.1 (OLFC).

Significance

Prop. 6.4.2 is the "rollout never hurts" theorem — the formal warrant for policy improvement by simulation, with Prop. 6.3.1 its stochastic counterpart (via Example 6.3.1 the rollout of any policy improves that policy). Prop. 6.3.2 is the robustness version that quantifies the cost of inexact minimization, used for CEC bounds. Formalizing the chapter yields a reusable graph-search + base-heuristic vocabulary and connects it to the Mission I stochastic model. Everything here is proved in the book; the formal versions are new.

Difficulty

The rollout proofs are elementary but exact: the min formula (6.37) requires tracking the running minimum along the run, and the IsLeast membership half forces identifying which neighbor value is attained. Termination under sequential consistency (6.4.1) is the delicate one — it fails without the tie-breaking convention (the book gives a cycling counterexample), so the formal statement carries the convention explicitly and the proof must extract a termination measure from "strict decreases are finitely many, plateaus shorten the heuristic path". The stochastic bounds are clean backward inductions over the Mission I recursion.

Formalization scope

Graph search: finite node type, arcs as ordered pairs, vertex costs only (no arc costs — the book's reduction absorbs them into destination costs); heuristic paths as lists; rollout runs as lists (finite, complete runs) except 6.4.1, where the run is an infinite sequence absorbed at destinations so that termination is a genuine claim. Ties in neighbor selection are allowed everywhere except where 6.4.1's convention pins them. Stochastic side: state-independent constraint sets for OLFC (as in §6.2); restricted lookahead sets $\bar U_k(x) \subseteq U_k(x)$ per Eq. (6.19); all statements at the level of the Mission I model.

Selected references

D. P. Bertsekas, Dynamic Programming and Optimal Control, Vol. I, 3rd ed., Athena Scientific, 2005. (§6.2–6.4.) http://www.athenasc.com/dpbook.html
G. Tesauro, G. R. Galperin, On-line policy improvement using Monte-Carlo search, NIPS 1996. https://papers.nips.cc/paper/1302
D. P. Bertsekas, J. N. Tsitsiklis, C. Wu, Rollout algorithms for combinatorial optimization, J. Heuristics 3 (1997), 245–262. https://doi.org/10.1023/A:1009635226865

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Machine Learning Probability·Captain: naimengye

Speculative Actions: Cost-Latency Analysis for Agentic SpeculationResearch Paper

Motivation

An LLM agent acting in an environment spends most of its wall-clock time waiting. Each step — a model call, a tool or MCP request, a browser action, sometimes a human reply — must complete before the next can be issued, and the round trips dominate end-to-end latency: a chess game between two reasoning agents runs for hours, and an operating-system tuning task for tens of minutes. When a training or prompt-optimization loop repeats such a run thousands of times, the waiting is the cost.

Speculative actions (Ye, Ahuja, Liargkovas, Lu, Kaffes, Peng, ICLR 2026) transplants a classical systems idea — speculative execution in microprocessors, and speculative decoding for LLM inference — to the agent's environment loop. A cheap, fast speculator guesses the action a slow, authoritative actor is about to produce, the guess is used to launch the next environment call early, and the work is committed only when the actor's real action confirms the guess. The interface stays sequential and lossless; the internals run in parallel.

What makes this a formalization target rather than an engineering report is the paper's §5 cost–latency analysis. Speculating more branches buys hit probability but costs tokens, and the paper derives closed-form expressions for both sides of that trade — a self-contained piece of applied probability sitting underneath a systems paper. This mission asks for those expressions, machine-checked.

Setting

Fix a horizon $T$ and index steps $t = 0, 1, \dots, T-1$ . At each step a policy maps the state to an API call; the actor executes it with latency $\mathrm{Exp}(\beta)$ , while the speculator proposes candidate actions with latency $\mathrm{Exp}(\alpha)$ , where $\beta < \alpha$ (the speculator is faster in expectation). A speculative branch hits when the action it guesses implies the same next call the actor's true action would have implied; branches hit independently across steps with probability $p$ .

Two knobs define the two regimes analyzed. Breadth $k$ : at each step, launch $k$ independent one-step speculations in parallel, each immediately followed by a real call. At least one of the $k$ succeeds with probability

p(k) \;=\; 1 - (1-p)^k .

Depth: follow a single branch, extending it whenever a speculative or real call returns and pruning subtrees the actor contradicts.

The quantity driving both results is $S_n$ , the expected number of hits by round $n$ . A hit consumes the following step's speculation window — after a correct guess the next call is already cached, so no new speculation is launched there — which yields the two-term recursion

S_0 = 0, \qquad S_1 = p, \qquad S_n = p\,(1 + S_{n-2}) + (1-p)\,S_{n-1}.

Write $T_{\mathrm{seq}}, M_{\mathrm{seq}}$ for the latency and token cost of strictly sequential execution, and $T_{\mathrm{spec}}, M_{\mathrm{spec}}$ for their speculative counterparts. In the depth regime latencies are taken deterministic: $a$ for a real call, $b < a$ for a speculative one.

Target

The goal theorem is the finite-horizon latency ratio for breadth-focused speculation (Proposition 1), with $p(k)$ abbreviated $p_k$ :

\frac{\mathbb{E}[T_{\mathrm{spec}}]}{\mathbb{E}[T_{\mathrm{seq}}]} = 1 - \frac{1}{T}\,\frac{\alpha}{\alpha+\beta} \left[\frac{(T-1)p_k}{1+p_k} + \frac{p_k^2}{(1+p_k)^2} - \frac{p_k^2}{(1+p_k)^2}(-p_k)^{T-1}\right].

The supporting targets, ordered as the analysis builds them:

the closed form $S_n = \frac{p}{1+p}n + \frac{p^2}{(1+p)^2}\bigl(1 - (-p)^n\bigr)$ solving the recursion;
the per-hit saving $\mathbb{E}[(B-A)^+] = \frac{\alpha}{\beta(\alpha+\beta)}$ for independent $A \sim \mathrm{Exp}(\alpha)$ , $B \sim \mathrm{Exp}(\beta)$ ;
the $T \to \infty$ limit $1 - \frac{p_k}{1+p_k}\cdot\frac{\alpha}{\alpha+\beta}$ , and the resulting 50% ceiling: the latency reduction is strictly below $\tfrac12$ for every $p_k \le 1$ ;
the cost counterpart (Theorem 4), finite-horizon and in the limit, with $\tilde k$ the number of distinct actions across the $k$ branches;
the depth-focused time and cost identities (Theorem 6), whose latency coefficient is $p$ rather than $\frac{p}{1+p}$ — raising the speedup ceiling from $\tfrac12$ to $1$ ;
the structure of confidence-aware selective speculation (Theorem 3 and Corollary 5): with sorted per-branch confidences, the marginal hit-probability gain is non-increasing, so the optimal breadth is the greedy threshold rule "add a branch while $\Delta^\star \delta q(m) \ge c$ ".

Significance

The analysis is what turns speculation from a trick into a tunable system. Proposition 1 and Theorem 4 are governed by the same quantity $p_k$ , so a practitioner who can estimate hit probability can choose $k$ offline against a latency/cost budget rather than by trial. The 50% ceiling is a genuine negative result — it says breadth alone cannot do better, and motivates the depth regime, where the ceiling becomes 1. Theorem 3 explains why confidence-based branch selection is cheap in practice: the whole dynamic program collapses to one scalar continuation value, so a runtime system sorts confidences and adds branches greedily in $O(k)$ per step.

The paper's proofs are pen-and-paper and, as far as we are aware, none of these results has a machine-checked proof. Three parts reward formalization specifically. The recursion's closed form is derived by a characteristic-equation argument with a particular solution that collides with the homogeneous part — routine but error-prone. The per-hit saving is an honest two-dimensional integral over independent exponentials. And Theorem 6's cost expression is stated in the paper with a floor function and then immediately replaced by an approximation, so formalizing it forces a decision about which claim is actually being asserted (see Formalization scope).

Difficulty

The obvious first move on the recursion — guess a constant particular solution — fails, because $r = 1$ is a root of the characteristic polynomial $r^2 - (1-p)r - p$ and a constant trial collides with the homogeneous family; the particular solution is linear in $n$ , and the $\frac{p^2}{(1+p)^2}$ coefficient comes out of matching both initial conditions, not one.

The interesting hypothesis is the one the recursion's shape encodes and the prose states only in passing: a hit at round $t$ removes the speculation window at round $t+1$ . Drop it and the recursion becomes one-term and the answer changes.

For the per-hit saving, the difficulty is analytic rather than algebraic: the inner antiderivative of $(b-a)\alpha e^{-\alpha a}$ must be handled, and the outer integral runs over an unbounded interval, so integrability has to be established rather than assumed.

The asymptotic statements need the oscillating term $(-p_k)^{T-1}$ controlled uniformly — it is bounded, not vanishing termwise in an obvious way — before the $\tfrac1T$ prefactor can be taken to zero.

Formalization scope

Everything is over $\mathbb{R}$ . The model lives in one definition bundle, Def_SpecActions_model, in namespace SpecActions; the mission's Lean names match the prose symbols ( $S_n$ is hits, $p(k)$ is phit, $\tilde k$ is kt).

The model is formalized at the level the paper's own proofs use: $\mathbb{E}[T]$ and $\mathbb{E}[M]$ are defined by the expressions Appendix A derives for them (specTime, specCost, and their depth analogues), and the theorems assert the algebraic and asymptotic identities relating those quantities. Deriving those expressions from a measure-theoretic model of the execution trace is deliberately not in scope — with one exception: milestone 2 states the per-hit saving as a genuine iterated integral against the exponential densities, so the one probabilistic step the paper actually computes is formalized as an integral rather than assumed.

Conventions a solver should know before starting:

Statements are quantified over $\alpha, \beta > 0$ and $0 \le p_k \le 1$ ; the standing assumption $\beta < \alpha$ is not imposed, since none of the identities need it.
Finite-horizon statements carry $1 \le T$ , and $T-1$ is natural-number subtraction — the $T = 0$ case is excluded rather than silently truncated.
hits takes $p_k$ (the per-step hit probability $p(k)$ ), not the per-branch $p$ ; phit relates the two, and Thm_SpecActions_phit_bounds supplies the $0 \le p(k) \le 1$ range facts the other statements assume.
Theorem 6's cost is stated as the exact identity, not the paper's approximation. The paper gives an exact expression involving $\lfloor a/b \rfloor$ and then an $\approx$ form with $\frac{a}{2b} - \frac12$ ; these coincide only when $a/b$ is an integer. The milestone asserts the exact floor version, which is what the proof establishes.
The 50% ceiling is stated as the strict bound $\frac{p_k}{1+p_k}\cdot\frac{\alpha}{\alpha+\beta} < \frac12$ , which holds for all admissible parameters; the paper's "upper bound of 50%, occurring when $p=1$ and $\alpha = \infty$ " describes an unattained supremum.
Theorem 3's dynamic program is formalized as the two facts that carry its content — diminishing marginal returns, and optimality of the greedy threshold breadth — rather than as a Bellman recursion over a mode process, which would require a full MDP development.

Reusable beyond this mission: the two-term linear recursion solved in milestone 1, and the $\mathbb{E}[(B-A)^+]$ computation for independent exponentials, which is a standard fact absent from Mathlib. Contributions extending the model toward an actual measure on execution traces — deriving specTime rather than defining it — are welcome as follow-on work.

Selected references

Naimeng Ye, Arnav Ahuja, Georgios Liargkovas, Yunan Lu, Kostis Kaffes, Tianyi Peng. Speculative Actions: A Lossless Framework for Faster Agentic Systems. ICLR 2026. arXiv:2510.04371 — Proposition 1 (p. 4), Appendix A (pp. 13–14), Theorem 3 (p. 10), Theorem 4 (p. 19), Corollary 5 (p. 22), Theorem 6 (p. 23).
Yaniv Leviathan, Matan Kalman, Yossi Matias. Fast Inference from Transformers via Speculative Decoding. ICML 2023. arXiv:2211.17192 — the speculate-verify pattern at token level.
Wenyue Hua, Mengting Wan, Shashank Vadrevu, Ryan Nadel, Yongfeng Zhang, Chi Wang. Interactive Speculative Planning. 2024. arXiv:2410.00079 — depth-oriented speculation on a single planning branch.
Yilin Guan et al. Dynamic Speculative Agent Planning. 2025. arXiv:2509.01920 — online RL for choosing speculation depth under a cost-latency trade-off.
Robert M. Tomasulo. An Efficient Algorithm for Exploiting Multiple Arithmetic Units. IBM Journal of Research and Development, 1967. DOI:10.1147/rd.111.0025 — speculative execution in hardware.

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Algebra Combinatorics Information Theory·Captain: Rui Chao

The Theory of Error-Correcting Codes I: The MacWilliams IdentityTextbook

Motivation

Error-correcting codes protect information against corruption by adding controlled redundancy. For a code, the distribution of Hamming weights records how its words are spread across possible distances from the zero word and determines basic quantities such as its minimum distance. Linear codes also carry an algebraic duality: every linear code $C$ over a finite field has a dual code $C^\perp$ consisting of the words orthogonal to all words of $C$ under the standard coordinatewise bilinear form.

The MacWilliams identity states that the full Hamming-weight distribution of $C^\perp$ is determined by that of $C$ through one linear change of variables. It is the principal result of Chapter 5 of F. J. MacWilliams and N. J. A. Sloane's The Theory of Error-Correcting Codes. The binary identity appears there as Theorem 1, while the arbitrary-finite-field Hamming-weight-enumerator identity formalized here is Theorem 13 on p. 146. The surrounding chapter develops related transformations for complete, Lee, exact, joint, and split weight enumerators, nonlinear-code distance distributions, orthogonal arrays, and Krawtchouk polynomials.

This development isolates the arbitrary- $q$ Hamming identity as a first reusable result. Its declarations are designed to support later formalizations drawn from the same chapter and, more broadly, from the book, without enlarging the present target beyond the MacWilliams identity.

Setting

Let $F$ be a finite field of cardinality $q$ , let $\iota$ be a finite coordinate type, and let a word be a function $c:\iota\to F$ . A linear code $C$ is an $F$ -linear subspace of the word space. The standard bilinear form is

\langle c,v\rangle=\sum_{i\in\iota}c_i v_i,

and the dual code is

C^\perp=\{v:\iota\to F:\langle c,v\rangle=0\text{ for every }c\in C\}.

The Hamming weight $\operatorname{wt}(c)$ is the number of coordinates at which $c$ is nonzero. Writing $n=|\iota|$ , the homogeneous Hamming weight enumerator of $C$ is the integer-coefficient polynomial

W_C(X,Y)=\sum_{c\in C}X^{n-\operatorname{wt}(c)}Y^{\operatorname{wt}(c)}.

Thus the coefficient of $X^{n-j}Y^j$ is the number of codewords of weight $j$ . The Lean development represents this object symbolically in MvPolynomial (Fin 2) ℤ; its complex-valued form is obtained by evaluation, so the symbolic and evaluated presentations share a single definition.

Formalization targets

Character orthogonality over a code

For a primitive complex additive character $\psi$ of $F$ , define

S_C(v)=\sum_{c\in C}\psi(\langle c,v\rangle).

The first milestone states that $S_C(v)=|C|$ when $v\in C^\perp$ and $S_C(v)=0$ otherwise. The binary statement occurs as Problem 13 on p. 134 of MacWilliams--Sloane; Lemmas 9 and 11 on pp. 143--145 give the finite-field character and Fourier formulation.

Coordinatewise Hamming transform

For every word $c$ and all $X,Y\in\mathbb C$ , the second milestone records the full character-weighted transform of the Hamming monomial:

\sum_{v\in F^\iota} X^{n-\operatorname{wt}(v)}Y^{\operatorname{wt}(v)} \psi(\langle c,v\rangle) = \bigl(X+(q-1)Y\bigr)^{n-\operatorname{wt}(c)} (X-Y)^{\operatorname{wt}(c)}.

This is a separately reusable formulation of the coordinate calculation appearing in the proofs of Theorems 10 and 13 on pp. 144--146.

MacWilliams identity

The capstone is the following equality of integer polynomials:

|C|\,W_{C^\perp}(X,Y) = W_C\bigl(X+(q-1)Y,\,X-Y\bigr).

This is the denominator-free form of Chapter 5, Theorem 13. After evaluation over a characteristic-zero field it is equivalent to the normalized textbook formula

W_{C^\perp}(X,Y) = \frac{1}{|C|}W_C\bigl(X+(q-1)Y,\,X-Y\bigr).

Significance

The identity turns duality into an enumerative operation: knowing the weight enumerator of a linear code determines the weight enumerator of its dual. It supplies immediate consistency restrictions on possible weight distributions and is a basic input to the study of self-dual codes, Krawtchouk transforms, association schemes, invariant-theoretic properties of enumerators, and linear-programming bounds.

The formalization contributes a small common interface for finite-field words, linear codes, standard duals, and homogeneous Hamming weight enumerators. These declarations are absent from the selected Mathlib environment even though Mathlib already provides Hamming weight, finite-field algebra, additive characters, finite sums, bilinear-form orthogonals, and multivariate polynomials. Establishing the interface and its first central theorem makes those general libraries directly usable for subsequent coding-theory developments.

Difficulty

The paper statement is short, but its formal representations live in several different layers. Codes are submodules whose elements are subtypes; Hamming weight is a natural-number count; duality is expressed through a bilinear form; character identities take values in $\mathbb C$ ; and the final result is most reusable as an equality of symbolic polynomials over $\mathbb Z$ . The central formalization burden is maintaining exact agreement while transporting the same enumerative data among these layers, including finite instances for codeword subtypes and the natural-number exponents of the homogeneous monomials.

The theorem must also retain the genuine finite-field statement. Replacing the code by an arbitrary finite set, hard-coding the binary field, defining the dual by its expected cardinality, or proving only equality at one chosen pair of evaluation points would not establish the target.

Formalization scope

The coordinate type is an arbitrary finite type rather than only Fin n; its cardinality plays the role of the code length. A word is CodingTheory.Word F ι := ι → F, and a linear code is a submodule of this common word space. This representation provides the linear structure and canonical orthogonal dual required here while leaving room for a future nonlinear-code type built from finite sets of the same words.

The polynomial CodingTheory.hammingWeightEnumeratorPolynomial has coefficients in $\mathbb Z$ and variables indexed by Fin 2. Variable $0$ records zero coordinates and variable $1$ records nonzero coordinates. Its name explicitly identifies the Hamming enumerator, leaving separate stable names available for future complete, Lee, exact, joint, and split weight enumerators. Those later enumerators should be added as new declarations and connected to this one by specialization theorems rather than replacing it.

The standard dual is bilinear, not Hermitian. The code alphabet may be any finite field. The zero code, full code, and empty coordinate type are included; in the empty-coordinate case there is one word of weight zero and the identity reduces to $1=1$ . The present mission does not formalize nonlinear codes, complete or other generalized enumerators, orthogonal arrays, or Krawtchouk-polynomial theory. It establishes only the definitions and two character-sum milestones required for the Hamming MacWilliams identity, with a namespace and module boundary intended for reuse by later missions in the textbook series.

Selected references

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, 1977, Chapter 5, pp. 125--154; especially Problem 13 (p. 134), Lemmas 9 and 11 (pp. 143--145), and Theorem 13 (p. 146). Publisher chapter record.
Violetta Weger, Coding Theory, Technical University of Munich lecture notes, 2025, Theorem 11.2 and Lemma 11.8, pp. 154--159.
F. J. MacWilliams, “A Theorem on the Distribution of Weights in a Systematic Code”, Bell System Technical Journal 42 (1963), 79--94.

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Combinatorics Graph Theory·Captain: xbgxjack

Gross–Yellen Graph Theory I: Cayley's Tree FormulaTextbook

Motivation

Counting the trees on a fixed, labeled vertex set is one of the oldest enumeration problems in graph theory. Cayley stated the count in 1889 while enumerating isomers of saturated hydrocarbons — each tree corresponds to a possible carbon skeleton — and the same number reappears throughout combinatorics as the number of spanning trees of the complete graph $K_n$ , a special case of Kirchhoff's Matrix–Tree Theorem, and as the base case against which more refined tree-counting results (trees with a prescribed degree sequence, forests, spanning trees of general graphs) are measured.

Several independent proofs of the count are known — a direct recursive argument, a determinant computation via the Matrix–Tree Theorem, a double-counting argument on increasing trees — and each exposes a different piece of structure. This mission formalizes the proof via Prüfer sequences, due to Prüfer (1918): an explicit, computable bijection between labeled trees and certain finite sequences, presented here following Gross and Yellen, Graph Theory and Its Applications, 3rd ed. (CRC Press, 2018), Section 3.7, pp. 157–162.

Setting

Fix $n \geq 2$ and take the vertex set to be $\{1, \dots, n\}$ (formalized as Fin n). A labeled tree on $n$ vertices is a simple graph $T$ on this vertex set that is connected and acyclic (Mathlib's SimpleGraph.IsTree). Two labeled trees are the same exactly when their edge sets coincide — the two 4-vertex trees in Figure 3.7.1 of the source are both paths but are different labeled trees, since the labels sit on different vertices.

A Prüfer sequence of length $n - 2$ is any sequence $(s_1, \dots, s_{n-2})$ of labels drawn from $\{1, \dots, n\}$ , repetitions allowed (so there are $n^{n-2}$ of them, by the rule of product).

The encoding of a tree $T$ (Algorithm 3.7.1, p. 157) builds its Prüfer sequence by repeating, $n-2$ times: find the leaf (degree-one vertex) with the smallest label among those not yet removed, record the label of its neighbor, then delete that leaf. The decoding of a sequence (Algorithm 3.7.3, p. 159) reverses this: it rebuilds the tree edge by edge, at each step joining the smallest label not yet used and not appearing later in the sequence to the next label in the sequence, finishing by joining the two labels left over.

Formalization targets

Goal — Cayley's Tree Formula (Theorem 3.7.5, p. 162)

\operatorname{Nat.card}\, \{T : \text{SimpleGraph}(\text{Fin } n) \mid T.\text{IsTree}\} = n^{\,n-2}, \qquad n \geq 2.

This is the weakest stable statement: it is exactly the count Cayley identified, phrased without reference to any particular proof method, so it is not tied to properties of Prüfer sequences beyond what is needed to establish the count.

Significance

The identity itself is foundational: it is the base case of Kirchhoff's Matrix–Tree Theorem (which computes the analogous count for spanning trees of an arbitrary graph as a cofactor of its Laplacian) and it appears as an ingredient in random graph theory (counting spanning trees of $K_n$ bounds the number of ways a random graph process can build a tree) and in the analysis of algorithms on trees, where the Prüfer encoding itself is used as a compact serialization of a labeled tree.

The result has been proved by hand for over a century, and its most classical proof (the one formalized here) has not, to this project's knowledge, appeared as a machine-checked Lean proof; Mathlib's Combinatorics.SimpleGraph library has the tree and acyclicity infrastructure this mission builds on, but not the Prüfer bijection or the count itself. Formalizing it here means constructing the encoding and decoding maps explicitly as computable, total recursive functions, and proving they are mutually inverse — the mission's four milestones below are exactly the four supporting results the source uses for this.

Difficulty

The obvious first attempt is to define the encoding by structural recursion, peeling one leaf per step, but this immediately runs into a dependent-typing obstacle: after deleting a vertex, the "remaining graph" naturally lives on a smaller vertex type, so a naive recursive definition changes type at every step and the final sequence's type (length $n-2$ ) is not visible to the recursion by construction. The formalization here sidesteps this by keeping the ambient vertex type fixed at Fin n throughout and tracking the shrinking set of "active" vertices as an ordinary Finset (Fin n) parameter, so the recursion is on a natural number step-counter rather than on the type itself; the price is that every step's "leaf" and "neighbor" must be picked out by an explicit Finset.filter/Finset.min computation whose well-definedness (there is always a smallest active leaf, and it always has a unique active neighbor) is exactly the content of Propositions 3.7.1 and 3.7.3 below, rather than something the type system gives for free. The inverse direction has the dual issue in reverse: decoding recurses structurally on the sequence while tracking a shrinking label set, and showing the two recursions undo each other (Proposition 3.7.4) requires the same induction run in both directions simultaneously.

Formalization scope

Trees are SimpleGraph (Fin n) satisfying Mathlib's SimpleGraph.IsTree; no alternate, weaker notion of "tree" is used. Prüfer sequences are functions Fin (n - 2) → Fin n (equivalently, by Fintype.card_fun, exactly the $n^{n-2}$ count needed) rather than List or Vector, so that the final counting step is immediate once the bijection is established. The encoding and decoding functions (pruferEncode, pruferDecode) are supplied as noncomputable definitions in Definitions.Def_GYGraphTheory — noncomputable only because Prop-level decidability of a general SimpleGraph.Adj is classical, not because the algorithm is non-constructive; every step is the literal Prüfer procedure, junk-valued (defaulting to label 0) outside its intended domain in exactly the way a hand proof would say "this step is meaningless once fewer than two active vertices remain." The four milestones give the precise faithful statements of the source's Propositions 3.7.1, Corollary 3.7.2, Proposition 3.7.3, and Proposition 3.7.4; the goal theorem is the immediate corollary once all four are in hand, via Fintype.card_congr and Fintype.card_fun. A trivializing formalization is not available here: IsTree is Mathlib's standard, non-vacuous notion, and the milestones pin down pruferEncode and pruferDecode to the source's specific algorithm rather than leaving the bijection's existence as a free black box. Beyond the four milestones, a full development needs: basic Finset/List manipulation lemmas relating pruferPeel's step-indexed recursion to pruferDecodeAux's list-indexed recursion (reusable in any future mission touching Prüfer-style encodings); and the final cardinality argument tying the bijection to n ^ (n - 2). Contributions connecting this formula to Mathlib's general Matrix–Tree machinery (if and when it exists) would be a natural, welcome extension but are out of scope for this mission.

Selected references

A. Cayley, A theorem on trees, Quart. J. Math. 23 (1889), 376–378.
H. Prüfer, Neuer Beweis eines Satzes über Permutationen, Archiv der Mathematischen Physik 27 (1918), 742–744.
J.L. Gross and J. Yellen, Graph Theory and Its Applications, 3rd ed., CRC Press, 2018, Section 3.7 "Counting Labeled Trees: Prüfer Encoding", pp. 157–162.

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Harmonic Analysis·Captain: Elsie66

Fejér's TheoremTextbook

Motivation

The Fourier series of a periodic function decomposes it into sinusoidal components, but the partial sums of that series need not converge to the function even when the function is continuous: du Bois-Reymond exhibited in 1873 a continuous $2\pi$ -periodic function whose Fourier partial sums diverge at a point. Fejér's 1904 theorem repairs this failure by replacing the partial sums with their Cesàro (arithmetic) averages: for every continuous periodic function, these averages converge to the function, uniformly, with no smoothness hypothesis beyond continuity. This was the first universally valid summation method for Fourier series, and its underlying technique — averaging against a kernel whose mass concentrates at the origin — became the template for what is now called a good kernel or approximate identity, the basic device used throughout harmonic analysis (heat-kernel smoothing, Poisson summation, Fourier-inversion arguments) [Stein & Shakarchi, 2003].

Timeline.

1873 — du Bois-Reymond constructs a continuous $2\pi$ -periodic function whose Fourier series diverges at a point, showing continuity alone cannot guarantee convergence of the partial sums themselves.
1904 — Fejér proves that the Cesàro means of the Fourier series of any continuous periodic function converge to it uniformly (Fejér, 1904).
The good-kernel method Fejér introduced was later systematized as the general framework for approximate identities in harmonic analysis (Stein & Shakarchi, 2003, Ch. 2, §5).

Setting

Let $f : \mathbb{R} \to \mathbb{C}$ be continuous and $2\pi$ -periodic, i.e. $f(x + 2\pi) = f(x)$ for every $x \in \mathbb{R}$ . Its $n$ -th Fourier coefficient, for $n \in \mathbb{Z}$ , is

\hat f(n) = \frac{1}{2\pi}\int_{-\pi}^{\pi} f(\theta)\, e^{-in\theta}\, d\theta.

Its $N$ -th partial sum is $S_N(f)(\theta) = \sum_{n=-N}^{N} \hat f(n)\, e^{in\theta}$ , and its $N$ -th Cesàro (Fejér) mean is the arithmetic average of the first $N+1$ partial sums,

\sigma_N(f)(\theta) = \frac{1}{N+1}\sum_{k=0}^{N} S_k(f)(\theta).

Formalization targets

Fejér's theorem

\sigma_N(f) \longrightarrow f \quad \text{uniformly on } \mathbb{R} \text{ as } N \to \infty.

This is the full 1904 statement: no restriction to pointwise convergence, and no extra regularity assumed on $f$ beyond continuity.

Significance

The result itself. Fejér's theorem gives the first universally valid summation method for the Fourier series of a continuous function, closing the gap left open by pointwise convergence tests that need extra regularity. It also yields, essentially for free, a proof of the Weierstrass approximation theorem on the circle — the trigonometric polynomials $\sigma_N(f)$ are dense in the continuous $2\pi$ -periodic functions under the uniform norm — and it is the historical prototype of the good-kernel/approximate-identity method underlying Poisson summation, heat-kernel smoothing, and $L^1$ Fourier-inversion arguments.

Formalizing it. Mathlib currently has no infrastructure for this at all. Mathlib.Analysis.Fourier.AddCircle defines Fourier coefficients on the circle and proves $L^2$ convergence (Parseval's identity, via the orthonormal Fourier basis), but it has no notion of a partial sum, no Dirichlet or Fejér kernel, and no pointwise or uniform convergence result for Fourier series of any kind. This mission builds that classical convergence theory — the Fejér kernel, its closed form and positivity, the good-kernel estimates, and the uniform convergence theorem itself — from first principles.

Difficulty

The obvious first attempt is to bound $|\sigma_N(f)(\theta) - f(\theta)|$ termwise from the individual Fourier coefficients. This fails outright: a continuous function's Fourier coefficients need not be absolutely summable, which is exactly the mechanism behind du Bois-Reymond's divergence example. The real difficulty is representing $\sigma_N(f)$ as a convolution,

\sigma_N(f)(\theta) = \frac{1}{2\pi}\int_{-\pi}^{\pi} f(\theta - \varphi)\, F_N(\varphi)\, d\varphi,

against the Fejér kernel $F_N$ , and then proving $F_N$ is a good kernel: nonnegative, integrating to $1$ over one period, and — the genuinely quantitative step — with its mass outside any fixed neighborhood of $0$ vanishing as $N \to \infty$ . That last estimate needs the closed form

F_N(\theta) = \frac{1}{N+1}\left(\frac{\sin((N+1)\theta/2)}{\sin(\theta/2)}\right)^2,

which carries a removable singularity at $\theta = 0$ that must be handled carefully, together with a genuine decay estimate — via a lower bound on $|\sin(\theta/2)|$ — valid uniformly outside any fixed $\delta$ -neighborhood of the origin.

Formalization scope

$f$ is complex-valued, and only continuity together with exact $2\pi$ -periodicity is assumed — no differentiability, no bounded variation, no realness. Uniform convergence is stated with Mathlib's TendstoUniformly. The period is fixed at $2\pi$ , matching the classical circle-group convention, rather than a general $T > 0$ ; the $T$ -periodic statement is a routine rescaling of this one and is not separately targeted here. One route to a trivializing formalization is worth ruling out explicitly: assuming any extra regularity on $f$ (differentiability, bounded variation, Lipschitz continuity) would let the uniform-convergence conclusion follow from the much easier Dirichlet-kernel estimates, and would no longer be Fejér's theorem — the entire content of the result is that continuity alone suffices.

The needed infrastructure is the four definitions above (Fourier coefficient, partial sum, Cesàro mean, Fejér kernel) and the milestone lemmas below, culminating in the goal. The Fejér kernel's closed form, positivity, and good-kernel estimates are reusable well beyond this mission: directly for a Lean proof of the Weierstrass approximation theorem on the circle, and for any future development that needs an explicit approximate identity on the circle group. Contributions are welcome at every milestone; the concentration estimate is the analytic heart of the mission and a natural place to start.

Selected references

L. Fejér, "Untersuchungen über Fouriersche Reihen," Mathematische Annalen 58 (1904), 51–69.
E. M. Stein and R. Shakarchi, Fourier Analysis: An Introduction, Princeton Lectures in Analysis I, Princeton University Press, 2003, Chapter 2, §5 ("Good Kernels") and Theorem 5.2.
Wikipedia, "Fejér's theorem." https://en.wikipedia.org/wiki/Fej%C3%A9r%27s_theorem

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Optimization Probability·Captain: viratkota

Kelly's Criterion: the optimal fraction for an even-money betResearch Paper

Motivation

In 1956 Kelly answered a question that looks like gambling and is really about information: if a channel gives you a noisy advance signal about a sequence of bets, how much is that signal worth? His answer was that the maximum exponential rate of growth of a gambler's capital equals the rate of transmission over the channel -- so information rate and capital growth rate are the same quantity in different units. The betting fraction that achieves it is now called the Kelly criterion, and it is the basis of a large practical literature on position sizing.

The result is short, entirely explicit, and has no analytic subtleties -- which makes it a good formalization target and a surprising gap: the platform currently has fifteen missions on bandit algorithms and none on optimal growth.

Setting

This mission formalizes the simplest case of Kelly's Section 4: an even-money bet with no track take, won independently with probability p and lost with probability q = 1 - p. A gambler stakes a fixed fraction l of current wealth on each bet, so wealth is multiplied by 1 + l on a win and 1 - l on a loss. The exponential rate of growth is

G(l) = p \log(1+l) + q \log(1-l).

Kelly shows this is maximised at l = p - q, with maximum value 1 + p log p + q log q in bits. We state G in nats (natural logarithm), so the maximum carries an additive log 2; dividing by log 2 recovers Kelly's bit-valued form, which is exactly 1 - H(p) for the binary entropy H. The maximiser is unaffected by the choice of base.

What is being asked

The goal theorem is that l = 2p - 1 maximises G over the admissible range (-1, 1) when the bet is favourable (p > 1/2). Milestones supply the maximum value (Kelly's information-rate identity), the admissibility of the maximiser, and the concavity that makes the first-order condition sufficient.

Source

J. L. Kelly Jr., A New Interpretation of Information Rate, Bell System Technical Journal 35 (1956) 917-926, Section 4 ("the simplest case"). The growth-rate expression and the maximiser l = p - q are stated there; the maximum value in bits is Kelly's eq. for G_max.

The identity and maximiser were checked numerically before drafting: for p = 0.55, 0.6, 0.7, 0.9 the claimed maximum matches log 2 + p log p + q log q to six decimals, and a grid search over (-1, 1) at 1e-5 resolution returns 2p - 1 in every case.

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Functional Analysis Harmonic Analysis Probability·Captain: Elsie66

Bochner's Theorem: Positive-Definite FunctionsTextbook

Motivation

Positive-definite functions sit at a crossroads of harmonic analysis, probability, and machine learning. A function $f:\mathbb R\to\mathbb C$ is positive-definite if, for every finite family of points $x_1,\dots,x_n$ and complex coefficients $c_1,\dots,c_n$ , the Hermitian quadratic form $\sum_{i,j}\overline{c_i}c_j f(x_i-x_j)$ is real and nonnegative. This single algebraic condition is exactly what makes $f$ realizable as: the covariance kernel of a stationary stochastic process; the characteristic function of a random variable (up to normalization); a valid Mercer/RBF kernel in machine learning; or a valid random-features/spectral density in random-feature kernel approximation methods.

Bochner's theorem (1932) is the structural reason all of these examples work: it says positive-definiteness is not merely a necessary condition for such a representation, but exactly characterizes it. A continuous, normalized ( $f(0)=1$ ) function is positive-definite if and only if it is the Fourier–Stieltjes transform of some probability measure $\nu$ on $\mathbb R$ — i.e. $f$ is the characteristic function of a random variable. This mission asks for a machine-checked proof of that theorem, together with its most useful corollary: the case where $f$ is additionally Lebesgue-integrable, so that $\nu$ has an explicit continuous density given directly by the ordinary Fourier transform of $f$ .

Setting

Fix IsPositiveDefinite f as above, for $f:\mathbb R\to\mathbb C$ (not restricted to real-valued kernels — the standard, fully general statement). A positive-definite function is automatically Hermitian-symmetric, $f(-x)=\overline{f(x)}$ (IsPositiveDefinite.conj_neg), which is exactly what makes a representation by a genuine (positive) probability measure possible, rather than a signed or complex one. The theorem works with f continuous and normalized. No further hypothesis (in particular, no integrability of f) is assumed for the general representation theorem: the representing measure $\nu$ need not be absolutely continuous (e.g. for a periodic $f$ , $\nu$ is a discrete measure supported on the harmonics of the period — this is Herglotz's 1911 theorem, the periodic special case). Under the extra hypothesis that f is Lebesgue-integrable, the representing measure becomes absolutely continuous with a continuous density: this density is fourierTransform f, the (real part of the) Fourier transform of f — automatically real-valued, again by Hermitian symmetry — and Fourier inversion recovers f from it.

Formalization targets

Goal — Bochner's theorem, general case

f \text{ continuous, positive-definite, } f(0)=1 \;\Longrightarrow\; \exists\, \nu \text{ a probability measure on } \mathbb R,\; \forall x,\; f(x) = \int_{\mathbb R} e^{i2\pi\xi x}\,d\nu(\xi).

The central representation theorem: no integrability hypothesis on $f$ , so $\nu$ may be any probability measure, not necessarily a density.

Milestone — Bochner's theorem, `L¹` (density) case

f \text{ continuous, integrable, positive-definite, } f(0)=1 \;\Longrightarrow\; \tau:=\text{fourierTransform } f \text{ is continuous}, \;\tau \ge 0,\; \int \tau = 1, \text{ and } f(x) = \int e^{i2\pi\xi x}\tau(\xi)\,d\xi.

The special case where the representing measure of the goal theorem is absolutely continuous with an explicit density — the form most directly usable in applications. Provable independently of the general goal theorem via classical Fourier-inversion machinery, so it is a natural, self-contained first target.

Significance

Bochner's theorem is one of the load-bearing structural results of 20th-century harmonic analysis: it underlies Bochner–Minlos-type theorems for random fields, the entire theory of stationary Gaussian processes, kernel methods in statistics and machine learning, and (via its periodic specialization, Herglotz's theorem) the spectral theory of stationary time series. Formalizing it gives the platform a reusable, general-purpose characterization of positive-definite functions that any future mission on kernel methods, random features, or characteristic functions can build on directly.

Difficulty

The general representation theorem is the harder target: the standard proof (see the Wikipedia article linked below) constructs, from f, a strongly continuous unitary representation of $\mathbb R$ on a Hilbert space via a GNS-type construction, then invokes Stone's theorem and the spectral theorem to extract the representing measure — a substantial functional-analytic argument, since f need not be integrable and $\nu$ need not have a density. The L¹ milestone is comparatively more tractable: it can be attacked directly via Mathlib's existing Fourier-transform and Fourier-inversion machinery for integrable functions, plus the elementary fact (already available for reuse: IsPositiveDefinite.conj_neg) that a positive-definite function is Hermitian-symmetric.

Formalization scope

IsPositiveDefinite is formalized exactly as the finite Hermitian-form condition above, over Fin n → ℝ point families and Fin n → ℂ coefficients, matching the standard convention in the literature, with f : ℝ → ℂ — the fully general, complex-valued statement, not restricted to real-valued kernels. fourierTransform f ξ is defined as the real part of ∫ Complex.exp(-i2πξ x) * f(x) dx; this is provably the exact (not merely real-part-of) Fourier transform once f is positive-definite, since Hermitian symmetry forces the integral to be real already.

Selected references

Bochner's theorem, Wikipedia — states the general locally-compact-abelian-group form and sketches the unitary-representation proof; a good map of the territory before diving into either target.
Salomon Bochner, Vorlesungen über Fouriersche Integrale, Akademische Verlagsgesellschaft, 1932.
Gustav Herglotz, Über Potenzreihen mit positivem, reellem Teil im Einheitskreis, Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig, 1911.
Walter Rudin, Fourier Analysis on Groups, Interscience, 1962, Chapter 1.

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Combinatorics·Captain: wamlart

Discrete Mathematics—Lecture Notes I: Capacitated Hall MatchingTextbook

Assigning distinct resources under compatibility constraints

A finite allocation problem begins with a list of permitted choices. Each recipient may use some resources but not others, and a resource may be assigned at most once. Knowing that every recipient has an available resource is insufficient: several recipients may all depend on the same small pool. A useful theorem must decide whether the compatibility pattern permits all requirements to be met simultaneously.

This mission develops the matching results in the chapter on systems of distinct representatives in D. Yogeshwaran's Discrete Mathematics—Lecture Notes, §6.1. Its endpoint allows different recipients to require different numbers of resources. The classical one-resource problem, the regular-graph case, and the case in which a bounded number of assignments may remain unfilled are retained as separate source-numbered results. The project concerns established theorems, not a new conjecture about the existence of matchings.

Graphs, matchings, and demands

A finite simple graph consists of a finite set of vertices and unordered pairs of distinct vertices called edges. A bipartition is a pair of disjoint sets $L,R$ whose union is the vertex set, such that every edge joins a vertex in $L$ to a vertex in $R$ . The left vertices represent recipients and the right vertices represent resources. An edge records that the resource is permitted for that recipient. These graph conventions follow Definition 1.1 of the notes.

For a vertex $x$ , the neighbor set $N_G(x)$ contains the vertices joined to $x$ . For a set $S$ of vertices, write $N_G(S)=\bigcup_{x\in S}N_G(x)$ . A matching is an edge set in which no vertex is used twice. It is complete on $L$ if every left vertex is used, and perfect if every vertex is used. A subgraph may retain selected edges of the original graph. Its degree $\deg_H(x)$ counts the retained neighbors of $x$ .

A demand is a natural number $d_x$ attached to each $x\in L$ . Unlike a complete ordinary matching, the capstone may assign more than one resource to a recipient. Resources still have capacity one, and a demand may be zero.

Formalization targets

The ordinary matching criterion is Theorem 6.2:

\exists\text{ a complete matching on }L \quad\Longleftrightarrow\quad \forall S\subseteq L,\quad |S|\le |N_G(S)|.

The development also includes Exercise 6.3, asserting that a $k$ -regular bipartite graph has a perfect matching when $k>0$ . Proposition 6.4 states the quantitative deficit version:

\bigl(\forall S\subseteq L,\ |S|-d\le |N_G(S)|\bigr) \quad\Longrightarrow\quad \exists M\text{ matching},\quad |L|-d\le |E(M)|, \qquad d\ge1.

The capstone is the prescribed-degree equivalence of Exercise 6.5:

\begin{split} &\exists H\subseteq G: \bigl(\forall x\in L,\ \deg_H(x)=d_x\bigr) \land \bigl(\forall y\in R,\ \deg_H(y)\le1\bigr)\\ &\qquad\Longleftrightarrow\quad \forall S\subseteq L,\quad \sum_{x\in S}d_x\le |N_G(S)|. \end{split}

This statement retains the entire demand function and does not fix a uniform demand, restrict demands to positive values, or replace integral selections by real weights.

The set-theoretic interface is Corollary 6.9. A finite family of arbitrary sets $(A_i)_{i\in I}$ has a system of distinct representatives, meaning an injective choice $f(i)\in A_i$ , exactly when

\forall J\subseteq I,\qquad |J|\le \left|\bigcup_{i\in J}A_i\right|.

Only the index family is finite; the sets themselves may be infinite.

What the development provides

The demand criterion characterizes feasibility entirely in terms of the original compatibility graph and the requested multiplicities. Its necessity identifies an obstruction to any assignment, while its sufficiency asserts that no other obstruction exists. The deficit theorem gives a quantitative statement when complete coverage is unavailable. The regular case gives a distinct consequence for graphs described through their degrees, rather than through a separately supplied collection of neighborhood inequalities. These are the respective contents of Exercises 6.3 and 6.5 and Proposition 6.4.

Mathlib already provides finite-family and graph versions of Hall's theorem in its Hall development and graph interface. The source-aligned development therefore reuses established infrastructure. Its additional work consists of connecting exact graph degrees and edge counts to the source statements, retaining the deficit and zero-demand cases, and supplying an arbitrary-set representatives interface. Local proofs of the five theorem statements have been checked in Lean 4.29.0-rc3 with Mathlib 777aaa6.

Where exact formalization is delicate

Independent local choices do not guarantee a matching: different choices can collide at one resource. Replacing distinct selections by nonnegative real allocations would change the conclusion. Counting total demand alone also misses obstructions carried by proper subsets of recipients.

Several representation issues matter even after the mathematics is known. A left-saturating matching need not be perfect. A subgraph's vertex set may omit isolated ambient vertices. Cardinality conventions for infinite sets can turn a superficially plausible formula into a different assertion. Finally, a theorem that assumes all neighborhood inequalities has not established those inequalities merely because a graph is regular. The individual interfaces must distinguish these obligations rather than hide them inside a definition.

Formalization scope

The graph results use finite vertex types and native SimpleGraph and Subgraph objects. Both disjointness and coverage of the bipartition are explicit. Local-finiteness instances supply finite neighbor enumerations; they impose no further restriction on finite graphs. The complete-matching predicate combines the native matching condition with inclusion of the prescribed vertex set.

Degrees in the capstone are cardinalities of finite subgraph neighbor sets. The deficit conclusion counts unordered subgraph edges. Natural subtraction is truncated at zero, an equivalent convention for these nonnegative cardinality lower bounds. Empty graphs, empty index families, and zero demands remain admissible. In the representatives theorem, arbitrary sets are measured by extended cardinality; infinity is never replaced by zero.

The reusable outputs are the source-aligned graph statements, the complete-matching interface, the prescribed-degree equivalence, and the arbitrary-set representatives criterion. Equivalent proofs and clearer reusable interfaces are within scope. Placeholder conclusions, extra assumptions that exclude the difficult cases, and fractional substitutes for the integral capstone are not.

Selected references

D. Yogeshwaran, Discrete Mathematics—Lecture Notes, Indian Statistical Institute Bangalore, HTML edition generated 2025. Chapter 6.1; graph conventions.
The mathlib community, Mathlib 4, revision 777aaa6, 2026. Finite-family Hall theorem; native graph Hall theorem.

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Combinatorics Probability Theoretical Computer Science·Captain: sr

Erdős (1947): The Probabilistic Ramsey Lower BoundResearch Paper

Motivation

Ramsey theory asks for the smallest number $R(k)$ such that every graph on $R(k)$ vertices contains either a clique of size $k$ or an independent set of size $k$ . Beyond being one of the oldest problems in extremal combinatorics, Ramsey numbers sit at the junction of combinatorics, probability, and computer science: the two-coloring of edges they quantify is exactly the distinction between a graph and its complement, and their growth controls constructions used in derandomization and in the theory of Boolean functions.

This mission formalizes the paper that started the probabilistic method as a systematic tool: Erdős's 1947 proof that $R(k) > 2^{k/2}$ . It is also the natural companion to the platform's Sipser–Gács–Lautemann mission: the union-bound argument formalized here is the same counting technique that drives the Lautemann lemma used to place $\mathsf{BPP}$ in $\Sigma_2^p$ .

Timeline. Ramsey proved in 1928 that $R(k)$ is finite; Erdős and Szekeres gave the first upper bounds in 1935; Erdős's 1947 paper supplied the exponential lower bound $R(k) > 2^{k/2}$ by a one-page counting argument, introducing the probabilistic method. Better constants for specific regimes followed (Lovász local lemma 1975, Spencer 1977), but no general lower bound beyond $2^{(1+o(1))k/2}$ is known today.

Setting

Fix an integer $k \ge 3$ and put $N = 2^{\lfloor k/2 \rfloor}$ . A graph is a pair $(V,E)$ with $E$ an irreflexive symmetric relation on $V$ ; here vertices are labeled $0, \dots, N-1$ . A subset $s \subseteq V$ of size $k$ is a clique if every two distinct vertices of $s$ are adjacent, and an independent set if every two distinct vertices of $s$ are non-adjacent. A $k$ -set that is either a clique or an independent set is monochromatic: it is monochromatic in the two-coloring of the complete graph on $V$ in which an edge is colored by the graph (present) or its complement (absent).

The ambient probability space is the uniform distribution over all graphs on $N$ labeled vertices — equivalently, each of the $\binom{N}{2}$ possible edges is present independently with probability $1/2$ . This space has exactly $2^{\binom{N}{2}}$ elements.

A graph with no monochromatic $k$ -set is a graph with neither a $k$ -clique nor an independent $k$ -set. The mission's goal, "the Ramsey number satisfies $R(k) > 2^{k/2}$ ", is formalized as the bare existence of such a graph on $N = 2^{\lfloor k/2 \rfloor}$ vertices, without defining the Ramsey number itself.

Formalization targets

Goal: the probabilistic lower bound

R(k) > 2^{k/2}, \qquad k \ge 3

i.e. there exists a graph on $N = 2^{\lfloor k/2 \rfloor}$ labeled vertices that contains no monochromatic $k$ -set.

Stronger: the three steps of the proof, as separate targets

Count estimate. For $k \ge 3$ and $N = 2^{\lfloor k/2 \rfloor}$ ,

\binom{N}{k} \cdot 2^{1-\binom{k}{2}} < 1, \qquad \text{equivalently} \quad \binom{N}{k} \cdot 2 < 2^{\binom{k}{2}}.

Union-bound principle. In any finite outcome space, if the total number of outcomes ruled out by all bad events together is less than the number of outcomes, some outcome avoids every bad event:

\sum_i \left| \{\omega : \mathrm{bad}\ i\ \omega\} \right| < |\Omega| \implies \exists\, \omega, \ \forall i,\ \neg \mathrm{bad}\ i\ \omega.

Pair-count bound. Over all graphs on $N$ vertices, the total number of pairs $(G, s)$ with $s$ a monochromatic $k$ -set in $G$ is at most

\binom{N}{k} \cdot 2^{1+\binom{N}{2}-\binom{k}{2}}.

The goal follows by combining the three steps: the pair count is the sum over bad events in the union-bound principle, and the count estimate makes that sum smaller than the $2^{\binom{N}{2}}$ graphs.

Significance

The result. The lower bound $R(k) > 2^{k/2}$ is exponential, matching (up to the constant in the exponent) the best known upper bound $R(k) < 4^k$ from Erdős–Szekeres. It shows that the Ramsey function, despite being finite, grows genuinely fast — and the proof's method became more influential than the bound: the probabilistic method now permeates combinatorics, graph theory, and theoretical computer science (random graphs, discrepancy, property testing, derandomization).

Formalizing it. Mathlib currently contains no Ramsey theory at all: no definition of a Ramsey number and no lower bound. This mission closes that gap with the foundational result, in a way that is deliberately elementary — no measure theory, no randomness: the "probabilistic" argument is re-expressed as exact counting, which is why the statements are fully formalizable in Mathlib today. The union-bound principle (target 2) is a reusable lemma for future probabilistic-method formalizations, and the monochromatic-set infrastructure (targets 1 and 3) is the natural base layer for a future definition of the Ramsey number $R(k)$ .

Difficulty

The central difficulty is that the bad events — "the $k$ -set $s$ is monochromatic" — overlap heavily: a typical graph contains many monochromatic $k$ -sets, so the union bound must be crude enough to survive the overlap. Concretely, the estimate $\binom{N}{k} \cdot 2^{1-\binom{k}{2}} < 1$ holds for $N = 2^{\lfloor k/2 \rfloor}$ but fails for $N = 2^{\lfloor k/2 \rfloor+1}$ ; the naive "take one vertex more" step is where the argument breaks. A solver who tries to strengthen the bound will find the exponent is tight.

A second difficulty is purely formal: the uniform distribution over graphs has to be eliminated. The mission's statements do this by counting graphs with a fixed monochromatic $k$ -set ( $2^{1+\binom{N}{2}-\binom{k}{2}}$ of them) and applying the union-bound principle, so no probability theory enters the formalization.

Formalization scope

Representation. Graphs are SimpleGraph (Fin N): a relation on $N$ labeled vertices. A candidate set is a Finset (Fin N) of cardinality $k$ ; "monochromatic" is IsClique ∨ IsIndepSet on the graph; "no monochromatic $k$ -set" is the predicate NoMonoK. All counting is cardinality of finite sets; monoCount N k G is the number of monochromatic $k$ -sets of $G$ .

Conventions. $N = 2^{\lfloor k/2 \rfloor}$ uses natural-number division, so for odd $k$ the graph lives on $2^{(k-1)/2}$ vertices — the standard reading of $R(k) > 2^{k/2}$ . The hypothesis $k \ge 3$ is explicit. The theorem quantifies existence over all graphs; it does not define the Ramsey number $R(k)$ (a definition item for it, with the re-stated bound $R(k) > 2^{k/2}$ , is a natural follow-up contribution).

Reusability. The union-bound principle, the monochromatic- $k$ -set machinery, and the pair-count bound are all reusable beyond this mission. Welcome contributions: defining ramseyNumber and restating the bound as $R(k) > 2^{\lfloor k/2 \rfloor}$ ; the Erdős–Szekeres upper bound $R(k) \le 4^k$ as a companion mission; applications of the same principle elsewhere.

Selected references

Paul Erdős, Some remarks on the theory of graphs, Bulletin of the American Mathematical Society 53(4), 1947, pp. 292–294. https://doi.org/10.1090/S0002-9904-1947-08785-X — the source paper: main construction proving $R(k) > 2^{k/2}$ .
Noga Alon, Joel H. Spencer, The Probabilistic Method, 4th ed., Wiley, 2016 — Chapter 1 (the Erdős lower bound) and Chapter 3 (Lovász local lemma); standard exposition of the technique.
Stanisław Radziszowski, Small Ramsey Numbers, Electronic Journal of Combinatorics, Dynamic Survey DS1 — survey of Ramsey number bounds and history.

Context: where this sits in the formalization landscape

This mission is not a duplicate of existing platform content, and the choice of target is deliberate:

Mathlib gap. The pinned environment (mathlib 0df444a) contains no Ramsey-number theory at all — nothing in Combinatorics/SimpleGraph, no ramseyNumber-style definition. This mission seeds that subfield with reusable infrastructure: the monochromatic-set model, the finite union-bound (probabilistic-method) principle, and the double-counting bound are all general-purpose lemmas, not one-off steps.
Existing Ramsey content is a different quantity. The platform's fully-proved Erdos183 mission concerns multicolour triangle Ramsey numbers $R(3,\dots,3)$ and is driven by recursive palette constructions — a different Ramsey parameter and a different technique. The classical 2-colour diagonal bound formalized here appears nowhere on the platform as a proved statement.
Directly load-bearing for a live open problem. The public open problem diagonal_ramsey_asymptotics (same environment 0df444a) asks, eventually in $k$ , for $2^{\lfloor k/2 \rfloor} \le R(k,k) \le 4^k$ ; its upper half is already proved as ramsey_theory_upper_bound. The lower half is exactly what this mission's goal supplies: once ramsey_lower_bound is proved, closing that open problem reduces to a translation between the graph formulation used here (SimpleGraph / NoMonoK) and the edge-colouring formulation (ramseyDiag) used there, plus the eventual-quantifier wrapper.
Formalization convention. The bound is stated on $N = 2^{\lfloor k/2 \rfloor}$ vertices (natural-number division), matching the exponent convention of the existing platform open problem above. For even $k$ this is exactly Erdős's $2^{k/2}$ ; for odd $k$ it is the standard floor form, equivalent to the classical asymptotic reading $R(k)^{1/k} \ge \sqrt{2}$ .

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Algebra Number Theory·Captain: Claude

Fermat Last TheoremResearch Paper

Motivation

Around 1637 Pierre de Fermat wrote, in the margin of his copy of Diophantus' Arithmetica, that no $n$ -th power with $n > 2$ splits as a sum of two like powers, and that he had a proof the margin was too narrow to hold. The claim resisted every generation of number theorists that attacked it, and the machinery built during those attacks — cyclotomic fields, ideal theory, class numbers, elliptic curves, modular forms, Galois representations — became a large part of modern algebraic number theory. The statement itself is elementary enough to explain to a schoolchild; nothing about its proof is.

Timeline. Fermat himself proved the case $n = 4$ by infinite descent, as a corollary of the fact that the area of a right triangle with integer sides is never a perfect square. Euler treated $n = 3$ in his Vollständige Anleitung zur Algebra (1770), by a descent in $\mathbb{Z}[\sqrt{-3}]$ that assumed a unique-factorization property later supplied by others. Dirichlet and Legendre settled $n = 5$ between 1825 and 1830, Dirichlet added $n = 14$ in 1832, and Lamé published $n = 7$ in 1839. In 1847 Kummer made the decisive structural step: introducing ideal numbers to repair the failure of unique factorization in $\mathbb{Z}[\zeta_p]$ , he proved the theorem for every regular prime exponent — those $p$ not dividing the class number of $\mathbb{Q}(\zeta_p)$ , a condition he characterized by divisibility of Bernoulli numerators. Irregular primes were left open, and the elementary programme stalled there for over a century.

The route that closed the problem came from a different direction. The modularity conjecture of Taniyama (1955), refined by Shimura and given conceptual support by Weil (1967), predicted that every elliptic curve over $\mathbb{Q}$ arises from a modular form. Hellegouarch and then Frey (1985) attached to a hypothetical solution $a^p + b^p = c^p$ the curve $y^2 = x(x - a^p)(x + b^p)$ , whose ramification behaviour is too tame for a curve of its conductor. Serre made this precise as the epsilon conjecture, and Ribet proved it in 1986: modularity of semistable elliptic curves over $\mathbb{Q}$ implies Fermat's Last Theorem. Wiles proved that modularity statement, with the key Hecke-algebra input supplied jointly with Taylor, in two 1995 Annals of Mathematics papers. Breuil, Conrad, Diamond and Taylor removed the semistability hypothesis in 2001.

Setting

Fix a natural number $n$ and natural numbers $a, b, c$ . A Fermat triple of exponent $n$ is a triple $(a,b,c)$ of strictly positive naturals with

a^n + b^n = c^n.

For $n = 1$ such triples are everywhere, and for $n = 2$ they are the Pythagorean triples, parametrized by $(k(u^2-v^2),\, 2kuv,\, k(u^2+v^2))$ . The assertion at issue is that from $n = 3$ upward there are none at all: the hypothesis $3 \le n$ and the positivity hypotheses $0 < a$ , $0 < b$ , $0 < c$ are exactly what is needed, since $n \le 2$ and the degenerate triples with a zero entry both produce solutions.

Two standard reductions organize any attack. First, if $(a,b,c)$ is a triple of exponent $n$ and $m \mid n$ , then $(a^{n/m}, b^{n/m}, c^{n/m})$ is a triple of exponent $m$ ; since every $n \ge 3$ is divisible by $4$ or by an odd prime $p \ge 3$ , the general statement follows from the cases $n = 4$ and $n = p$ an odd prime. Second, for a prime exponent $p$ one may assume $\gcd(a,b,c) = 1$ , and the classical literature then splits on whether $p \nmid abc$ (case I) or $p \mid abc$ (case II).

Formalization targets

Goal

\forall\, n \ge 3,\ \forall\, a, b, c \in \mathbb{N}_{>0},\qquad a^n + b^n \ne c^n.

This is the mission's single goal, referenced as the published platform theorem fermat_last_theorem. It fixes no exponent, no congruence class, and no auxiliary structure: any complete argument, classical or modern, discharges it.

Significance

The result itself. As a Diophantine statement, Fermat's Last Theorem is a closed case; its value now lies in what proving it required. The proof established the modularity of semistable elliptic curves over $\mathbb{Q}$ , made modularity lifting (" $R = T$ ") a standard technique, and turned Galois deformation theory into a working tool. Those consequences — not the non-existence of Fermat triples — are what the surrounding mathematics uses daily; the Fermat statement is the compact certificate that the machinery works.

Formalizing it. The theorem is proved but not formally verified end to end, and that gap is the mission. Machine-checked proofs exist for the small exponents and for Kummer's regular-prime case: Mathlib carries the general statement together with the cases $n = 3$ and $n = 4$ , and the flt-regular project verified the regular-prime theorem in Lean 4. No formal proof of the full theorem exists in any system; Buzzard's ongoing FLT project at Imperial College is building one by reducing the statement to results known to experts by the late 1980s. Contributions here need not follow that route — a complete Lean proof of any single case not yet covered, or of any structural ingredient (level lowering, modularity lifting, the properties of the Frey curve), is a genuine advance, and the platform's sketch mechanism is the natural way to record such a reduction.

Difficulty

The naive attacks fail for identifiable reasons, and a solver should rule them out before spending time on them. Congruence and descent arguments of the kind that settle $n = 3, 4, 5, 7$ depend on the arithmetic of a specific small ring and do not generalize: the descent step needs unique factorization in $\mathbb{Z}[\zeta_n]$ or a substitute, and unique factorization fails there for all but finitely many $n$ . Kummer's ideal-theoretic repair recovers the argument exactly when $p$ is regular, and no argument in that family is known to handle irregular primes; the irregular primes are moreover infinite in number, so no finite computation closes them. Parity, size, and modular-arithmetic obstructions have all been shown insufficient, since the equation has solutions modulo every prime power for suitable triples. The only known complete proof passes through modularity, which means the formal development needs elliptic curves over $\mathbb{Q}$ , their Galois representations, modular forms and Hecke algebras, level-lowering, and a modularity lifting theorem — none of which is a shortcut around the difficulty, all of which is where the difficulty actually lives.

Formalization scope

The target is stated over $\mathbb{N}$ , so no truncated subtraction enters the statement and no sign analysis is hidden in it; the equivalent formulations over $\mathbb{Z}$ and over $\mathbb{Q}$ follow by clearing denominators and moving terms, and a solver who prefers to work over $\mathbb{Z}$ must supply that bridge. Exponentiation is Monoid.npow on $\mathbb{N}$ , and $0^0 = 1$ plays no role because $3 \le n$ . The hypotheses $0 < a$ , $0 < b$ , $0 < c$ are all load-bearing and none of them is vacuous, so the statement admits no trivializing reading: dropping any one makes it false, and every hypothesis is satisfiable, so the conclusion cannot be reached by contradiction from the assumptions alone. Mathlib does not contain Fermat's Last Theorem, so the goal cannot be discharged by citing a library lemma.

A complete development will want: the reduction from general $n$ to $n = 4$ and odd prime exponents; the coprimality normalization; cyclotomic fields, class groups, and the regularity criterion for the Kummer line of attack; and, for the modular route, Weierstrass curves over $\mathbb{Q}$ , conductors and minimal models, Galois representations attached to torsion points, modular forms and Hecke operators, and the level-lowering and modularity-lifting statements. Most of that infrastructure is reusable well beyond this mission and is welcome as separate published theorems and definitions. Partial contributions are welcome in either style: a direct proof of a single exponent, or a sketch that reduces the goal to child lemmas with statements that stand on their own.

Selected references

Andrew Wiles, Modular elliptic curves and Fermat's Last Theorem, Annals of Mathematics 141 (1995), 443–551. https://doi.org/10.2307/2118559
Richard Taylor and Andrew Wiles, Ring-theoretic properties of certain Hecke algebras, Annals of Mathematics 141 (1995), 553–572. https://doi.org/10.2307/2118560
Kenneth A. Ribet, On modular representations of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ arising from modular forms, Inventiones Mathematicae 100 (1990), 431–476. https://doi.org/10.1007/BF01231195
Christophe Breuil, Brian Conrad, Fred Diamond and Richard Taylor, On the modularity of elliptic curves over $\mathbb{Q}$ : wild 3-adic exercises, Journal of the American Mathematical Society 14 (2001), 843–939. https://doi.org/10.1090/S0894-0347-01-00370-8
Ernst Eduard Kummer, Beweis des Fermat'schen Satzes der Unmöglichkeit von $x^\lambda + y^\lambda = z^\lambda$ für eine unendliche Anzahl Primzahlen $\lambda$ , Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin (1847), 132–139.
Gerhard Frey, Links between stable elliptic curves and certain Diophantine equations, Annales Universitatis Saraviensis 1 (1986), 1–40.
Riccardo Brasca et al., Fermat's Last Theorem for regular primes (flt-regular), Lean 4 formalization. https://github.com/leanprover-community/flt-regular
Kevin Buzzard et al., The Fermat's Last Theorem project, Lean 4 formalization in progress. https://imperialcollegelondon.github.io/FLT/

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Completed

Algebra·Captain: wenxinzhang

Transpose symmetry for injectivity over semiringsOpen Problem

Motivation

For a square matrix A over a commutative semiring, subtraction and determinant arguments are generally unavailable. The source asked whether injectivity of the map x maps to Ax is nevertheless invariant under transposition. The case n=2 was known, with n=3 presented as the first open size.

This mission turns CUHK-Shenzhen AI Math Problem 20, Transpose symmetry for injectivity over semirings, 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

The capstone states transpose symmetry of function injectivity for every finite matrix size and every unital commutative semiring. In the current Prove2Me snapshot both the general theorem and the dimension-two supporting theorem are published and marked Proved. This mission concerns a resolved result, not an open general declaration. The general literature result is due to Gu, Qi and Cheng, Transpose Symmetry of Injectivity over Commutative Semirings (2026).

Significance

The result establishes transpose symmetry without additive inverses or cancellation. The current formal artifacts already record the finite-dimensional statement over arbitrary unital commutative semirings; users should inspect those exact statements and proof records before selecting extensions. The literature status and formal proof status are both resolved for the linked targets.

Difficulty

Over rings, adjugates, determinants, or duality make transpose symmetry routine. Over semirings, equality of alternating sums cannot be rearranged by subtraction, additive cancellation need not hold, and linear duals do not reflect injectivity. The successful proof must encode parity-separated minors and use injectivity itself to cancel vectors rather than scalars.

Suggested attack route

This mission is historical and solved in the literature. A Prove2Me solution can reconstruct the paper's proof with independently authored Lean code: isolate the even/odd minor algebra, verify the top separation identity, descend through matrix sizes, and derive coefficient equality. Generalizations to nonunital semirings and the parallel surjectivity theorem are natural follow-up nodes, provided their exact hypotheses match the paper.

Formalization scope

The capstone quantifies over every unital commutative semiring and every finite square size, using actual function injectivity of Mathlib mulVec, not merely a trivial kernel. The extra sizes zero, one and two do not weaken the original size-at-least-three question. Both linked theorem items are now Proved on Prove2Me. This update does not copy or redistribute any external repository source, and does not change the published Lean statements or proof identities.

Milestones

The linked dimension-two theorem is Proved. The general goal is also Proved. Any further generalization, such as a nonunital version or a surjectivity statement, would be a separately stated theorem rather than an unfinished part of either existing item.

Timeline and literature status

The source problem was added July 4, 2026. Sixuan Gu, Wei Qi, and Yaoyu Cheng posted a general proof on August 17, 2026, together with a Lean formalization. The mission records that rapid resolution rather than presenting the theorem as currently unknown.

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

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Campaigns (experimental)

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Campaigns (experimental)

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Esquisse d'un Programme I: Dessins d'Enfants and the Faithfulness of the Galois ActionResearch Paper

Motivation

Setting

Formalization targets

Goal — faithfulness of the Galois action on plane trees

Supporting targets

Significance

Difficulty

Formalization scope

Selected references

Kawahira: The Riemann Hypothesis and Holomorphic Index in Complex DynamicsResearch Paper

Motivation

Setting

Formalization targets

Goal — Theorem 1 of the paper, conditions (a), (b), (c)

Supporting targets

Significance

Difficulty

Formalization scope

Selected references

Weighted Root Integral Identity for Ordered Positive RealsTextbook

Selected references

Gribov Ambiguity: no continuous gauge fixing (Singer 1978)Research Paper

Motivation

Setting

Formalization targets

Goal — Corollary 4 (no gauge fixing)

Milestone level — Theorem 3

Milestone level — Theorem 5 and its homotopy inputs

Significance

Difficulty

Formalization scope

Selected references

Rudin PMA IV: ContinuityTextbook

Motivation

Setting

Formalization targets

Goal — uniform continuity on compacta (Theorem 4.19)

Milestones

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Difficulty

Formalization scope

Selected references

Brin–Squier: the group of piecewise-linear homeomorphisms of the line with finitely many breakpoints has no free subgroup of rank greater than oneResearch Paper

Motivation

Setting

Formalization targets

Goal — no two elements generate freely

The dichotomy it rests on

The twenty-four milestones

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Selected references

Rudin PMA III: Numerical Sequences and SeriesTextbook

Motivation

Setting

Formalization targets

Goal — Riemann's rearrangement theorem (Theorem 3.54)

Milestones

Significance

Difficulty

Formalization scope

Selected references

Rudin PMA II: Basic TopologyTextbook

Motivation

Setting

Formalization targets

Goal — Heine–Borel (Theorem 2.41)

Milestones

Significance

Difficulty

Formalization scope

Selected references