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Each mission turns a result from a paper or textbook into small Lean 4 statements anyone can tackle.
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.
Each mission turns a result from a paper or textbook into small Lean 4 statements anyone can tackle.
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.
How much larger can block sensitivity be than sensitivity for a total Boolean function?
For a total Boolean function , a block of coordinates is sensitive at if flipping every bit in that block changes . Let count sensitive singletons and be the maximum number of pairwise disjoint sensitive blocks. Define and .
This campaign seeks larger exponents for which there exists an infinite family of total Boolean functions , indexed by , satisfying
Here denotes a quantity tending to zero as . Higher is better. The classical exponent is , and Huang's bound gives a ceiling of .
OpenAI's construction achieves an exponent strictly greater than .
One sufficient way to establish an exponent is to find a single total Boolean function with and . Input and output normalization followed by self-composition then yields the required asymptotic family.
Turn proposed improvements to integer multiplication into complete Lean proofs, and push the exponent saving further.
Harvey and van der Hoeven established an algorithm in 2021. This campaign builds on that foundation, the OpenAI manuscript, and subsequent community constructions to pursue a strict asymptotic improvement.
For two -bit integers, the target is
A positive beats asymptotically; larger 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.
Classical algorithms solve 3SUM in time. In a 2026 breakthrough, Alman and Vassilevska Williams gave a deterministic 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 -bit words, and pursues smaller exponents.
Classical algorithms solve all-pairs shortest paths in time. In a 2026 breakthrough, Alman and Vassilevska Williams refuted the APSP conjecture with a deterministic algorithm. How low can the exponent go?
Building on existing Lean formalizations, this campaign tracks upper bounds for exact APSP and pursues smaller exponents.
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.
The sharp Hlawka inequality for Schatten -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 . We conjecture that the same formula holds for all .
What is the smallest cutoff for which this formula holds for every real ?
References:
Is every odd number a sum of primes? This campaign tracks formalized proofs of the smallest that suffices.
Schnirelmann (1930) showed some finite works. Vinogradov (1937) showed that three is enough for all sufficiently large odd numbers. Tao (2012) proved unconditionally. Helfgott (2013) proved that every odd number greater than is a sum of three primes, though the proof is still unrefereed. Ideally, we can formalize this statement here. Note that three is optimal: is neither prime nor + prime.
Schoolbook matrix multiplication takes operations. The exponent is the infimum of all such that two matrices can be multiplied in arithmetic operations; trivially , and is conjectured but open.
Strassen gave the first nontrivial bound, , in 1969, and introduced the laser method in 1986 to reach . Coppersmith and Winograd's 1990 bound of stood for two decades. Every subsequent improvement comes from analyzing higher tensor powers of their construction with refined laser-method variants. That line reached in 2025, and the current record is , from August 2026. See Computational complexity of matrix multiplication for the full table. Can we formalize these results and even improve on them?
A reservoir computer learns a dynamical input/output relation with a recurrent network whose internal weights are fixed once and never trained; only a linear readout on the state is fitted. The method works in practice — it is a standard tool for learning chaotic dynamics — but its justification requires an approximation theorem: the family of reservoirs used must be rich enough to reach any reasonable target system.
The target class is fixed by fading memory, the continuity notion Boyd and Chua introduced in 1985 for the approximation of nonlinear operators: a filter has fading memory when inputs that agree on the recent past produce nearby present outputs, however much they differ long ago. The question is then which reservoir families are dense in that class.
Non-homogeneous state-affine systems are the family that answers it. They are affine in the state, with coefficients depending polynomially on the input, and the density result proved for them is what every later universality theorem for reservoir computing rests on — including the one for echo state networks, whose proof approximates a target filter by a state-affine system first and only then by a network.
Timeline.
Time is indexed by the nonpositive integers, so an input has an infinite past and a present. Inputs are real-valued and bounded by one: the set of sequences with .
A non-homogeneous state-affine system is the reservoir
where is a polynomial with matrix coefficients, a polynomial with -vector coefficients, and the linear readout. Writing , the system is affine in the state and polynomial in the input.
Two constants govern it: and . When the state map contracts, the system has the echo state property — exactly one bounded state sequence per input — and the states obey . The induced map from input history to present output is the SAS functional .
Any fading memory filter on uniformly bounded scalar inputs is approximated, uniformly over all such inputs, by a state-affine system read out linearly.
The result itself. It is the density theorem of reservoir computing. Without it, nothing guarantees that a reservoir family can represent the system one is trying to learn, and the practice of fitting only a linear readout has no theoretical backing. It is also the input to the universality theorem for echo state networks: that proof replaces the target filter by a state-affine system before replacing it by a network, so the present result is a prerequisite rather than a parallel statement.
Formalizing it. The supporting target is a specialization of a result already published on this platform: a state-affine system is a contracting reservoir map, so its echo state property follows from the abstract contraction theorem rather than from a new argument. What this mission adds beyond that is the density statement itself, which is of a different nature — an approximation theorem in a function space, not a fixed point argument.
The obvious approach to the goal is to exhibit an approximating system directly, and it fails: the target is an arbitrary fading memory filter, given by no formula, so no construction can be read off it. The proof is not constructive in that sense. It proceeds instead by showing that the family of SAS functionals is a polynomial algebra which separates points and contains the constants, and by applying a Stone-Weierstrass argument on a space of input sequences made compact by the weighted topology.
Two points resist. The compactness is not that of the supremum norm — the space of uniformly bounded sequences is not compact for it — but of the weighted norm, and it is that topology in which the approximation is obtained. And the algebra property is delicate: the product of two SAS functionals must again be one, which is what forces the non-homogeneous form. The corresponding statement fails for linear reservoirs, whose products leave the family.
Time is indexed by , index denoting the instant steps into the past and the present; the system equation reads . This is a relabelling of the source's indexing, not a weakening.
Inputs are scalar, as in the source's Section 3, where the restriction is made explicit and the multidimensional extension deferred to a remark. Polynomials are given by their coefficient families, and evaluated as ; the bounds and are stated as explicit operator and norm bounds valid on rather than through a maximum, so that any valid bound may be supplied.
The fading memory property of the target is the one already published on this platform, stated for a functional rather than a filter: the two are in linear bijection, so nothing is lost and causality and time-invariance need not be formalized separately.
One trivialization is ruled out. The goal quantifies over state sequences satisfying the system equation, and the supporting target is what guarantees such a sequence exists and is unique under the stated bounds; without it, the approximation claim could be read as vacuous.
A complete development needs the Stone-Weierstrass theorem, available in Mathlib, together with compactness of the weighted sequence space, which is not and has to be built. Contributions are welcome on both targets.
How short can a code for a data source be, if the code must still be uniquely decodable — if every string of concatenated codewords can be unambiguously split back into the original symbols? Shannon's 1948 source coding theorem answers this exactly: the entropy of the source is a hard lower bound on the average codeword length of any uniquely decodable code, and it is also achievable up to a one-symbol slack. Entropy is not just a measure of "average surprise" — it is the literal, tight answer to a combinatorial question about how densely symbols can be packed into strings without losing decodability. This is the theorem that gives Shannon's entropy its operational meaning, and it underlies every practical lossless compression scheme (Huffman coding, arithmetic coding, Lempel–Ziv) as the benchmark they approach.
Timeline.
A source has a finite alphabet of symbols , with at least two symbols, and probability distribution (, ). A code assigns to each symbol a codeword , a finite string over a -ary code alphabet (); the code is uniquely decodable if every finite sequence of codewords is determined by its concatenation. The entropy of in base is
The expected codeword length of under is .
(For a source with symbols — see Formalization scope for why the single-symbol case must be excluded.)
The result itself. This theorem is the reason entropy is called entropy in an information-theoretic sense at all: it converts a quantity defined by an abstract formula () into the exact answer to an operational question (minimum achievable expected code length), with a slack no worse than one symbol. It is the founding theorem of lossless source coding and the benchmark every practical compressor is measured against.
Formalizing it. Mathlib recently gained genuine information-theoretic coding content: InformationTheory.UniquelyDecodable and the necessity direction of the Kraft–McMillan inequality (McMillan's 1956 result: a uniquely decodable code's lengths satisfy ) are already proved, via a counting argument on concatenations of codewords. This mission builds directly on that foundation rather than duplicating it. What Mathlib does not have — and what this mission's milestones supply — is Kraft's original 1949 sufficiency direction (existence of a uniquely decodable code realizing any length assignment satisfying the Kraft sum bound), any notion of Shannon entropy for a general finite distribution, and the source coding theorem itself.
The lower bound () is the easier half: it follows from the Kraft–McMillan inequality (already in Mathlib) via Gibbs'/Jensen's inequality applied to the two probability-like sequences and (where is the Kraft sum) — a short, self-contained convexity argument.
The achievability half is the genuine construction. Given the ideal (generally non-integer) lengths , one rounds up to (Shannon–Fano–Elias lengths); a one-line estimate shows , so the Kraft sum of the rounded lengths is still , and the bound gives immediately once a code with exactly these lengths is shown to exist. Producing that code is Kraft's sufficiency direction, and it needs an explicit construction: order the lengths, and assign to symbol the first digits of the -ary expansion of the cumulative sum . Verifying this assignment is injective, has the prescribed lengths, and is uniquely decodable (indeed prefix-free) is a careful but standard combinatorial argument — the main open piece of this mission.
The source alphabet must have at least two symbols (), not merely be nonempty. A single-symbol source forces and entropy , so the achievability conjunct would demand a codeword of length — but a uniquely decodable code can never contain the empty codeword (InformationTheory.UniquelyDecodable.epsilon_not_mem, provable from the definition: the empty string decodes ambiguously as zero or two copies of itself), so no admissible code exists and the theorem would be false, not merely hard, at . The same defect breaks Kraft's sufficiency direction (Milestone 2) whenever any prescribed length is , independent of ; that milestone accordingly requires every length strictly positive. With and full support, every strictly, so the Shannon–Fano lengths are automatically all , and the achievability construction only ever needs Milestone 2 at positive lengths. The code alphabet is likewise an arbitrary finite type (matching Mathlib's own Fintype/Nonempty conventions for the Kraft–McMillan file), with required to keep Real.logb non-degenerate. The source distribution is required strictly positive () — the standard simplifying assumption (zero-probability symbols can always be dropped without loss). Unique decodability is stated exactly as Mathlib's InformationTheory.UniquelyDecodable, not re-derived from a "prefix code" definition, so the mission's results transport directly onto Mathlib's existing Kraft–McMillan file. A trivializing route to rule out: proving only the lower bound (citing Mathlib's inequality) while leaving the existential achievability half unaddressed would not be Shannon's theorem — the sandwich is the theorem's actual content, and the lower bound alone (already essentially free from Mathlib) is not a novel contribution on its own.
Reusable output: the Kraft sufficiency construction (Milestone 2) is directly reusable for any future formalization of Huffman coding optimality, arithmetic coding, or the general "Kraft-inequality-achieving code exists" fact used throughout coding theory. Contributions are welcome starting from Milestone 2 (the open construction) or Milestone 3 (the Gibbs'-inequality lower bound, which only needs Milestone 1, already available via Mathlib).
Mathlib.InformationTheory.Coding.UniquelyDecodable and Mathlib.InformationTheory.Coding.KraftMcMillan (2026).The separation theorem — certainty equivalence for linear-quadratic control with imperfect state information — is one of the celebrated structural results of stochastic control: the optimal controller splits into a least-squares estimator and the deterministic LQR actuator, designed independently. It underlies every LQG autopilot and Kalman-filter-based regulator. Section 5.2 of Bertsekas, Dynamic Programming and Optimal Control, Vol. I (3rd ed., 2005) proves it from the DP algorithm over information vectors, with Lemma 5.2.1 supplying the key fact that the estimation error is beyond the controller's influence. No formal analogue exists in Mathlib.
Linear dynamics and measurements
with quadratic cost , , . The initial state and the zero-mean disturbances/noises are independent with finite ranges; independence is structural — the sample space is the product of an initial-state coordinate and per-stage noise coordinates (BertsekasLQGModel, BertsekasLQGSample, BertsekasLQGProb). A policy maps the realized measurement history to ; the closed-loop process is BertsekasLQGTraj, the expected cost BertsekasLQGCost. The estimator is an explicit conditional average (BertsekasCondExpVec, BertsekasLQGEstimate); the gains come from the time-varying Riccati recursion (BertsekasLQGRiccati, BertsekasLQGGain).
— BertsekasDP.lqg_certainty_equivalence (goal). Milestone: Lemma 5.2.1 in pointwise form — the error is the same under any two policies, outcome by outcome (lqg_estimation_error_policy_independent).
This is the theorem that justifies designing estimator and controller separately — remove it and the entire LQG methodology loses its warrant. The formalization also yields the first machine-checked instance of the informational decomposition (control-dependent part + policy-independent error) that recurs throughout imperfect-information control. Notably the result needs no Gaussian assumption — only zero mean and independence — and the finite-support model makes that generality exact. The result is classical (Joseph–Tou 1961, Gunckel–Franklin 1963; the book's §5.2); the formal proof is new.
The heart is Lemma 5.2.1: showing the estimation error coincides, sample by sample, with the error of the control-free system — which requires proving that the observation-history σ-events under any policy coincide with those of the control-free system (controls are determined by the history, so they shift observations by a known amount). Then the DP argument over information histories must carry the quadratic decomposition through the backward recursion. Bookkeeping over histories-as-lists is the main formal burden; probability theory stays finite.
Finite-support randomness (all expectations are finite sums); conditional expectation with the explicit junk value 0 on zero-probability events — the goal's hypothesis is accordingly restricted to outcomes of positive probability. Policies are functions of the measurement list only (equivalent to the book's information vector for deterministic policies, since past controls are recoverable from past measurements). Matrices are time-varying; positive definiteness of makes every matrix inverse in the gains genuine. Measurement noise covariance is not assumed positive definite — the estimator is the abstract conditional expectation, not the Kalman filter (whose recursive form, §5.2.1, would be a natural follow-up mission).
Label correcting methods are the workhorse family of shortest-path algorithms — Dijkstra's method, Bellman–Ford, SLF/LLL variants and A* all fit the template analyzed in §2.3.1 of Bertsekas, Dynamic Programming and Optimal Control, Vol. I (3rd ed., 2005), where shortest paths appear as the purely deterministic face of dynamic programming. The correctness proof (Prop. 2.3.1) is short on paper but genuinely nondeterministic — any node may be removed from the candidate list, children processed in any order — so a formal proof certifies a whole family of concrete algorithms at once.
A finite directed graph with arc set , real arc lengths , origin and destination (BertsekasSPGraph). Walks are nonempty node lists whose consecutive pairs are arcs (BertsekasIsWalkFrom), with length the sum of arc lengths (BertsekasWalkLength); the shortest distance is the infimum of walk lengths in the extended reals, if no walk exists (BertsekasShortestDistance). The standing assumption of §2.3: every cycle has nonnegative length (negative arcs allowed).
The algorithm state (BertsekasLCState) carries labels , the scalar UPPER, and the candidate list OPEN. Initially , all other labels , UPPER , OPEN . One iteration (BertsekasLCStep, nondeterministic): remove any from OPEN; for each child of in any order, if set , and put in OPEN if , or update UPPER if . The algorithm terminates when OPEN is empty.
for every execution, under the nonnegative arc length assumption of §2.3 ( for every arc) — BertsekasDP.label_correcting_correctness_of_nonneg_arcs (goal). Milestones: termination — no infinite execution exists, which needs only the weaker nonnegative-cycle assumption (label_correcting_terminates) — and the workhorse invariant that every finite label is the length of an actual walk from , which needs neither (label_correcting_invariant).
The nonnegative-arc hypothesis is essential and not a formalization artifact: the algorithm prunes with the test , and with a negative arc a longer prefix can still reach more cheaply, so the pruned node is never entered into OPEN. An earlier version of this mission's goal carried only the nonnegative-cycle assumption of §2.1 and was disproved by the counterexample , , , , (a graph with no cycles at all), where the algorithm terminates with while the shortest distance is . Exercise 2.7 of the source treats the nonnegative-cycle case, which requires a modified algorithm.
Prop. 2.3.1 certifies simultaneously breadth-first search, Dijkstra (best-first), depth-first and small-label-first variants — every removal discipline is one refinement of the nondeterministic relation. Formally, the development contributes a reusable small-step framework for label-setting/correcting algorithms on which sharper results (Dijkstra's single-pass property, A* admissibility, §2.3.3) can later be built. The result is classical; the formal content is the induction along the nondeterministic step relation.
Termination is the subtle half: labels do not decrease monotonically along the run in an obvious well-founded way; the book's argument counts the finitely many distinct walk lengths below a bound — this needs the nonnegative-cycle assumption and a careful bound relating labels to simple-path lengths. The invariant proof must thread through the fold over children within a single step.
Finite node type with decidable equality; arcs as a Finset of ordered pairs; lengths total on (only arc values matter). The step relation is fully nondeterministic in pivot choice and child order (a permutation quantifier); correctness quantifies over all reachable terminal states — there is no fixed schedule to exploit. Distances live in EReal, so the no-path case is the honest empty infimum, not a sentinel. The trivializing risk of restricting to nonnegative arcs is avoided: only cycles are constrained.
Compressing an operator to a closed subspace keeps the information visible within that subspace but can discard interactions with its orthogonal complement. The compression-rigidity question asks whether a particular equality detects that no such interactions were present. Its inputs are two commuting positive contractions and an ordinary strictly convex function of two real variables. The issue is the equality case, not the existence of a general operator inequality for every convex function.
The question was contributed by Boris Bilich to the CUHK-Shenzhen AI Math Problems collection and added on June 1, 2026. The original problem asks about operators on a Hilbert space without imposing finite dimension. The first formal target in this mission treated matrices with supplied joint spectral data. That finite-dimensional declaration has a verified proof on Prove2Me, but it does not settle the unrestricted Hilbert-space question. The September 2026 correction restores arbitrary complex Hilbert spaces as the main target and preserves the earlier result as a supporting artifact.
Let be a complete complex Hilbert space, and let be its algebra of bounded complex-linear operators. The multiplication means composition, with acting first, and denotes the adjoint. A positive contraction is self-adjoint, satisfies for every , and has operator norm at most one. Both zero and the identity are permitted.
Write . Let be positive contractions satisfying . Their joint continuous functional calculus assigns an operator to each continuous real function on . It is characterized as a continuous unital real star-algebra homomorphism from to that maps the two coordinate functions to and . The domain has the uniform norm and the codomain the operator norm. The existence and uniqueness of this calculus follow from the standard joint spectral theorem; the relevant source is Dereziński, Lemma 6.6, printed page 43.
An orthogonal projection is an operator satisfying and . The compressed operators and are still viewed as operators on the original space , not as operators on a separately chosen finite-dimensional range. The question includes the additional hypothesis that these compressed operators commute. The original commutation of and does not remove the need to state this hypothesis.
The main target is the entire compression implication from the source. Let be continuous and strictly convex: for distinct and , its value at is strictly less than . For as above, the question is whether
Both outer projections on the right are part of the original assertion. There is no hypothesis that . Commutation of with each coordinate operator is exactly the reducing-subspace conclusion asked for in the source.
The accompanying standard infrastructure target asserts that, for every commuting pair of positive contractions on ,
where is continuous, unital, real-linear, multiplicative and star-preserving. This is the unit-square, real-valued-function specialization of Lemma 6.6, not a claim that this known theorem is a new research conjecture. Its Lean proof is a separate supporting obligation. The earlier two-dimensional milestone and the proved finite-dimensional capstone remain available; neither replaces the new main target.
A positive resolution would show that exact preservation of one strictly convex functional-calculus value, under the specified commuting-compression hypothesis, forces both operators to preserve the projection's range and its orthogonal complement. A negative resolution would require an actual Hilbert space, operators and strictly convex function satisfying every hypothesis while at least one of the two reducing identities fails.
The formal development separates this research question from the standard spectral infrastructure needed to express it. The new main declaration is an open proof obligation. The joint-calculus existence-and-uniqueness declaration is also unproved in this contribution, although mathematically standard. Local compilation and server publication check the declarations' well-formedness; they are not proofs of either statement. Only the earlier finite-dimensional result is being reported here as already proved.
Arbitrary bounded commuting self-adjoint operators need not have a joint eigenbasis. Consequently a matrix formulation that records finitely many joint spectral atoms cannot serve as the general operator model. The compressed pair may also have different spectral data from the original pair. The equality involves these two different functional calculi, with a projection on either side of each value.
Strict convexity in this question is ordinary scalar strict convexity on the square. Operator convexity, finite rank of the projection, compactness of the coordinate operators, and a multivariable operator Jensen inequality are not additional assumptions. Introducing any of them would change the requested question. The general formulation must also retain boundary cases rather than exclude them to simplify an argument.
The Lean model uses actual bounded complex-linear maps on an arbitrary complete inner-product space. It imposes no finite-dimensionality, separability, common-eigenbasis or nonzero-space assumption. It includes , and the zero Hilbert space. The scalar field convention is complex; a real-Hilbert-space transfer is not separately formalized here.
The function is stored on the ambient real plane, but continuity, strict convexity and evaluation use only its restriction to . Values outside are irrelevant, and continuity outside the square is not required. Thus storing an ambient function does not exclude any continuous function originally defined only on the square.
Joint evaluation is a total definition. If a representing continuous unital real star-algebra homomorphism exists, it chooses one for the operator pair and then evaluates the supplied function. Otherwise it returns zero. The separate existence-and-uniqueness target establishes that this fallback is inapplicable to commuting positive contractions and that the choice is immaterial. No field in the model assumes the compression-rigidity conclusion. The supporting standard theorem must also apply to the compressed pair using the original projection and positivity hypotheses, without adding representation existence as a new restriction on the main question.
The replacement definition and both statements were built at Lean 4.30.0 with supported Mathlib revision c5ea00351c28e24afc9f0f84379aa41082b1188f, and their final versions received independent blind readbacks. Published older declarations and their proof identities are preserved. They should be cited with their finite-dimensional scope, not described as a solution of the arbitrary-Hilbert-space target.
The fundamental group is the first algebraic invariant a student of topology meets, and is the first computation of it that carries real content. Allen Hatcher's Algebraic Topology (Cambridge University Press, 2002; freely available at pi.math.cornell.edu/~hatcher/AT/AT.pdf) is the standard text on the subject. Its Chapter 1 opens with exactly this computation (Theorem 1.7, p. 29) and immediately draws three classical consequences from it: the Fundamental Theorem of Algebra (Theorem 1.8), the Brouwer fixed point theorem for the disk (Theorem 1.9), and the Borsuk–Ulam theorem for the sphere (Theorem 1.10).
This mission is the opening entry in a series that formalizes Hatcher's book capstone by capstone. It covers the subsection "The Fundamental Group of the Circle" of Section 1.1 (pp. 29–33): the covering-space lifting properties that drive the proof, the theorem itself, and its three applications. Later entries in the series (van Kampen's theorem, the classification of covering spaces, simplicial and singular homology) will build on the declarations introduced here, which all live in the shared Lean namespace Hatcher.
A path in a topological space is a continuous map , where . A homotopy of paths is a family , , such that the endpoints and are independent of and the associated map , , is continuous. A loop at a basepoint is a path with . The set of homotopy classes of loops at is the fundamental group ; its product is , where traverses and then , each at double speed (Hatcher, Proposition 1.3).
The circle is realised as the unit circle of , so the point is and the basepoint is . Hatcher's map
is Hatcher.circleCover. The loops
based at are Hatcher.omegaLoopN n, and is Hatcher.omegaLoop; its class is Hatcher.omegaClass.
A covering space of is a space together with a map such that every has an open neighbourhood for which is a disjoint union of open sets each mapped homeomorphically onto by (Hatcher's condition , p. 29; such a is evenly covered). A lift of a map is a map with .
is an infinite cyclic group generated by . In the form stated in Lean:
Surjectivity of says generates; uniqueness of says the group is infinite cyclic rather than finite.
The result itself. The computation assigns to every loop in the circle an integer, its winding number, and shows that this integer is the only homotopy invariant of the loop. It is the seed of degree theory, and in Hatcher's text it is the starting point for every later computation of fundamental groups (products, van Kampen, covering spaces). The three applications are the standard demonstration that a single algebraic invariant can settle purely geometric or algebraic existence questions.
Formalizing it. Mathlib (revision 0df444a) already contains the covering-space infrastructure: IsCoveringMap, path lifting (IsCoveringMap.liftPath, eq_liftPath_iff'), homotopy lifting (IsCoveringMap.liftHomotopy, eq_liftHomotopy_iff'), monodromy, and the fact that Circle.exp is a covering map (Circle.isCoveringMap_exp). It also has FundamentalGroup X x as the endomorphism group of the fundamental groupoid. It does not contain the computation , nor the two-dimensional Brouwer and Borsuk–Ulam theorems. The Fundamental Theorem of Algebra is in Mathlib as Complex.exists_root (proved by Liouville's theorem rather than by Hatcher's argument); it is kept as a milestone because it is one of the section's stated theorems, and a solver may close it directly from Mathlib. Milestones 2–4 are also within reach of the existing lifting API, but they are the lemmas Hatcher states and uses, and a faithful record of them in the mission's own namespace is what later entries in the series will import.
The obvious first idea for the goal is to define the winding number of a loop through the complex argument. That fails because is discontinuous on ; the integer has to be produced by lifting the loop through and reading off the endpoint of the lift, which is only well defined because of the uniqueness in the path lifting property. The second difficulty is uniqueness of : this needs lifting of homotopies (milestone 4), not just of paths, together with the observation that a lifted homotopy of paths has constant endpoints.
Connecting the concrete loops to Mathlib's abstract is its own obstacle. FundamentalGroup Circle 1 multiplies by composing morphisms of the fundamental groupoid, so identifying with the class of the explicit loop (milestone 6) requires reparametrization arguments for concatenated paths, for negative as well as positive.
For Theorem 1.9 the difficulty is the construction and continuity of the retraction from a fixed-point-free map, and then the non-existence of a retraction, which uses that . For Theorem 1.10 Hatcher's proof lifts a loop through and shows the lift changes by an odd integer over half a turn; making that parity argument rigorous in Lean is the substance of the milestone.
Circle (the unit circle in ) with basepoint 1; is Metric.closedBall (0 : EuclideanSpace ℝ (Fin 2)) 1; is Metric.sphere (0 : EuclideanSpace ℝ (Fin 3)) 1, with the antipodal point.IsCoveringMap p. This agrees with Hatcher's condition ; neither requires to be surjective.C(I, X) or Mathlib Paths; for homotopies of paths, the square is written I × I with Hatcher's coordinate order : the first coordinate is the path parameter, the second the homotopy parameter. In the general homotopy lifting property the domain is Y × I with an arbitrary topological space, as in Hatcher.FundamentalGroup Circle 1, and is FundamentalGroup.fromPath ⟦omegaLoop⟧. Because the goal quantifies over integer powers of a single element, the order of multiplication in FundamentalGroup is immaterial to its truth.FundamentalGroup Circle 1 ≃* Multiplicative ℤ sending to is an immediate corollary and a welcome contribution.0 < f.degree, which excludes both the zero polynomial and nonzero constants.Contributions welcome: proofs of the milestones from Mathlib's lifting API, a degree homomorphism packaged for reuse, and any lemma about concatenation and reparametrization of loops in Circle that later chapters of the series can import.
Mathlib/Topology/Homotopy/Lifting.lean (path and homotopy lifting for covering maps). https://github.com/leanprover-community/mathlib4/blob/0df444a360eaa60ab8c11dca51a86af692955474/Mathlib/Topology/Homotopy/Lifting.leanMathlib/AlgebraicTopology/FundamentalGroupoid/FundamentalGroup.lean (the fundamental group). https://github.com/leanprover-community/mathlib4/blob/0df444a360eaa60ab8c11dca51a86af692955474/Mathlib/AlgebraicTopology/FundamentalGroupoid/FundamentalGroup.leanThe strategic-form game is the basic object of noncooperative game theory, and the Nash equilibrium — a profile of randomized strategies from which no player benefits by deviating unilaterally — is its central solution concept. Nash proved in 1951 that every game with finitely many players and finite strategy sets has such an equilibrium (Nash, Non-cooperative games, Ann. Math. 54 (1951)); this single existence theorem is the reason the concept organizes the rest of the field, from the computational complexity of finding equilibria to the price of anarchy. The theorem is stated as Theorem 1.8 of Nisan–Roughgarden–Tardos–Vazirani (eds.), Algorithmic Game Theory (Cambridge, 2007), the source text of this mission series, whose first chapter (Tardos–Vazirani) also treats the two special cases that admit direct algorithmic proofs: two-person zero-sum games, where equilibria are exactly the optimal solutions of a dual pair of linear programs (von Neumann 1928; Theorem 1.11), and a simple linear market, where equilibrium prices are computed by an ascending tight-set algorithm (Theorem 1.17).
A timeline of the existence theorem: von Neumann (1928) proved the minimax theorem for two-person zero-sum games; Nash (1950, 1951) extended existence to arbitrary finite games, first via Kakutani's fixed-point theorem and then via Brouwer's. All known proofs of the general theorem pass through a fixed-point principle, and this is not an artifact: computing a Nash equilibrium is PPAD-complete (Daskalakis–Goldberg–Papadimitriou 2009; Chen–Deng–Teng 2009), and PPAD is precisely the complexity class of the fixed-point arguments.
A finite strategic-form game consists of a finite set of players, for each player a finite nonempty set of pure strategies, and for each player a payoff function ; all players are utility maximizers. A mixed strategy for player is a probability distribution on , represented as a weight function with and (a lottery). Players randomize independently, so a mixed profile induces the product distribution on pure strategy vectors, and player 's expected payoff is
A mixed profile is a (mixed) Nash equilibrium if for every player and every lottery on , replacing by does not increase .
A two-person zero-sum game is given by a matrix : the row player picks a row distribution , the column player a column distribution , and the column player pays the row player in expectation.
The market of §1.8.1 of the source has finitely many divisible goods, good in units, and finitely many buyers, buyer bringing budget and interested in a nonempty set of goods; utilities are linear 0/1, so a buyer wants any goods from her interest set and none other. Market-clearing prices are positive prices under which each buyer can spend her whole budget on cheapest goods in her interest set while every good sells out exactly.
Stated for an arbitrary finite family of finite nonempty strategy types; no bound on the number of players, no genericity assumptions.
Mathlib currently has no form of Brouwer's theorem; every known proof of Theorem 1.8 needs it (or an equivalent), so it enters the mission as an explicit milestone rather than an assumed library fact.
of the explicit two-player game with payoffs to the row player and to the column player. The source states the result as: optimal solutions of a dual pair of LPs form a Nash equilibrium of the zero-sum game; the first two conjuncts are the saddle point that LP optimality amounts to, and the third states the Nash-equilibrium clause against the mission's own game vocabulary, so "zero-sum" is formal (the two payoffs sum to zero) rather than implicit in the shape of the statement.
The source proves this by an ascending-price algorithm and also bounds its running time; the complexity half has no formal counterpart in this mission.
The capstone is the foundation of the whole mission series: correlated equilibria, price-of-anarchy bounds, and mechanism-design characterizations in later missions all quantify over or compare against Nash equilibria, and the series inherits its game vocabulary (IsLottery, IsMixedProfile, expectedPayoff, IsMixedNash) from this mission.
Formalizing it produces the first Brouwer fixed-point theorem in this environment — a well-known gap in mathlib with reuse value far beyond game theory (every degree-theoretic and equilibrium-existence argument needs it). The zero-sum milestone yields the minimax theorem, reusable for the learning-dynamics mission that follows. All results here are classical and proved on paper; the work requested is machine-checked proof, not new mathematics.
The central difficulty is Brouwer. The standard routes are (i) Sperner's lemma plus a limit argument, which needs a formal theory of simplicial subdivisions that does not exist in mathlib; (ii) algebraic topology (no retraction of the ball onto the sphere), for which mathlib has singular homology but not yet the homology of spheres in usable form; (iii) analytic proofs (Milnor–Rogers). None is short; the milestone is deliberately stated for a general nonempty compact convex set in a finite-dimensional normed space so that any route serves, and so the lemma lands in reusable generality.
Given Brouwer, Theorem 1.8 still requires Nash's gain-function construction on the product of simplices and the verification that fixed points are equilibria — bookkeeping-heavy but standard. Theorem 1.11 does not need Brouwer: mathlib's Sion minimax theorem (Mathlib.Topology.Sion) applies to the bilinear payoff on the product of standard simplices, or one can argue by LP duality directly. Theorem 1.17 needs the tight-set/max-flow argument of Lemmas 1.15–1.16 or any direct construction of the equilibrium.
Games are presented concretely: players form a finite index type, strategies a finite type per player, payoffs are functions into ; mixed strategies are weight functions with a IsLottery predicate, not measure-theoretic distributions. Deviations in the equilibrium definition range over all lotteries (not only pure strategies): the pure-deviation reduction is a lemma a solver may prove, not part of the definition. Strategy sets are assumed nonempty in the capstone; the player set need not be. In the zero-sum milestone both dimensions are positive (Fin (m+1), Fin (n+1)), payoffs flow from the column player to the row player, stdSimplex plays the role of the mixed-strategy space, and the Nash-equilibrium conjunct is stated for the Boolean-indexed two-player game built by matrixGameStrat/zeroSumPayoff/matrixGameProfile from the definitions bundle. In the market milestone all supplies and budgets are positive, every buyer's interest set is nonempty, and every good has an interested buyer, matching the standing assumptions of §1.8.1; allocations are recorded as money spent, so the clearing condition is with no division anywhere.
Trivializing readings are ruled out: the empty simplex has no lotteries, so nonemptiness hypotheses appear exactly where their absence would make an existence claim false (Brouwer on the empty set, games with an empty strategy set, zero-dimensional matrix games).
The matrix-multiplication exponent measures the asymptotic arithmetic cost of multiplying square matrices. An upper bound means that, over the field under consideration, matrices can be multiplied using arithmetic operations for every . Improvements to are a central benchmark in algebraic complexity because matrix multiplication is also a basic subroutine in linear algebra, graph algorithms, and symbolic computation.
The existing Prove2Me mission formalizes Schönhage's bound from a concrete two-summand tensor degeneration. The present mission advances the same formal development to the next clean historical construction. Pan and Winograd found a simultaneous approximate algorithm for three matrix products; Romani recorded its tensor form and the parameter choice , , which gives . Schönhage's 1981 paper reports the equivalent bound in the arbitrary-field setting. The exact formal target here is the slightly weaker rational inequality .
For a field , the matrix-multiplication tensor encodes multiplication of an matrix by a matrix:
A direct sum places several such tensors in disjoint coordinate blocks. A tensor has border rank at most when it is a polynomial degeneration of the diagonal tensor . In the Lean development this relation is Degenerates T (TensorObj.diagObj K 3 r). The argument order matters: the first tensor is the target and the diagonal tensor is the source.
The platform already defines ordinary tensor rank, asymptotic tensor rank, the tensor-rank exponent matMulExp K, the equivalent Strassen-preorder exponent matMulExp_strassen K, and Schönhage's asymptotic sum inequality. This mission reuses those declarations. No alternative definition of is introduced.
The goal has exactly the same quantified proposition as the existing mission, with only the rational endpoint changed:
The source construction to be formalized is
Each summand has volume :
The milestone chain records the degeneration, its asymptotic-rank consequence, the exact numerical implication
and the resulting Strassen-form exponent bound. The public goal then transfers the bound to matMulExp K through the already established equality of the two exponent definitions.
Mathematically, this construction improves the concrete exponent certified by the existing mission from to without changing the surrounding theory. It isolates the first genuinely new ingredient after the accepted Schönhage example: a larger simultaneous tensor degeneration rather than a sharper numerical estimate for the old witness.
For formalization, the mission tests whether the current polynomial-degeneration API can express a historically important trilinear aggregation at realistic scale. Once the explicit witness is available, the remaining declarations form a reusable template for later bounds: a source tensor degeneration, an asymptotic-rank bound, a specialization of the asymptotic sum inequality, and a final exponent transfer. This creates a trustworthy stepping stone toward the Coppersmith--Winograd tensor and later laser-method analyses.
The theorem is known mathematically; the open work is its machine-checked Lean formalization. The exact numerical endpoint and every downstream bridge from the degeneration have already been checked locally. The explicit Pan--Winograd degeneration remains the substantive open milestone.
The central difficulty is not the logarithmic comparison. It is constructing and verifying the polynomial family whose leading nonzero coefficient is exactly the tagged direct sum of the three matrix-multiplication tensors and whose earlier coefficients vanish. The family has diagonal source slots and many indexed target coordinates. A proof must account for all mixed-coordinate terms and all cancellations uniformly over an arbitrary field.
Romani's published summary states the approximate-rank inequality but does not spell out a Lean-ready map between its trilinear forms and the platform's TensorObj.bigAdd coordinate spaces. A solver must therefore recover the source indexing carefully and prove that the resulting modewise linear maps have the required coefficients. Reversing the degeneration direction, conflating tensor rank with asymptotic rank, or silently assuming a characteristic-zero scalar identity would invalidate the result.
All theorems quantify over an arbitrary type with [Field K], matching the existing Schönhage goal and the arbitrary-field statement of the source bound. Tensor spaces are finite-dimensional function spaces already packaged by MMObj; the three products are combined with TensorObj.bigAdd. Border rank is represented by the existing finitely supported polynomial-family predicate Degenerates. Because the source summary specifies approximate rank but not a leading order, the main degeneration milestone existentially quantifies that order instead of hard-coding one.
The mission includes no placeholder laser-value definition and makes no claim about the later analysis. It also excludes Schönhage's additional microscopic symmetrization improvement beyond . A valid solution must construct the stated degeneration itself; a vacuous hypothesis or a redefinition of matMulExp is outside scope.
Reusable contributions include coefficient lemmas for polynomial tensor families, finite-index equivalences for direct sums, and generic aggregation identities that specialize to the , witness. Contributions that merely restate the target under stronger field hypotheses do not close the arbitrary-field milestone.
The coupling method of Mission III asks for a coupling of two copies of a chain from every pair of starting states — often painful to construct globally. Chapter 14 of Levin–Peres–Wilmer, Markov Chains and Mixing Times (AMS, 2009) replaces that global demand by a local one. The path coupling technique of Bubley and Dyer says: put a connected graph structure on the state space, and couple one step of the chain only across edges; if each edge contracts in expectation, contraction propagates automatically along paths to arbitrary pairs of distributions. The bookkeeping runs through the transportation metric (Kantorovich distance) between distributions, whose theory — attainment by an optimal coupling, the triangle inequality — is developed on the way. The chapter's payoff is the sharpest elementary bound for sampling proper colorings (Theorem 14.8: the Glauber dynamics mixes in steps once ), and, through the sampling-to-counting reduction of Jerrum–Valiant–Vazirani, a polynomial-time approximation algorithm for counting colorings — the paradigm of the Markov chain Monte Carlo method as an algorithmic tool.
All chains live on a finite state space with a transition matrix ; is the time- distribution from , the total variation distance, the worst-case distance to the stationary distribution , and the mixing time. A coupling of distributions is a distribution on with marginals and .
Given a metric-like cost on pairs of states, the transportation metric between two distributions is the cheapest expected cost of moving one onto the other:
Given a connected graph structure on the state space with edge lengths , the path metric is the least total length of a -path from to .
For the colorings application: a -coloring of the vertices of a graph is proper when adjacent vertices receive distinct colors, and the Glauber dynamics on proper colorings picks a uniform vertex and re-samples its color uniformly among the colors legal there; its stationary distribution is uniform on the proper colorings. Throughout, is the number of vertices and the maximum degree of the graph being colored.
Theorem 14.8, the capstone of Chapter 14: for the Glauber dynamics on proper -colorings, if then
The results. Path coupling converted the coupling method from an art into a calculus: one bounds a single-edge contraction constant, and the machinery does the rest. It is the standard tool for Glauber dynamics on colorings, independent sets, and other constraint-satisfaction models, and the colorings bound is its flagship application. Theorem 14.12 is the discrete embodiment of the Jerrum–Valiant–Vazirani equivalence between approximate counting and sampling — the conceptual foundation of the entire MCMC approach to -hard counting problems.
Formalizing them. Mathlib has no transportation/Kantorovich metric in the finite setting, no path coupling, and nothing on approximate counting. The transportation-metric layer (optimal couplings, triangle inequality) is reusable far beyond this mission — it is the finite Wasserstein distance. The path-coupling theorem feeds directly into Mission IX (Ising) and is quoted throughout modern mixing literature.
The transportation metric asks for minimization over the (compact) polytope of couplings: attainment is a finite-dimensional compactness argument, and the triangle inequality requires gluing two optimal couplings along their common marginal — the classic construction that must be carried out with explicit finite sums here. Path coupling itself is an induction along geodesics of the path metric, with the subtlety that the composite coupling produced along a path need not be optimal, only admissible; the bookkeeping of the contraction constant through the induction is exactly the kind of argument Lean keeps honest. Theorem 14.8 instantiates the machinery: the single-edge coupling for colorings needs a careful case analysis of the proposed recolorings at the two endpoints (matching legal colors bijectively), and the contraction constant emerges from counting disagreeing proposals. Theorem 14.12 layers a probabilistic-amplification argument (medians of means over independent runs) on top of the mixing bound; its combinatorial core — expressing as a telescoping product of marginal probabilities — is elementary but notation-heavy, and the formal statement quantifies over explicit seed spaces, so the whole estimator is a finite object.
The transportation metric is an sInf over coupling costs (the coupling polytope is nonempty for genuine distributions, and attainment is part of the milestone, so the junk value never propagates); the path metric is an sInf over walk lengths in a connected graph. The Glauber dynamics on colorings is the restriction to proper colorings of the single-site heat-bath chain of Mission II, matching §3.3 of the book; its state space is the subtype of proper colorings, nonempty whenever (a fact the hypotheses of the goal supply). Mixing-time upper bounds are stated with the book's explicit ceilings, so no rounding slack is hidden. In Theorem 14.12 the estimator is presented concretely as a function of finitely many uniform seeds, and "with probability " is a counting inequality over the seed space — no measure theory enters. Contributions of intermediate lemmas (optimal-coupling gluing, geodesic decompositions, colorings edge-coupling) are welcome and will be reused by Mission IX.
A reversible Markov chain is an electrical network: states are nodes, and the conductance turns hitting probabilities into voltages and hitting times into resistances. This dictionary, going back to Kakutani and popularized by Doyle and Snell, converts probabilistic estimates into the physical laws of circuits — series/parallel reduction, energy minimization, monotonicity under edge removal. Chapters 9–11 of Levin–Peres–Wilmer, Markov Chains and Mixing Times (AMS, 2009) develop the dictionary and its two crown results: the commute time identity of Chandra–Raghavan–Ruzzo–Smolensky–Tiwari, , and the Matthews method bounding cover times by hitting times with harmonic-number precision.
A network is a symmetric nonnegative conductance function on pairs of vertices; the associated walk moves with probabilities
where , and . A function is harmonic at if . The voltage with boundary values at and at is , the harmonic extension of its boundary data; the current flowing out of has strength , and the effective resistance is . A flow from to is an antisymmetric edge function obeying the node law off ; its energy is . Hitting times , their expectations, the Green's function , the maximal hitting time , and the cover time (expected time to visit every state, maximized over starts) all use the trajectory calculus of Mission I.
This is Proposition 10.6, the commute time identity — the exact bridge between the probabilistic and electrical sides, and the engine of the transience/recurrence theory of Mission XII.
Reversibility and stationarity of the network walk with (§9.1); Proposition 9.1 (existence and uniqueness of harmonic extensions with given boundary values, ); Lemma 9.6 (the Green's function identity ); Theorem 9.10 (Thomson's principle: is the minimal energy of a unit flow, attained); Theorem 9.12 (Rayleigh monotonicity: lowering conductances raises effective resistance); Lemma 10.1 (the random target lemma: does not depend on ); Corollary 10.8 (the resistance triangle inequality); Theorem 11.2 (Matthews: ); Proposition 11.4 (the matching Matthews lower bound over subsets).
The results. The commute time identity computes hitting times from circuit reductions — this is how hitting times on trees, tori and glued graphs are actually evaluated — and, through Thomson and Rayleigh, makes them monotone under graph operations, something invisible probabilistically. The Matthews bounds pin cover times up to a factor in complete generality; they are the tool behind cover-time results for lamplighter groups in Mission XI's sequel. Green's function identities feed Mission XII's recurrence theory, where decides transience.
Formalizing them. Mathlib has graph Laplacians but no electrical network theory: no effective resistance, no flows, no energy, no Thomson/Rayleigh, no hitting or cover times. This mission publishes that layer over the trajectory calculus of Mission I. It is the most reusable single block of the series outside Missions I–II: effective resistance on finite networks is of independent interest to combinatorics (spanning trees, spectral sparsification) well beyond mixing times.
The identity chain behind the goal runs: Green's function of the stopped walk escape probability (via harmonic uniqueness) the Aldous–Fill occupation identity for stopping times with — each step is a manipulation of infinite series of trajectory sums whose exchange steps (splitting a path at its first visit, last-exit decompositions) need summability from Mission I's Lemma 1.13. Thomson's principle is a finite-dimensional convex minimization: existence of the minimizer needs a compactness or completing-the-square argument, and the identification of the minimizer with the current flow needs the cycle law; the naive "differentiate the energy" route must be made exact. Matthews' method is a clean but genuinely clever argument — a uniformly random ordering of targets and the harmonic-number telescoping; the formal cost is the exchangeability of the randomized order against the chain, handled combinatorially.
Networks are functions with a symmetry-and-nonnegativity predicate; loops are permitted; connectivity enters as irreducibility of the induced walk. The voltage is defined probabilistically as (the book's harmonic characterization is Proposition 9.1); is the reciprocal of the explicit current strength, with total division junk when are disconnected — statements carry irreducibility so this does not arise. Energy counts each undirected edge once, formalized as half the ordered double sum, and handles absent edges. Cover times are tail sums of the explicit event "some state unvisited". The Matthews lower bound is stated with an arbitrary lower bound for the pairwise hitting times of the subset — equivalent to the book's min over pairs and easier to instantiate.
Welcome contributions: series/parallel reduction laws, the cycle and node law API for flows, escape-probability lemmas — all reused in Mission XII's infinite-network arguments.
Modern A/B tests must infer lifetime treatment effects — e.g. customer lifetime value under a new feature — from short-horizon experiment data. Chen, Simchi-Levi and Wang (arXiv:2407.19618) model the experiment as a Markov decision process and exploit a structural fact of many practical interventions: the treatment is local, modifying the system at a single crucial state only. This mission formalizes the core asymptotic theory of the paper: for any differentiable estimator built from the experiment's transition and reward statistics, information sharing — pooling across test arms the samples collected away from the treated state — keeps the estimator asymptotically normal with the same asymptotic bias and never increases its asymptotic variance (Theorem 9), and is asymptotically efficient among unbiased estimators (Theorem 5). The route runs through a Markov chain central limit theorem with the asymptotic variance identified as the autocovariance series, and the linearization/delta method for functionals of chain statistics.
Finite Markov chains are the basic model for memoryless random dynamics: card shuffles, random walks on graphs and groups, Monte Carlo samplers, and queueing systems are all chains on a finite state space. The single most used fact about them is that an irreducible chain has exactly one stationary distribution — a probability vector with — and that this is strictly positive and encodes the long-run behaviour of the chain through the return-time identity . Every later result in the theory of mixing times (convergence theorems, coupling bounds, spectral methods, cutoff) is a statement about the distance of the chain from this , so nothing in the subject can be formalized before this mission is.
This mission is the first in a series formalizing D. A. Levin, Y. Peres and E. L. Wilmer, Markov Chains and Mixing Times (AMS, 2009), covering Chapters 1–2: the basic vocabulary of finite chains (stochastic matrices, irreducibility, period, reversibility, time reversal, random walks on graphs and groups) and the classical examples of Chapter 2 (gambler's ruin, coupon collecting, the reflection principle for simple random walk on ). Later missions in the series build on the definitions published here.
A chain on a finite state space is presented by its transition matrix, a matrix with nonnegative entries whose rows sum to . A distribution is a row vector with nonnegative entries summing to ; one step of the chain carries to , and the -step transition probabilities are the entries of the matrix power .
The chain is irreducible if for all states there is a with . The period of a state is where , and the chain is aperiodic if every state has period . A distribution is stationary if , and and are in detailed balance (the chain is reversible) if for all .
Trajectory events over a finite horizon are finite sums of path weights: a length- trajectory is a function , with weight conditional on its starting state. The tail probability of the first hitting time is the sum of the weights of the trajectories from that avoid at times , and expectations of hitting times are recovered by the tail-sum formula , formalized as a tsum over .
This is Corollary 1.17 of the book. It asserts only existence and uniqueness, leaving the finer structure of to the milestones; it is the weakest statement on which the rest of the series can stand, which is why it is the goal.
The milestone list follows the book's own route: well-definedness of the period (Lemma 1.6), positivity of some matrix power for irreducible aperiodic chains (Proposition 1.7), finiteness of expected hitting times (Lemma 1.13), existence of a positive stationary distribution together with
(Proposition 1.14), constancy of harmonic functions (Lemma 1.16), stationarity from detailed balance (Proposition 1.19), the stationary and reversible measure of simple random walk on a graph (Examples 1.12 and 1.20), the time reversal and its path-reversal identity (Proposition 1.22), and the random walks on finite groups of Section 2.6 (Propositions 2.12–2.14). From Chapter 2 the list adds the gambler's ruin formulas and (Proposition 2.1), the coupon collector expectation and tail bound (Propositions 2.3 and 2.4), and the reflection principle and the bound for simple random walk on (Lemma 2.18 and Theorem 2.17).
The result itself. Existence and uniqueness of is the pivot on which the entire quantitative theory turns: it defines the target of convergence, and the identity ties the stationary measure to return times, which later missions use for hitting-time and cover-time results. Detailed balance is the practical tool by which stationary measures of graph and group walks are computed, and the Chapter 2 examples (gambler's ruin, coupon collecting, reflection) are the standard building blocks reused throughout the book — the coupon collector bound, for instance, is exactly the estimate behind the analysis of the top-to-random shuffle in a later mission of this series.
Formalizing it. Mathlib currently has no theory of finite Markov chains: no stochastic-matrix predicate, no stationary distribution, no periodicity, no hitting times. Everything proved in this mission is new formal mathematics, and the definition layer published here (mm_basic, mm_path, mm_classical) is the shared foundation that all twelve subsequent missions of the series import. All results are classical and have textbook proofs; none has a machine-checked proof.
The delicate point is the existence proof. The natural first idea — extract from an eigenvector of for eigenvalue , or invoke a fixed-point theorem — either does not give positivity and nonnegativity without further work, or uses compactness machinery (Brouwer) that is unavailable. The book's proof instead builds and verifies by reindexing trajectory sums; formalizing it requires managing infinite series of path sums (summability from the geometric tail bound of Lemma 1.13, exchanging tsum with finite sums, splitting a trajectory at its last step). The uniqueness half is linear algebra via constancy of harmonic functions (Lemma 1.16), which is elementary but requires a maximum-principle argument over a finite state space. The reflection principle and Theorem 2.17 are finite combinatorics on paths — the bijection is easy to describe and fiddly to implement.
States form a Fintype with decidable equality; chains are Matrix V V ℝ with the row-stochasticity predicate IsStochastic; distributions are functions V → ℝ with the predicate IsDist. Everything is distribution-side: no probability space or measure theory is used. The period is formalized as , which equals when and takes the junk value otherwise. Expectations of hitting times are tsums of tail probabilities, with the usual junk value for non-summable families — the statements are arranged (e.g. multiplicatively, ) so that junk values cannot make them vacuously true. Existence statements carry a Nonempty V hypothesis; irreducibility on the empty space is vacuous, and without nonemptiness the goal would be false, not trivial. The coupon collector and the walk on are presented directly by their driving randomness (uniform draws Fin t → Fin n, uniform sign strings Fin r → Bool), so those probabilities are elementary counting; in particular the reflection principle is stated as an equality of cardinalities of sets of sign strings — this is equivalent to the probabilistic statement because all strings are equally likely.
Contributions welcome beyond the milestone list: simp lemmas for the definition layer, the taboo-matrix representation of avoidance probabilities (useful for Lemma 1.13), and any interface lemmas connecting pathWeight sums to matrix powers — these will be reused by every later mission in the series.
The classical convergence theory of smooth convex minimization. For a function that is -strongly convex and -smooth (), gradient descent converges linearly, while Newton's method exhibits its famous two phases: a damped phase in which every backtracking step decreases the objective by a fixed amount , and a quadratically convergent phase in which the scaled gradient norm squares at each step, . Together they give the iteration count of B&V (9.36),
with the Lipschitz constant of the Hessian and the backtracking parameters. This mission formalizes Chapters 9–10 of Boyd & Vandenberghe with every constant exactly as printed — a quantitative theory entirely absent from Mathlib.
The Karush–Kuhn–Tucker conditions are the central result of convex optimization: for a convex differentiable problem satisfying Slater's condition, a point is optimal exactly when primal feasibility, dual feasibility, complementary slackness and Lagrangian stationarity hold. This mission formalizes Chapters 4–5 of Boyd & Vandenberghe end to end — the first-order optimality criterion, concavity of the Lagrange dual, weak duality, Slater's strong-duality theorem with dual attainment (via the separating-hyperplane argument of §5.3.2), the saddle-point characterization, sensitivity bounds and Pareto scalarization — culminating in the full KKT characterization.
Linear programming has a complete duality theory; integer programming does not — and the Lagrangean dual measures exactly how far duality reaches. This mission formalizes the duality theory of integer programming from Section 11.4 of Bertsimas–Tsitsiklis, built on the general linear programming duality of Section 4.10. For the integer program
with integer data, the complicating constraints are dualized with multipliers over the tractable set : the dual function is
and the Lagrangean dual is . Weak duality (Theorem 11.2) always holds, but strong duality can fail. The convex hull of the integer points of a polyhedron with integer data is itself a polyhedron (Theorem 11.3, Meyer's theorem), and the capstone — Theorem 11.4, the central result of Section 11.4 — identifies the Lagrangean dual exactly: equals the optimal cost of the linear program
. This is the geometric explanation of the strength of Lagrangean relaxation, yields the bound ordering , and Corollary 11.1 characterizes exactly when the bounds collapse. The polyhedral engine is the general weak/strong duality pair (Theorems 4.17/4.18) over a primal s.t. , , and the formulation-strength comparison of Theorem 10.1 supplies the motivating principle that tighter relaxations of the same integer set give sharper bounds.
How can an unbounded polyhedron be described by finitely many geometric objects? Sections 4.8-4.9 of Bertsimas-Tsitsiklis build the cone machinery: recession cones and their rays, extreme rays (defined, like basic solutions, by linearly independent active constraints), the pointedness criterion (Theorem 4.12: is an extreme point of a polyhedral cone iff the cone contains no line iff of the constraint vectors are linearly independent), and the characterization of unbounded linear programs (Theorems 4.13-4.14: over a pointed polyhedral cone, and then over any polyhedron with an extreme point, the optimal cost is iff some extreme ray has ). The capstone is the resolution theorem (Theorem 4.15): a nonempty polyhedron with at least one extreme point equals — the convex hull of its extreme points plus the cone generated by a complete set of its extreme rays. It specializes to Theorem 2.9 / Corollary 4.4 (a nonempty bounded polyhedron is the convex hull of its extreme points) and Corollary 4.5 (a pointed polyhedral cone is generated by its extreme rays). The converse, Theorem 4.16, states that every finitely generated set is a polyhedron — in particular the convex hull of finitely many vectors is a polyhedron. Together these form the Minkowski-Weyl equivalence of the two representations of polyhedra, verified absent from Mathlib and the genuine content of this mission.
Matrix completion — recovering a low-rank matrix from a small random subset of its entries — powers recommender systems and collaborative filtering. In practice it is solved by running (stochastic) gradient descent on the non-convex objective
where and is a certain regularizer. from a random starting point, and it just works.
Ge, Lee and Ma (NeurIPS 2016 Best student paper award) explained why: the regularized objective has no spurious local minima — every local minimum is global and exactly recovers . This mission formalizes that landmark theorem in Lean 4, in its strongest known form and along its simplest known proof: the unified landscape analysis of Ge–Jin–Zheng (ICML 2017) and an improved sampling bound in Chen–Li (JMLR 2019). Conditional on an explicit good-sample predicate (which holds with high probability under Bernoulli sampling), every local minimum of satisfies .
Bandit feedback is only one point on a spectrum: a learner might see more than its own loss (full information) or less (a spam filter never learns what happened to mail it deleted). Chapter 37 of Lattimore–Szepesvári studies finite adversarial games where the loss matrix and the feedback matrix are decoupled. The goal theorem is the celebrated classification theorem: every finite partial-monitoring game has minimax regret exactly , , or — determined by two purely combinatorial conditions, global and local observability, on the game's neighbourhood structure. A single geometric dichotomy thus governs the price of information in every online decision problem with finite actions and feedback.
When actions are feature vectors and the mean reward is linear — — a bandit can generalize across arms: pulling one arm reveals information about all of them. Chapter 19 of Lattimore–Szepesvári carries the optimism principle into this setting: LinUCB (a.k.a. OFUL) plays the action maximizing over the confidence ellipsoid of Mission IX. The goal theorem: with probability , — regret independent of the number of actions. The combinatorial engine is the elliptical potential lemma, bounding how many times adaptively chosen directions can be surprising. Chapter 22's phased elimination with G-optimal design (Mission IX) sharpens this to for finite action sets.
Least-squares estimation from adaptively collected data is the statistical heart of linear bandits: the actions depend on past noise, so classical fixed-design theory does not apply. Chapter 20 of Lattimore–Szepesvári resolves this with the method of mixtures: the process is a supermartingale, and integrating over a Gaussian mixture yields the self-normalized bound — the goal theorem — , valid uniformly over all times. The resulting confidence ellipsoids for the regularized least-squares estimator (Abbasi-Yadkori et al.) calibrate every algorithm of Mission X. The mission also formalizes the Kiefer–Wolfowitz theorem of Chapter 21: G-optimal and D-optimal experimental designs coincide, with optimal value exactly — the classical equivalence theorem of optimal design theory.
Coding theorems and (strong) converses for memoryless quantum communication channels and quantum sources are proved: for the quantum source the coding theorem is reviewed, and the strong converse proven. For classical information transmission via quantum channels we give a new proof of the coding theorem, and prove the strong converse, even under the extended model of nonstationary channels. As a by-product we obtain a new proof of the famous Holevo bound. Then multi-user systems are investigated, and the capacity region for the quantum multiple access channel is determined. The last chapter contains a preliminary discussion of some models of compression of correlated quantum sources, and a proposal for a program to obtain operational meaning for quantum conditional entropy. An appendix features the introduction of a notation and calculus of entropy in quantum systems.
This mission is the statement-level formalization of Coding Theorems of Quantum Information Theory (arXiv:quant-ph/9907077v1). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.
76 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Theorem I.1: Dbar²≤1−Fbar≤Dbar; for quantum encoders also 1−Fbar≤1−Fe. (Theorem I.1, PDF p. 8). The milestones are the other numbered results, in the paper's order, each quoting the paper.
The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.
Provenance: prove2quantum's pipeline drafted each statement (gpt-6.1-sol on Stanford's Sherlock cluster). A blind read-back, written without seeing the paper, is attached to every item. An independent moderator compared each statement against the paper and passed the whole paper without edits.
9 results of the paper are not stated yet, because the library lacks an interface they need:
In previous papers, we expressed the Entanglement of Formation in terms of Conditional Mutual Information (CMI). In this brief paper, we express the Entanglement of Distillation in terms of CMI.
This mission is the statement-level formalization of Entanglement of Distillation and Conditional Mutual Information (arXiv:quant-ph/0202144v2). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.
9 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Common-cause classical laws have zero CMI (§2, Eq.(5), PDF p. 3). The milestones are the other numbered results, in the paper's order, each quoting the paper.
The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.
Provenance: prove2quantum's pipeline drafted each statement (gpt-6.1-sol on Stanford's Sherlock cluster). A blind read-back, written without seeing the paper, is attached to every item. An independent moderator compared each statement against the paper and passed the whole paper without edits.
1 results of the paper are not stated yet, because the library lacks an interface they need:
These results of the paper are already proved in Mathlib, so they are not open goals here: InformationTheory.klDiv_comp_right_le.
We investigate entanglement measures in the infinite-dimensional regime. First, we discuss the peculiarities that may occur if the Hilbert space of a bi-partite system is infinite-dimensional, most notably the fact that the set of states with infinite entropy of entanglement is trace-norm dense in state space, implying that in any neighborhood of every product state there lies an arbitrarily strongly entangled state. The starting point for a clarification of this counterintuitive property is the observation that if one imposes the natural and physically reasonable constraint that the mean energy is bounded from above, then the entropy of entanglement becomes a trace-norm continuous functional. The considerations will then be extended to the asymptotic limit, and we will prove some asymptotic continuity properties. We proceed by investigating the entanglement of formation and the relative entropy of entanglement in the infinite-dimensional setting. Finally, we show that the set of entangled states is still trace-norm dense in state space, even under the constraint of a finite mean energy.
This mission is the statement-level formalization of On the quantification of entanglement in infinite-dimensional quantum systems (arXiv:quant-ph/0112064v3). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.
13 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Proposition 10: no separable ball under a strict mean-energy bound (Proposition 10: no separable ball under a strict mean-energy bound (seed paper result), PDF p. 13). The milestones are the other numbered results, in the paper's order, each quoting the paper.
The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.
Provenance: prove2quantum's pipeline drafted each statement (gpt-6.1-sol on Stanford's Sherlock cluster). A blind read-back, written without seeing the paper, is attached to every item. An independent moderator compared each statement against the paper and passed the whole paper without edits.
2 results of the paper are not stated yet, because the library lacks an interface they need:
The symmetric traveling salesman polytope is the convex hull of the incidence vectors of all Hamiltonian cycles of the complete graph . Minimizing a linear cost over it is the traveling salesman problem, and polyhedral methods for the TSP (cutting planes, branch and cut, LP relaxations and their integrality gaps) depend on linear inequalities that are valid for and, ideally, define its facets. Known facet classes include the comb, clique-tree, path and crown inequalities; Naddef's survey (Naddef 2007) collects them.
Circulant TSP instances are those whose edge costs depend only on the edge's length around a circle of equally spaced cities. Gutekunst and Williamson (SIAM J. Discrete Math. 2019) showed that the subtour LP has integrality gap exactly on circulant instances; the witnessing family of LP solutions is cut off by several known facets, but not robustly under small modifications. They conjectured an inequality, now called the circlet inequality, that would cut off the whole family. The paper of this mission (arXiv:2012.12363v1) proves that conjecture and shows the inequality is facet defining when . Its coefficients are themselves circulant, a property shared among classical facets only by the crown inequalities of Naddef and Rinaldi (Math. Oper. Res. 1992).
Timeline. 2019: Gutekunst and Williamson characterize the subtour LP gap on circulant instances and conjecture Inequality (2). 2020: Gutekunst and Williamson prove validity (Theorem 3.1) and the facet property (Theorem 5.1) for , and compute the inequality's strength.
Fix cities , edges of , and . The length of the edge is
an integer in . For a vector and , is the total weight on edges of length . A Hamiltonian cycle visits every city once and returns; its incidence vector marks its edges, and
For the circlet coefficients are for odd and for even (for : ), and the circlet inequality is
An inequality is valid for if every point of satisfies it, and facet defining if moreover the face has .
In the Lean development, cities are Fin n, a Hamiltonian cycle is an ordering π : Equiv.Perm (Fin n), its incidence vector is MetricTSP.tourVec π, STSP n is the convex hull of these vectors, lenCount n i x is , coeff n i is and circletForm n x is .
For every positive with :
Every Hamiltonian cycle on cities, , satisfies (2). The proof is by minimal counterexample and runs through the milestones Claim 3.2 (the base case ), Lemma 3.3 (a violating cycle has ), Lemma 3.4 (a minimal violating cycle contains none of three local edge patterns), and the window count with .
Lemma 3.4 is proved by contracting four vertices of a cycle containing one of the patterns and showing the cost drops by at least : Claims 4.1 and 4.2 (symmetries), Proposition 4.3 (surviving edges never get more expensive), Propositions 4.4 and 4.5 (the five contracted edges), and the resulting contraction step.
The dimension (cited on p. 19 from Naddef and Rinaldi, Theorem 3.7), and a family of tight Hamiltonian cycles with linearly independent incidence vectors.
The circlet inequality is a facet of whose coefficients depend only on edge length, so it is invariant under rotation of the city labels. It resolves the circulant TSP conjecture of Gutekunst and Williamson and removes from the subtour LP every member of the canonical family of solutions that witnesses the integrality gap on circulant instances, a class of TSP instances for which relatively little is known. Since any relabelling of the cities gives another valid inequality, it supplies on the order of facets of . The validity proof also gives a self-contained example of a minimal-counterexample argument by vertex contraction, a technique common in polyhedral combinatorics.
The result is proved in the paper; it has not been formalized. This mission produces machine-checked statements of the validity theorem, the facet theorem, and every intermediate result their proofs use, together with a reusable formal model of , edge lengths on a cycle, and facets of polytopes built from tour incidence vectors. The dimension of is a classical cited result that, to the best of current knowledge, has no formal proof either; it is included as a target in its own right.
Validity is not a local statement. A Hamiltonian cycle may use edges of any length, and the coefficients alternate between and with parity, so no edge-by-edge or degree-by-degree bound works: the inequality is tight on cycles that mix lengths , and one other length. The paper's argument needs the global structure of the cycle twice, once to rule out the patterns of Figure 4 in a minimal counterexample (where a parity argument on a four-group quotient graph handles the hardest equality case of Proposition 4.5) and once to count windows. The contraction step requires checking that the contracted object is again a Hamiltonian cycle and tracking costs on two different vertex sets with different moduli and .
For the facet half, exhibiting the tight tours is explicit, but proving the linear independence of incidence vectors for all and proving the dimension of are substantial linear-algebra arguments. The paper's own justification of independence for the first family is inaccurate (a length- edge is shared by two rotations), so a correct argument has to be supplied.
Fin n with 0-based labels; lengths and costs depend only on label differences, so the shift from changes nothing. The contraction results (Propositions 4.3–4.5, the contraction step) use the paper's 1-based labels in ℕ, with integer lengths and costs, and refer to a cycle's edges through the label of each vertex ().Fin n → Fin n → ℝ (ordered pairs); the incidence vector is symmetric with zero diagonal, and affine dimension is the same as in . counts each unordered edge once.Disjunctive.Polarity.IsFacet (extreme, nonempty, dimension one less), and the goal conjoins validity, since a face of codimension one is also cut out by the reversed inequality.We present in this paper a set of routines constructed to compute the rank of a matrix Lie algebra and also to determine a Cartan subalgebra from a given list of elements
This mission is the statement-level formalization of Algorithm to compute the rank and a Cartan subalgebra of a matrix Lie algebra with Mathematica (arXiv:math/0205235v1). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.
16 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is so(n): BSo ordered basis (so(n): BSo ordered basis, PDF p. 5). The milestones are the other numbered results, in the paper's order, each quoting the paper.
The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.
Provenance: prove2quantum's pipeline drafted each statement (gpt-6.1-sol on Stanford's Sherlock cluster). A blind read-back, written without seeing the paper, is attached to every item. An independent moderator compared each statement against the paper and passed the whole paper without edits.
2 results of the paper are not stated yet, because the library lacks an interface they need: