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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?
In the predict-then-optimize framework, a machine-learning model predicts the cost vector of a linear optimization problem , and a decision is made by solving that problem with the prediction. Elmachtoub and Grigas (Smart "Predict, then Optimize", Management Science 2022) proposed judging predictions by the SPO loss, the excess cost of the decision induced by when the true cost is . El Balghiti, Elmachtoub, Grigas and Tewari (arXiv:1905.11488v3) develop generalization bounds for learning with the SPO loss.
The SPO loss is discontinuous in : its value jumps where the optimization problem has several optimal solutions. The paper's sharper bounds (its Theorems 4 and 5) therefore replace the SPO loss by a margin SPO loss that is Lipschitz, and they hold whenever the feasible region satisfies a geometric condition called the strength property. This mission formalizes the paper's first class of feasible regions for which that condition holds: strongly convex sets, such as Euclidean balls and balls with .
Let be a finite-dimensional real vector space with a norm (the paper's with a generic norm). A cost vector is a linear functional on ; its value at is written , and its dual norm is . The closed ball of radius around is .
The feasible region is nonempty, compact and convex. An optimization oracle is any map with and for all ; no tie-breaking rule is fixed.
If is compact, not a singleton, and -strongly convex for some , then for every oracle ,
The strength parameter equals the strong convexity constant.
Theorem 7 is what makes the paper's margin-based bounds usable for a concrete family of feasible regions. It says two things: the strength property holds with , so the Lipschitz constants in the margin analysis are explicit; and , so the margin of a prediction, and with it the empirical margin SPO loss, is as easy to compute as a dual norm. Combined with bounds on the multivariate Rademacher complexity, this gives generalization bounds for strongly convex regions whose dependence on the dimension improves on the paper's Natarajan-dimension bound. In dimension one it recovers the classical margin bounds for binary classification (Example 7, p. 24).
The results are proved in the paper; Proposition 1 is due to Vial (1983). None of them has a machine-checked proof known to this mission, and Mathlib has no notion of a strongly convex set (its StrongConvexOn concerns functions). The mission produces a formal definition of strongly convex sets for a general norm, the normal-cone characterization, and the connection to the predict-then-optimize strength property. It is one of four missions on this paper; the margin-based generalization bound itself is the subject of mission II, and polyhedral regions of mission IV.
Definition 5 speaks about balls around convex combinations, while Proposition 1 is a pointwise inequality with the exact constant . Evaluating the ball inclusion at any single convex combination loses that constant, since the admissible radius and the displacement of the centre both shrink with the mixing weight. Relating a ball to a linear functional also requires the maximum of over a ball to be attained and equal to , a fact about dual norms whose attainment depends on finite dimensionality.
For , comparing with gives only the inequality ; equality needs every nonzero cost vector to have a unique minimizer over . The oracle minimizes, whereas the normal cone is written for maximizers, so the signs in (5) and (8) do not match directly and are a common source of error.
StrongDual ℝ E), so is c w and the operator norm is the dual norm; balls are Metric.closedBall.S.Nontrivial. Proposition 1 and the ball identity carry no compactness hypothesis, as in the paper.Metric.infDist to the degenerate set; the parameter conditions and are hypotheses of the theorems, not parts of the predicates.Facility location asks where to open service points (warehouses, plants, servers) and how to connect customers to them so that the total opening cost plus the total connection cost is minimum. In the capacitated version each facility can serve only a limited number of customers, which is the situation in most applications: a warehouse has a floor area, a server a bandwidth. The problem is NP-hard, and the algorithms used in practice for it are often simple local search heuristics: start from a solution and repeatedly apply a small change that lowers the cost, until no such change exists.
The quality of such a heuristic is measured by its locality gap: the largest possible ratio between the cost of a solution that no allowed change can improve and the cost of an optimum solution. Arya, Garg, Khandekar, Meyerson, Munagala and Pandit (SIAM J. Comput. 33(3), 2004) gave locality-gap analyses for k-median, uncapacitated facility location, and the capacitated problem in which several copies of a facility may be opened. This mission formalizes their §5, the capacitated case.
Timeline (as surveyed on pp. 545–546 of the paper). For the variant 1-CFL, where at most one facility may be opened at each location, Korupolu, Plaxton and Rajaraman (1998) showed that local search with add, drop and swap moves has locality gap at most 8 when capacities are uniform; Chudak and Williamson (IPCO 1999) refined this to 6, and Pál, Tardos and Wexler gave a local search with gap 9 for nonuniform capacities. For ∞-CFL, the variant with copies studied here, the known algorithms were LP-based: a 3-approximation of Chudak and Shmoys (1999) for uniform capacities, a 4-approximation of Jain and Vazirani for nonuniform capacities, and a 2-approximation of Mahdian, Ye and Zhang. Arya et al. (2004) analysed local search for ∞-CFL with nonuniform capacities: with a new move that drops any set of open copies and opens several copies of one facility, the locality gap is at most 4 (Theorem 5.5), and scaling the facility costs gives (p. 561). The tight example for uncapacitated facility location (§4.3) also shows a locally optimum solution of cost 3 times the optimum, so the locality gap of the procedure lies between 3 and 4; its exact value was left open (§6).
An instance consists of a finite set of clients, a set of facilities and a distance on that is nonnegative, symmetric and satisfies the triangle inequality; is the cost of serving client from facility . Every facility has an opening cost and an integer capacity . Any number of copies of a facility may be opened; each copy of costs and serves at most clients.
A solution opens a finite list of copies, copy being a copy of facility , and assigns every client to a copy so that each copy serves at most clients. Write for the set of clients served by copy and for the clients served by a set of copies. Its costs are
The neighbourhood (9) of a solution whose multiset of open facilities is consists of
is locally optimum if no neighbour, with any feasible assignment of the clients, has smaller cost.
For every instance with at least one client, every locally optimum solution and every solution ,
In the order the paper's proof uses them:
A companion item states the scaled bound of p. 561: a local optimum for facility costs has cost at most in the original instance.
The result. Theorem 5.5 gives a constant locality gap for a local search procedure for capacitated facility location with copies and nonuniform capacities, a variant previously approached through LP-based algorithms; the drop-add move it analyses is the paper's new operation for this problem. The scaled bound gives an approximation algorithm once local search is run to approximate local optimality. The paper also shows (Figure 13, the procedure T-hunt) that the exponentially large neighbourhood can be searched with a knapsack oracle, so the analysis applies to an implementable algorithm.
Formalizing it. The result is proved in the paper; to our knowledge no machine-checked version exists. The mission produces a checked model of capacitated facility location with copies (solutions, costs, the multiset neighbourhood) and of the locality-gap argument. The per-copy model and the flow comparison of Lemma 5.4 are reusable for other capacitated location problems and for local search analyses that compare a local optimum with an optimum through a flow or a matching.
The obvious attempt imitates the uncapacitated analysis: close one copy of and send its clients to a nearby copy of . With capacities this fails, since that copy of may be too small to absorb them, and single-copy moves do not certify a constant bound. With the drop-add move a single copy of must be charged for a whole group of copies of , and the groups must be chosen so that the charges add up to a constant times ; the rounding of the number of new copies costs an additional that has to be absorbed as well.
CFLSol Cl Fa u: a number n of open copies, a map loc : Fin n → Fa giving the facility of each copy, and an assignment σ : Cl → Fin n with the capacity constraint for every copy. Copies are separate indices because is per copy. Its multiset of facilities is the image multiset of loc.Contributions are welcome at every level: proofs of the milestones, alternative proofs of Lemma 5.4 or (10) (for instance through a matching argument instead of flows), and general infrastructure for multiset neighbourhoods and assignment problems.
The k-median problem asks where to open facilities so that the total distance from a set of clients to their nearest open facility is as small as possible. It is a basic model of facility location in operations research (placing depots, warehouses or servers) and of clustering with representative centres, and it is NP-hard, so the question of interest is how close a polynomial-time method can come to the optimum.
Local search is among the most widely used heuristics for it: start from any facilities and repeatedly exchange one open facility for a closed one while the cost decreases. Arya, Garg, Khandekar, Meyerson, Munagala and Pandit (SIAM J. Comput. 33(3), 2004) gave the first constant-factor guarantee for this heuristic on metric instances: every local optimum of the single-swap local search costs at most five times any solution with facilities. This mission formalizes that result.
Timeline of the relevant bounds:
A metric instance consists of a finite set of clients, a finite set of facilities and a distance on that is nonnegative, symmetric and satisfies the triangle inequality. Write for the cost of serving client by facility .
For a nonempty set of open facilities every client is served by its nearest open facility, and the cost of is
The k-median problem asks for a set of at most facilities of minimum cost.
A swap closes a facility and opens a facility , giving . The neighbourhood of is
and is locally optimum if for every . The local search starts from an arbitrary set of facilities and applies improving swaps until none exists; swaps preserve the number of facilities, so it stops at a locally optimum set of exactly facilities. The locality gap is the supremum, over instances, of the ratio between the cost of a worst local optimum and the optimal cost.
The analysis uses the following notation. For a solution , let assign each client to a nearest facility of , let be the service cost of client , and let be the set of clients served by . For two solutions and put . A facility captures if ; is bad if it captures some and good otherwise.
For every metric instance, every , every locally optimum set of exactly facilities and every nonempty set of at most facilities,
The comparison solution is arbitrary, not an optimum; the statement is the locality gap bound in the form the proof gives.
Theorem 3.2 shows that the simplest exchange heuristic for k-median is a constant-factor approximation on every metric instance, and the paper states that the analysis is tight: its example of §3.5, given for swaps of two facilities, is said to generalize to swaps of facilities, where the bound is 5 for . Combined with the standard device of accepting only swaps that improve the cost by a factor , it yields a polynomial-time -approximation (p. 548). The same capture-and-reassignment argument is reused for multi-swap k-median, for uncapacitated and capacitated facility location in the same paper, and in later work on k-means and on local search for clustering; its milestones (the capture graph and the mapping ) are the reusable part.
The result has been proved since 2001 and is textbook material (Williamson and Shmoys, The Design of Approximation Algorithms, 2011, Chapter 9). No machine-checked proof of it is known; Mathlib has no k-median problem and no locality-gap result for any clustering objective. The work remaining is to formalize the known proof.
The obvious argument adds up the inequalities over a pairing of with , rerouting the clients of the closed facility to the nearest remaining facility. This fails when a single facility of serves most clients of several facilities of : closing it leaves those clients with no nearby open facility, and no bound in terms of follows. The analysis must choose which swaps to consider so that such facilities are never closed, and must reroute the displaced clients of the facilities it does close to a facility other than the closed one while paying only a constant multiple of their own service costs. Both choices must work for arbitrary ties in the nearest-facility assignments and when and share facilities.
Namespace LocalSearchFL.KMedian. Clients and facilities are types Cl, Fa with Fintype and DecidableEq; the distance is a real-valued function on Cl ⊕ Fa with fields for nonnegativity, symmetry and the triangle inequality, and is not assumed. Solutions are Finset Fa. The cost is defined only for nonempty sets, from a nonemptiness proof, so no value is assigned to the empty solution; the goal takes nonempty with S.card = k, which is the paper's . Local optimality quantifies over every swap with and , exactly the neighbourhood of Theorem 3.2, and not only over the swaps with that the proof uses. The inequality is stated multiplied out, , so it is meaningful when .
The milestones quantify over nearest-facility assignments , with arbitrary ties. Capture is stated in integers as . The bijection of is a permutation of all clients fixing every client outside ; in inequality (2) it is a single permutation preserving every . Milestones 1–3 are purely combinatorial and are stated for arbitrary assignments, which contains the paper's case.
A formalization in which local optimality ranges over the swaps , , only, or in which or optimal is assumed, or in which the cost of the empty set is , is a different statement and is ruled out.
A complete development needs the finite-sum and Finset.inf' API of Mathlib, permutations (Equiv.Perm) and finite counting. The capture machinery and the mapping are reusable for the multi-swap and facility location missions of this series. Proofs of individual milestones are welcome independently of the goal.
A coherent risk measure assigns to a future financial position the smallest amount of capital that makes it acceptable to a supervisor. Artzner, Delbaen, Eber and Heath characterised the one-period version: every coherent risk measure has the form for a set of test probabilities (ADEH 1999). Regulators, insurers and banks, however, assess positions that evolve over several periods and whose risk is re-evaluated as information arrives. A multiperiod measurement should then be time consistent: the value assigned today should agree with the values the same method assigns tomorrow, so that it can be computed by backward induction, as in dynamic programming.
Artzner, Delbaen, Eber, Heath and Ku (Ann. Oper. Res. 2007) identify exactly which sets of test probabilities give time-consistent multiperiod risk-adjusted values. The condition, stability under pasting, also appears as "rectangularity" in the recursive multiple-priors model of decision theory (Epstein and Schneider 2003), and as m-stability in the theory of risk-neutral measures (Delbaen, The structure of m-stable sets, Séminaire de Probabilités XXXIX, 2006). Riedel treated dynamic coherent risk measures on finite state spaces (Riedel 2004); the continuous-time case is in Delbaen's m-stable paper and Cheridito, Delbaen and Kupper 2004.
Let be a probability space with a filtration and a horizon . A value process is an adapted process with every essentially bounded; the class of value processes is . All stopping times take values in , and is the σ-algebra of the stopping time .
A set of test probabilities is a closed convex set of probabilities on , each absolutely continuous with respect to . Its elements are identified with their densities , and "closed" refers to . denotes the elements equivalent to . Each has the density martingale .
Pasting. For with density martingales and a stopping time , the pasted martingale is for and for : the pasted probability follows up to and afterwards. is stable (Definition 3.1) if every such pasting is again in .
Two risk-adjusted values. For a value process and a stopping time ,
the worst conditional expected value over all test probabilities and all later stopping times. The generalized Snell envelope is the backward recursion
For a stopping time let for and for , and for and for .
Assume is -trivial and . Then the following are equivalent:
In attack order, all stated for any set with unless stability is named:
A supporting item states that the essential infimum defining exists.
The result. Theorem 4.2 characterises the sets of test probabilities for which the natural worst-case risk-adjusted value is computable by dynamic programming. Stability is thereby the structural condition behind time-consistent coherent risk measurement, recursive multiple-priors utility and backward-induction pricing under ambiguity. Without it, the worst-case value computed today may disagree with the value obtained by first computing tomorrow's worst case and then today's. The equivalence with the submartingale property says that stability is also exactly what makes the largest submartingale minorant of Theorem 4.1. Theorem 4.3 and the recursivity results for final values in Section 5 are corollaries.
Formalizing it. The paper's proof is complete apart from Lemma 4.1 and its Corollary, whose proofs are left to the reader. No machine-checked version of this result, of the generalized Snell envelope or of essential infima of families of random variables is known to exist. A formal proof would provide a reusable development of discrete-time optimal stopping under a set of probabilities, the essential-infimum calculus of Neveu, and density-martingale pasting — infrastructure that many results on robust optimal stopping, dynamic risk measures and robust Markov decision processes need.
The direction from stability to Bellman's principle needs an essential infimum to be exchanged with a conditional expectation under another probability (step (3) of the proof). This is false for a general family: an essential infimum of conditional expectations is not the conditional expectation of an essential infimum. The exchange works only because stability makes the family closed under minima (Lemma 4.1), so that it is directed downward and its essential infimum is the limit of a decreasing sequence, and because Lemma 3.1 lets the test probabilities used before and after be chosen independently. The converse, from the submartingale property to stability, is not a computation: it uses the separation theorem in against a pasted density assumed outside , which is where convexity and -closedness of are used. Dropping either hypothesis breaks that direction.
Lean represents by its set of densities in : -measurable, a.s. nonnegative, integrable, of mass one, convex, sequentially closed in the seminorm and saturated under a.s. equality. Test probabilities are , and is Mathlib's conditional expectation under . Time is , and every stopping time is bounded by . A -submartingale on is Mathlib's Submartingale of the process frozen after . The essential infimum of a family is defined in the mission (Mathlib has only that of a single function); a supporting item shows that it exists, so its fallback value is never used. All identities between risk-adjusted values hold -almost surely.
Conventions made explicit:
Bellman's principle must be stated with on both sides and for all stopping times . Replacing by , or restricting to deterministic times, turns the goal into a property of the recursion and is not the theorem.
Needed infrastructure, reusable beyond this mission: existence and directedness of essential infima of families; conditional expectations under equivalent measures and the Bayes formula; optional sampling for bounded stopping times under each ; the – separation theorem. Contributions of any of these, and proofs of the milestones in any order, are welcome.
The bilinear program
with and polyhedra, is one of the recurring nonconvex problems of mathematical programming. It arises from constrained bimatrix games (Mangasarian 1964), dynamic Markovian assignment, multicommodity network flow and certain dynamic production problems (Konno 1971, 1976). Its classical structural fact is that, because the constraints on and on are separate and the objective is linear in each block, an optimal solution can be found at an extreme point of (Falk 1973, doi:10.1007/BF01580119). Vertex-enumeration, extreme-point ranking and cutting-plane methods for the problem rest on that fact.
Al-Khayyal and Falk (Math. Oper. Res. 8(2), 1983) consider the jointly constrained version, in which the feasible region is an arbitrary set of pairs , so that constraints may couple and . They observe that the extreme-point property is then lost, and replace it with a weaker structural fact that survives: a minimum is attained on the boundary of the feasible region. This mission formalizes that result, Theorem 1 of the paper, together with the two examples on p. 274 that delimit it.
Write points of as , and let be a nonempty compact set. No convexity of is assumed. Let be continuous on .
The function is biconcave over (Lean: BiconcaveOn S φ) when both partial functions are concave wherever they live inside : for every fixed , the map is concave on every convex set with ; and for every fixed , the map is concave on every convex set with . Equivalently, is concave along every segment of that is parallel to the -block or to the -block. A bilinear objective with and concave is biconcave; joint concavity of is not required.
The boundary is the topological frontier of in : the closure of minus its interior. A solution of is a point with for all (Lean: IsMinOn φ S z). The Euclidean distance on the product (Lean: eucDist) is .
If () is nonempty and compact, and is continuous on and biconcave over , then
The conclusion is existence of a boundary minimizer. It does not assert that every minimizer lies on (a constant is a counterexample to that).
Theorem 1 is the structural statement that separates jointly constrained bilinear and biconcave programs from their separably constrained special case. It says where a global search may restrict attention, namely to , and example 3 shows that this cannot be sharpened to extreme points once the constraints couple the blocks. Example 4 records that such problems have proper local minima, so a local method alone does not solve them; this motivates the branch-and-bound algorithm of the same paper, which is the subject of the companion mission.
The result is proved in the paper; this mission produces its machine-checked statement and proof. Mathlib contains the Bauer-type principle for jointly concave functions on compact convex sets, but no statement of this kind for biconcave functions on nonconvex sets, and no formalization of Theorem 1 is known. The two examples are small but exact computations, and they certify that the definitions admit the intended instances.
The first idea is to apply the concave-minimization principle: a concave function on a compact convex set attains its minimum at an extreme point. It does not apply. need not be convex, so it has no useful extreme-point structure, and is concave only along segments parallel to one block, so it is not concave along the segment from an interior point to a boundary point in a general direction. Example 3 shows that the conclusion "extreme point" is actually false here.
What remains is local geometry around an interior minimizer: one must find points of around it at which biconcavity can be applied in both blocks, and control their membership in without convexity. The feasibility of the mixed points and , which the paper uses without comment, is where the choice of the Euclidean distance matters, and it is recorded as a separate milestone.
EuclideanSpace ℝ (Fin p) × EuclideanSpace ℝ (Fin q). The boundary is Mathlib's frontier in this product; it does not depend on the norm. Only milestone 1 refers to a distance, and it uses eucDist, the Euclidean distance written out, because Mathlib's default metric on a product is the maximum of the block distances.IsCompact S, S.Nonempty, ContinuousOn φ S (continuity only on ), and BiconcaveOn S φ. One hypothesis is added: . For the space is a point, whose only nonempty subset has empty frontier, so the conclusion fails; the paper works in positive dimension throughout.IsLocalMinOn relative to the box, and non-globality of records the paper's word "proper".IsCompact.exists_mem_frontier_infDist_compl_eq_dist, stated for the ambient metric), the fact that a closed ball about an interior point whose radius is the distance to the frontier lies in the set, and concavity on segments. A lemma that works for an arbitrary norm on the product would be reusable. Contributions of alternative proofs are welcome.A recent conjecture proposes a holographic duality between a static patch of Jackiw–Teitelboim (JT) gravity with positive cosmological constant, a two-dimensional reduction of three-dimensional de Sitter gravity, and the double-scaled SYK model at infinite temperature (DSSYK). The conjecture comes from Susskind (arXiv:2109.14104, arXiv:2205.00315), Lin–Susskind (arXiv:2206.01083) and Rahman. In the paper arXiv:2209.09999, Susskind looks at the semiclassical limit of de Sitter space, where the horizon entropy diverges. There physics splits into a cosmic sector, sensitive to the de Sitter radius , and a microscopic (string) sector, which does not see it. The paper argues that the limit of DSSYK shows the same split, with the two sectors separated by a factor of . This parallel is offered as evidence for the duality.
Most of the paper's arguments are heuristic scaling estimates. Several of them rest on explicit closed-form expressions: the epidemic scrambling function, the large- correlators, and the second moments of the Hamiltonians. These can be stated and proved as precise limit theorems. This mission collects them.
The SYK model has Majorana fermions with , coupled by -body interactions with Gaussian random couplings. The cosmic Hamiltonian is , where the couplings have variance (eq. (3.4)), and is a fixed energy. The string Hamiltonian is (eq. (3.7)). Cosmic time and string time are related by .
The double-scaled limit is a sequence of models with
In particular . Infinite temperature means expectations are taken in the maximally mixed state.
In DSSYK scrambling is described by the epidemic scrambling probability (eq. (8.1))
The infinite-temperature second moments of the two Hamiltonians are
For a double-scaled sequence and :
Early scrambling is ordinary fast scrambling in string units (8.3). Late scrambling is a quasinormal-mode decay in cosmic units (8.6).
The goal theorem is the paper's clearest quantitative instance of the separation of scales. The same function (8.1) shows a finite-rate exponential onset only in string time, and a finite-rate exponential saturation only in cosmic time. Neither regime survives in the other's units. Milestones 6–7 are the quantitative input to the paper's claim, attributed to H. Lin, that the cosmic Hamiltonian is not in the cosmic algebra while the string Hamiltonian passes the corresponding fluctuation test. Milestones 4–5 make precise the emergence of the string scale and the confinement of fermions near the horizon.
On formalization: the paper derives these limits by informal asymptotics, and some are stated only up to "". The formal statements fix the exact constants. Two of them differ from the paper's displayed formulas: the factor in (11.1)–(11.2), which comes from when , and the factor in the exponent of (9.4). The paper's qualitative conclusions are unchanged. At the time of drafting, a search of the platform found no existing formal statements of these results.
Every statement is a limit in which two large parameters, and , compete inside an exponent. The naive move of expanding to first order is valid only when . In the goal's late-time regime diverges, and at the end of the fast window stays finite. Each regime needs its own uniform control of . The Hamiltonian milestones need the product asymptotics under , which is a second-order effect that first-order Stirling-type estimates miss.
Filter.Tendsto, Real.rpow, Real.log, Real.cosh) and an improper integral of . Reusable lemmas about in the regime are welcome.General relativity describes gravitation through the curvature of the Levi-Civita (Christoffel) connection of a spacetime metric. An alternative description, going back to Einstein's 1928-1930 papers on distant parallelism and developed in modern form as the teleparallel equivalent of general relativity, uses instead a tetrad field and the Weitzenböck connection it defines, a connection with vanishing curvature but non-zero torsion. The paper of Aldrovandi, Pereira and Vu (Braz. J. Phys. 34 (2004) 1374) reviews this formulation as a gauge theory of the translation group, and uses it to discuss the weak equivalence principle and a global (phase-factor) formulation of gravitation.
The claim that both descriptions are physically equivalent rests on a small number of exact tensor identities: the Weitzenböck connection splits as the Christoffel connection plus the contortion tensor, its curvature vanishes identically, and the teleparallel Lagrangian differs from the Einstein-Hilbert Lagrangian only by a total divergence. This mission asks for machine-checked proofs of these identities, in coordinates, exactly as used in Sections 2-3 of the paper.
Timeline (for orientation): Einstein introduced distant parallelism in 1928 (translations in the Delphenich collection Selected Papers on Teleparallelism); Weitzenböck studied the differential invariants of the theory the same year; the modern teleparallel equivalent of general relativity was developed from the 1960s-1970s onward, and the gauge-theoretic reading used here is the one reviewed by Aldrovandi-Pereira-Vu (2004).
Work in a single global chart and write for coordinate partial derivatives. Tangent-space (Latin) indices are raised and lowered with the Minkowski metric .
Curvature of a connection is , Ricci , scalar (Landau-Lifshitz conventions, consistent with the paper's Eq. (15)). The Einstein-Hilbert density is .
For a smooth, positively oriented tetrad (, so ):
The paper states only "up to a divergence"; the divergence term is written explicitly because an unspecified divergence would make the statement empty (every function on is a divergence).
The goal identity is the precise sense in which teleparallel gravity is "equivalent" to general relativity: the two actions differ by a boundary term, so they give the same field equations (the paper's Eqs. (12) and (15)). Milestone 4 is what turns the teleparallel force equation into the geodesic equation (Eq. (20)), and milestone 2 is the dictionary between the gauge field strength and torsion used throughout Sections 3-4. The results are classical and proved on paper; to our knowledge they are not formalized in Mathlib, which has no coordinate tensor calculus of this kind. A formal development would provide reusable coordinate lemmas (derivative of an inverse matrix field, symmetry of second derivatives in index form, Christoffel symbols of ).
Each identity is elementary on paper, but the index computations are long, and in Lean every derivative carries differentiability side conditions: derivatives of inverse matrices and of determinants, and symmetry of mixed partial derivatives, must be justified from smoothness and invertibility of the tetrad. The goal identity involves second derivatives of the inverse metric and a careful bookkeeping of about a hundred index contractions.
Spacetime is as Fin 4 → ℝ with one global chart; partial derivatives are Fréchet derivatives applied to coordinate basis vectors. A tetrad is a family of real functions h a μ, assumed with invertible matrix at every point (IsTetrad). Inverses are matrix inverses (which would be for singular matrices; the invertibility hypothesis excludes this). The physical constants are arbitrary reals. All definitions live in the single definition file of the mission, namespace TeleparallelGravity.
At one loop, the worldsheet of a closed string is a torus, and the one-loop vacuum amplitude of the bosonic string is an integral over all conformally inequivalent tori. Two tori with modular parameters and are equivalent exactly when for an integer matrix of determinant one, so a consistent string amplitude must be built from -invariant data. In David Tong's Cambridge Part III lecture notes (Section 6.4), the partition function of the bosonic string is computed in lightcone gauge and written in terms of the Dedekind eta function; the notes then observe that the measure and the integrand are individually modular invariant. This invariance is the mechanism by which string theory avoids the ultraviolet divergences of point-particle loops: the integration region is a fundamental domain that never reaches (Polchinski 1986).
Write . A modular transformation is the map
generated by and . With the Dedekind eta function is
convergent and non-vanishing on . The partition function of a single free scalar field on the torus is, up to constant factors, (6.20)
and the bosonic string partition function is
where is the standard fundamental domain.
For every and all with ,
The result. Modular invariance of the integrand is what makes the one-loop string amplitude well defined as an integral over the moduli space of tori, rather than over the whole upper half-plane: restricting to is only consistent because nothing depends on the representative. It is the first instance of a consistency condition that, in the superstring, constrains the allowed gauge groups and spectra.
Formalizing it. The eta transformation laws are classical (Apostol 1990, Ch. 3); Mathlib contains the definition of the eta function and results on and its fundamental domain, but, to the mission author's knowledge, not the -transformation law of . The mission produces a self-contained formal statement of the physics claim "both the measure and the integrand are individually modular invariant" and asks for its proof; milestones 1–2 are expected to follow from existing Mathlib results, milestone 6 is the analytic core.
The -law is immediate from the product. The -law is not visible from the product formula at all: it requires either Poisson summation / the Jacobi triple product for theta functions, or a Mellin-transform / Eisenstein-series argument ( is an Eichler integral of ), together with care about the branch of the square root. Passing from generators to all of requires an induction on words in and in which the multiplier must be tracked.
Points of are represented as complex numbers with the hypothesis ; a modular transformation is given by four integers with and acts by TongString.modularAction a b c d τ = (aτ+b)/(cτ+d). The eta function is defined directly by Tong's product formula as TongString.dedekindEta, using Mathlib's unconditional infinite product tprod. The square root in the -law is the principal branch, , which is the one intended since . The invariant measure is formalized as the measure on with density with respect to Lebesgue measure restricted to , and invariance as equality of push-forward measures. Constant prefactors that Tong explicitly drops (powers of , ) are omitted. Because every statement is restricted to , where has no zeros, the reciprocal is never a division by zero, so no statement is vacuous.
Mission Free Energy Principle I published the core variational step of the free energy principle (FEP): whatever recognition density a system carries, the posterior-form variational free energy never undercuts the data's surprisal, the bound is exact at the Bayesian posterior, and equality characterizes the posterior. That mission's shared finite substrate — normalized finite laws, finite kernels, entropy/cross-entropy/KL, and the finite generative model — now exists as platform definitions in the namespace FreeEnergyPrinciple.
The Free Energy Principle II mission formalizes the four structures the FEP literature builds on top of that core, all already machine-checked in the source repository fep_lean / fep_formal (Active Inference Institute):
Timeline of the mathematical content this mission formalizes:
fep_formal (Active Inference Institute): a machine-checked Lean 4 catalogue of 155 FEP topics compiled with zero proof holes against a pinned Mathlib. This mission transcribes the proved modules behind expected free energy, native Markov blankets, the scalar Gaussian filter/VFE, and Bayesian model reduction onto the platform.Two carriers, both fully machine-checked in the source repository:
Definitions.Def_fep_finite_laws, Def_fep_finite_information, and Def_fep_generative_model — no substrate is re-published. Zero-mass atoms are handled by the same totalized conventions as FEP-I: entropy uses Real.negMulLog (so exactly), KL is the nonnegative klFun integrand, and division premises are explicit.gaussianReal/gaussianPDF at a fixed strictly positive variance; the scalar OU prediction and the closed filter update give the posterior mean/variance the recognition family varies over. The native KL between two family members is exactly — proved against Mathlib's log-likelihood-ratio definition.The four definition items of this mission package exactly these carriers:
Def_fep2_expected_free_energy — the predicted state-outcome joint, preference risk, likelihood ambiguity, epistemic value, pragmatic cost, expected free energy (epistemic sign fixed by definition), the full-support contract, and the marginal/product/conditional-entropy/mutual-information lemmas the decomposition needs. Imports FEP-I.Def_fep2_gaussian_vfe — fixed-variance Gaussian family with its exact KL, scalar OU parameters, the exact scalar Gaussian filter (prediction, observation kernel, gain, closed posterior, evidence law), evidence surprisal, and the posterior-form Gaussian variational free energy. Self-contained.Def_fep2_bayesian_model_reduction — posterior odds, Bayes factor, and the model-odds update , with totalized division boundaries kept explicit. Imports FEP-I.Def_fep2_native_blanket — the static blanket factorization, the Dirac-mass embedding of finite laws into native measures, blanket/internal/external coordinates, the conditional-pair kernel, and the marginal/composition identifications the independence proof needs. Imports FEP-I.For every finite generative model, every policy , and every model with full support:
The epistemic-value sign is fixed by definition ( pragmatic cost epistemic value); the decomposition follows from two entropy identities: epistemic value is predicted outcome entropy minus ambiguity, and risk is cross-entropy minus the same entropy (Gibbs' inequality under full reference support). Nonnegativity of follows as a corollary — but the decomposition, not the bound, is the target.
For the exact scalar Gaussian filter, the posterior-form variational free energy at recognition mean is
the exact fixed-variance Gaussian KL (the recognition-to-posterior gap) plus the density-relative evidence surprisal. Equality with the surprisal holds exactly at the posterior mean — the Gaussian analogue of FEP-I's exactness theorem.
Bayes' rule in odds form: at positive evidence,
with the reference prior mass and reference likelihood as exact division premises. The multiplicative Bayes-factor structure (topic fep-120: factorized evidence ratios multiply; sequential model-odds updates agree with one update by product evidence) is available from the same definition layer as a further target.
The embedded static blanket factorization satisfies Mathlib's native CondIndepFun predicate: internal coordinates are conditionally independent of external coordinates given the blanket coordinate. The result is obtained by identifying the authored finite conditional kernels with Mathlib conditional distributions (the embedding preserves marginals and joints exactly on discrete carriers) — not by a finite mutual-information argument.
These four results are the load-bearing extensions of FEP-I's bound: expected free energy converts the variational principle into a theory of action selection; Markov blankets make "internal states" and "external states" well-defined relative to a blanket, which is what lets the FEP talk about self-organizing systems at all; the Gaussian filter is the tractable regime in which the variational machinery becomes the Kalman update; and Bayesian model reduction is the learning/comparison step that updates structure, not just parameters.
Formalizing them. All four families are proved with zero proof holes in the source repository, against a pinned Mathlib; the definition layer here is faithful (same carriers, same totalized conventions, same support contracts made explicit) and every item below compiles locally against the platform environment. The mission's value is reusable community infrastructure: the definition items publish the EFE layer, the Gaussian filter, the odds layer, and the native-blanket embedding in the shared namespace FreeEnergyPrinciple, so later missions (policy trees, collective inference, predictive coding) can import them instead of re-deriving. Status honesty: all eight items below are formalized and machine-checked locally against the platform environment; each is an open problem on the platform only in the sense that no proof has yet been submitted to it.
Committed conventions of this mission's Lean development:
Free Energy Principle I finite substrate (namespace FreeEnergyPrinciple, definitions Def_fep_finite_laws, Def_fep_finite_information, Def_fep_generative_model); this mission adds definition items Def_fep2_expected_free_energy, Def_fep2_gaussian_vfe, Def_fep2_bayesian_model_reduction, and Def_fep2_native_blanket, all in the same namespace.gaussianReal/gaussianPDF, no finite substrate, no manifold geometry, no singular (zero-variance) branch.nextStaticModel construction.How much fluid does a pipe carry for a given pressure drop? For slow, viscous flow the answer is Poiseuille's law, . It is the standard first quantitative model of blood flow in vessels and appears in Lecture 15 ("The Mathematics of Water") of the Drexel biophysics course PHYS 461/561 (Fall 2011), where it is derived from the Navier–Stokes equation and then used to estimate flow speeds in capillaries. The fourth-power dependence on the diameter is what makes small changes in vessel radius matter so much physiologically.
A Newtonian fluid of viscosity flows steadily through a straight cylindrical pipe of diameter and length under a pressure drop . By cylindrical symmetry the velocity is axial, , where is the distance to the axis. The force balance on a thin cylindrical shell (slide 14), integrated along the pipe (slide 15), gives the radial equation
with boundary conditions: finite at the axis, and no slip, , at the wall. The flow rate is and the average velocity is .
Milestones, in order:
Goal (Poiseuille's law): for every pipe flow as above,
The result is classical and completely known; the value of the mission is a clean, machine-checked version of the full chain from the radial ODE and its boundary conditions to the flow-rate formula, rather than just the final integral. The ODE-uniqueness step (milestones 1–2), which uses boundedness at the axis to exclude the logarithmic solution, is the part lecture notes usually wave through.
Milestones 3 and 4 are calculus exercises. The real work is milestones 1–2: turning "integrate twice" into a rigorous statement on an open interval (a function with zero derivative on an interval is constant), and then showing that boundedness near forces (because ) and that continuity plus no-slip at the wall fixes .
All quantities are real. The ODE is imposed pointwise on the open interval with explicit differentiability; "" is encoded as boundedness of on , and the wall condition as plus continuity from inside. are assumed positive in every theorem; is an arbitrary real. The pressure field itself is not modelled: the statements start from the integrated radial equation of slide 15. Two typos on the slides are corrected: slide 15 writes the homogeneous term as (correct: ), and slide 16 prints (correct, as used on slide 17: ). All definitions are in the single definition file MathematicsOfWater_PipeFlow; all declarations use the namespace MathematicsOfWater.
Neutrinos are produced and detected in charged-current weak interactions as flavour eigenstates , but they propagate as superpositions of mass eigenstates . When the masses differ, the mass components acquire different quantum-mechanical phases and the flavour content of the beam changes periodically along its path: neutrino flavour oscillations, first proposed by Pontecorvo. Oscillations of atmospheric and solar neutrinos are the experimental evidence that neutrinos are massive and that leptons mix, and the vacuum oscillation formula is what experiments fit to extract mass-squared differences, mixing angles and the CP-violating phase.
The formulas are elementary quantum mechanics, but the standard three-flavour form requires careful bookkeeping of phases, of unitarity relations, and of the sign conventions for the CP-odd term. The source for this mission, Section 4 of P. Lipari's lecture notes Introduction to Neutrino Physics, derives the whole chain in a few pages (Eqs. (44)–(73)). The mission turns that chain into machine-checkable statements.
Fix flavours and mass eigenstates. The mixing matrix is a complex matrix, rows indexed by flavour and columns by mass eigenstate , with ; physically is unitary. Let be the squared masses, the distance travelled and the neutrino energy. In the relativistic approximation (, ) the transition amplitude and transition probability are
The antineutrino probability is the same expression with replaced by its entrywise complex conjugate . Write and define the CP-even amplitudes and the CP/T-odd coefficients
For two flavours the mixing matrix is the rotation ; for three flavours the standard parametrization is with mixing angles and CP phase (Eq. (57)).
For unitary and flavours ,
The source writes the last term as ; the formal goal fixes the sign by writing it in terms of .
The goal is the formula used to interpret long-baseline accelerator and reactor oscillation data: the three amplitudes govern the CP-conserving oscillation pattern, and the single product term isolates all CP- and T-violating effects in vacuum. Together with milestones 3–6 it shows that one real number, the Jarlskog parameter, controls CP violation, and that such violation requires a nontrivial phase, three nonzero mixing angles and three active oscillation frequencies.
These results are textbook material and their proofs are known; what this mission adds is a machine-checked derivation with every convention explicit. Two points in the printed source are resolved by the formal statements: the constant term of Eq. (61) is printed with exponent where squaring the amplitude gives exponent , and with the source's own Eqs. (57) and (69), the value of has the opposite overall sign from the one printed in Eq. (70). Milestone 6 is stated with the sign that follows from (57) and (69).
No step is conceptually deep. The work is in the algebra: expanding a squared modulus of a sum of three complex exponentials, and then using unitarity to remove the constant term and to identify the sine coefficients with one another. The unitarity relations among the products (footnote 8 of the source) have to be combined in the right order, and it is easy to lose a sign or a conjugate. Eq. (70) needs an explicit expansion of a product of three complex matrices.
Matrix (Fin n) (Fin n) ℂ; flavours and mass labels are the indices 0, 1, 2. Unitarity is membership in Matrix.unitaryGroup (Fin 3) ℂ, assumed only where the source uses it (Eqs. (67), (68), (71)).m2 : Fin n → ℝ; and are arbitrary reals. No positivity of is imposed: all statements remain true at , where Lean's convention makes every phase vanish.Complex.normSq of the amplitude. Antineutrino probabilities use the entrywise conjugate U.map star, not the conjugate transpose.Chapter 26 bounded the rate of uniform convergence by the Rademacher complexity; Chapter 27 of Shalev-Shwartz and Ben-David, Understanding Machine Learning: From Theory to Algorithms (doi:10.1017/CBO9781107298019), introduces a second, metric measure of the size of a set of vectors, its covering numbers , the smallest number of Euclidean balls of radius needed to cover , and connects the two through Dudley's chaining. Covering numbers behave well under scaling and under coordinatewise Lipschitz maps (Lemmas 27.2–27.3), they are easily bounded for sets lying in a low-dimensional subspace (Example 27.1), and the chaining lemma turns a bound on at all scales into a bound on (Lemma 27.4), with the clean corollary when (Lemma 27.5). The chapter's example recovers for sets in a -dimensional subspace, the technique that the book says would sharpen the fundamental theorem's sample complexity from to .
For with the Euclidean metric, is an -cover of if every is within distance of some , and is the cardinality of the smallest -cover (Definition 27.1). The Rademacher complexity is Mission XX's. Chaining is run at the scales , , where is a radius of a ball containing , the book's being the smallest such radius.
For a nonempty , , contained in the ball of radius about some , and every integer ,
Example 27.1 (the grid -cover of a set of norm at most in a -dimensional subspace, of size ); Lemma 27.2 (scaling and translation); Lemma 27.3 (the contraction principle); Lemma 27.5 (the corollary of chaining). Further item: Example 27.2 ( for sets in a -dimensional subspace).
Chaining is the standard way to get sharp uniform convergence rates: a single-scale union bound (Massart's lemma at one resolution) loses a logarithmic factor, and summing Massart bounds over a geometric sequence of scales, applied to the increments between successive nearest cover points, recovers it. Lemma 27.4 is the discrete Dudley integral, and Lemma 27.5 is the form in which it is used: any polynomial-in- covering number gives -type bounds without the extra logarithm. On the platform these items complete the complexity toolbox begun in Mission XX and provide covering numbers as a reusable notion; the contraction and scaling lemmas mirror their Rademacher counterparts.
Lemmas 27.2 and 27.3 are immediate: the image of an -cover under the affine map is an -cover, and under a coordinatewise -Lipschitz map a -cover, since ; formally they are manipulations of the infimum in . Example 27.1 needs an orthonormal basis of the subspace (Gram–Schmidt, or Mathlib's orthonormal bases of finite-dimensional inner product subspaces of with the Euclidean structure) and the rounding of coordinates to a grid. Lemma 27.4 is the real work: after centering, take minimal -covers , the near-maximizer of (which depends on ), its nearest points , the telescoping , the bound , and Massart's lemma (Mission XX) on the sets of increments, of cardinality at most ; a formal proof must handle the supremum not being attained (approximate maximizers) and the dependence of all choices on inside the finite average. Lemma 27.5 lets using and . Example 27.2 combines Example 27.1 at the scales with Lemma 27.5, with the book's constant . The book's derivation uses the count without , so a proof needs the volumetric covering bound for and a direct count for .
Vectors are Fin m → ℝ with an explicit Euclidean norm, because Mathlib's norm on that type is the sup norm; covers are arbitrary finsets of and is an infimum in , so no junk value arises when no finite cover exists, and the chaining statements read through ENat.toNat for the bounded sets they concern, where it is finite. Subspaces are Mathlib Submodules with finrank = d. Two statements are given with the constants their proofs support, and the item texts say so. Example 27.1's grid has points per coordinate, so the cover has size , not , which is less than for and cannot bound a covering number of a nonempty set; Example 27.2 correspondingly has in place of . Lemma 27.4 is stated for any enclosing radius about any center, since the proof only uses that is a -cover of ; the book's minimal radius is the special case, and this is the form Example 27.2 needs (with and ). Lemma 27.5 keeps the book's .
Not stated: nothing else is in the chapter beyond the bibliographic remarks.
The book is discriminative almost throughout: it learns predictors, not distributions, following Vapnik's advice not to solve a more general problem as an intermediate step. Chapter 24 of Shalev-Shwartz and Ben-David, Understanding Machine Learning: From Theory to Algorithms (doi:10.1017/CBO9781107298019), presents the generative alternative: assume a parametric form for the data distribution and estimate its parameters. The maximum likelihood principle is introduced on Bernoulli and Gaussian samples, shown to be empirical risk minimization for the log-loss, and analyzed through the decomposition of the true log-loss risk into a relative entropy plus an entropy (24.5), which explains both its consistency under a correct model and its overfitting on small samples. Naive Bayes and linear discriminant analysis show how generative assumptions reduce the number of parameters and make the Bayes classifier linear (24.8). The chapter's main theorem concerns the Expectation-Maximization algorithm of Dempster, Laird and Rubin for latent-variable models such as Gaussian mixtures: EM never decreases the log-likelihood (Theorem 24.3), because it is an alternate maximization of a lower bound that touches the likelihood at the posterior (Lemma 24.2). The chapter ends with Bayesian reasoning and the rule of succession.
A Bernoulli sample has log-likelihood and estimator (24.1); a Gaussian sample has . The log-loss is (24.4); on a finite domain, and . A latent-variable model is a parametric joint , , with ; , over the set of row-stochastic matrices, and EM alternates the E-step (24.10) with the M-step (24.11).
For a positive parametric joint , a sample , and any run of EM (each M-step returning some maximizer of ), the log-likelihood never decreases:
Equation (24.2) (Hoeffding for the Bernoulli estimator); the Gaussian maximum likelihood estimates of §24.1.1; Equation (24.5) (the risk decomposition ); Equation (24.8) (the LDA log-likelihood ratio is affine); Lemma 24.2 (EM as alternate maximization of , with and equality at the posterior). Further items: Gibbs' inequality, the Bernoulli maximum likelihood estimator (24.1)/(24.3), Exercise 1 (the biased variance estimate), Equation (24.6), the overfitting example of §24.1.3, Exercise 3 / (24.14), the weighted-centroid M-step (24.13), and the rule of succession of §24.5.
Theorem 24.3 is the guarantee that makes EM a sensible algorithm: it does not find the maximum likelihood estimate, but it climbs monotonically, and Lemma 24.2 identifies why, the E-step chooses the tightest lower bound at the current parameter and the M-step maximizes it. This variational view underlies a large part of modern latent-variable inference. Equation (24.5) is the information-theoretic content of maximum likelihood: the true risk is the entropy of the data plus the relative entropy to the model, so the best parameter is a projection of the data distribution onto the model class, and Gibbs' inequality is what makes that projection meaningful. The Bernoulli and Gaussian computations are the standard first examples, and Equation (24.8) is the reason linear classifiers appear in generative modeling. On the platform, the mission adds the relative entropy on finite domains, the EM objects, and Gaussian-integral identities that later probabilistic work can reuse.
The Bernoulli and Gaussian maximum likelihood facts are calculus, but as global maximization statements they need the concavity of and an explicit completion of squares rather than the book's stationary-point argument; the Gaussian case reduces to minimizing . Equation (24.5) is a finite-sum identity; Gibbs' inequality is Jensen for with the equality case, or the elementary . Lemma 24.2 is Jensen's inequality applied row by row to , with care at entries , where the convention is exactly Lean's junk value; Theorem 24.3 chains the lemma's three parts as the book does. The Gaussian expectation identities (Exercise 1 and (24.6)) require the moments of gaussianReal and Fubini over the product law. Hoeffding's inequality (24.2) is Mission II's Theorem for Bernoulli variables; the overfitting example is the inequality on . The rule of succession is a Beta-function identity provable by integration by parts.
Parametric families are functions from a parameter type to real-valued probabilities or densities, following the book's convention (p. 344) that denotes either; no measure-theoretic densities are needed except in the two Gaussian-integral items, which use gaussianReal and the i.i.d. law of Mission I, and in the two Bernoulli probability items, which use the Bernoulli law of Mission XIV. Lean's log 0 = 0 is handled explicitly: the EM items assume a positive joint, since with junk logarithms Theorem 24.3 is false (the M-step could pick a parameter with a zero component and inflated ), while the entropy terms use the convention that the book intends; the Bernoulli maximum likelihood statement ranges over ; the log-loss decomposition and Gibbs' inequality take the second distribution positive. The M-step is a predicate ("some maximizer"), so Assumption 24.1 is not modeled, and an EM run is any sequence of such steps. The Gaussian maximum likelihood statement requires a nonconstant sample, without which the likelihood is unbounded; the overfitting example is stated for , the range on which the book's inequality holds. Equation (24.8) is stated as a matrix identity for any symmetric in place of ; the soft k-means M-step is stated as the weighted-centroid minimization it amounts to.
Not stated: Naive Bayes (24.7), which is a rewriting of Bayes' rule; the mixture density itself and the E-step formula (24.12); the Bayesian derivations (24.16) and maximum a posteriori estimation; Exercise 2.
Dimensionality reduction maps data in to , , by a linear map , for computational reasons, for generalization (Chapter 19's curse of dimensionality) and for interpretability. Chapter 23 of Shalev-Shwartz and Ben-David, Understanding Machine Learning: From Theory to Algorithms (doi:10.1017/CBO9781107298019), studies three ways to choose . Principal Component Analysis chooses the pair of compression and recovery matrices that minimizes the total squared reconstruction error, and the answer is the eigenvectors of for the largest eigenvalues (Theorem 23.2). Random projections choose with independent Gaussian entries, and the Johnson–Lindenstrauss lemma says that the norms of any finite set of vectors are then preserved up to with (Lemma 23.4). Compressed sensing exploits sparsity: a matrix with the restricted isometry property compresses every -sparse vector losslessly (Theorem 23.6), the reconstruction can be done by minimization, a linear program, with an error bound that degrades gracefully for approximately sparse inputs (Theorem 23.8, due to Candès), and Gaussian random matrices with rows are RIP with high probability (Theorem 23.9).
Vectors are functions with , and . The PCA problem (23.1) is , and . A random matrix has independent entries, in Lemma 23.3 and afterwards. is -RIP if for every with (Definition 23.5); is restricted to an index set .
Let , , and let be eigenvectors of for its largest eigenvalues, formalized as the first columns of a spectral decomposition with and nonincreasing. Then with minimizes (23.1): for every ,
Lemma 23.1 (the reduction of (23.1) to orthonormal and ); Lemma 23.4 (Johnson–Lindenstrauss); Theorem 23.6 (exact recovery under RIP); Theorem 23.8 (Candès' recovery bound); Theorem 23.9 (Gaussian matrices are RIP). Further items: Equation (23.3), Exercise 2, Remark 23.1 (the optimal value ), the eigenvector transfer of §23.1.1, Lemma 23.3, Theorem 23.7, Lemma 23.10, Lemma 23.11 and Lemma 23.12.
Theorem 23.2 is the Eckart–Young–Mirsky theorem in the form the book states it: PCA is the optimal linear compression-and-recovery scheme in the least-squares sense, and its solution is spectral. The Johnson–Lindenstrauss lemma is the basic tool of randomized dimensionality reduction, with a bound independent of , and the book's variant with explicit constants is what later chapters and the compressed-sensing proofs use. Theorems 23.6–23.9 together are the three "surprising results" of compressed sensing: information-theoretic recoverability from RIP, efficient recovery by convex relaxation, and the existence of RIP matrices by randomness; their proofs, Candès' cone argument and Baraniuk–Davenport–DeVore–Wakin's net-plus-union-bound, are among the cleanest in applied mathematics and are natural formalization targets. On the platform, the mission introduces Gaussian random matrices as product measures and the RIP predicate, usable by later work on sparse recovery.
Lemma 23.1 requires building an orthonormal basis of the range of , padded to vectors when the range has smaller dimension, and the identity ; Equation (23.3) is a trace computation. Theorem 23.2 combines (23.3), the change of basis with , the bound from extending to an orthogonal matrix, and Exercise 2, a rearrangement inequality; Remark 23.1 adds . Lemma 23.3 is the concentration of a variable (Lemma B.12), which must itself be established from the Gaussian moment generating function; the Johnson–Lindenstrauss lemma is then a union bound. Theorem 23.6 is a two-line contradiction with RIP applied to . Theorem 23.8 is the substantial one: the partition of into blocks of largest remaining entries, the bound , the -minimality inequality (23.8), Lemma 23.10, and the two claims combined through (23.5); a formal proof must handle the last, possibly shorter block, which the book's "assume is an integer" sidesteps. Lemma 23.11 is a volumetric net bound; Lemma 23.12 applies the Johnson–Lindenstrauss lemma to the image of an -net of the unit sphere of and closes the gap by the "smallest " argument, and Theorem 23.9 is a union bound over index sets.
Vectors are plain functions Fin d → ℝ with explicit norms, and matrices are Mathlib matrices, so the objectives are finite sums with no coercions between normed spaces. Random matrices are functions Fin n → Fin d → ℝ with the product of Gaussian laws gaussianReal 0 v, applied through Matrix.of; probability statements bound the outer measure of the failure event, and the failure events of Lemmas 23.4 and 23.12 are written with so that the book's strict conclusions follow. "Eigenvectors corresponding to the largest eigenvalues" is formalized as the first columns of a spectral decomposition with nonincreasing diagonal, which is exactly the set of such systems and avoids Mathlib's eigenvalue ordering conventions. Minimizers (, , ) are arbitrary elements of the argmin.
Five statements are given as their proofs support them, and the item texts say so. Lemma 23.3 and the Johnson–Lindenstrauss lemma are stated for : the printed range (and ) is false, since the upper tail decays like , slower than for (at it fails for ); the audit found this. Lemma 23.1 as printed, "every solution has orthonormal columns and ", is false, since has the same objective as ; the item states what the proof shows, that every is dominated by some with , which is all that (23.2) needs. Theorem 23.9 is stated with : Lemma 23.12 with (so that lies within ) and , followed by a union bound over the at most index sets, gives these constants, and the printed and are not reached by the argument. Theorem 23.8's proof assumes is an integer for simplicity; the statement is given without that assumption, since only the last block of the partition can be short and the block inequality still holds. Lemma 23.3 has , and the Johnson–Lindenstrauss lemma , since for its is and the conclusion fails.
Not stated: §23.1.2 (implementation), Remarks 23.2–23.3, §23.4 (the comparison of PCA and compressed sensing), Exercises 1 and 3–6.
Clustering is the most widely used tool of exploratory data analysis and, at the same time, the least well defined: similar points should share a cluster and dissimilar points should not, but similarity is not transitive while cluster membership is, and without labels there is no ground truth against which to evaluate a proposed grouping. Chapter 22 of Shalev-Shwartz and Ben-David, Understanding Machine Learning: From Theory to Algorithms (doi:10.1017/CBO9781107298019), surveys the main paradigms, linkage-based algorithms, cost minimization with the k-means family, spectral relaxations of graph cuts, and the information bottleneck, and then returns to the question of what clustering is through Kleinberg's axioms. Its one theorem about that question is negative: no clustering function is simultaneously scale invariant, rich and consistent (Theorem 22.4). The mission formalizes this impossibility together with the chapter's positive facts: an iteration of the k-means algorithm never increases the k-means objective (Lemma 22.1), the RatioCut objective is the trace of a quadratic form of the graph Laplacian over cluster indicator vectors (Lemma 22.3), and the farthest-first traversal is a 2-approximation for the k-diam objective (Exercise 3).
A clustering of a finite set is a partition . For the k-means objective is with the centroid of , equivalently (22.1); the k-means algorithm alternately reassigns each point to a nearest centroid and recomputes the centroids. For a similarity matrix , the degree matrix is , the unnormalized graph Laplacian is (Definition 22.2), and . Kleinberg's setting is a clustering function that takes a dissimilarity over , symmetric, zero on the diagonal and positive off it, and returns a partition; the three axioms are Scale Invariance (), Richness (every partition is some ) and Consistency (shrinking within-cluster and expanding between-cluster dissimilarities leaves unchanged). The k-diam objective is , and the farthest-first traversal picks arbitrarily and maximizing , then clusters by nearest center.
For a finite domain with at least two points, there is no function from dissimilarities over to partitions of satisfying Scale Invariance, Richness and Consistency.
Lemma 22.1 (a k-means iteration does not increase ); the Laplacian identity from the proof of Lemma 22.3; Lemma 22.3 ( and for ); Exercise 3 (farthest-first traversal is a 2-approximation for k-diam). Further item: the centroid minimizes , the content of (22.1)–(22.3).
Kleinberg's theorem is the chapter's conceptual center: it says there is no ideal clustering function, only trade-offs, and the choice of a method must encode prior knowledge about the task, the unsupervised analogue of the No-Free-Lunch theorem. Its proof is short but delicate about what a dissimilarity is, and formalizing it fixes the exact hypotheses. Lemma 22.1 is the only guarantee the book offers for Lloyd's algorithm, and it is the reason the algorithm terminates on finite data. Lemma 22.3 is the bridge from a combinatorial cut objective to the spectrum of the Laplacian, the starting point of spectral clustering and of the PCA-type argument used in Chapter 23. The farthest-first result of Exercise 3 is Gonzalez's classical 2-approximation for k-center-type objectives, stated here for the diameter objective, and it is tight in the sense that no better constant is possible unless P = NP.
Theorem 22.4 follows the book: Richness gives with all-singleton output and with a different output; positivity lets one scale above pointwise, and Scale Invariance and Consistency then force two different values for . Formally the work is in building the scaled dissimilarity and in comparing Setoids. Lemma 22.1 is two inequalities: the nearest-centroid reassignment does not increase for the old centroids, because it minimizes it pointwise over assignments, and recomputing centroids does not increase it either, because the centroid minimizes the within-cluster sum of squares; the latter is the separate centroid item, a completing-the-square computation in an inner product space. The Laplacian identity is a finite double-sum manipulation that uses the symmetry of ; Lemma 22.3 applies it to the columns of and computes from the partition structure. Exercise 3 is the hint's argument: let be the distance from the next farthest-first point to the chosen centers; every point is within of its center, so every cluster of the algorithm has diameter at most , while the points are pairwise at distance at least , so two of them share a cluster of any -clustering, whose diameter is then at least . When the argument degenerates but the statement stays trivially true.
Partitions are -indexed families of finsets covering each point of the data exactly once, and nearest-center assignments and farthest-first centers are predicates rather than functions, so every tie-breaking rule is covered. The k-means items live in as EuclideanSpace; the centroid of an empty cluster is , which never enters any sum. The spectral items use Mathlib matrices over Fin m, Matrix.diagonal, Matrix.trace, the root-namespace dotProduct, and require symmetric, which the identity needs and which every similarity matrix satisfies; Lemma 22.3 requires nonempty clusters, without which has a zero column. Kleinberg's function is formalized on a fixed finite domain, as a map from Dissimilarity X to Setoid X, dissimilarities being positive on distinct points as in Kleinberg (2003): the book's model of p. 309 only asks for , but the scaling step of the proof of Theorem 22.4 requires positivity, and the theorem is stated for domains with at least two points, since the proof uses two partitions only. The k-diam theorem is stated without a maximum: every cluster of the algorithm has diameter at most twice the diameter of some cluster of the competitor, which is without conventions for empty index sets, and Metric.diam gives on sets of fewer than two points, the exercise's convention.
Not stated: the linkage-based algorithms and dendrograms of §22.1 (no theorem is stated about them), the k-medoids and k-median objectives, the spectral clustering algorithm itself, the information bottleneck of §22.4, Exercises 1, 2 and 4–6.
When a charged particle scatters, it can radiate photons of arbitrarily low energy. In perturbative quantum electrodynamics this shows up as infrared divergences: loop integrals over very soft virtual photons diverge logarithmically, and so do the rates for emitting soft real photons. The standard resolution, due to Bloch and Nordsieck and systematized by Yennie, Frautschi and Suura, is that the divergences cancel in any rate that sums over undetectable soft emissions below an energy resolution .
S. Weinberg's paper Infrared Photons and Gravitons (Phys. Rev. 140, B516 (1965)) shows that the same mechanism works for gravitons, derives explicit formulas for the exponents that control the soft factors, and proves that the extra "collinear" divergences produced by massless hard particles cancel for gravitons but not for photons. The soft-graviton factor (2.5)–(2.7) used there is the factor that appears in Weinberg's soft graviton theorem, and the spectrum formula is used in the paper to estimate the gravitational radiation from thermal collisions in the sun.
Timeline.
Work in units with the paper's metric: for four-vectors and , . A process has finitely many external lines ; line has mass , three-momentum , on-shell energy , charge , and sign (outgoing) or (incoming). The relative velocity of lines is
For a unit vector (the direction of a soft quantum) define the photon and graviton angular functions
and the infrared exponents and , integrals over the unit sphere against surface measure. With infrared cutoff and dividing point , virtual soft quanta multiply a rate by or , and summing over real soft emission of total energy at most gives the cutoff-free factors and of Eqs. (2.51)–(2.52).
For masses ,
with for the diagonal terms. This is the formula for graviton bremsstrahlung in an arbitrary collision.
The exponents and are the only process-dependent quantities in the soft factors: once they are known in closed form, the infrared behaviour of any rate, the shape or of the soft spectrum, and the soft power spectrum (2.53) follow. The Sec. III cancellation is the statement that gravitation, unlike massless electrodynamics or Yang–Mills theory, is free of unremovable collinear divergences at this level; the paper uses the contrast to argue that charged massless particles are problematic. The expansion (4.9) feeds the nonrelativistic quadrupole formula (4.13) behind the solar estimate.
The paper's arguments are informal physics derivations. This mission makes the mathematical content of those derivations precise and machine-checkable: the angular integrals, the sign statements, the limiting behaviour as a mass goes to zero, and the combinatorial factorization.
The closed forms need the solid-angle integral of a product of two denominators for two arbitrary, non-collinear momenta; the one-denominator case is a textbook integral in polar coordinates, but the two-denominator case does not reduce to it by an obvious choice of axis. The positivity statements are false without the conservation laws, so any proof must use them in an essential way. The massless-limit statements concern the interplay of a logarithmically divergent term with a coefficient that vanishes only at the limiting configuration, so a termwise limit fails.
Three-momenta live in EuclideanSpace ℝ (Fin 3); energies are the on-shell values and are not free variables. Solid-angle integrals are Bochner integrals on the unit sphere against Mathlib's volume.toSphere (total mass ). External lines are indexed by an arbitrary Fintype; the signs are real numbers constrained to by hypothesis. Sums over run over all ordered pairs, including , where ; the kernels and are extended by their limits and at , so the formulas are not trivialized by Lean's convention . The printed Eq. (4.6) has an exponent on the logarithm's argument that contradicts (2.26), (4.5) and (4.9); the definition of drops it. The prescriptions are suppressed. The integration in (2.13) is carried out, leaving an integral over the shell . Positivity is stated as , because in degenerate configurations. Sec. IV beyond (4.9) (the velocity expansion (4.8), the quadrupole formula (4.13), the solar estimate) and the Coulomb phases of Sec. V are out of scope for this mission.
A complete development needs solid-angle integration on (a spherical-coordinates change of variables for volume.toSphere), integration in polar coordinates on , and elementary inequalities for Minkowski products of timelike and null vectors. These are reusable beyond this mission, and contributions of them as separate lemmas are welcome.
Solitons (localized, finite-energy, non-dissipating solutions of classical field equations) are the simplest examples of the topological objects that dominate the non-perturbative physics of gauge theories: vortices, magnetic monopoles and instantons. G. 't Hooft's lecture notes Monopoles, Instantons and Confinement (arXiv:hep-th/0010225) open with the two textbook examples in one space and one time dimension, the φ⁴ kink and the sine-Gordon soliton, because every later construction in the notes (the Abrikosov–Nielsen–Olesen vortex, the 't Hooft–Polyakov monopole, the BPST instanton) repeats the same pattern: a degenerate vacuum, a static solution interpolating between two vacua, and an energy (or action) bounded below by a topological quantity — the Bogomol'nyi bound — which the soliton saturates.
This mission formalizes that pattern in its cleanest setting, Chapter 1 (Sections 1.1–1.2) together with Exercise (i) of the notes.
A static real scalar field is a function . Its energy (eq. (1.5)) for a potential is
Two potentials are considered, with parameters :
The static field equation is . The notes give the explicit solutions
which interpolate between and (case (a)) and between and (case (b)).
For every : in case (a) the kink has energy and no differentiable configuration with , has smaller energy; in case (b) the soliton has energy and no differentiable configuration with , has smaller energy:
The result identifies the soliton mass with a boundary term, with , so the mass depends only on the pair of vacua joined, and shows the kink is a global energy minimizer in its topological sector — the classical stability statement behind treating the kink as a particle of mass , heavy in the weak-coupling regime.
The physics is classical and fully known. What this mission adds is a machine-checked account: explicit verification of the closed-form solutions, exact evaluation of improper energy integrals, and a rigorous Bogomol'nyi argument for configurations that are merely differentiable, with limits at rather than compact support. No existing platform theorem matching these statements was found by a library search at drafting time.
The pointwise inequality is elementary; the difficulty lies in turning it into a statement about improper integrals over for arbitrary differentiable fields. One must handle configurations of infinite energy, pass from integrals over to using only the boundary limits, and deal with being non-smooth at the vacua of and at every vacuum of (a configuration may overshoot a vacuum). The explicit energy computations require exact evaluation of and type integrals as Lebesgue integrals.
deriv. Every competitor in the bounds is assumed differentiable everywhere, and boundary values are expressed as limits at and .MonopolesInstantonsConfinement, intended to be shared by later missions drawn from the same notes.Lecture 5 of the Les Houches lectures on deep learning at large and infinite width (arXiv:2309.01592, Section 5, lectures by B. Hanin) shows that a random ReLU network evaluated at a single input can be solved exactly. The statistics of the network depend on depth and width through the inverse temperature . The input–output Jacobian is the simplest quantity where this can be seen. Its second moment does not depend on depth or width. Its fourth moment grows like , which gives a quantitative sense in which the regime where both and are large is controlled by rather than by or alone. The lectures derive this through a combinatorial sum-over-paths formalism developed in Hanin's earlier papers (references [20–22] of the notes).
Fix a depth and widths . Let be a probability measure on that has a density with respect to Lebesgue measure, is symmetric (), and has variance . The weights are
all biases are , and the preactivations at an input are
The network output is . The input–output Jacobian has entries . A path is a tuple with , and is the set of paths with and . Along a path, and .
For every fixed input and all ,
Identity (124) says that, with the initialization and zero biases, the typical size of the input–output Jacobian does not depend on depth or width. This is the exact form of the "criticality" of ReLU at . The fourth-moment statement shows that the fluctuations are not controlled in this way: they grow exponentially in . Together with Proposition 5.2, these results turn questions about deep random ReLU networks at one input into questions about products of random matrices with dropout. As far as the drafter knows, these statements have not been machine-checked before. The mission asks for formal proofs of the lecture's exact identities.
The ReLU network is not a linear function of the weights. Its activation pattern depends on the weights of all earlier layers, so the path sum cannot be averaged term by term without first showing that the activation pattern is, in distribution, independent of the weights. That is the content of Proposition 5.2, which is argued only in sketch form in the notes. On top of that, the Jacobian must be identified with the path sum almost surely: at inputs where a preactivation vanishes the network is not differentiable. This requires the density assumption on and a separate treatment of layers in which every neuron is inactive.
n : ℕ → ℕ; only are used. The normalized weights are coordinates of the product measure on a finite index set (weightLaw). The Bernoulli masks are coordinates of the uniform product measure on Bool (maskLaw).fderiv) of at applied to the -th basis vector. Mathlib returns at points of non-differentiability, which form a null event for .At infinite width a randomly initialized network is a Gaussian process (Mission I of this series). Real networks have finite width , and the leading departure from Gaussianity is measured by the connected four-point function . It captures both correlations between neurons and non-Gaussian fluctuations. Lecture 4 of the Les Houches lectures (arXiv:2309.01592, lectures by B. Hanin) states the central finite-width result, Theorem 4.2: is of order and obeys an explicit layer-to-layer recursion up to . At criticality this gives the effective depth as the parameter controlling finite-width effects. The result was first derived at a physics level of rigor by Yaida (2020) and in Roberts–Yaida–Hanin (2022), and later derived more mathematically by Hanin (reference [19] of the notes).
A network of depth with widths and nonlinearity has preactivations and . The parameters are independent, with and , where and (eqs. (118)–(119)). At a single input , write for the average of against . The infinite-width kernel is and (eq. (120)). The parallel susceptibility is . The normalized connected four-point function is
If the hidden widths satisfy , then and
with constants independent of the widths.
Theorem 4.2 is the first quantitative statement that finite-width networks at initialization are not Gaussian processes. The size of the deviation is per layer, and it accumulates linearly in depth at criticality. This is the basis for the claim of Lecture 4 that controls correlations between neurons, fluctuations and, in later lectures, feature learning. As far as the drafter knows these statements have not been machine-checked. Lemma 4.4 and the covariance exercise are exact finite-width identities and are natural first targets.
The next layer is Gaussian only conditionally, with a random variance that is an average over dependent neurons. Establishing the recursion to order requires expanding Gaussian averages around the mean of and controlling all higher cumulants of this collective observable uniformly in the widths. The nonlinearity is only assumed polynomially bounded, so smoothness must come from Gaussian averaging, not from .
LesHouchesWidth_GaussianMLP: the network mlpZ, stdGaussianParams, nngpKernel, uniformWidths) are reused. Mission I must be launched first, and its definition then added to this proposal as a reference item.LesHouchesWidth_FiniteWidth): gaussAvg, gaussAvgVec, gaussVarSq, chiParallel, PolyBounded, kappa4, dressedTwoPoint, collectiveSigma.A fully connected neural network with random Gaussian weights defines a random function of its input. Lecture 1 of the Les Houches lectures on deep learning at large and infinite width (arXiv:2309.01592, lectures by Y. Bahri) explains that, when the hidden layers become infinitely wide, this random function becomes a Gaussian process (the "neural network Gaussian process", NNGP). Its covariance kernel is computed by an explicit layer-to-layer recursion. The observation goes back to Neal (1996) for one hidden layer. It was extended to deep networks by Matthews et al. and Lee et al. (2018). It underlies Bayesian inference with infinitely wide networks (Section 1.6) and the analysis of signal propagation at large depth (Section 1.7). Lecture 2 introduces Wick's theorem, the tool for computing moments of Gaussian vectors that the lectures then use for finite-width corrections.
A network of depth with widths and nonlinearity maps an input to preactivations
with independent and (eqs. (1)–(3) and (5); layers are indexed as in Lectures 4–5, so of Lecture 1 is here). For a covariance write (eq. (15)). The NNGP kernel is
A pairing of is a partition into two-element blocks.
For a network with one hidden layer of width , fixed inputs and output width , as the vector converges in distribution to a centered Gaussian with covariance
A further item states the deep version of the limit, eqs. (13)–(14), in the simultaneous-width limit. It is included as a supporting theorem rather than a milestone.
Result 1 and its deep extension identify the prior over functions induced by random initialization. They also make the NNGP kernel the central computational object of the infinite-width theory. The finite-width covariance identities (9)–(11) are exact and explain where the recursion comes from. Formula (16) makes the recursion explicit for ReLU. Wick's theorem is the basic tool of the finite-width perturbation theory of later lectures. These are classical results. The mission asks for their formal proofs against a single shared model of random networks that the later missions of this series reuse.
Result 1 is a multivariate central limit theorem for sums of i.i.d. vectors whose entries are products of a Gaussian weight and a nonlinear function of Gaussian first-layer preactivations. No assumption beyond square-integrability of against the relevant Gaussians is imposed, so the CLT must be applied in its form. The deep limit is harder: for the hidden preactivations are not Gaussian at finite width, and one must control a triangular array in which the widths of all layers grow together. The ReLU formula (16) is an explicit but delicate Gaussian integral over a cone.
stdGaussianParams), scaled by and (mlpBias, mlpWeight). This is equality in law with the prior (5).multivariateGaussian. Convergence in distribution is stated with bounded continuous test functions: for every bounded continuous .Big-bang nucleosynthesis and the acoustic peaks of the cosmic microwave background fix the baryon density of the universe to about 5% of its total energy density, but in the low-redshift universe only some 10% of those baryons are seen in galaxies, with another 10% in the circumgalactic and intracluster medium. Simulations place the remainder — the missing baryons — in a diffuse warm-hot intergalactic medium (WHIM) at – K, spread along the filaments of the cosmic web. Absorption-line studies and X-ray emission from individual filaments probe the cool and the hot ends of that range respectively, leaving the bulk poorly constrained.
de Graaff, Cai, Heymans & Peacock (2019) attack the problem statistically, by stacking the Planck Compton- map over pairs of CMASS galaxies and measuring the residual signal in the bridge between each pair. Interpreting that residual requires an analytic model of a filament, and that model — a cylinder with a Gaussian cross-section, convolved with the instrument beam — is what this mission formalizes. The same model, applied to the CMB lensing convergence map, converts the measured signals into a gas density and temperature, and hence into a baryon fraction.
A filament is modelled as an infinite cylinder seen side-on. In the plane perpendicular to the filament axis, write for the line-of-sight coordinate and for the transverse coordinate on the sky. The electron density is taken to be a two-dimensional Gaussian
with central density , intrinsic width and beam width ; the transverse direction, unlike the line of sight, is smoothed by the instrument beam, which is why appears only there.
The Compton parameter of a line of sight is the optical-depth-weighted temperature of the scattering electrons,
with the Boltzmann constant, the Thomson cross-section, the electron rest energy and the electron temperature, assumed constant across the filament. The convergence of CMB lensing is the analogous projection of the total matter density contrast ,
with the Hubble constant, the matter density parameter, the scale factor and the comoving distances of lens and source. For a structure thin compared with the lensing kernel, the geometric factor is constant across it and reduces to a line-of-sight integral of against a fixed prefactor.
The significance of the measurement is assessed with a second, statistical layer. A stack of binned profiles ( the map index, the bin index) has mean profile and covariance estimator ; the deviation of the mean profile from zero is measured by . Because the stacked maps overlap on the sky, the paper replaces by a jackknife covariance built from sky sub-samples, , and multiplies the inverse by the Hartlap factor , with the number of bins.
Measurements enter through two numbers: the mean Compton parameter and the mean convergence over the boxed filament region, together with the empirical observation that the peak of each profile is close to times its mean. Every statement in this mission imposes that calibration as a hypothesis on the model.
This is the quantity the paper's baryon budget is computed from: given the measured , an assumed temperature and the beam width, it fixes the number of electrons in a filament of length .
Both and are symmetric and positive semidefinite, for every data set and every sample size.
The four displayed identities are the entire inferential chain from two stacked maps to a baryon fraction. Eq. (A.2) says what the model predicts for the observable; Eq. (A.3) inverts it at the filament axis; Eq. (A.4) turns the inversion into a total electron count, from which the paper obtains gas at and K, accounting for of the cosmic baryon budget; Eq. (A.5) supplies the independent lensing constraint that breaks the density–temperature degeneracy of the SZ measurement alone. The statistical milestones cover the other half of the analysis: the covariance estimators of Eqs. (3) and (5)-(6) that turn a stack into an error bar, the nonnegativity of the of Eq. (4) that is converted into the quoted , and the Hartlap rescaling of Eq. (7). The beam-dominance milestone quantifies the paper's claim that the result barely depends on the one free shape parameter, the intrinsic width , which is taken from simulations rather than measured.
Formalizing them contributes a machine-checked derivation layer for a widely used observational technique: the projection of a Gaussian cylinder onto a Compton- or convergence map, and the inversion of that projection, recur throughout stacked-SZ and stacked-lensing analyses. What this mission adds over the paper is a statement of each identity with all its hypotheses exposed — which positivity conditions are needed, which parameters genuinely drop out — rather than a new physical result. The physics is not re-derived and the measurement is not re-analysed; the empirical inputs , and the peak-to-mean ratio enter as hypotheses, not as claims.
The identities are Gaussian integrals, so the mathematical depth is modest; the difficulty is bookkeeping and faithfulness. Three specific traps. First, the line-of-sight and transverse directions have different widths — against — and the asymmetry is what makes the final answer depend on the beam; a formalization that symmetrizes them is a different theorem. Second, the paper's is not free: it is pinned by the calibration , so the goal must carry that equation as a hypothesis rather than substituting a closed form for silently. Third, the total electron count is an iterated integral over the plane, and the inner and outer variables must be kept in the paper's order.
Everything is stated over with no units, so every physical constant is an explicit real variable; integrals are Bochner integrals over the whole real line with Lebesgue measure, which return on a non-integrable integrand. The definition bundle fixes the model once — the SZ prefactor, the Gaussian cylinder profile, the Compton parameter, the thin-lens convergence prefactor, the convergence and the total electron content — and every statement is phrased against it. The convergence definition is the thin-lens specialization of Eq. (2): the geometric kernel is a constant prefactor, not an integral over lens distance, and the mission does not claim the reduction from the full Eq. (2) to that form.
Degenerate readings are ruled out as follows. The calibration hypotheses are satisfiable for every admissible choice of constants, so no statement is vacuous; conversely they genuinely constrain the central amplitude, so no statement holds for a trivial reason. The positivity hypotheses are the minimal ones that make the denominators nonzero, and the widths enter as with unrestricted, so the beam-free case is included rather than excluded. The statistical layer is finite-dimensional: profiles are real arrays indexed by a finite bin type, covariances are real matrices, and matrix inversion is the nonsingular inverse, which returns the zero matrix on a singular argument — hence the positive-definiteness hypothesis in the milestone. Counts are natural numbers but all arithmetic in the prefactors is performed after casting to the reals, so is a genuine real subtraction and may be negative. The statistical statements assert the structural properties of the estimators as written; they do not assert unbiasedness, nor the reduction for the covariance of a mean, nor any property of the resampling scheme. Only Mathlib's Gaussian-integral, measure-theory and positive-semidefinite-matrix API is needed; no new infrastructure is required, and the resulting projection lemmas are reusable for any other stacked-profile analysis.
The Rydberg constant is the constant of spectroscopy that sets the scale of the hydrogen spectrum. It first appeared as an empirical fitting parameter in J. Rydberg's formula for the hydrogen spectral series; N. Bohr later showed that its value can be computed from more fundamental constants within his model of the atom. Before the 2019 revision of the SI it was, together with the electron spin -factor, among the most accurately measured physical constants (relative standard uncertainty about , CODATA 2022). This mission formalizes the algebraic content of the standard account of as presented in the Wikipedia article Rydberg constant: its closed form, its reduced-mass correction, its alternative expressions through the fine-structure constant, and the Bohr-model derivation of the Rydberg formula.
Fix five strictly positive real numbers: the electron rest mass , the elementary charge , the vacuum permittivity , the Planck constant and the speed of light . From them define
the Compton wavelength , Compton frequency , Compton angular frequency , Bohr radius , classical electron radius , and the Rydberg unit of energy .
For a nucleus of mass the reduced mass is and the corrected Rydberg constant is ; for hydrogen and .
A Bohr orbit with principal quantum number of a particle of mass and charge about a fixed charge is a circular orbit of radius and speed such that the Coulomb force supplies the centripetal force and the angular momentum is quantized:
Its energy is . The infinite-nuclear-mass model uses ; the reduced-mass model uses .
For distinct positive integers and Bohr orbits of mass with quantum numbers , the wavenumber of the photon emitted in the transition satisfies
These identities are the standard bridge between the empirical Rydberg formula and the constants : they explain why a single constant governs all hydrogen series, why isotopes such as deuterium show shifted lines (the shift led to the discovery of deuterium), and how relates to , and the Compton scales. The results are classical and not open; the mission's contribution is a machine-checked, reusable layer of definitions (Bohr orbits, reduced mass, the named atomic length and energy scales) on which later atomic-physics formalizations can build.
The content is elementary algebra with positivity side conditions. The main point to handle carefully is that the Bohr-orbit conditions determine and only implicitly; the energy must be computed from the two defining equations rather than from explicit formulas, and every division must be justified by positivity of the constants.
All constants are real numbers bundled in a structure with strict positivity fields; no SI numerical values (CODATA figures, , etc.) are formalized, and the precision-measurement and QED discussion of the source is out of scope. The Bohr model is formalized for a single particle of mass orbiting a fixed charge (hydrogen-like, ); the reduced-mass correction is modelled, as in the source, by substituting for . The photon wavenumber is taken to be the orbital energy difference divided by . The Bohr-orbit hypotheses are satisfiable for every and every , so the Bohr-model statements are not vacuous. Contributions of alternative proofs and of generalizations (nuclear charge , explicit orbit radii and speeds) are welcome.
Since the 2019 revision of the International System of Units (SI), the elementary charge is no longer a measured quantity: it is one of the seven defining constants of the SI and is fixed by definition at
Before that revision, had to be extracted from experiment, and the metrology literature accumulated several independent routes to it: Faraday's laws of electrolysis combined with the Avogadro constant, Millikan and Fletcher's oil-drop experiment (1909), shot-noise analysis, and — since the 1980s — the combination of the Josephson effect and the quantum Hall effect. Each route is a short algebraic recipe that turns other measured constants into . The 2019 redefinition did not delete those recipes; it inverted their role. With , the Planck constant , the Avogadro constant and the speed of light all fixed exactly, the recipes become exact arithmetic identities between defining constants, and the residual measurement uncertainty migrates to the constants that are no longer fixed (the vacuum magnetic permeability , equivalently the fine-structure constant ).
This mission formalizes exactly those identities, as they are stated in the source article, together with the two quantization statements the same article records: Dirac's 1931 argument that a magnetic monopole forces charge quantization, and the closure property behind the observation that although quarks carry charges in multiples of , isolatable particles carry integer multiples of .
All quantities are real numbers carrying SI units implicitly; the Lean development works in ℝ and fixes the numerical values as exact rationals, never as floating-point approximations.
The defining constants used here are (coulomb), (per mole), (joule second) and (metre per second).
From these the article builds four derived constants, each of which is a definition in this mission rather than an axiom:
A fifth definition captures the natural unit of charge of those natural unit systems in which , with the electric constant and the reduced Planck constant.
The goal theorem asserts that the three determination routes recorded in the source return one and the same number, the fixed SI value of , and that the electrolysis route's intermediate constant has its exact SI value:
The milestones are the individual identities, each stated for arbitrary positive values of the constants rather than only at the SI values, plus the two quantization statements:
The result itself. The content is the coherence of the SI's electrical sector: the same comes out of an electrochemical measurement chain, a quantum-electrical one and the CODATA relation, and the arithmetic that links them is exact. Milestones 6 and 7 record the two quantization statements in the source: the Dirac argument is a genuine implication (a monopole forces the charge spectrum into a discrete subgroup), while milestone 7 isolates the algebraic half of "stable groupings of quarks carry integer charge" — closure of a quantized spectrum under composition.
Formalizing it. What this mission produces is a small, faithful, reusable encoding of the 2019 SI defining constants and of the standard derived constants, with the exact rational values rather than floating-point ones, and with the degenerate cases made explicit. Nothing here is an open problem, and the mission should not be read as one: the value is a checked reference layer for physics-flavoured formalization, not a research advance.
Low, and stated as such deliberately. Milestones 1–3 are field arithmetic; 4 and 5 are field arithmetic plus Real.sqrt of a square, which needs the positivity hypotheses that are carried explicitly; 6 and 7 are elementary manipulations of ℤ-multiples and of AddSubgroup.closure. The only places a solver can go wrong are the degenerate ones: division by a constant that has not been assumed nonzero, and Real.sqrt of a quantity not known to be nonnegative — which is why every statement carries positivity or nonvanishing hypotheses rather than leaving Lean's junk values to decide the outcome.
Conventions this mission commits to:
eSI = 1602176634 / 10 ^ 28), so the numeric milestone is an exact identity, provable by norm_num, not a floating-point comparison.Real.sqrt a nonnegativity or positivity hypothesis, so no statement is discharged by a Lean junk value.ℝ, using only Mathlib.Contributions welcome beyond the current list: the shot-noise and oil-drop routes, which the source describes qualitatively rather than by a formula, would need a probabilistic or mechanical model before they can be stated, and that model would be a worthwhile addition.
The Boltzmann constant is the proportionality factor relating the average thermal energy of the particles of a system to its thermodynamic temperature. It appears wherever a microscopic energy has to be compared with a macroscopic temperature: in the ideal gas law written per molecule, in the equipartition theorem, in the Boltzmann factor that governs equilibrium occupation probabilities, in Boltzmann's entropy formula , in the thermal voltage of a – junction, and in the Johnson noise of a resistor.
The constant also has an unusual documentary history. Boltzmann linked entropy and probability in 1877 but never introduced a constant; Max Planck first wrote , and gave the first numerical value ( J/K, about 2.5% below the modern figure), in his 1900–1901 derivation of the black-body law — the terse on Boltzmann's tombstone is Planck's formulation. For most of the twentieth century was a measured quantity; the 2017 acoustic gas thermometry campaign reached a relative uncertainty of 0.2 ppm, and as part of the 2019 revision of the SI the constant was defined to the exact value J·K⁻¹, which is now what fixes the kelvin.
This mission formalizes the elementary quantitative content of that picture: the identities that make a conversion factor, and the exact arithmetic that the 2019 SI definitions make possible.
Work throughout with real numbers; units are carried in the prose, not in the types. Fix the exact post-2019 SI values
and define the molar gas constant as the product .
A gas sample is described by its pressure , volume , absolute temperature , amount of substance , molecule count , particle mass , and mean square particle speed . A system with a finite set of microstates is described by an energy function ; its partition function is and its Boltzmann probabilities are . For a distribution on , the Gibbs entropy is , the Shannon entropy (in nats) is , and for equiprobable microstates Boltzmann's entropy is . The thermal voltage is . All logarithms are natural.
The goal theorem is the statement that fixes the physical meaning of : kinetic theory plus the per-molecule gas law determine the mean translational kinetic energy. From
conclude
The milestones are the surrounding identities, in the order of the source: the equivalence ; equipartition, per degree of freedom; the ratio law ; normalisation of the Boltzmann factors, ; the reduction of the Gibbs entropy on the uniform distribution to ; the identification Shannon entropy; the energy of one nat of rescaled entropy; and three numerical facts — mV, eV/K, and the exact value J·K⁻¹·mol⁻¹.
The results. Individually these are the standard first facts of kinetic theory and statistical mechanics; together they pin down what the constant does. The equipartition chain converts a temperature into a velocity distribution scale and explains the measured room-temperature rms speeds from helium (about 1370 m/s) to xenon (about 240 m/s). The entropy statements make precise the claim that thermodynamic entropy is Shannon entropy in energy units, with as the exchange rate — the statement behind the natural-unit convention . The numerical milestones exercise the consequence of the 2019 SI revision that unit conversions involving are now exact rational arithmetic rather than propagated measurement uncertainty.
Formalizing them. All of these are settled physics; nothing here is open. What the mission produces is a small, reusable Lean development of the SI defining constants and of the elementary statistical-mechanical vocabulary (Boltzmann weights, partition function, Gibbs/Shannon/Boltzmann entropies), with each textbook identity stated against that shared model rather than re-derived ad hoc. It is a suitable entry-level mission and a foundation other physics missions can import.
The mathematics is elementary; the difficulty is entirely one of faithful modelling, and solvers should expect the friction to be there rather than in the proofs. Three specific points. First, Lean's division and logarithm are total: is defined at and evaluates to , and for , so statements must carry the positivity hypotheses that keep them physically meaningful rather than relying on the definitions to exclude bad inputs. Second, the entropy definitions are stated for an arbitrary real-valued weight function, not for a distribution; normalisation is a hypothesis where it is needed, never an assumption baked into the type. Third, the numerical milestones are claims about specific decimal numerals, and the tolerance in each is part of the statement: two of them are genuine approximations and one, the value of , is an exact identity, which is only true because both factors are exact SI numerals.
Quantities are real numbers (ℝ); no dimensional-analysis layer is used, and unit
correctness is a convention of the prose. Finite state spaces are Finsets over an
arbitrary index type, with nonemptiness stated as a hypothesis where the partition
function must be nonzero. Entropies use the natural logarithm, so they are measured in
nats, and the Shannon entropy is the Gibbs entropy divided by by construction.
Temperatures, masses, volumes and particle counts are positive reals where the physics
requires it. The mean square speed is a free real variable constrained only by the
stated equations; it is not built from a velocity distribution, and extending the
development to an actual Maxwell–Boltzmann distribution is out of scope here and a
natural follow-up mission.
The statements are deliberately non-vacuous: every hypothesis set in the mission is
satisfiable (positive temperature, positive mass, nonempty state space), and the
numerical items admit no trivializing reading since they are closed claims about fixed
numerals. Only Mathlib is required — Real.exp, Real.log, Real.sqrt, and
Finset.sum. The definition file is intended to be reusable: any later mission about
thermodynamics, black-body radiation, or the Shockley diode equation can import the SI
constants and entropy vocabulary unchanged.
How large can the power set of an infinite set be? For a regular cardinal (one that is not the supremum of fewer than smaller ordinals) the answer is: almost anything. Easton's theorem (1970) shows that the function on regular cardinals can be prescribed arbitrarily in any model of ZFC, subject only to monotonicity and König's inequality . For a long time it was expected that singular cardinals — those that are such a supremum, like — would behave the same way.
They do not. In 1974 Jack Silver proved that the Generalized Continuum Hypothesis cannot fail for the first time at a singular cardinal of uncountable cofinality: if for every infinite and , then . This was the first ZFC theorem constraining the continuum function at singular cardinals, and it opened the area now called the singular cardinal problem.
A short timeline:
This mission is the first in a series formalizing Thomas Jech, Set Theory (Third Millennium Edition, Springer 2003). It covers Chapter 8, "Stationary Sets" (pp. 91–98).
Fix a regular uncountable cardinal and regard it as the well-ordered set of ordinals below it. A set is closed unbounded, or a club, if it is unbounded in and contains all of its limit points below (an ordinal is a limit point of when ). A set is stationary if for every club . Clubs are closed under intersections of fewer than of them, so they generate a -complete filter, the club filter; its dual is the nonstationary ideal.
The club filter has a second closure property with no analogue for ordinary filters. The diagonal intersection of a -indexed family is
and a filter closed under diagonal intersections is called normal. A function defined on is regressive if for all nonzero .
For cardinal arithmetic, denotes the cofinality of (the least length of an unbounded sequence in ), the cardinal successor, and is singular when . The Singular Cardinal Hypothesis (SCH) is the assertion that for every singular with . A sequence of cardinals is normal if it is strictly increasing and continuous at limits.
This is the weakest statement of the chapter that still needs the full machinery: it fixes no particular and no particular cofinality, and it stays correct no matter how the singular cardinal problem develops above it.
Silver's theorem is the boundary between the two halves of cardinal arithmetic. Above it sit the ZFC theorems of pcf theory; below it sit the consistency results (Magidor, Prikry, Radin forcing) that show how much freedom is left, and they are confined to cofinality precisely because Theorems 8.12 and 8.13 close off everything else. Its proof also packages tools used throughout set theory: the normality of the club filter, Fodor's pressing-down lemma, and the technique of bounding almost disjoint families of functions by a stationary-set argument.
Formalization status: Mathlib already has clubs and stationary sets in an arbitrary well-ordered type (IsClub, IsStationary), with the finite and -indexed intersection lemmas — Jech's Lemma 8.2 and Theorem 8.3. It does not have diagonal intersections, Fodor's theorem, Solovay's splitting theorem, or any singular cardinal arithmetic beyond the definitions of regular and singular cardinals. Each milestone below is therefore a genuine addition, and the first three are reusable well outside this mission.
The naive route to the goal — induct on and pass to the limit — fails immediately: for singular is not determined by the values for in any elementary way; that is exactly the content of the independence results. What the proof must do instead is bound the number of functions on , and it gets that bound from a stationary set rather than from a club: uncountable cofinality is what makes the set of relevant stages stationary, and stationarity is what survives the diagonal argument. Cofinality breaks this at the first step, since every subset of that is unbounded is already a club and Fodor's theorem is empty.
The two hard pieces are Lemma 8.14 and, inside it, Lemma 8.16: given an almost disjoint family of functions with and on a stationary set of , one must show by assigning to each a pair (stationary set, bounded restriction) and checking the assignment is injective. That argument uses Fodor's theorem on a set of functions, and the bookkeeping does not simplify.
The ambient order is Mathlib's type of ordinals below a cardinal, k.ord.ToType (written Below k in the mission's definition bundle); clubs and stationary sets are Mathlib's IsClub and IsStationary on that type, so a set is closed in the sense of being closed under suprema of directed subsets — equivalent, for a well-order with the order topology, to Jech's "contains its limit points". Families indexed by "" are functions out of that same type, which is what makes the diagonal intersection typecheck without a side condition. Cardinal exponentiation, cofinality (Ordinal.cof of k.ord) and the successor cardinal (Order.succ) are Mathlib's.
One trivializing formalization is ruled out explicitly: the GCH hypothesis of the goal is stated for infinite cardinals only. Quantified over all cardinals it would be unsatisfiable — — and Silver's theorem would become vacuous.
A complete development needs: diagonal intersections and the normality of the club filter; Fodor's theorem; the sets and their stationarity; Solovay's splitting theorem via Lemmas 8.8 and 8.9; almost disjoint families of ordinal functions and the counting Lemmas 8.15 and 8.16; and Theorem 5.22(ii) on cardinal powers, which Jech's proof of Theorem 8.13 cites. Contributions of any of these as separate reductions are welcome, as are alternative proofs of the goal (for instance via a generic elementary embedding) that bypass some of the chain.