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

Campaigns (experimental)

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

Block Sensitivity vs. Sensitivity

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

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

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

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

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

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

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

≥ 2.1308255793978239Formalized record≥ 2.15Open frontier

9 provers on it17 of 18 missions formalized

Integer Multiplication Below n log n

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

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

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

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

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

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

8 provers on it4 of 7 missions formalized

3SUM Exponent

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

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

≤ 1.960614Formalized record

4 provers on it7 of 7 missions formalized

All-Pairs Shortest Paths (APSP) Exponent

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

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

≤ 2.995561Formalized record

3 provers on it5 of 5 missions formalized

The irrationality measure of π

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

≤ 7.101862832357Formalized record≤ 2Open frontier

17 provers on it8 of 9 missions formalized

Sharp diagonal Hlawka constant

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

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

References:

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

≤ 21Formalized record≤ 16Open frontier

5 provers on it12 of 13 missions formalized

Odd numbers as sums of primes

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

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

≤ 27Formalized record≤ 5Open frontier

38 provers on it13 of 15 missions formalized

Matrix multiplication exponent

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

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

≤ 2.25Formalized record

16 provers on it9 of 9 missions formalized

All missions

Get started

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

Discover

Find your next mission.

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

Campaigns (experimental)

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

All missions

Get started

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

Discover

Find your next mission.

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

Campaigns (experimental)

See all campaigns →

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

Block Sensitivity vs. Sensitivity

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

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

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

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

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

≥ 2.1308255793978239Formalized record≥ 2.15Open frontier

9 provers on it17 of 18 missions formalized

Integer Multiplication Below n log n

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

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

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

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

8 provers on it4 of 7 missions formalized

3SUM Exponent

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

≤ 1.960614Formalized record

4 provers on it7 of 7 missions formalized

All-Pairs Shortest Paths (APSP) Exponent

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

≤ 2.995561Formalized record

3 provers on it5 of 5 missions formalized

The irrationality measure of π

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

≤ 7.101862832357Formalized record≤ 2Open frontier

17 provers on it8 of 9 missions formalized

Sharp diagonal Hlawka constant

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

References:

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

≤ 21Formalized record≤ 16Open frontier

5 provers on it12 of 13 missions formalized

Odd numbers as sums of primes

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

≤ 27Formalized record≤ 5Open frontier

38 provers on it13 of 15 missions formalized

Matrix multiplication exponent

≤ 2.25Formalized record

16 provers on it9 of 9 missions formalized

All missions

Machine Learning Probability Theoretical Computer Science·Captain: mikedeng1

What Can We Learn Privately? II: Parity Functions Are Privately PAC Learnable from O(d log(1/β)/(εα)) ExamplesResearch Paper

Learning parities under differential privacy

Differential privacy (Dwork, McSherry, Nissim and Smith, 2006) requires a randomized algorithm's output distribution to change little when one individual's record is changed. PAC learning asks an algorithm to output, from i.i.d. labeled examples, a hypothesis whose error is small with high probability. Kasiviswanathan, Lee, Nissim, Raskhodnikova and Smith (arXiv:0803.0924v3; SIAM J. Comput. 40(3) (2011) 793–826, DOI 10.1137/090756090) asked which concept classes can be learned under both requirements at once. All theorem, page and display numbers in this mission are those of the arXiv v3.

Their generic learner (§3, the subject of mission I of this series) learns every finite class privately with a polynomial number of examples, but its running time is exponential in the representation size for classes such as parity. Parity functions are the standard example of a class that is learnable efficiently by linear algebra and not learnable with statistical queries (Kearns, 1998; Blum et al., 1994), a model that was then the main known source of efficient private learners (Blum, Dwork, McSherry and Nissim, 2005). Section 4 of the paper gives an efficient private PAC learner for parity. The result separates efficient private learning from statistical-query learning, and the local (randomized-response) model from the centralized one (missions III–V).

Setting

For $d \ge 1$ and $r \in \{0,1\}^d$ , the parity function $c_r(x) = r \odot x$ is the inner product of $r$ and $x$ modulo 2; PARITY is $\{c_r : r \in \{0,1\}^d\}$ . A database is $z = (z_1,\dots,z_n)$ with $z_i = (x_i, y_i) \in \{0,1\}^d \times \{0,1\}$ . Two databases are neighbors if they differ in exactly one entry. A randomized algorithm $A$ , identified with the law $A(z)$ of its output, is $\varepsilon$ -differentially private if $\Pr[A(z) \in S] \le e^{\varepsilon}\Pr[A(z') \in S]$ for all neighbors $z, z'$ and all sets $S$ of outputs. $\mathrm{Lap}(s)$ is the Laplace distribution with density $\frac{1}{2s}e^{-|x|/s}$ .

For $S \subseteq [n]$ , $V_S \subseteq \mathbb Z_2^d$ is the solution space of the linear system $\{x_i \odot r = y_i : i \in S\}$ . The basic learner $\mathcal A(z,\varepsilon)$ outputs $\bot$ with probability $1/2$ ; otherwise it puts each $i \in [n]$ into $S$ independently with probability $p = \varepsilon/4$ and outputs $c_{r^*}$ for $r^*$ uniform in $V_S$ , or $\bot$ if $V_S = \emptyset$ .

The amplified learner $\mathcal A^*(z,\varepsilon,\alpha,\beta)$ sets $\beta' = \beta/2$ , $\alpha' = \alpha/5$ , $k = \lceil\log_{4/3}(1/\beta')\rceil$ , $n' = \lceil cd/(\varepsilon\alpha')\rceil$ , $s = \lceil (c'k/(\alpha'\varepsilon))\log_2(k/\beta')\rceil$ for constants $c, c'$ . If $n \le kn' + s$ it returns "insufficient samples". Otherwise it runs $\mathcal A$ on $k$ disjoint blocks of $n'$ examples, obtaining $h_1,\dots,h_k$ . It perturbs each training error on the next $s$ examples by independent $\mathrm{Lap}(k/(s\varepsilon))$ noise and outputs the $h_j$ with the least perturbed error.

For a distribution $\mathcal X$ on $\{0,1\}^d$ and a target $c_r$ , the error of $c_{r'}$ is $\mathrm{err}(c_{r'}) = \Pr_{x\sim\mathcal X}[c_{r'}(x) \ne c_r(x)]$ , and probabilities of the learner's success are over $x_1,\dots,x_n \sim \mathcal X$ i.i.d., $z_i = (x_i, c_r(x_i))$ , and the learner's coins.

Formalization targets

Goal: Theorem 4.4

There are constants $c, c', C > 0$ such that for all $d \ge 1$ , $0 < \varepsilon \le 1/2$ , $\alpha, \beta \in (0,1/2)$ : $\mathcal A^*(\cdot,\varepsilon,\alpha,\beta)$ is $\varepsilon$ -differentially private on databases of every size, and

n \ge C\,\frac{(d + \log_2(1/\beta))\log_2(1/\beta)}{\varepsilon\alpha} \implies \Pr\big[\mathcal A^* \text{ outputs } h^* \text{ with } \mathrm{err}(h^*) \le \alpha\big] \ge 1-\beta

for every distribution $\mathcal X$ and every target $c_r$ . When $\log_2(1/\beta) \le d$ the bound is the paper's $O(d\log(1/\beta)/(\varepsilon\alpha))$ . The constants are fixed before all other parameters. The goal is literally the conjunction of Lemma 4.5 and Lemma 4.6.

Milestones

\text{Thm 2.3 (Laplace mechanism)},\quad \text{Claim 2.2 (composition)},\quad \text{Thm A.1 (Chernoff)},\quad \text{Occam's razor for } |H| = 2^d,

\text{Lemma 4.1: } n \ge \tfrac{8}{\varepsilon\alpha}(d\ln 2 + \ln 4) \Rightarrow \Pr[\mathcal A(z,\varepsilon) = h,\ \mathrm{err}(h)\le\alpha] \ge \tfrac14,

|V_T| \le 2|V_{T\cup\{i\}}| \text{ if } V_{T\cup\{i\}} \ne \emptyset,\quad \text{Claim 4.3},\quad \text{Lemma 4.2 (}\mathcal A\text{ private)},\quad \text{Lemma 4.5 (}\mathcal A^*\text{ private)},\quad \text{Lemma 4.6 (}\mathcal A^*\text{ learns)}.

Significance

The theorem shows that the restriction to private algorithms does not separate efficient learnability from statistical-query learnability: parity is efficiently and privately PAC learnable, yet not SQ learnable. Its sample complexity exceeds the non-private $O((d + \ln(1/\beta))/\alpha)$ by a factor $O(\ln(1/\beta)/\varepsilon)$ (remark, p. 18). The private amplification step (private model selection with Laplace-perturbed test errors) is a general technique that converts any private learner with constant success probability into a private PAC learner.

The result is proved in the paper; none of it is machine-checked as far as we know. The mission produces a formal library of the basic privacy calculus (Laplace mechanism, composition and post-processing), the $\mathbb Z_2$ solution-space estimate behind the privacy of $\mathcal A$ , and the analysis of both learners. Formalizing it also repairs the paper's statement: the printed sample bound fails when $\log(1/\beta)$ is large compared with $d$ (see the scope section), and the base of the logarithm defining $k$ is misprinted.

Difficulty

The obvious private learner, "solve the linear system on all examples and output a random solution", is not private. One changed example can make the system inconsistent and turn every hypothesis output into failure, so the output law on neighboring databases differs by an unbounded factor. Two devices restore privacy: subsampling each example with probability $\varepsilon/4$ , and failing with probability $1/2$ unconditionally. The privacy proof then needs a quantitative fact about $\mathbb Z_2$ : adding one consistent equation at most halves the solution space. The proof must also handle databases labeled by no parity at all, where solution spaces are empty.

Utility has its own difficulty. The amplified learner selects among candidates through noisy test errors, so three failure events (no good candidate, unrepresentative test set, large Laplace noise) must be bounded together. The sizes $k$ , $n'$ , $s$ must also be shown to fit inside the claimed sample bound.

Formalization scope

Bits are ZMod 2; $\{0,1\}^d$ is Fin d → ZMod 2; $c_r(x)$ is dotProduct r x. Databases are Fin n → (Fin d → ZMod 2) × ZMod 2, with 0-based indices.
An algorithm is the law of its output. $\mathcal A$ 's output is Option (Fin d → ZMod 2) (none = $\bot$ ). $\mathcal A^*$ 's output is "insufficient samples", $\bot$ , or a hypothesis $c_r$ . Every set of outputs is measurable, and differential privacy is required for all measurable sets.
The distribution $\mathcal X$ is a PMF on the finite set $\{0,1\}^d$ . The probability over the examples is a finite sum weighted by $\prod_i \mathcal X(x_i)$ , and $\bot$ has error 1.
Corrections and explicit hypotheses. (1) $k = \lceil\log_{4/3}(1/\beta')\rceil$ : the printed base $3/4$ gives $k \le 0$ , and the proof needs $(3/4)^k \le \beta'$ . (2) Sizes are ceilings; $\log$ is base 2 and $\ln$ is natural, as on p. 8. (3) Privacy is stated for $0 < \varepsilon \le 1/2$ : the proof of Lemma 4.2 uses $\varepsilon \le 1/2$ , and its step $2p/(1-p) + 1 \le \varepsilon + 1$ fails beyond that. Utility statements need only $0 < \varepsilon \le 4$ , so that $p = \varepsilon/4$ is a probability. (4) The printed bound $O(d\log(1/\beta)/(\varepsilon\alpha))$ is false when $\log(1/\beta) \gg d$ , because $\mathcal A^*$ refuses unless $n > kn' + s$ and $s$ grows like $\log^2(1/\beta)/(\varepsilon\alpha)$ . Utility is therefore stated with the bound $C(d + \log_2(1/\beta))\log_2(1/\beta)/(\varepsilon\alpha)$ , the order of $kn' + s$ , which equals the printed order whenever $\log_2(1/\beta) \le d$ . The privacy conjunct does not depend on the sample bound. (5) A $\bot$ candidate does not take part in the argmin, and ties go to the least index. (6) Lemma 4.1 assumes $\alpha \le 1/2$ , which its proof uses. (7) Claim 4.3 is stated in product form (no division). (8) Theorem 2.3 takes any positive bound $\Delta$ on the global sensitivity.
"Efficiently" (polynomial running time) is not modelled.
Ruling out trivial readings. The constants $c, c', C$ are existentially quantified once, outside every other quantifier. A constant chosen after $d$ , $\varepsilon$ , $\alpha$ , $\beta$ would make the utility claim trivial. Success requires an actual hypothesis output, so "insufficient samples" and $\bot$ are failures. Privacy quantifies over all databases, not only consistently labeled ones.
Reused results: ComputationalLearning.chernoff_bounds (Theorem A.1) and ComputationalLearning.occam_cardinality (Occam's razor), both proved on the platform.
Contributions welcome: proofs of the privacy calculus (Theorem 2.3, Claim 2.2) for general use, the $\mathbb Z_2$ halving lemma, and the utility analysis. The first two are reusable well beyond this mission.

Selected references

S. P. Kasiviswanathan, H. K. Lee, K. Nissim, S. Raskhodnikova, A. Smith, What Can We Learn Privately?, SIAM J. Comput. 40(3) (2011) 793–826. arXiv:0803.0924v3, DOI 10.1137/090756090
C. Dwork, F. McSherry, K. Nissim, A. Smith, Calibrating Noise to Sensitivity in Private Data Analysis, TCC 2006. DOI 10.1007/11681878_14
A. Blum, C. Dwork, F. McSherry, K. Nissim, Practical Privacy: The SuLQ Framework, PODS 2005. DOI 10.1145/1065167.1065184
M. Kearns, Efficient Noise-Tolerant Learning from Statistical Queries, J. ACM 45(6) (1998) 983–1006. DOI 10.1145/293347.293351
A. Blum, M. Furst, J. Jackson, M. Kearns, Y. Mansour, S. Rudich, Weakly Learning DNF and Characterizing Statistical Query Learning Using Fourier Analysis, STOC 1994. DOI 10.1145/195058.195147
M. Kearns, U. Vazirani, An Introduction to Computational Learning Theory, MIT Press, 1994. DOI 10.7551/mitpress/3897.001.0001

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Complexity Theory Linear Optimization Operations Research+1·Captain: mikedeng1

An O(√nL)-Iteration Homogeneous and Self-Dual Linear Programming Algorithm 1: In O(√nL) Predictor-Corrector Iterations It Finds LP and Dual Optima or Detects InfeasibilityResearch Paper

Motivation

Interior-point methods solve linear programs in polynomially many iterations, but the classical analyses assume something about the instance before the algorithm starts: a known strictly feasible primal–dual pair, a nonempty interior, or a big- $M$ bound on the feasible region of order $2^L$ , where $L$ is the bit length of the data. In practice none of this is known in advance, and an infeasible or unbounded instance should be reported rather than assumed away.

Ye, Todd and Mizuno (1994) removed these assumptions. They embed a standard-form linear program and its dual into a single homogeneous and self-dual artificial program (HLP), which always has a known interior starting point, and apply the Mizuno–Todd–Ye predictor–corrector method to it. The resulting algorithm needs no regularity assumption, no big- $M$ parameter, and an $O(\sqrt n L)$ iteration bound, the best known for interior-point methods. Self-dual embeddings of this kind are now the standard initialization in interior-point solvers for linear and conic programs.

Timeline. Karmarkar (1984) gave the first polynomial interior-point method on a homogeneous canonical form. Mizuno, Todd and Ye (1990) introduced the predictor–corrector method with $O(\sqrt nL)$ iterations under a known interior start. Kojima, Mizuno and Yoshise (1993) built a self-dual big- $M$ artificial problem. Ye, Todd and Mizuno (1994) combined homogeneity and self-duality without big $M$ , obtaining the result formalized here.

Setting

Given $A\in\mathbb R^{m\times n}$ , $b\in\mathbb R^m$ , $c\in\mathbb R^n$ , the primal problem is (LP) $\min\{c^Tx : Ax=b,\ x\ge0\}$ and its dual is (LD) $\max\{b^Ty : A^Ty\le c\}$ , with dual slacks $s=c-A^Ty$ .

Fix $x^0>0$ , $s^0>0$ , $y^0$ and put $\bar b=b-Ax^0$ , $\bar c=c-A^Ty^0-s^0$ , $\bar z=c^Tx^0+1-b^Ty^0$ . The homogeneous self-dual program (HLP) in the variables $(y,x,\tau,\theta,s,\kappa)$ is

\begin{aligned} \min\ & ((x^0)^Ts^0+1)\,\theta\\ \text{s.t. } & Ax-b\tau+\bar b\theta=0,\qquad s=-A^Ty+c\tau-\bar c\theta\ge0,\qquad \kappa=b^Ty-c^Tx+\bar z\theta\ge0,\\ & -\bar b^Ty+\bar c^Tx-\bar z\tau=-(x^0)^Ts^0-1,\qquad x\ge0,\ \tau\ge0 . \end{aligned}

Its feasible set is $\mathcal F_h$ , and $\mathcal F_h^0$ is the subset with $(x,\tau,s,\kappa)>0$ . A strictly self-complementary solution is an optimal solution with $\theta=0$ and $(x+s;\tau+\kappa)>0$ . The paper then fixes $y^0=0$ , $x^0=s^0=e$ , so the objective is $(n+1)\theta$ and $(0,e,1,1,e,1)\in\mathcal F_h^0$ .

With $\mu=(x^Ts+\tau\kappa)/(n+1)$ , the neighborhood of the central path is $\mathcal N(\beta)=\{z\in\mathcal F_h^0 : \|(Xs;\tau\kappa)-\mu e\|\le\beta\mu\}$ (Euclidean norm). The predictor–corrector algorithm starts at $(0,e,1,1,e,1)$ . At even steps it solves a Newton system with centering parameter $\gamma=0$ and moves by the largest step keeping the iterate in $\mathcal N(1/2)$ . At odd steps it takes a full Newton step with $\gamma=1$ . A termination projection then solves a least-squares problem on the columns where $x_j^k\ge s_j^k$ and produces a candidate exact solution.

Formalization targets

Goal: Corollary 7

For integer data with bit length $L$ and $n\ge1$ , there is an absolute constant $C$ and a finite predictor–corrector run that reaches, within

k\le C\sqrt n\,L

iterations, an iterate whose termination output $u=(y^*,x^*,\tau^*,\theta^*,s^*,\kappa^*)$ either has $\tau^*>0$ with $x^*/\tau^*$ optimal for (LP) and $(y^*/\tau^*,s^*/\tau^*)$ optimal for (LD), or has $\tau^*=0$ with $c^Tx^*<0$ and (LD) infeasible, or $-b^Ty^*<0$ and (LP) infeasible.

Theorem 6

The same bound, with the conclusion that the termination output, rescaled to the normalization $e^Tx+e^Ts+\tau+\kappa-(n+1)\theta=n+1$ , is a strictly self-complementary solution of (HLP).

Milestones

Proposition 1 (self-duality of skew-symmetric programs), Theorem 2 (ii), (iv), (v), Theorem 5 (ii), the two Mizuno–Todd–Ye estimates in the proof of Theorem 6, the predictor-step and corrector-step claims, the termination claim, the integrality and $O(L)$ size of the (HLP) data, and Theorem 3 (i), (ii).

Significance

The result. Corollary 7 gives a single polynomial algorithm that decides the status of an arbitrary linear program. It returns optimal primal and dual solutions when they exist and an infeasibility certificate otherwise. It needs no Phase I, no big- $M$ constant and no interior assumption, and its $O(\sqrt nL)$ iteration count matches the best bound known under those assumptions. Theorem 3 is the bridge: an exact strictly self-complementary solution of (HLP) determines the status of (LP) and (LD).

Formalizing it. The paper's result is proved; no machine-checked proof of it is known. The mission produces a formal account of the self-dual embedding, the central-path neighborhood analysis for (HLP), an exact finite-termination step from an approximate iterate, and the reading of a bit-length complexity bound with explicit quantifier order. The neighborhood estimates and the bit-length machinery are reusable for other interior-point and complexity results.

Difficulty

The per-step estimates are short calculations once the orthogonality $d_x^Td_s+d_\tau d_\kappa=0$ is available. The step that resists is termination. Geometric decrease of the gap reaches $2^{-O(L)}$ in $O(\sqrt nL)$ iterations, but turning such an iterate into an exact strictly self-complementary solution requires a quantitative separation: on $\mathcal N(1/2)$ , once the gap is below $2^{-O(L)}$ , the comparison $x_j\ge s_j$ must identify the optimal partition and the least-squares projection must land in the positive orthant. That separation bound rests on determinant estimates for integer matrices (Hadamard-type bounds in terms of $L$ ) and on the strict complementarity of (HLP), neither of which comes from the iteration analysis itself. A second subtlety is that the algorithm is not a function: without a rank assumption the Newton system leaves $d_y$ undetermined.

Formalization scope

Data for the complexity statements are integer matrices and vectors, cast to $\mathbb R$ ; all optimization objects are real. $L$ is Schrijver's encoding size, $\sum(1+\lceil\log_2(|v|+1)\rceil)$ over the entries plus the dimensions, so $L\ge1$ whenever $n\ge1$ .
Every $O(\cdot)$ is pinned with an absolute constant chosen before $m$ , $n$ and the data: $O(\sqrt nL)$ as $C\sqrt n\,L$ , $2^{-O(L)}$ as $2^{-C'L}$ , and $O(L)$ as $C\cdot L$ . The hypothesis $n\ge1$ , implicit in the paper, is explicit.
The algorithm is a relation on finite runs: any solution of the Newton system may be used, and each predictor step taken before termination must attain the maximum in (13). The conclusion asserts a successful run and its output. Stopping at the successful iterate avoids requiring a later step whose maximum may not exist.
The termination projection is defined relationally (an output is built from a minimizer) and its existence is part of the conclusion. Theorem 6 is stated for the output rescaled to the normalization (9): the projection imposes only homogeneous constraints, so the unscaled output need not be feasible for (HLP).
"(LP) has a solution (feasible and bounded)" is read as "(LP) has an optimal solution". "An indication that (LP) or (LD) is infeasible" is the negative sign of $c^Tx^*$ or $-b^Ty^*$ together with the infeasibility it certifies, not the bare disjunction "one of them is infeasible".
"Optimal for (HLP)" means minimizing the objective over $\mathcal F_h$ . It is not defined as $\theta=0$ , which would make Theorem 2 (iv) definitional.
Norms are Euclidean and written out explicitly, since Lean's default norm on $\mathbb R^n$ is the sup norm.
(LP) and (LD) are built on the published general-form LP definitions (LinearOptimization_DualLP). The self-dual objects of the Vanderbei and Matoušek series are different embeddings and are not reused.

A complete development needs: skew-symmetric LP duality and strict complementarity (Goldman–Tucker) for (HLP), the Mizuno–Todd–Ye neighborhood calculus, and Hadamard-type bounds on integer determinants. Contributions to any milestone, or to these supporting libraries, are welcome.

Selected references

Y. Ye, M. J. Todd, S. Mizuno, An $O(\sqrt nL)$ -iteration homogeneous and self-dual linear programming algorithm, Mathematics of Operations Research 19(1), 53–67, 1994. https://doi.org/10.1287/moor.19.1.53
S. Mizuno, M. J. Todd, Y. Ye, On adaptive-step primal-dual interior-point algorithms for linear programming, Mathematics of Operations Research 18(4), 964–981, 1993. https://doi.org/10.1287/moor.18.4.964
S. Mehrotra, Y. Ye, Finding an interior point in the optimal face of linear programs, Mathematical Programming 62, 497–515, 1993. https://doi.org/10.1007/BF01585180
M. Kojima, S. Mizuno, A. Yoshise, A little theorem of the big M in interior point algorithms, Mathematical Programming 59, 361–375, 1993. https://doi.org/10.1007/BF01581253
N. Karmarkar, A new polynomial-time algorithm for linear programming, Combinatorica 4, 373–395, 1984. https://doi.org/10.1007/BF02579150
A. Schrijver, Theory of Linear and Integer Programming, Wiley, 1986.

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Complexity Theory Linear Optimization Operations Research+1·Captain: mikedeng1

On Adaptive-Step Primal-Dual Interior-Point Algorithms for Linear Programming 3: The Potential-Reduction Algorithm in N∞⁻(β) with ρ − n of Order n² Attains Precision t in O(nt) IterationsResearch Paper

Motivation

Primal–dual interior-point methods solve a linear program by following, approximately, the central path of strictly feasible primal–dual pairs. Two families dominate the theory. Path-following methods keep the iterates in a neighbourhood of the path and measure progress by the duality gap. Potential-reduction methods measure progress by a merit function, the primal–dual potential of Todd and Ye (Todd–Ye 1990), and choose the step to decrease it.

The potential carries a parameter $\rho > n$ . The classical analyses (Ye 1991, Freund, Anstreicher–Bosch, Kojima–Mizuno–Yoshise) give $O(\sqrt n\,t)$ iterations with $\rho = n + \Theta(\sqrt n)$ . Practical codes, however, performed better with much larger $\rho$ (McShane–Monma–Shanno, Lustig–Marsten–Shanno). Mizuno, Todd and Ye (Cornell ORIE Technical Report No. 944, 1990, rev. 1991; journal version Math. Oper. Res. 18(4), 1993, doi:10.1287/moor.18.4.964) supply a worst-case justification for a large parameter. With $\rho = n + \Theta(n^2)$ and the step chosen to minimise the potential inside the wide neighbourhood $\mathcal N_\infty^-(\beta)$ , the method needs $O(nt)$ iterations. That matches the adaptive-step path-following method in the same neighbourhood (their Theorem 2).

Timeline:

1990: Todd and Ye introduce the primal–dual potential function.
1989–1991: Kojima–Mizuno–Yoshise, Ye, Freund and others prove $O(\sqrt n\,t)$ bounds for $\rho = n + \Theta(\sqrt n)$ .
1990–1991: Mizuno, Todd and Ye prove the $O(nt)$ bound for $\rho = n + \Theta(n^2)$ (Theorem 3 of the report formalized here).

Setting

Fix $m \ge 0$ , $n \ge 1$ , $A \in \mathbb R^{m\times n}$ , $b \in \mathbb R^m$ , $c \in \mathbb R^n$ . The primal problem is $\min c^Tx$ subject to $Ax = b$ , $x \ge 0$ . The dual is $\max b^Ty$ subject to $A^Ty + s = c$ , $s \ge 0$ . The strictly feasible set $\mathcal F^0$ is the set of pairs $(x, s)$ with $x > 0$ , $s > 0$ , $Ax = b$ and $A^Ty + s = c$ for some $y$ . The duality measure is $\mu = x^Ts/n$ .

For $z \in \mathbb R^n$ put $(z^-)_j = \min\{z_j, 0\}$ and $\|z\|_\infty^- = \|z^-\|_\infty$ . For $\beta \in (0,1)$ , the wide neighbourhood is

\mathcal N_\infty^-(\beta) = \{(x,s) \in \mathcal F^0 : \|Xs - \mu e\|_\infty^- \le \beta\mu\},

i.e. $x_js_j \ge (1-\beta)\mu$ for all $j$ , where $X = \mathrm{diag}(x)$ and $e$ is the all-ones vector.

The potential is

\psi(x,s) = \rho\log(x^Ts) - \sum_{j=1}^n \log(x_js_j), \qquad \rho > n,

with the natural logarithm. At $(x,s)$ and for $\gamma \in (0,1)$ , a search direction $d = (d_x, d_y, d_s)$ solves

S d_x + X d_s = \gamma\mu e - Xs,\qquad A d_x = 0,\qquad A^Td_y + d_s = 0,

and the iterates move along $x(\theta) = x + \theta d_x$ , $s(\theta) = s + \theta d_s$ . The scaled vectors are $p_j = \sqrt{s_j/x_j}\,(d_x)_j$ and $q_j = \sqrt{x_j/s_j}\,(d_s)_j$ , $Pq = (p_jq_j)_j$ , and $\theta_2^- = \min\{1, \beta\gamma\mu/\|Pq\|_\infty^-\}$ .

Algorithm 3. Start from $(x^0,s^0) \in \mathcal N_\infty^-(\beta)$ with $(x^0)^Ts^0 \le 2^t$ and $\psi(x^0,s^0) \le (\rho-n)t + n\log n$ . While $(x^k)^Ts^k > 2^{-t}$ , compute $d$ at $(x^k, s^k)$ , choose $\bar\theta$ with $(x(\bar\theta), s(\bar\theta)) \in \mathcal N_\infty^-(\beta)$ and $\psi(x(\bar\theta), s(\bar\theta)) \le \psi(x(\theta), s(\theta))$ for every $\theta$ with $(x(\theta), s(\theta)) \in \mathcal N_\infty^-(\beta)$ , and move to $(x(\bar\theta), s(\bar\theta))$ .

Formalization targets

Goal: Theorem 3 (p. 11)

Let $\beta,\gamma \in (0,1)$ with $\gamma \le 2(1-\beta)$ and

\rho = n + \Bigl(\frac{3}{\beta\gamma(1-\gamma)}\log\frac{1}{1-\beta}\Bigr)n^2.

For every $t > 0$ and every run of Algorithm 3,

\exists\, k \le \Bigl\lceil \frac{6nt}{11\beta\gamma(1-\gamma)} \Bigr\rceil:\quad (x^k)^Ts^k \le 2^{-t}.

The paper states " $O(nt)$ iterations". The explicit count is the one its own proof yields, and it is stated so that the goal is not an existential over a constant.

Milestones

§5, p. 11. $\psi(x,s) = (\rho-n)\log(x^Ts) - \sum_j\log\frac{x_js_j}{x^Ts/n} + n\log n \ge (\rho-n)\log(x^Ts) + n\log n$ . Hence reducing $\psi$ to $-(\rho-n)t + n\log n$ gives $x^Ts \le 2^{-t}$ .
(16), p. 12. One step of Algorithm 3 satisfies $\psi(x(\bar\theta),s(\bar\theta)) - \psi(x,s) \le -\bigl(\frac{3n^2\theta_2^-}{\beta\gamma} - n\bigr)\log\frac{1}{1-\beta}$ .
Proof of Theorem 3, p. 12. For $n \ge 1$ , every step decreases $\psi$ by at least $11n\log\frac{1}{1-\beta}$ .

Significance

The result shows that a potential-reduction method with $\rho - n$ of order $n^2$ keeps a polynomial worst-case bound. The earlier analyses cited by the report used $\rho - n$ of order $\sqrt n$ . It thereby gives a worst-case justification for the large parameters that worked in practice. It also connects the two families: the direction is the path-following direction, and only the step rule changes. Corollary 3 of the paper adds that iterations with a small second-order term $Pq$ decrease the potential far more.

The result is proved in the report. As far as the platform's index shows, no part of it has been machine-checked. The platform already holds the potential function (Bertsimas–Tsitsiklis §9.3) and a proved bookkeeping theorem converting a constant per-iteration decrease into an iteration count (Bertsimas–Tsitsiklis Theorem 9.4). This mission adds the analysis of a concrete algorithm: the admissibility of the step $\theta_2^-$ in $\mathcal N_\infty^-(\beta)$ , the potential decrease (16), and the complexity bound.

Difficulty

The step $\bar\theta$ is defined by minimisation, not by an explicit formula, so its decrease must be bounded by comparison with a step whose effect can be computed. That reference step must be shown to stay in the neighbourhood. This requires controlling the second-order term $Pq$ of the Newton direction in the one-sided norm $\|\cdot\|_\infty^-$ , together with the centrality of the current iterate. A fixed decrease of order $n$ arises only because $\rho - n$ is of order $n^2$ . Below that scale, the duality-gap term of the potential no longer dominates the loss in centrality from a long step.

A second obstacle is specific to formal work. Algorithm 3's rule ranges over every real $\theta$ whose point lies in the neighbourhood, and a minimiser need not exist. The goal must hold for every run, including the degenerate ones.

Formalization scope

Vectors are Fin n → ℝ, with index $j = 1,\dots,n$ mapped to Fin n; $\|\cdot\|_\infty$ is Mathlib's sup norm. The general standard-form LP pair is a separate definition, and the neighbourhood and algorithm build on its strict feasibility predicate. $\psi$ is the published definition LinearOptimization.interiorPointPotential with $q := \rho$ . It equals (1) at every strictly positive pair, and it is evaluated only at points of $\mathcal N_\infty^-(\beta)$ . All logarithms are natural, and $2^{\pm t}$ are real powers with real $t > 0$ . A run is a pair of sequences satisfying the start conditions and, whenever the current gap exceeds $2^{-t}$ , the step relation of Algorithm 3. Iterates after termination are unconstrained.

Conventions and repairs:

$\theta_2^- := 1$ when $\|Pq\|_\infty^- = 0$ , reading $\beta\gamma\mu/0$ as $+\infty$ .
The per-iteration milestone keeps the page's $n \ge 1$ . The paper's step $\theta_2^- \ge 4\beta\gamma/n$ needs $4\beta\gamma \le n$ , which can fail for $n = 1$ ; but for $n = 1$ the potential is unbounded below along the line inside the neighbourhood, so no step exists and the statement holds vacuously there. The goal keeps $n \ge 1$ as well.
The minimisation in Algorithm 3 ranges over all real $\theta$ , as printed. It does not range over a segment, and the step is not the largest admissible step of Algorithm 2.
Both start conditions are kept: $(x^0)^Ts^0 \le 2^t$ from Algorithm 2 and the potential bound of §5.

A formalization in which no run can exist for any instance, or in which $\rho$ or the iteration count absorbs an instance-dependent constant, would trivialize the goal. The step relation above is satisfiable for $n \ge 2$ whenever the potential attains its minimum along the line, and the count is the explicit expression above.

Needed infrastructure: the algebra of system (2), namely $d_x^Td_s = 0$ , the identity $\mu(\theta) = (1-(1-\gamma)\theta)\mu$ , and $X(\theta)s(\theta) - \mu(\theta)e = (1-\theta)(Xs - \mu e) + \theta^2 Pq$ . Also needed are the bounds $\|Pq\|_\infty \le \|r\|^2/4$ and $\|r\|^2 \le n\mu$ on $\mathcal N_\infty^-(\beta)$ when $\gamma \le 2(1-\beta)$ , and the arithmetic–geometric mean inequality for $\psi$ . These facts are shared with the sibling missions on the predictor–corrector and wide-neighbourhood algorithms of the same paper, and lemmas proved here are welcome there. The proved Bertsimas–Tsitsiklis theorem LinearOptimization.interior_point_potential_reduction is included as a reference item for the final counting step. Its hypotheses quantify over all iterations and need explicit dual iterates, so it does not apply verbatim.

Selected references

S. Mizuno, M. J. Todd, Y. Ye, On adaptive-step primal-dual interior-point algorithms for linear programming, Cornell ORIE Technical Report No. 944, 1990, rev. 1991; Mathematics of Operations Research 18(4):964–981, 1993. https://doi.org/10.1287/moor.18.4.964
M. J. Todd, Y. Ye, A centered projective algorithm for linear programming, Mathematics of Operations Research 15(3):508–529, 1990. https://doi.org/10.1287/moor.15.3.508

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Machine Learning Operations Research Optimization+1·Captain: mikedeng1

A Nonparametric Asymptotic Analysis of Inventory Planning with Censored Demand 1: AIM's Average Expected Cost Is Within O(1/√T) of the Newsvendor Cost, Perishable and Non-PerishableResearch Paper

Motivation

A retailer who restocks a product every period must choose an order-up-to level without knowing the demand distribution, and in most retail settings it observes only sales, the minimum of demand and stock. Demand above the stock is lost and never recorded. This is censored demand: the data needed to estimate the distribution are truncated exactly at the decision. Classical inventory theory assumes the distribution is known, and then the optimal policy in the stationary case is to order up to the newsvendor quantity in every period.

Huh and Rusmevichientong (Math. Oper. Res. 34(1), 2009) give a nonparametric policy that learns from censored sales alone, the Adaptive Inventory Management (AIM) algorithm, and prove that its average expected cost approaches the newsvendor cost at rate $1/\sqrt T$ . The perishable case reduces to online convex optimization in the sense of Zinkevich (2003) and Flaxman, Kalai and McMahan (2005). The non-perishable case is new: leftover stock constrains the next decision, and the paper controls this constraint through a single-server queue.

Setting

Demands $D_1, D_2, \dots \ge 0$ are independent and identically distributed with an unknown law of finite mean; $D$ denotes a generic demand and $F$ its distribution function. A holding cost $h > 0$ is paid per unit left over and a penalty $b > 0$ per unit of lost demand. The newsvendor cost of a level $y \ge 0$ is

Q(y) = h\,E[y-D]^+ + b\,E[D-y]^+ ,

and the newsvendor quantity is $y^{NV} = \inf\{y \ge 0 \mid F(y) \ge b/(b+h)\}$ .

In period $t$ the manager holds $x_t \ge 0$ units ( $x_1 = 0$ ), raises the stock to $y_t \ge x_t$ , and observes $\min\{D_t, y_t\}$ . Excess inventory is either perishable ( $x_{t+1} = 0$ ) or non-perishable ( $x_{t+1} = [y_t - D_t]^+$ ).

Assumption 1 gives the manager a bound $\bar y \ge y^{NV}$ , and in the non-perishable case a bound $\rho$ with $0 < \rho < E[D_1]$ . AIM keeps a target $\hat y_t$ and an implemented level $y_t$ : $y_1 = \hat y_1 \in [0, \bar y]$ , and for $t \ge 1$

\hat y_{t+1} = P_{[0,\bar y]}\big(\hat y_t - \epsilon_t H_t(\hat y_t)\big), \qquad y_{t+1} = \max\{\hat y_{t+1}, x_{t+1}\},

with the projection $P_{[0,\bar y]}(z) = \max\{\min\{z,\bar y\},0\}$ , steps $\epsilon_t = \gamma\bar y/(\max\{b,h\}\sqrt t)$ for a scale $\gamma > 0$ , and the censored gradient estimate $H_t(\hat y_t) = h$ if $D_t < \hat y_t$ and $-b$ if $D_t \ge \hat y_t$ . The manager can compute $H_t$ from the observed sales.

For $\theta > 0$ the waiting times $W_1(\theta) = 0$ , $W_{t+1}(\theta) = [W_t(\theta) + \theta - D_t]^+$ describe a GI/D/1 queue with inter-arrival times $D_t$ and service time $\theta$ . Its busy periods $J_i(\theta)$ are the stretches between successive returns to $0$ . The auxiliary queue $Z_1(\theta) = 0$ , $Z_{t+1}(\theta) = [Z_t(\theta) + \theta/\sqrt t - D_t]^+$ has shrinking service times.

Formalization targets

Goal: Theorem 2

For every $T \ge 1$ , in the perishable case

E\Big[\frac1T\sum_{t=1}^T Q(y_t)\Big] - Q(y^{NV}) \le \Big(\gamma + \frac1\gamma\Big)\frac{\bar y\max\{b,h\}}{\sqrt T},

and in the non-perishable case, if $E[D_1^6] < \infty$ and $\gamma \le \rho\max\{b,h\}/(h\bar y)$ , there is a constant $C$ , independent of $T$ , with

E\Big[\frac1T\sum_{t=1}^T Q(y_t)\Big] - Q(y^{NV}) \le \frac{C}{\sqrt T}.

The non-perishable bullet asserts only the existence of $C$ , because the paper's explicit formula (9) depends on Proposition 4 (i), which is false as printed.

Milestones

§2.4: $Q$ is convex with left derivative $h P\{D<y\} - bP\{D\ge y\}$ , and $E[H_t \mid \hat y_t]$ equals it.
Lemma 3: projected stochastic subgradient descent with steps $\gamma\operatorname{diam}(S)/(\bar B\sqrt t)$ has average error at most $(\gamma + 1/\gamma)\operatorname{diam}(S)\bar B/\sqrt T$ .
§2.4: $Q(y) - Q(\hat y) \le h(y - \hat y)$ for $0 \le \hat y \le y$ .
Theorem 6: if $\gamma \le \rho\max\{b,h\}/(h\bar y)$ , then $y_t - \hat y_t \le Z_t(\rho)$ .
Lemma 13: $E|J_{i(\theta,t)}(\theta)| \le E[|J_1(\theta)|^2]$ .
Lemma 5: $E[\sum_{t=1}^T Z_t(\theta)] \le 2\theta E[|J_1(\theta)|^2]\sqrt T$ .
Lemma 12: $E[(\sum_{i\le n} X_i)^6] \le 21 n^3 E X_1^6$ for i.i.d. mean-zero $X_i$ (and the fourth-moment analogue).
Proposition 4 (i), corrected: $E[|J_1(\theta)|^2] \le 1 + \tfrac{21\pi^2}{2}E[(D_1 - ED_1)^6]/(ED_1 - \theta)^6$ for $0 < \theta < E[D_1]$ .

Significance

Theorem 2 shows that a manager who never observes lost sales can still reach the cost of a manager who knows the distribution, at a dimension-free rate, under only a sixth-moment condition in the non-perishable case. The example of §2.5 of the paper shows that the rate $1/\sqrt T$ cannot be improved in general for AIM. Theorem 6, the coupling of the excess inventory $y_t - \hat y_t$ with a queue, is the step that handles decisions constrained by past decisions, and the same paper reuses it when $\rho$ is unknown (§3.2).

The results are proved in the paper, and none of them is formalized as far as is known. This mission formalizes them and also corrects them: Proposition 4 (i) fails for deterministic demand, its explicit constant has to be replaced, and Eq. (2) labels a right derivative as a left one. Lemma 3 for decreasing steps and Lemma 12 are standard tools that are not in Mathlib.

Difficulty

The perishable case is a stochastic subgradient method, but its estimates are conditionally unbiased only given the current iterate, and $Q$ is merely convex, not differentiable, so the textbook smooth argument does not apply directly. In the non-perishable case the implemented levels $y_t$ are not the gradient iterates, and $y_t - \hat y_t$ is not small along a path: carried-over stock can keep $y_t$ above the target for many periods. A bound on $E[y_t - \hat y_t]$ for each $t$ does not follow from a bound on one period. The queue $Z_t(\rho)$ dominates the whole excess process, but it is non-stationary. Its sum has to be compared with the stationary queue $W_t(\rho)$ , whose busy period covering a fixed time is length-biased, so second moments of busy periods enter.

Formalization scope

All declarations are in the namespace AIMInventory.Known. The demand law and the i.i.d. path measure are the published CappedBaseStock.DemandLaw and demandPathLaw, whose law has finite and strictly positive mean. $Q$ and $y^{NV}$ are the published backorderCost and newsvendorLevel of LostSalesOrderUpTo.Asymptotic.Backorder. The optimality of $y^{NV}$ is the already posed LostSalesOrderUpTo.Asymptotic.newsvendor_optimal and is included as a reference.

Conventions:

The demand path is zero-based ( $D_t$ is d (t - 1)), and all paper indices $t \ge 1$ are kept.
The AIM run, $W$ , $Z$ , the renewal epochs and the busy periods are functions of the demand path, defined by recursion exactly as printed.
The queue indexing is $W_1 = Z_1 = 0$ and $\tau_0 = 1$ (the paper's $W_0$ , $Z_0$ need an undefined $D_0$ and $\theta/\sqrt 0$ ). Period $1$ lies in no busy period, so Lemma 13 is stated for $t \ge 2$ .
Busy-period lengths are ℕ∞-valued. Every expectation that can be infinite is a lower Lebesgue integral in $[0,\infty]$ . The expectations in Theorem 2 have bounded integrands.
Readings of loose phrases:
- "for some $\gamma > 0$ " means for every $\gamma > 0$ ;
- "any value in $[0,\bar y]$ " means for every $\hat y_1 \in [0, \bar y]$ ;
- "with probability one" in Theorem 6 means for every demand path;
- "There is a constant $C$ " means $\exists C,\ \forall T \ge 1$ .
The implicit positivity of $h$ , $b$ , $\bar y$ (they appear in denominators) is added as hypotheses.
Lemma 3 is stated on $\mathbb R^n$ for every comparator $u \in S$ , with $E[H(z)\mid z]$ read as conditioning on the σ-algebra of the current iterate.
Proposition 4 (ii) and the explicit constant (9) are not stated.

The mission does not accept a formalization that sums over arbitrary sequences with $y_t \ge \hat y_t$ instead of the AIM run, uses an uncensored gradient, or lets a non-integrable expectation default to $0$ .

Contributions toward a stochastic subgradient method with anytime steps, moment bounds for i.i.d. sums, and busy periods of Lindley's recursion are reusable beyond this mission.

Selected references

W. T. Huh, P. Rusmevichientong, A Nonparametric Asymptotic Analysis of Inventory Planning with Censored Demand, Mathematics of Operations Research 34(1), 2009. https://doi.org/10.1287/moor.1080.0355
M. Zinkevich, Online Convex Programming and Generalized Infinitesimal Gradient Ascent, ICML 2003. https://dl.acm.org/doi/10.5555/3041838.3041955
A. Flaxman, A. T. Kalai, H. B. McMahan, Online Convex Optimization in the Bandit Setting: Gradient Descent without a Gradient, SODA 2005. https://arxiv.org/abs/cs/0408007
E. Hazan, A. Agarwal, S. Kale, Logarithmic Regret Algorithms for Online Convex Optimization, Machine Learning 69, 2007. https://doi.org/10.1007/s10994-007-5016-8

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Bandit Algorithms Operations Research Stochastic Systems·Captain: mikedeng1

On the Gittins Index for Multiarmed Bandits 1: A Policy Falls Short of the Gittins Policy by at Most Its Expected Discounted Excess of the Greatest Prevailing Charge over Its RewardResearch Paper

Motivation

The multiarmed bandit problem asks how to allocate a single resource, one play per period, among several alternative projects whose states evolve only while they are worked on. It models the scheduling of jobs on one machine, the sequencing of research projects, and the design of sequential clinical trials. Gittins and Jones (1974) showed that the problem with discounted rewards is solved by an index policy: compute for every project a number depending only on that project's current state, and always work on a project of greatest index. Weber's paper (Ann. Appl. Probab. 2 (1992)) gives a short proof of this theorem through fair and prevailing charges, and uses the same accounting to derive bounds on how much any other policy loses.

In practice the optimal policy is often not the one used: indices may be expensive to compute, or a simpler heuristic may be preferred. A bound on the loss of an arbitrary policy, computable along that policy's own trajectory, tells a practitioner how far from optimal the chosen heuristic is. This mission formalizes the first such bound of the paper, Theorem 2, which Weber attributes to Glazebrook (1990).

Timeline. Gittins and Jones (1974) and Gittins (1979) proved optimality of the index policy by an interchange argument. Whittle (1980) gave a proof through a retirement option. Glazebrook (1982) bounded the suboptimality of a policy in terms of Gittins indices only; Glazebrook (1990) gave the sharper bound stated here as Theorem 2. Weber (1992) rederived it from the prevailing-charge accounting and added the index-gap bound (Theorem 3) and the submodularity of the optimal value (Theorem 4).

Setting

There are $k$ bandits. Bandit $i$ has a state $x_i(t)$ at each epoch $t = 0, 1, 2, \dots$ . At each epoch the gambler plays one bandit $j(t)$ , receives the reward $r(x_{j(t)}(t))$ , and the state of bandit $j(t)$ moves according to a Markov kernel $P$ ; the states of all other bandits stay unchanged. Rewards are nonnegative and uniformly bounded, and future rewards are discounted by a factor $\beta$ with $0<\beta<1$ . A policy $\pi$ chooses $j(t)$ , possibly at random, as a function of everything observed so far. Its value from the initial states $x$ is

V_\pi(x) = \mathbb{E}_\pi\Big[\sum_{t=0}^\infty \beta^t\, r\big(x_{j(t)}(t)\big) \,\Big|\, x(0)=x\Big]. \tag{1}

The fair charge $\gamma(y)$ of a state $y$ is the per-play charge at which a gambler holding a single bandit in state $y$ , free to stop at any time, would only just break even. Equivalently it is the largest ratio of expected discounted reward to expected discounted time over stopping times $\tau \ge 1$ . Weber's Gittins index is $G(y) = \gamma(y)/(1-\beta)$ . The prevailing charge of bandit $i$ at time $t$ is the least fair charge seen so far on that bandit, $\min_{0\le s\le t}\gamma(x_i(s))$ . A Gittins policy $\pi^*$ plays at each epoch a bandit of greatest index; Theorem 1 of the paper says it is optimal.

Formalization targets

Goal: Theorem 2, inequality (6)

For a Gittins policy $\pi^*$ , every policy $\pi$ and every initial state vector $x$ ,

V_{\pi^*}(x) - V_\pi(x) \le (1-\beta)\,\mathbb{E}_\pi\Big[\sum_{t=0}^\infty \beta^t\Big(\max_i \min_{0\le s\le t} G_i(x_i(s)) - \frac{R_{j(t)}(x_{j(t)}(t))}{1-\beta}\Big) \,\Big|\, x(0)=x\Big].

Milestones

Eq. (5), from Remark 1: for every policy, $V_\pi(x) \le \mathbb{E}_\pi\big[\sum_t \beta^t \min_{0\le s\le t}\gamma_{j(t)}(x_{j(t)}(s))\big]$ .
The greatest prevailing charge on a fixed realisation (p. 1028): along any fixed realisation of the bandits' state sequences, the greatest prevailing charge at time $t$ under any policy is at least its value under the Gittins policy.
The bound is achieved by the Gittins policy (proof of Theorem 1): $V_{\pi^*}(x) = \mathbb{E}_{\pi^*}\big[\sum_t \beta^t \max_i \min_{0\le s\le t}\gamma(x_i(s))\big]$ .

Significance

The result. Inequality (6) bounds the loss of an arbitrary policy by a quantity observed along that policy's own trajectory: at each epoch, how far the reward collected falls below the greatest prevailing charge. A policy that plays a bandit of greatest prevailing charge whenever possible and collects its charge as reward loses nothing, which recovers Theorem 1. The bound is the basis of Glazebrook's analysis of near-optimal heuristics for bandit and stochastic-scheduling problems, and of the index-gap bound (7) of Theorem 3, which is the second mission of this series.

Formalizing it. Theorem 1 of the paper (optimality of the Gittins policy) is already formalized and proved on Prove2Me as BanditAlgorithm.gittins_index_theorem (Shuze Chen, following Lattimore and Szepesvári, Theorem 35.9), together with the finite-horizon prevailing-charge accounting and the charge-stack interleaving lemmas (Harry_Xu). Theorem 2 itself has no machine-checked proof. This mission produces the infinite-horizon forms of the charge accounting, (5) and its equality case for the Gittins policy, and the realisation-wise comparison of greatest prevailing charges, and combines them into (6). Missions 2 (Theorem 3, the index-gap bound) and 3 (Theorem 4, submodularity of the optimal value) of this series state the paper's other results on the same model; none of the three relies on the others.

Difficulty

The paper's proof is one sentence: combine (5) with the realisation-wise observation. The two pieces live on different probability spaces. Inequality (5) and the equality for the Gittins policy are expectations under the law of each policy's own history, while the comparison of greatest prevailing charges is made on a fixed realisation of each bandit's state sequence. Transferring the deterministic comparison into an inequality between expectations under two different policies needs a coupling under which every policy sees the same per-bandit state sequences, so that the prevailing-charge stacks are the same random objects under $\pi$ and $\pi^*$ . The naive approach, comparing the two laws round by round, fails: the round- $t$ histories of $\pi$ and $\pi^*$ have different distributions, and no pointwise inequality holds between them. The infinite-horizon statements also require controlling the terminal retirement potential, which vanishes only in the discounted limit.

Formalization scope

The development uses the published BanditAlgorithm model (GittinsIndex, GittinsTerminalPotential, GittinsPrevailingChargeValue, GittinsChargeInterleaving). Conventions committed to:

Weber's $n$ bandits with separate state spaces are one state space $S$ with one kernel $P$ and reward $r$ (take the disjoint union). $S$ is a standard Borel space and $r$ is measurable: regularity the paper leaves implicit.
$r$ is the conditional mean of Weber's random reward $R_j(x_j(t))$ ; only the mean enters (1), (5) and (6).
The standing assumptions of Section 1, "rewards are nonnegative and uniformly bounded" and $0<\beta<1$ , are hypotheses of the goal and of milestones 1 and 3.
Every $\mathbb{E}_\pi[\sum_t \beta^t(\cdot)]$ is written as $\sum_t \beta^t\,\mathbb{E}_\pi[\cdot]$ , the convention of the published value $V_\pi$ ; these agree for bounded rewards.
Time starts at $0$ . At round $t$ the integrand is evaluated on the history of the first $t$ rounds together with the current state vector $x(t)$ , so every $\min_{0\le s\le t}$ includes the current state.
$\gamma$ is the published gittinsIndex (the ratio (4)); $G = \gamma/(1-\beta)$ is a new definition weberIndex. The goal uses $G$ and the factor $1-\beta$ , as in (6); milestones 1 and 3 use $\gamma$ , as the paper does.
" $\pi^*$ is the optimal (Gittins) policy" is any policy satisfying the published IsGittinsIndexPolicy (with probability one it plays an arm of greatest index, ties arbitrary). The goal does not assume such a policy exists; a one-armed instance shows the hypotheses can be met.
"On a fixed realisation" (milestone 2) is encoded deterministically: arbitrary nonincreasing sequences $H_i$ as the prevailing-charge stacks, any action sequence as a policy, and any greedy interleaving as the Gittins policy.
$\max_i$ is a supremum over $\mathrm{Fin}\,k$ ; at $k=0$ no policy exists.

Replacing $G$ by $\gamma$ in (6) would weaken the bound by the factor $1-\beta$ , and dropping the boundedness of rewards would let the published value be a junk $0$ ; neither is the paper's statement, and both are ruled out by the conventions above.

Contributions welcome: the infinite-horizon limits of the published finite accounting, a coupling of all policies on independent per-bandit state sequences, and the deterministic comparison of greatest exposed charges, which is reusable for any interleaving argument on nonincreasing sequences.

Selected references

R. Weber, On the Gittins index for multiarmed bandits, Annals of Applied Probability 2(4), 1024–1033, 1992. https://doi.org/10.1214/aoap/1177005588
K. D. Glazebrook, Procedures for the evaluation of strategies for resource allocation in a stochastic environment, Journal of Applied Probability 27(1), 215–220, 1990.
K. D. Glazebrook, On the evaluation of suboptimal strategies for families of alternative bandit processes, Journal of Applied Probability 19(3), 716–722, 1982.
J. C. Gittins and D. M. Jones, A dynamic allocation index for the sequential design of experiments, in Progress in Statistics (J. Gani et al., eds.), North-Holland, 241–266, 1974.
P. Whittle, Multi-armed bandits and the Gittins index, Journal of the Royal Statistical Society B 42(2), 143–149, 1980. https://doi.org/10.1111/j.2517-6161.1980.tb01111.x
T. Lattimore and Cs. Szepesvári, Bandit Algorithms, Cambridge University Press, 2020, Chapter 35. https://doi.org/10.1017/9781108571401

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Analysis Operations Research Probability·Captain: mikedeng1

The Data-Driven Newsvendor Problem: New Bounds and Insights II: Every Log-Concave Demand Has Weighted Mean Spread at Least min(b,h)/(b+h)Research Paper

Motivation

The newsvendor problem is the basic single-period inventory model: a decision maker orders $q$ units before a random demand $D$ is observed, pays $b>0$ per unit of unmet demand and $h>0$ per unit left over, and minimizes the expected cost $C(q)=E[b(D-q)^+ + h(q-D)^+]$ . When the distribution of $D$ is known, the optimal order is the $b/(b+h)$ quantile of $D$ . In practice the distribution is unknown and only a sample of past demands is available; the standard data-driven order is the sample average approximation (SAA), the $b/(b+h)$ quantile of the empirical distribution.

R. Levi, G. Perakis and J. Uichanco, The Data-Driven Newsvendor Problem: New Bounds and Insights (Operations Research 63(6), 2015), ask how many samples guarantee that the SAA order is $\epsilon$ -optimal with high probability. Levi, Roundy and Shmoys (2007) gave a distribution-free bound. Levi, Perakis and Uichanco show that, asymptotically in $\epsilon$ , the right measure of difficulty is a single scalar of the demand law, the weighted mean spread at the critical quantile, and that for the class of log-concave demand distributions (normal, uniform, exponential, logistic, Laplace and many others used in inventory theory) this scalar is bounded below by $\min(b,h)/(b+h)$ . That uniform bound turns a distribution-dependent sample-size guarantee into one that holds for every log-concave demand, and is markedly tighter than the distribution-free one.

This mission formalizes the bound (Proposition 2) and the chain of lemmas the paper uses to prove it. The source is the authors' accepted manuscript of the paper (MIT DSpace), cited by its own page numbers.

Setting

Fix costs $b>0$ , $h>0$ and the critical ratio $\beta=b/(b+h)\in(0,1)$ . A demand law is given by a probability density $f:\mathbb R\to\mathbb R$ : measurable, nonnegative, with $\int f=1$ . Its cdf is $F(t)=\int_{(-\infty,t]}f$ , and its $\beta$ quantile is

q^*=\inf\{q : F(q)\ge \beta\}.

The absolute mean spread (AMS, Definition 1) at $t$ is the gap between the conditional means above and below $t$ ,

\Delta(t)=E(D\mid D\ge t)-E(D\mid D\le t)=\frac{\int_{[t,\infty)}x f(x)\,dx}{1-F(t)}-\frac{\int_{(-\infty,t]}x f(x)\,dx}{F(t)},

and the weighted mean spread (WMS, Definition 2) is $\Delta(q^*)f(q^*)$ .

A density is log-concave (Definition 3) if $\log f$ is concave on its support. Equivalently, $f(x)^a f(y)^{1-a}\le f(ax+(1-a)y)$ for all $x,y$ and $a\in[0,1]$ ; this form allows $f$ to vanish outside an interval.

For a log-concave $f$ and a point $t$ with $f(t)>0$ , the number $\gamma_1$ is a supergradient of $\log f$ at $t$ , written $\gamma_1\in\partial\log f(t)$ , if $\log f(x)\le\log f(t)+\gamma_1(x-t)$ for all $x$ with $f(x)>0$ . The paper splits the class $\mathbb L$ of log-concave densities into the subclasses $\mathbb L_{q^*,\gamma_0,\gamma_1}$ of densities with quantile $q^*$ , $f(q^*)=\gamma_0$ and $\gamma_1\in\partial\log f(q^*)$ , and minimizes the AMS over each subclass (problem (11)). The minimizer is the truncated exponential density

\tilde f(x)=\gamma_0e^{\gamma_1(x-q^*)}\ \text{ on }[\underline x,\overline x],\qquad \underline x=q^*+\tfrac1{\gamma_1}\log\!\big(1-\tfrac{\gamma_1}{\gamma_0}\beta\big),\quad \overline x=q^*+\tfrac1{\gamma_1}\log\!\big(1+\tfrac{\gamma_1}{\gamma_0}(1-\beta)\big),

and zero elsewhere (display (12)), with AMS $z^*_{q^*,\gamma_0,\gamma_1}$ in closed form.

The Lean development uses the names IsPdf, cdfOf, quantileOf, ams, IsLogSupergradient, tildeF and zStar for these objects, in the namespace DataDrivenNV.WMS.

Formalization targets

Goal: Proposition 2

For every log-concave density $f$ ,

\Delta(q^*)\,f(q^*)\ \ge\ \frac{\min(b,h)}{b+h}.

The statement has no further hypothesis: no moment condition, no continuity, no restriction on $\gamma_0,\gamma_1$ .

Milestones, in the order of the paper's proof

Lemma 1: at the quantile, $-\frac{b+h}{h}\le\frac{\gamma_1}{\gamma_0}\le\frac{b+h}{b}$ .
Lemma 2: $f(x)\le\gamma_0e^{\gamma_1(x-t)}$ for every $x$ .
Lemma 3 (Domination Lemma): a density dominated on the support of another, with the same mass left of $t$ , has the larger AMS at $t$ .
Proposition 1: $\tilde f$ belongs to $\mathbb L_{q,\gamma_0,\gamma_1}$ and minimizes the AMS there.
The closed form: $\Delta_{\tilde f}(q)=z^*_{q,\gamma_0,\gamma_1}=\frac{\gamma_0}{\gamma_1^2}\big[(\frac{b+h}{h}+\frac{\gamma_1}{\gamma_0})\log(1+\frac{\gamma_1}{\gamma_0}\frac{h}{b+h})+(\frac{b+h}{b}-\frac{\gamma_1}{\gamma_0})\log(1-\frac{\gamma_1}{\gamma_0}\frac{b}{b+h})\big]$ .
Lemma 4: three elementary inequalities in $\beta\in(0,1)$ and $\eta\in(-\frac1{1-\beta},\frac1\beta)$ .

Significance

The proposition is what makes the paper's log-concave sample-size bound (Theorem 4) parameter-free: the probability that the biased SAA order is $\epsilon$ -optimal is, asymptotically, at least $1-2\exp(-\frac14N\epsilon\,\Delta(q^*)f(q^*))$ (Theorem 3), and Proposition 2 replaces the unknown $\Delta(q^*)f(q^*)$ by $\min(b,h)/(b+h)$ . A manager who only knows that demand is log-concave can then size a sample without estimating any parameter of the law.

The result is proved in the paper; nothing about it is open. As far as is known it has no machine-checked proof. Formalizing it produces a checked lower bound for an inventory-theory quantity together with reusable facts about log-concave densities on the line: exponential envelopes from a supergradient, the quantile-and-slope constraints of Lemma 1, and the comparison of conditional means behind the Domination Lemma.

Difficulty

The definitions are elementary, but the proof passes through an infinite-dimensional optimization problem over a class of densities. The obvious first step, "the AMS is minimized by the most concentrated density", has no direct meaning without fixing the density value and the slope of $\log f$ at the quantile; after fixing them, one needs the envelope of Lemma 2, a stochastic comparison (Lemma 3) and an explicit integral of a truncated exponential. The Domination Lemma as printed is false (a density whose support has a gap to the right of $t$ is a counterexample), so it is formalized under the hypothesis that the dominating density is positive exactly on an interval around $t$ , which is how Proposition 1 uses it. The case $\gamma_1=0$ (uniform minimizer) and the two boundary values of $\gamma_1/\gamma_0$ (one-sided exponential minimizers) are not covered by the closed form (12) and must be handled separately in a proof of the goal. Finally, Lemma 1 rests on the monotonicity of the failure rate $f/(1-F)$ and the reversed hazard rate $f/F$ of log-concave laws, which must be proved from log-concavity.

Formalization scope

Densities are functions ℝ → ℝ with IsPdf f (measurable, nonnegative everywhere, integrable, total integral 1); the law of $D$ is never introduced separately. Log-concavity is the published definition ConvexOptimization.LogConcaveOn Set.univ f (power form, zeros allowed). Concavity of Real.log ∘ f is not used, because Real.log 0 = 0 would treat $\log f$ as $0$ off the support. The quantile is an sInf; for $\beta\in(0,1)$ its defining set is nonempty and bounded below. The AMS is the difference of two ratios of integrals. At the quantile both denominators are positive, and the goal and Proposition 1 assume no integrability of $xf(x)$ , since log-concave densities have exponential tails. The value $f(q^*)$ is taken pointwise; $q^*$ lies in the interior of the support, where a log-concave density is continuous.

Conventions and disclosed departures from the page:

" $\gamma_1\in\partial\log f$ , the set of all subgradients" is read as the superdifferential of the concave $\log f$ on $\{f>0\}$ .
Lemma 3 carries three added hypotheses: $f_2$ vanishes outside some $[l,u]\ni t$ and is positive on $(l,u)$ ; $0<F_1(t)<1$ ; and $xf_1(x)$ , $xf_2(x)$ are integrable. The first repairs the printed statement; the other two are the conditions under which Definition 1 makes sense for general densities.
Proposition 1 and the closed form of $z^*$ are stated for $\gamma_1\neq0$ and $\gamma_1/\gamma_0$ strictly inside the interval of Lemma 1, where (12) is a finite interval.
The goal has no such restriction.

Assumption 1 of the paper (monotonicity of $f$ beyond $q^*$ ) and the continuity assumption of §3 are hypotheses of Theorems 3–4 only and are not used here. A formalization in which $f(q^*)$ or $\Delta(q^*)$ takes a junk value (a non-integrable $xf(x)$ , a zero denominator) would make the goal trivially true or false; the hypotheses above exclude this, and no statement assumes the conclusion of another.

Contributions welcome: proofs of the milestones in any order; general lemmas on log-concave densities on $\mathbb R$ (exponential tails, integrability of moments, monotone hazard rates, continuity in the interior of the support); and the boundary cases of problem (11).

Selected references

R. Levi, G. Perakis, J. Uichanco, The Data-Driven Newsvendor Problem: New Bounds and Insights, Operations Research 63(6):1294–1306, 2015. Authors' accepted manuscript, MIT DSpace. https://doi.org/10.1287/opre.2015.1422
R. Levi, R. O. Roundy, D. B. Shmoys, Provably Near-Optimal Sampling-Based Policies for Stochastic Inventory Control Models, Mathematics of Operations Research 32(4):821–839, 2007. https://doi.org/10.1287/moor.1070.0272

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Convex Optimization Operations Research Optimization+1·Captain: mikedeng1

Loss Networks 1: The Primal Problem Has a Unique Optimum x_r = ν_r ∏_j (1 − B_j)^{A_jr}; the Conditions on B Are Solvable, Uniquely if rank A = J, and Biject with Dual OptimaResearch Paper

Motivation

A loss network carries calls that require capacity on one or more links. A call is accepted only when every required link has enough free circuits; otherwise it is lost. This makes the traffic on different links dependent even when calls on different routes arrive independently. Network operators need to know where capacity constraints bind and how offered traffic is converted into carried traffic. Kelly's 1991 survey develops a continuous optimization problem that identifies a characteristic flow vector and gives a linkwise description of blocking in this setting. Kelly, Loss networks (1991), §§1.2 and 2.1.

The paper begins with the equilibrium distribution of the discrete network and uses a continuous relaxation when analyzing its likely state. The resulting primal problem is useful beyond the original probability calculation because it links route flows, capacity constraints, and a convex dual. Theorem 2.8 collects the existence, uniqueness, and correspondence statements for these descriptions. This mission targets that theorem and the intermediate claims Kelly states in §2.1. Kelly (1991), pp. 327–329.

Setting

There are finitely many links $j$ and finitely many routes $r$ . Link $j$ has $C_j$ circuits, and one call on route $r$ uses $A_{jr}$ circuits there, with $A_{jr}$ a nonnegative integer. Calls request route $r$ at positive offered rate $\nu_r$ . The matrix $A$ need not be a zero–one incidence matrix: one call may use more than one circuit on a link. Kelly's fixed-routing model and its general integer matrix are described in §1.2. Kelly (1991), p. 321.

A real route-flow vector $x=(x_r)$ is primal feasible if $x_r\ge0$ for every route and $\sum_r A_{jr}x_r\le C_j$ for every link. The objective is

F(x)=\sum_r\bigl(x_r\log\nu_r-x_r\log x_r+x_r\bigr),

with $x_r\log x_r=0$ at $x_r=0$ . The primal optimum maximizes $F$ over precisely this feasible set. This is the continuous problem (2.1), rather than an optimization over the integer-valued network state. Kelly (1991), p. 327, (2.1).

A nonnegative vector $y=(y_j)$ assigns one multiplier to each link. The dual objective is

D(y)=\sum_r\nu_r\exp\!\left(-\sum_jy_jA_{jr}\right)+\sum_jy_jC_j, \qquad y_j\ge0.

A blocking vector $B=(B_j)$ has $0\le B_j<1$ . Its carried route flow is $\nu_r\prod_i(1-B_i)^{A_{ir}}$ . The conditions on $B$ say that the total carried flow through link $j$ equals $C_j$ when $B_j>0$ , and is at most $C_j$ when $B_j=0$ . The multiplier and blocking coordinates are related by $B_j=1-e^{-y_j}$ , equivalently $y_j=-\log(1-B_j)$ . These are equations (2.3), (2.6), and (2.7). Kelly (1991), p. 328.

Formalization targets

Theorem 2.8: primal optimum, blocking solutions, and dual optima

The goal first asserts that the primal problem has exactly one optimizer. Every blocking solution must describe that same optimizer:

x_r=\nu_r\prod_j(1-B_j)^{A_{jr}}\quad\text{for every route }r.

It also asserts that a blocking solution exists, and that it is unique if $A$ has rank $J$ , the number of links. Finally, the transformations $B_j=1-e^{-y_j}$ and $y_j=-\log(1-B_j)$ give a bijection between all blocking solutions and all dual optima. The formal goal includes both directions and both inverse identities, even when the dual optimizer is not unique. Kelly (1991), p. 329, Theorem 2.8.

Claims leading to the theorem

The milestone list follows Kelly's stated progression: unique primal attainment; the Lagrangian maximum and the dual value in (2.2); feasibility and complementary slackness in (2.4)–(2.5); the substitution (2.6); dual attainment; both directions of the blocking–dual correspondence; and strict convexity plus uniqueness under full row rank. One direction of the correspondence is an existing, proved platform theorem, KellyStochasticNetworks.dual_optimum_conditions; it is referenced directly. The other direction remains a new target. Kelly (1991), pp. 327–329.

Significance

The theorem makes the carried flow uniquely determined even when several blocking vectors or dual optimizers describe it. Under full row rank, it also identifies a unique linkwise blocking vector. Its correspondence lets a statement in the flow language be translated to a statement in the multiplier language without selecting a preferred optimizer. The interpretation following Theorem 2.8 explains the linkwise condition: positive blocking occurs only on a saturated link. Kelly (1991), p. 329, discussion after Theorem 2.8.

The mathematical result was proved in the source paper. The work here is to obtain machine-checked proofs of its exact finite-dimensional statements. The published Kelly–Yudovina definition KellyStochasticNetworks.dualObjective already represents the dual formula, and its proved optimum-conditions theorem supplies one direction of the correspondence. The remaining development would give reusable facts about separable entropy objectives, nonnegative linear capacity constraints, complementary slackness, and exponential coordinate changes. Those facts can support later loss-network formalizations without changing this mission's target.

Difficulty

The boundary $x_r=0$ requires care: the objective has a continuous value there, but its logarithmic derivative does not. Thus a calculation that differentiates the objective throughout the closed nonnegative cone does not justify the printed differentiability sentence on p. 327. Dual existence is another boundary issue. A nonnegative capacity permits $C_j=0$ , while the paper's stated growth of the dual objective in every nonnegative coordinate can fail at that input. Any formal argument for the complete theorem must cover these domains accurately. Full row rank controls uniqueness of the dual coordinates; without it, the paper explicitly allows nonunique blocking solutions. Kelly (1991), pp. 327–329.

Formalization scope

Lean represents links by Fin J, routes by Fin R, $A_{jr}$ and $C_j$ by natural numbers, and $x_r$ , $y_j$ , $B_j$ , and $\nu_r$ by real numbers. The primal feasible set includes both coordinatewise nonnegativity and every link-capacity inequality. The dual optimum is minimal over the nonnegative orthant, using the published dual objective. The finite sums and products have their usual empty-index values; no hidden nonemptiness assumption is imposed. Matrix rank is taken over the reals and compared with $J$ , so the rank condition is full row rank.

The goal assumes $\nu_r>0$ and $C_j\ge1$ for every route and link. Positive rates give the logarithm in the primal objective its intended meaning. Positive capacities exclude a degeneracy in the dual-attainment and blocking parts of the theorem. The value at $x_r=0$ is the continuous extension $x_r\log x_r=0$ ; no differentiability assertion at that boundary is part of the formal target. Blocking values belong to $[0,1)$ , so the inverse logarithm is evaluated on a positive number. A flow optimum always ranges over the exact primal feasible set, and a blocking solution always ranges over the full conditions (2.7); neither can be replaced by an arbitrary vector with a convenient equation.

A complete proof may require finite-dimensional convex analysis, strict concavity of the entropy term, attainment on a closed feasible set, the dual first-order condition, and facts about exponentials, logarithms, products, and matrix rank. The definitions and theorems are kept separate so work on these pieces can be reused. Contributions proving either a milestone or a reusable supporting lemma are in scope.

Selected references

F. P. Kelly, Loss networks, The Annals of Applied Probability 1(3), 319–378, 1991. DOI: 10.1214/aoap/1177005872.

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Bandit Algorithms Dynamic Programming Operations Research+1·Captain: mikedeng1

Bandit Processes and Dynamic Allocation Indices: A Policy Is Optimal If and Only If It Almost Surely Continues a Bandit Process of Maximal Dynamic Allocation IndexResearch Paper

Motivation

The multi-armed bandit problem asks how to share effort sequentially among several independent projects, each of which evolves only while it is worked on. It models the scheduling of jobs on a single machine, the order in which to explore candidate sites, sequential clinical trials and the allocation of research effort among competing projects. Before 1972 the problem had no general solution: dynamic programming over the joint state of all projects is intractable as soon as there are more than a few of them.

J. C. Gittins and D. M. Jones (1974) announced, and Gittins (1979) set out in full, a solution of a striking form: each project can be assigned a number depending only on its own current state, its dynamic allocation index (DAI, now the Gittins index), and it is optimal to work on a project whose index is largest.

Timeline. Gittins and Jones presented the index result at the European Meeting of Statisticians in 1972 (proceedings 1974). Gittins (1979) gave the general discrete-time formulation, stated the Forwards Induction Theorem and the DAI Theorem without proof, and proved the attainment Lemma of its Section 4. Whittle (1980) gave a dynamic-programming proof through the retirement formulation, Weber (1992) the prevailing-charge interchange argument, and Tsitsiklis (1994) a short inductive proof. Gittins, Glazebrook and Weber (2011) is the standard monograph; Lattimore and Szepesvári (2020, Chapter 35) give a measure-theoretic treatment on general state spaces.

Setting

A bandit process $D$ is a Markov chain $x(0), x(1), \dots$ on a measurable state space $\Theta$ with transition kernel $P$ . Applying the continuation control at process time $t$ earns the discounted reward $a^t R(x(t))$ , with $0<a<1$ , and moves the chain one step; the freezing control leaves the state unchanged and earns nothing. For a stopping time $\tau$ of the chain (possibly infinite) write

R_\tau(D,x)=E\Big\{\sum_{t=0}^{\tau-1}a^tR(x(t))\,\Big|\,x(0)=x\Big\},\qquad W_\tau(D,x)=E\Big\{\sum_{t=0}^{\tau-1}a^t\,\Big|\,x(0)=x\Big\}.

The dynamic allocation index of $D$ in state $x$ is the largest reward per unit of discounted time obtainable by continuing for at least one step:

\nu(D,x)=\sup_{\tau>0}\nu_\tau(D,x),\qquad \nu_\tau(D,x)=\frac{R_\tau(D,x)}{W_\tau(D,x)}. \tag{2}

A simple family of alternative bandit processes consists of $k$ bandit processes. At each stage $n=0,1,2,\dots$ a policy chooses exactly one of them to continue, possibly at random and as a function of the whole past; the others are frozen. A policy is optimal if it attains the supremum of the total expected discounted reward over all policies.

The Lean development uses the published objects markovChainMeasure P x (the chain law), IsPositiveStoppingTime, stoppedRatio P r α τ x ( $\nu_\tau$ ), gittinsIndex P r α x ( $\nu$ ), MarkovBanditPolicy k S, markovBanditMeasure P π x n (the law of the first $n$ stages) and markovBanditDiscountedValue P r α π x (the total expected discounted reward), with $a$ written α and $R$ written r.

Formalization targets

Goal: the DAI Theorem (Section 3, p. 154)

For every policy $\pi$ and initial state vector $x$ ,

V_\pi(x)=\sup_{\pi'}V_{\pi'}(x)\iff \forall n,\ \text{a.s. under the play of }\pi\text{ from }x:\ \ \pi_n\Big(\big\{i:\nu(x_i(n))=\max_j\nu(x_j(n))\big\}\Big)=1 .

Milestones

Section 4, Lemma (p. 154). For $\Theta_0=\{y:\nu(D,y)<\nu(D,x)\}$ and $\tau$ the first $t\ge1$ with $x(t)\in\Theta_0$ ,

\nu_\tau(D,x)=\nu(D,x).

The "if" half of the DAI Theorem. A policy that selects a process of maximal index on every history is optimal: the published BanditAlgorithm.gittins_index_theorem, already Proved.
Section 4, Corollary 1 (p. 155). For the $M$ -horizon index $\nu^M(D,x)=\sup_{0<\tau\le M}\nu_\tau(D,x)$ (5), the rule that stops at the first $t\in\{1,\dots,M-1\}$ with $\nu^{M-t}(D,x(t))<\nu^M(D,x)$ , and at $M$ otherwise, attains $\nu^M(D,x)$ .

Significance

The result. The DAI Theorem replaces a dynamic program on the product of $k$ state spaces by $k$ one-dimensional optimal stopping problems. The "only if" half says more than that the index rule is one optimal policy: it characterizes every optimal policy, so any rule that departs from the index on a set of positive probability loses reward. The Lemma identifies the stopping rule that attains the index, which is how indices are computed in practice, and Corollary 1 is the finite-horizon version on which the paper's numerical algorithm (Section 7) rests.

Formalizing it. The sufficiency half is formalized: BanditAlgorithm.gittins_index_theorem (Lattimore–Szepesvári Theorem 35.9) is Proved on the platform, together with gittins_index_policy_dominates, gittins_stopping_ratio_le_index and gittins_epsilon_optimal_stopping. The attainment Lemma is Proved only for countable state spaces with bounded rewards (AllocationIndices.optimal_stopping_set, Gittins–Glazebrook–Weber Lemma 2.2), and the necessity half is on no platform item. This mission asks for the Lemma on general standard Borel state spaces with integrable rewards, the finite-horizon Corollary 1, and the full equivalence. The Forwards Induction Theorem of p. 153, which needs superprocesses and forwards induction policies, is outside its scope.

Difficulty

The "only if" direction is the new content, and it does not follow from the sufficiency theorem: knowing that one index policy is optimal says nothing about a policy that sometimes plays a lower-index process. The deviation must be shown to cost a strictly positive amount whenever it happens with positive probability, which requires a strict comparison that survives discounting, ties in the index and infinite stopping times. A second obstacle is measure-theoretic. On a general state space the index is a supremum over uncountably many stopping times, its measurability is not available, and the Lemma's stopping rule is a stopping time only if $\{y:\nu(D,y)<\nu(D,x)\}$ is measurable; the arguments of the countable case, which treat each state separately, do not transfer.

Formalization scope

All objects are the published discounted $k$ -armed Markov bandit of Def_GittinsIndex and the stopping-time quantities of Def_AllocationIndices_Index; time is indexed from $0$ , stopping times are $\mathbb N\cup\{\infty\}$ -valued and adapted to the coordinate filtration, and the index is the real supremum over stopping times $\tau\ge1$ . The explicit readings committed to are:

"Optimal" means attaining $\sup_{\pi'}V_{\pi'}(x)$ from the given initial state vector, over all randomized history-dependent policies; this supremum is finite because the values are bounded uniformly in the policy under the integrability hypothesis.
"At each stage … almost always" means: for every stage $n$ , almost surely under the law of the first $n$ stages of $\pi$ from $x$ , the selection kernel puts mass one on the processes of maximal current index. Requiring the index rule on every history would make "only if" false.
"The supremum of the total expected reward is finite" (p. 151) is read as $E_x\sum_t a^t|R(x(t))|<\infty$ for every state $x$ .
" $\nu(D,\cdot)$ is an $\mathcal X$ -measurable function" (p. 151) is an explicit hypothesis of the Lemma and of the goal; the measurability of every $\nu^m(D,\cdot)$ is a hypothesis of Corollary 1.
"The supremum is attained by setting $\Theta_0$ " means that the first hitting time of $\Theta_0$ from time $1$ on is a positive stopping time whose ratio equals the index.
The state space is standard Borel (the paper asks only for measurable singletons), and all processes share one kernel and one reward function; a family of different processes is encoded on the disjoint union of their state spaces.

A trivializing formalization is ruled out: the goal is not stated for index policies on every history (which would only restate the referenced theorem), the index is the published supremum over stopping times rather than a one-step reward or a finite-horizon quantity, competitors are not restricted to Markov or deterministic policies, and the Lemma's stopping rule starts at time $1$ , so it never stops at once. A one-point state space satisfies all hypotheses together and admits a policy.

Useful infrastructure, reusable beyond this mission: measurability of the Gittins index on standard Borel spaces, a measurable selection of a maximal-index arm, and an interchange argument for $k$ -armed Markov bandits on general state spaces. Proofs of any milestone, and of either half of the goal, are welcome.

Selected references

J. C. Gittins, Bandit Processes and Dynamic Allocation Indices, J. R. Statist. Soc. B 41(2), 148–177, 1979. https://doi.org/10.1111/j.2517-6161.1979.tb01068.x
J. C. Gittins and D. M. Jones, A dynamic allocation index for the sequential design of experiments, in Progress in Statistics (J. Gani et al., eds.), North-Holland, 241–266, 1974.
P. Whittle, Multi-armed bandits and the Gittins index, J. R. Statist. Soc. B 42(2), 143–149, 1980. https://doi.org/10.1111/j.2517-6161.1980.tb01111.x
R. Weber, On the Gittins index for multiarmed bandits, Ann. Appl. Probab. 2(4), 1024–1033, 1992. https://doi.org/10.1214/aoap/1177005588
J. N. Tsitsiklis, A short proof of the Gittins index theorem, Ann. Appl. Probab. 4(1), 194–199, 1994. https://doi.org/10.1214/aoap/1177005207
J. C. Gittins, K. D. Glazebrook and R. Weber, Multi-armed Bandit Allocation Indices, 2nd ed., Wiley, 2011. https://doi.org/10.1002/9780470980033
T. Lattimore and C. Szepesvári, Bandit Algorithms, Cambridge University Press, 2020, Chapter 35. https://doi.org/10.1017/9781108571401

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Complexity Theory Linear Optimization Optimization·Captain: mikedeng1

On Linear Characterizations of Combinatorial Optimization Problems II: A Small Polynomial-Time Generator of Violated Inequalities Puts the Decision Problem in PResearch Paper

Motivation

Many combinatorial optimization problems ask for the largest value of a linear objective over discrete feasible solutions. A linear program over the convex hull of those solutions has the same optimum, so one approach is to describe that hull by inequalities. For matching, for example, inequalities support efficient optimization; for other problems, the full family can be very large. A generator of violated inequalities offers a different interface: given a proposed point, it either certifies hull membership or produces one valid inequality that the point violates. Karp and Papadimitriou asked what the existence of an efficient generator implies for the underlying decision problem. Their answer is Theorem 2 of the technical report.

The report appeared as MIT/LCS/TM-154 in 1980 and later in SIAM Journal on Computing in 1982. Its first theorem addresses a small facial description whose inequality family belongs to NP. This mission concerns its second theorem, where the inequalities may be generated on demand in polynomial time. The conclusion is stronger: the decision language itself belongs to P. The report derives an NP-complete consequence as Corollary 2, conditional on the unresolved equality P = NP. The published article gives the later bibliographic record; all numbering here follows the technical report.

Setting

A combinatorial optimization problem $C=(L,n,S)$ consists of a binary input language $L$ , a nonnegative integer dimension $n(z)$ , and a set $S(z)\subseteq\mathbb Z_+^{n(z)}$ of feasible vectors for each $z\in L$ . The report requires polynomial-time recognition of $L$ , of correctly sized pairs $\langle z,y\rangle$ , and of feasible pairs $\langle z,x\rangle$ . An instance adds an integer objective $c\in\mathbb Z^{n(z)}$ . Its threshold decision language is

D(C)=\{\langle z,c,k\rangle:z\in L,\ k\in\mathbb Z,\ \exists x\in S(z),\ c\cdot x\ge k\}.

Write $\mathrm{CH}(S(z))$ for the convex hull of the feasible vectors. A generator $G(C)$ receives $z\in L$ and a rational vector $p\in\mathbb Q^{n(z)}$ . It returns “O.K.” exactly when $p$ belongs to the hull. Otherwise it returns an integer vector $f$ and integer $g$ for which $f\cdot p>g$ while $f\cdot x\le g$ for every $x\in S(z)$ . The family $F_G(C)$ contains every triple $\langle z,f,g\rangle$ that occurs as some output. It is a facial description because its inequalities describe the hull exactly. The generator is small if one polynomial $p_0$ bounds every coefficient in every output by $2^{p_0(|z|+n(z))}$ in absolute value. Its running time is measured on the full encoded query, including every rational coordinate. These definitions are from §§2 and 4 of the report.

Formalization targets

Theorem 2

The goal is the report's stated implication:

G(C)\text{ is a small polynomial-time generator}\quad\Longrightarrow\quad D(C)\in\mathrm P.

It allows any c.o.p. satisfying Definition 1, including an empty feasible set or dimension zero. It does not assert P = NP; that consequence needs the additional hypothesis that $D(C)$ is NP-complete. The goal is supported by five source claims: the output family is facial; feasibility of system (4) is equivalent to membership in $D(C)$ ; Lemma 4 bounds the distance from a rational point to a small hyperplane; its Corollary extends the bound to a real point on a flat; and Lemma 5 controls distance to a flat after one more hyperplane is added. The system (4) target is

\exists x\in\mathbb Q^{n(z)}:\quad c\cdot x\ge k,\quad f\cdot x\le g\quad(\langle z,f,g\rangle\in F_G(C))\quad\Longleftrightarrow\quad\langle z,c,k\rangle\in D(C).

The distance targets use the report's parameter $t$ and the bounds $2^{-(n(z)+1)t}$ and $(2^t-1)\delta$ , exactly as printed on pp. 12–13.

Significance

Theorem 2 turns an efficient separation interface into an efficient test for the threshold language. Consequently, if a c.o.p.'s decision language is NP-complete, a small polynomial-time generator would imply P = NP, as the report's Corollary 2 says. This gives a complexity barrier to a whole class of proposed cutting-plane interfaces, independently of how many inequalities the complete description contains. The result concerns the worst-case running time and output coefficient size; it does not rule out useful generators without those guarantees. Karp and Papadimitriou also point out positive applications to problems already known to be in P.

The mathematical theorem was proved in the report. The mission asks for a machine-checked version of that known result, with explicit interfaces for the generator, coefficient bounds, and the underlying complexity class. The local platform catalog already contains Cook's Turing-machine definitions and a reusable binary-integer encoding; it also contains point-to-hyperplane distance results and a Gram-determinant distance result in other settings. Theorem 2 and these particular source claims are new formalization targets here. The distance and encoding interfaces can serve later missions about separation oracles and bit complexity.

Difficulty

Calling the generator once does not by itself decide whether the objective threshold is attainable: the point supplied to it may be outside the hull, and a single returned inequality says only that this point is excluded. Repeatedly asking at an unchanged point may return the same inequality. The report must also control very small separations, coefficient sizes, and finite precision when an iterative linear-feasibility procedure uses the oracle. Lemmas 4 and 5 quantify the geometric gaps and movement that such a procedure needs. Its Lemmas 2, 3, and 6 address the surrounding perturbation, volume, and iteration arguments; Lemma 6 explicitly relies on another analysis for the finite-precision bound. These are substantive proof obligations even though their statements are not all separate milestones in this proposal. See §4 of the report.

Formalization scope

The report's footnote uses $R$ for the rationals. Thus $p$ , the hull, and system (4) live in $\mathbb Q^{n(z)}$ . Euclidean distances are evaluated after casting coordinates into $\mathrm{EuclideanSpace}\,\mathbb R\,(\mathrm{Fin}\ n)$ ; a hyperplane distance is the quotient $|f\cdot r-g|/\|f\|_2$ , and distance to a flat uses Euclidean infimum distance. The report does not define “affinely independent hyperplanes”; this mission reads it as linear independence of their normals, which ensures a common flat. Small hyperplanes have nonzero normals, avoiding a zero denominator.

The alphabet is $\{0,1,-,\#\}$ . Binary tuple codes contain $z$ , an explicit vector dimension, and every integer; a rational coordinate contains its reduced numerator and denominator. The generator's string function must match the mathematical oracle on every valid query and be polynomial time in the complete code length. A malformed string is outside $D(C)$ . The source's arbitrary polynomial bound is written $m^a+a$ for one exponent $a$ fixed before all instances. For the bit-length parameter $t$ , the printed logarithms of possibly nonpositive $c_i$ and $k$ are read as ceiling logarithms of absolute values, with $\operatorname{clog}_2(0)=0$ and the printed $+1$ terms retained. The source's formula $\varepsilon=2^{-(2n(z)+2)t}$ remains relevant to the eventual proof of Theorem 2, although no $\varepsilon$ system is a separate draft item here.

Every oracle answer must separate the queried point and remain valid on all of $S(z)$ ; these obligations prevent an arbitrary function from satisfying the goal. The c.o.p. structure retains all three polynomial-recognition conditions of Definition 1. The paper's displayed “min” in (3) is inconsistent with its threshold language and surrounding maximization formulas; $D(C)$ uses the printed $c\cdot x\ge k$ . The typed source's “ $x\notin H$ ” in Lemma 4 is read as the already named point $r\notin H$ .

Contributions can develop the rational-hull equivalence, the two quantitative hyperplane lemmas, the finite-precision ellipsoid procedure underlying Lemma 6, or the perturbation and volume claims. The complexity substrate is Cook's one-tape model of P and NP, and the binary integer codec is the published ProjSchedTW_Complexity_Encoding definition. The geometry lemmas are phrased so that existing Mathlib Euclidean-space results can be reused without changing the source's rational optimization domain.

Selected references

Richard M. Karp and Christos H. Papadimitriou, On Linear Characterizations of Combinatorial Optimization Problems, MIT/LCS/TM-154, Massachusetts Institute of Technology, 1980. Technical report.
Richard M. Karp and Christos H. Papadimitriou, “On Linear Characterizations of Combinatorial Optimization Problems,” SIAM Journal on Computing 11 (1982), 620–632. DOI.
Stephen A. Cook, “The Complexity of Theorem-Proving Procedures,” Proceedings of the Third Annual ACM Symposium on Theory of Computing, 1971. DOI.

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Complexity Theory Linear Optimization Operations Research·Captain: mikedeng1

On Linear Characterizations of Combinatorial Optimization Problems I: A Small Facial Description in NP Puts the Decision Problem in co-NPResearch Paper

Motivation

Polyhedral combinatorics attacks a combinatorial optimization problem by replacing its finite set of feasible solutions with the convex hull of that set and describing the hull by linear inequalities. Once such a description is known, the problem becomes a linear program, and linear programming duality supplies optimality certificates. For matchings, matroids and network flows this programme succeeded completely (Edmonds' matching polytope is the classical example). For the traveling salesman problem, set covering and integer programming, decades of work produced large families of valid and facet-defining inequalities but never a complete list.

R. M. Karp and C. H. Papadimitriou asked whether the failure is accidental. Their answer, in the MIT technical report On linear characterizations of combinatorial optimization problems (MIT/LCS/TM-154, February 1980; journal version SIAM J. Comput. 11 (1982) 620–632), is that for NP-complete problems a usable complete description cannot exist unless NP = co-NP. This mission formalizes the first of their two complexity results, Theorem 1, together with the steps of its proof and its Corollary 1.

Setting

A combinatorial optimization problem (c.o.p.) $C$ consists of a set $L\subseteq\{0,1\}^*$ of inputs, a function $n$ assigning to each input $z\in L$ a number of variables $n(z)\ge 0$ , and for each $z\in L$ a set $S(z)\subseteq(\mathbb Z^+)^{n(z)}$ of nonnegative integer vectors, the feasible solutions. The three languages $L$ , $\{\langle z,y\rangle : |y|=n(z)\}$ and $\{\langle z,x\rangle : x\in S(z)\}$ must be recognizable in polynomial time. An instance is a pair $\langle z,c\rangle$ with $z\in L$ and $c\in\mathbb Z^{n(z)}$ , and the decision problem of $C$ is

D(C)=\{\langle z,c,k\rangle : z\in L,\ c\in\mathbb Z^{n(z)},\ k\in\mathbb Z,\ \exists x\in S(z)\ \ c\cdot x\ge k\}.

Write $\mathrm{CH}(S(z))\subseteq\mathbb Q^{n(z)}$ for the convex hull of $S(z)$ over the rationals. A facial description of $C$ is a set $F(C)$ of triples $\langle z,f,g\rangle$ , with $z\in L$ , $f\in\mathbb Z^{n(z)}$ and $g\in\mathbb Z$ , such that for every $z\in L$ and every $x\in\mathbb Q^{n(z)}$

x\in\mathrm{CH}(S(z))\iff f\cdot x\le g\ \text{ for all }\langle z,f,g\rangle\in F(C).

It is small if for some polynomial $p$ every component of every triple $\langle z,f,g\rangle\in F(C)$ has absolute value at most $2^{p(|z|+n(z))}$ , so that each coefficient has polynomially many bits. The statement $F(C)\in\mathrm{NP}$ means that the language of the triples of $F(C)$ is in NP: membership of a listed inequality has short, checkable proofs.

Formalization targets

Goal: Theorem 1 (p. 7)

F(C)\ \text{a small facial description of }C,\quad F(C)\in\mathrm{NP}\ \Longrightarrow\ D(C)\in\text{co-NP}.

Milestones, in the order the proof uses them

(p. 5) For a small facial description and every $z\in L$ , only finitely many triples have first entry $z$ , and $\mathrm{CH}(S(z))$ is the intersection of finitely many half-spaces.
(pp. 7–8, (i)⇔(ii)) $\langle z,c,k\rangle\notin D(C)$ iff $c\cdot x<k$ for every $x\in\mathrm{CH}(S(z))$ .
(p. 8, (ii)⇔(iii)) For a facial description, the same holds with $\mathrm{CH}(S(z))$ replaced by the system $f\cdot x\le g$ , $\langle z,f,g\rangle\in F(C)$ .
(p. 8, (iii)⇔(vi)) For a small facial description and $S(z)\neq\emptyset$ : $c\cdot x<k$ on that system iff there are $n(z)$ triples $\langle z,f_i,g_i\rangle\in F(C)$ whose matrix $\mathbf F$ is nonsingular and the solution $y$ of $y^T\mathbf F=c$ satisfies $y\ge 0$ and $y^Tg<k$ .

After the goal: Corollary 1 (p. 8)

D(C)\ \text{NP-complete},\ F(C)\ \text{small facial},\ F(C)\in\mathrm{NP}\ \Longrightarrow\ \mathrm{NP}=\text{co-NP}.

Significance

Theorem 1 converts a question of polyhedral combinatorics into one of complexity theory. Corollary 1 then says that for the traveling salesman problem, Hamiltonian circuit, set covering, integer programming and every other c.o.p. with an NP-complete decision problem, no complete linear description of the hulls has both polynomially sized coefficients and polynomially verifiable membership, unless NP = co-NP. This explains why the search for complete descriptions of such polytopes has not succeeded and why research on them turned to partial descriptions and separation. The companion result of the same paper (Theorem 2, the subject of the second mission in this series) treats separation routines and P instead of facial descriptions and co-NP.

The result itself is proved, by a short argument in the paper. To our knowledge it has no machine-checked proof. Formalizing it requires connecting three things that the platform holds only separately: Cook's Turing-machine classes P, NP and co-NP (published as CookPvsNP_defs), linear programming duality over the rationals (Farkas' lemma is published, e.g. LinearOptimization.farkas_inequality_form_fintype, for real matrices), and polynomial-time arithmetic on binary-encoded rational data, such as solving a nonsingular integer linear system. The last ingredient, a polynomial-time verifier built from a polynomial-time recognizer, is reusable for every co-NP or NP membership proof on the platform.

Difficulty

The mathematical content of Theorem 1 is the duality chain (i)⇔(vi): the "no" answer is certified by $n(z)$ inequalities of $F(C)$ and a basic dual solution. Two points make the formal proof harder than the paper's page suggests.

First, the duality chain needs care at its edges. When $S(z)$ is empty the chain (iii)⇔(vi) can fail, so the verifier must also accept a different certificate (an infeasibility certificate from at most $n(z)+1$ triples), while milestone 4 carries the hypothesis $S(z)\neq\emptyset$ . Pointedness of the hull, which follows from $S(z)\subseteq(\mathbb Z^+)^{n(z)}$ , is what guarantees $n(z)$ linearly independent rows.

Second, "Algorithm B clearly runs in polynomial time" hides a complete complexity argument on Cook's one-tape machines: parsing the input, testing $z\in L$ and $|c|=n(z)$ with the recognizers of Definition 1, guessing the matrix within the size bound given by smallness, invoking the NP verifier of $F(C)$ $n(z)$ times, and solving $y^T\mathbf F=c$ exactly over $\mathbb Q$ with polynomially bounded bit sizes (Cramer's rule bounds the numerators and denominators of $y$ ). The certificate must be of length polynomial in the input, which uses the uniform polynomial in the definition of "small". The naive certificate, a single optimal vertex of the hull, does not work: it proves a "yes" answer, not a "no".

Formalization scope

All definitions live in the namespace KarpPapadimitriou.Facial. Strings are List Bool; $S(z)$ is a set of Fin (n z) → ℤ with nonnegative entries; $\mathrm{CH}(S(z))$ is convexHull ℚ of its image in Fin (n z) → ℚ, because the paper's "R" denotes the rationals (footnote, p. 3). Tuples are coded over the four-letter alphabet $\{0,1,-,\#\}$ of the published ProjSchedTW_Complexity_Encoding, integers in binary, vectors prefixed by their length so that codes are uniquely decodable. P, NP, co-NP and NP-completeness are the published CookPvsNP_defs; $D(C)$ and the triple language of $F(C)$ are sets of codes, so strings encoding no triple lie in the complement of $D(C)$ , as in Algorithm B's step (i). The three polynomial-time conditions of Definition 1 are fields of the structure COP.

Explicit readings of the paper's loose phrases:

"polynomial $p$ " in "small" is $m\mapsto m^k+k$ with one $k$ for all triples;
"has optimal value $<k$ " for (ii) and (iii) is "every feasible point has value $<k$ ", meaningful for infeasible and unbounded programs;
"the system $y^T\mathbf F=c$ has a unique nonnegative solution $y$ with $y^Tg<k$ " is " $\det\mathbf F\neq0$ and the solution satisfies $y\ge 0$ , $y^Tg<k$ ";
(iii)⇔(vi) carries the hypothesis $S(z)\neq\emptyset$ , tacit in the paper; Theorem 1 does not;
"NP = co-NP" in Corollary 1 holds for every finite nonempty alphabet, since Cook's classes are indexed by the alphabet.

Printed slips: display (3) reads "min" while $D(C)$ and the proof maximize; Algorithm B's step (v) reads " $y^Tb>k$ " for $y^Tg<k$ . Both are formalized in the corrected sense.

A trivializing formalization is ruled out: the goal quantifies over every c.o.p. and every small facial description, with complements taken over all strings, and no hypothesis restricts $S(z)$ or $F(C)$ beyond Definition 1 and the definitions of p. 4–5. Claim 1 (p. 9), announced without proof, is not part of the mission.

Contributions welcome: proofs of the four milestones (milestone 4 is an exercise in rational LP duality on finite systems), a reusable library of polynomial-time integer and rational arithmetic on Cook's machines, and the closure of NP under polynomial-time reductions across alphabets that Corollary 1 needs.

Selected references

R. M. Karp and C. H. Papadimitriou, On linear characterizations of combinatorial optimization problems, MIT/LCS/TM-154, February 1980. https://dspace.mit.edu/server/api/core/bitstreams/eb122126-c312-4445-a8d2-153e3e7d285f/content ; journal version SIAM J. Comput. 11 (1982) 620–632, https://doi.org/10.1137/0211053
S. A. Cook, The P versus NP problem, Clay Mathematics Institute Millennium Problem description, 2000. https://www.claymath.org/wp-content/uploads/2022/06/pvsnp.pdf
J. Edmonds, Maximum matching and a polyhedron with 0,1-vertices, J. Res. Nat. Bur. Standards 69B (1965) 125–130. https://doi.org/10.6028/jres.069B.013
A. Schrijver, Theory of Linear and Integer Programming, Wiley, 1986 (LP duality, basic solutions, sizes of solutions of linear systems). ISBN 978-0-471-98232-6
M. Grötschel and M. W. Padberg, On the symmetric travelling salesman problem I: Inequalities, Math. Programming 16 (1979) 265–280. https://doi.org/10.1007/BF01582116

10 thms4 active usersReviewed

Bandit Algorithms Linear algebra Machine Learning+1·Captain: mikedeng1

Feature-Based Dynamic Pricing: The EllipsoidPricing Algorithm Has Worst-Case Regret O(R d² ln(T/d))Research Paper

Motivation

Online marketplaces, ad exchanges and real-estate platforms sell items that are each described by a vector of features and are rarely seen twice. A seller who prices such items cannot learn a demand curve item by item. It must learn how features map to values, while pricing, from accept/reject feedback alone. Cohen, Lobel and Paes Leme (Management Science, 2020; SSRN 2737045) model this as feature-based dynamic pricing with adversarial features and a linear valuation. They show that a direct multi-dimensional binary search (PolytopePricing) can have regret exponential in the dimension (their Theorem 1). Their EllipsoidPricing algorithm, a pricing variant of Khachiyan's ellipsoid method (Khachiyan 1979), has worst-case regret quadratic in the dimension and logarithmic in the horizon (their Theorem 2). This mission formalizes Theorem 2 and the chain of lemmas behind it.

Setting

Fix a dimension $d \ge 2$ , a radius $R > 0$ and a horizon $T$ . Nature picks an unknown parameter $\theta \in \mathbb{R}^d$ with Euclidean norm $\|\theta\| \le R$ . In each period $t = 1, \dots, T$ a product arrives with a feature vector $x_t \in \mathbb{R}^d$ , $\|x_t\| \le 1$ , and market value $\theta' x_t$ . The seller sees $x_t$ , posts a price $p_t$ , and a sale occurs iff $p_t \le \theta' x_t$ , earning $p_t$ . The regret (Eq. (1)) is

\mathrm{Regret} = \sum_{t=1}^{T} \big[\theta' x_t - p_t\,\mathbb{I}\{\theta' x_t \ge p_t\}\big],

and the worst-case regret of a policy is its maximum over $\theta$ and over nature's (possibly adaptive) choice of features.

An ellipsoid with center $a$ and positive definite shape matrix $A$ is $E(A,a) = \{\theta : (\theta - a)' A^{-1} (\theta - a) \le 1\}$ . EllipsoidPricing with parameter $\epsilon > 0$ keeps an ellipsoid $E_t = E(A_t, a_t)$ , starting from the ball $E_1 = B(0,R)$ . In period $t$ it computes

\underline b_t = x_t' a_t - \sqrt{x_t' A_t x_t}, \qquad \bar b_t = x_t' a_t + \sqrt{x_t' A_t x_t},

the minimum and maximum of $\hat\theta' x_t$ over $E_t$ .

If $\bar b_t - \underline b_t \le \epsilon$ , it exploits: it posts $p_t = \underline b_t$ and keeps $E_{t+1} = E_t$ .
Otherwise it explores: it posts $p_t = \tfrac12(\bar b_t + \underline b_t)$ and replaces $E_t$ by the ellipsoid of Eq. (4) that covers the half of $E_t$ consistent with the feedback. With $b = A_t x_t / \sqrt{x_t' A_t x_t}$ , the new shape is $\tilde A = \frac{d^2}{d^2-1}\big(A_t - \frac{2}{d+1} b b'\big)$ and the new center is $a_t \pm \frac{1}{d+1} b$ , with $+$ after a sale.

Write $\lambda_d(A)$ for the smallest eigenvalue of a symmetric matrix $A$ .

Formalization targets

Goal: Theorem 2

There is a universal constant $C > 0$ such that for all $d \ge 2$ , $R > 0$ , $T \ge 2d$ , $\|\theta\| \le R$ and $\|x_t\| \le 1$ , EllipsoidPricing run with $\epsilon = R d^2 / T$ satisfies

\mathrm{Regret} \le C \, R \, d^2 \ln(T/d).

This is the paper's $O(R d^2 \ln(T/d))$ claim, with no constant fixed.

Milestones, in the order of the paper's argument

the closed forms of $\underline b_t, \bar b_t$ (§5.2);
the containment of each half-ellipsoid in the updated ellipsoid (Eq. (4), already proved on the platform);
$\theta \in E_t$ and $A_t \succ 0$ along the run;
the volume formula $\operatorname{Vol} E(A,a) = V_d \sqrt{\prod_i \lambda_i(A)}$ and the volume decrease $\operatorname{Vol} E(\tilde A) \le e^{-1/2d} \operatorname{Vol} E(A)$ (§5.1);
Lemma 2: if $z < \lambda_d(A)$ and $\det(A - \beta b b' - zI) \ge 0$ , then $\lambda_d(A - \beta b b') \ge z$ ;
Lemma 3: $\lambda_d(\tilde A) \ge \frac{d^2}{(d+1)^2} \lambda_d(A)$ ;
Lemma 4: if $\lambda_d(A) \le \epsilon^2/(400 d^2)$ and $x' A x > \epsilon^2/4$ , then $\lambda_d(\tilde A) \ge \lambda_d(A)$ ;
the eigenvalue floor $\lambda_d(A_t) \ge \epsilon^2 / (400 (d+1)^2)$ ;
Lemma 1: at most $2 d^2 \ln(20 R (d+1)/\epsilon)$ exploration periods;
an exploitation period sells and has regret at most $\epsilon$ .

Significance

The result. Theorem 2 shows that contextual posted-price learning with adversarial features costs only polynomially many "probing" sales in the dimension. A direct cutting-plane search over the polytope of consistent parameters (PolytopePricing) cannot do this: its worst-case regret is exponential in $d$ (Theorem 1). The ellipsoid analysis via the smallest eigenvalue (Lemmas 2–4) is reused in the paper's noisy-valuation extension (ShallowPricing, §6) and in later work on contextual pricing and contextual search.

Formalizing it. No part of the paper is machine-checked. The Eq. (4) containment is already proved on the platform (Bertsimas–Tsitsiklis Theorem 8.1). The volume factor there is $e^{-1/(2(d+1))}$ , weaker than the $e^{-1/2d}$ used here. The eigenvalue lemmas on rank-one perturbations (Lemmas 2–4) are new to the platform and reusable for any analysis of ellipsoid-type updates. A formal proof of the goal would also settle a gap in the printed proof, noted under the formalization scope.

Difficulty

The obvious argument is the textbook ellipsoid bound: volume shrinks by $e^{-1/2d}$ per exploration, so exploration must stop. This fails because the ellipsoid need not become small in every direction. The volume can go to zero while one axis stays long, and then a feature along that axis keeps triggering exploration. The paper replaces the volume argument by a lower bound on the smallest eigenvalue, which needs a sign analysis of the characteristic polynomial of a rank-one perturbation (Lemmas 2–4) together with the exploration test $x_t' A_t x_t > \epsilon^2/4$ . The constant $k = 1/(400 d^2)$ in Lemma 4 is chosen to make a specific inequality hold for every $d \ge 2$ .

The regret bound needs, in addition, a per-round bound for exploration periods. The printed proof (p. 16) uses "the trivial bound of regret $R$ per round". That bound is not immediate when an exploration price is accepted, since the center $a_t$ can leave $B(0,R)$ and the round's regret is then only bounded by $\sqrt{x_t' A_t x_t}$ . This is why the goal is the $O$ -form with an unspecified universal constant rather than the proof's explicit inequality.

Formalization scope

Types. Vectors are Fin d → ℝ; $\theta' x$ is the dot product θ ⬝ᵥ x. Both norms are Euclidean, encoded as $x\cdot x \le 1$ and $\theta\cdot\theta \le R^2$ ; Mathlib's ‖·‖ on Fin d → ℝ is the sup norm and is never used. Ellipsoids, the ball, and the update (4) are the published LinearOptimization.ellipsoid, ellipsoidBall, ellipsoidUpdateCenter and ellipsoidUpdateMatrix.
Eigenvalues and volume. $\lambda_d(A)$ is the infimum of the real spectrum, i.e. the minimum eigenvalue of a symmetric matrix. Volumes are Lebesgue measure on Fin d → ℝ in $[0, \infty]$ . $\varphi_D(z)$ is $\det(D - zI)$ as printed, not Mathlib's charpoly.
The algorithm. EllipsoidPricing is a deterministic recursion indexed from $0$ (Lean period $t$ is the paper's $t+1$ ), started from $E_1 = B(0,R)$ , which the paper allows ("or in fact any ellipsoid that contains $K_1$ ", p. 11). The paper's "smallest ellipsoid containing $H_{t+1}$ " is replaced by its closed form (4), which the paper gives on p. 14. Because the algorithm is deterministic, a closed-loop nature generates a fixed feature sequence, and the worst case is a universal quantifier over $\theta$ in the ball and over sequences.
Added hypotheses, all disclosed in the items.
- $d \ge 2$ everywhere, since (4) divides by $d^2 - 1$ .
- $T \ge 2d$ in the goal, since $\ln(T/d) \le 0$ for $T \le d$ .
- $\epsilon \le 20 R (d+1)$ in Lemma 1 and the eigenvalue floor: for larger $\epsilon$ the printed bound is negative.
- $\|x\| \le 1$ in Lemma 4, the §3 normalization that its proof uses.
Not taken from the page. E₁ is not the Löwner–John ellipsoid of a general $K_1$ : the proof's " $\tilde E_1$ lies in the ball of radius $R$ " fails for it. The proof's explicit inequality $\mathrm{Regret} \le NR + (T-N)\epsilon$ is not a milestone, because of the per-round gap above.
Ruled out. The goal may not be trivialized by any of the following:
- a constant $C$ that depends on $d$ , $R$ , $T$ or the instance;
- regret measured against a price the algorithm never posts, or an exploit price that is not accepted;
- features bounded in the sup norm;
- an explore test or update different from (4);
- a single fixed feature sequence.
Contributions welcome. Proofs of the volume formula and the $e^{-1/2d}$ decrease, which are reusable for any ellipsoid-method development; the eigenvalue lemmas; and the per-round bound for accepted exploration prices that the goal needs.

Selected references

M. C. Cohen, I. Lobel, R. Paes Leme, Feature-Based Dynamic Pricing, Management Science 66(11), 2020. https://doi.org/10.1287/mnsc.2019.3485 (authors' copy: https://ssrn.com/abstract=2737045)
L. G. Khachiyan, Polynomial algorithms in linear programming, USSR Comput. Math. Math. Phys. 20(1), 1980. https://doi.org/10.1016/0041-5553(80)90061-0
M. Grötschel, L. Lovász, A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer, 1993. https://doi.org/10.1007/978-3-642-78240-4
G. H. Golub, Some modified matrix eigenvalue problems, SIAM Review 15(2), 1973. https://doi.org/10.1137/1015032
D. Bertsimas, J. N. Tsitsiklis, Introduction to Linear Optimization, Athena Scientific, 1997 (Theorem 8.1, the platform's ellipsoid update).

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Algebraic Geometry Number Theory·Captain: Vas

Mazur’s Torsion Theorem: Full Classification and All Fifteen GroupsResearch Paper

Why rational torsion deserves a complete classification

An elliptic curve has rational solutions that can be added using a canonical group law. A torsion point is a rational solution that returns to the identity after a positive number of additions. These points form the finite part of the rational-point group and constrain which finite arithmetic configurations a curve can carry. Barry Mazur determined the possible groups uniformly over all elliptic curves defined over the rational numbers. The theorem is established mathematics; the remaining task is its complete formal proof, including the assertion that every permitted group occurs. The source is Theorem III.5.1 and its occurrence remark, printed page 156 of Modular curves and the Eisenstein ideal.

Curves, rational points, and finite additive order

A Weierstrass curve over the rational numbers is an equation

y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6,

where all five coefficients are rational. Its discriminant measures singularity. The curves in this mission have nonzero discriminant, so they are elliptic curves. The group $E(\mathbb Q)$ consists of its rational points together with the distinguished point at infinity, which is the additive identity. The rational torsion subgroup is

E(\mathbb Q)_{\mathrm{tors}}=\{P\in E(\mathbb Q):\exists n\ge1,\ nP=0\}.

This subgroup is formed using the actual rational-point group; it is not a separately supplied finite group or a collection of coordinate certificates.

The symbol $\mathbb Z/n\mathbb Z$ denotes a cyclic group of order $n$ , with its additive operation. A product of two such groups has componentwise addition. An additive group isomorphism is a bijection that preserves addition. Consequently, classifying a subgroup by an isomorphism identifies its complete group structure and its number of elements, rather than only the orders of selected points.

Formalization targets

For every elliptic curve $E/\mathbb Q$ , the torsion subgroup must be finite and isomorphic to one of

\mathbb Z/n\mathbb Z\quad\text{for }n\in\{1,2,3,4,5,6,7,8,9,10,12\},

\mathbb Z/2\mathbb Z\times\mathbb Z/(2m)\mathbb Z\quad\text{for }m\in\{1,2,3,4\}.

Conversely, for every parameter in either displayed range, there must exist an elliptic curve over $\mathbb Q$ whose full rational torsion subgroup is isomorphic to that group. The curve may depend on the parameter. The goal combines unconditional finiteness, classification, and all fifteen occurrence cases. The cyclic case $n=1$ includes curves with trivial rational torsion. Occurrence requires equality of the entire torsion group up to isomorphism; constructing an embedding or a single point of the desired order does not meet that target.

The supporting point-order milestone is Corollary III.5.2, on the same source page: every rational point of positive finite order has order in the eleven-element cyclic list. Infinite-order points are outside that milestone's hypothesis. The four other milestone clauses separate finiteness, group classification, cyclic occurrence, and bicyclic occurrence so their contributions can be reviewed individually.

What the formal proof will provide

The final classification theorem will be available in the standard Lean vocabulary for rational elliptic curves, finite-order subgroups, cyclic groups, products, and additive equivalences. Its main downstream consumer is the canonical classification endpoint of the MazurTheorem repository. Occurrence supplies the converse needed for an exact list of possible groups. Together these results also give a torsion-cardinality bound of sixteen and uniformly exclude rational torsion configurations outside the list.

The existing repository contains substantial checked prerequisites and reductions. A prerequisite receives credit according to its actual hypotheses, conclusion, and connection to the acceptance theorem. This campaign does not treat the full classification as already formalized, and it does not treat locally checked source as a platform-verified contribution before transfer and verification. Existing proofs should be reused with their attribution and dependency graph intact. Open definitions and unproved mathematical comparisons must remain visible as obligations.

The remaining difficulty

Finite-abelian-group classification alone cannot establish which groups arise from an arbitrary rational elliptic curve. The crucial arithmetic exclusions require connecting concrete rational points with genuine arithmetic-geometric objects. A theorem about a supplied quotient or Jacobian does not construct that quotient or Jacobian; an interface that assumes a missing comparison leaves that comparison unproved. The foundational work includes represented modular curves, Jacobians and Picard functors, integral models, specialization, and the arithmetic needed to exclude prohibited point orders. These requirements reflect the mathematical source and the repository's current proof architecture.

The occurrence direction has a different verification obligation. Explicit rational points and order computations must be supplemented by a proof that the curve is nonsingular and that its entire rational torsion subgroup has precisely the claimed shape. Machine arithmetic and external tables can guide the selection of examples, but the published occurrence theorem must have a checked proof.

Formalization scope and contribution boundaries

Curves are Mathlib's WeierstrassCurve ℚ with IsElliptic; rational points are (E⁄ℚ).Point. Torsion is AddCommGroup.torsion, the cyclic groups are the additive groups of ZMod, and isomorphisms are additive equivalences. The target definition bundle repeats the repository's fifteen-group vocabulary in a campaign namespace. A checked compatibility bridge identifies its classification predicate with the repository's canonical predicate. This definition boundary exists solely to state platform obligations faithfully; its named consumers are the campaign goal and the repository's classification endpoint.

The goal has no supplied finiteness, rank, reduction, modularity, semistability, point-order classification, or group-classification hypothesis. Finiteness is an explicit conclusion. The numerical expression Set.ncard ≤ 16 is only a corollary: Set.ncard is zero on infinite sets, so that expression alone cannot certify this mission's completion. Checked proofs must have no placeholders, custom axioms, unchecked computation, or proof-strengthening options. The permitted foundational dependencies are propext, Quot.sound, and Classical.choice; open targets are displayed separately from their eventual proofs.

The repository's execution ledger remains the source of contributor assignment. Its selected foundations are the coherent-cohomology-to-Jacobian lane and the represented-modular-curve lane. Contracts marked paused remain paused unless the maintainers explicitly revise their assignment. Platform milestones describe mathematical completion obligations, not automatic authorization to claim every repository task. Contributors can close occurrence examples, reuse checked arithmetic reductions, and advance selected foundational packages with a concrete downstream consumer. Progress must track integrated obligations rather than code volume or the number of generated statements. Licenses, source commits, original authors, and upstream notices must accompany all transferred proofs.

Selected references

Barry Mazur, Modular curves and the Eisenstein ideal, Publications Mathématiques de l'IHÉS 47 (1977), 33–186. Part III, §5, Theorem (5.1), the following occurrence remark, and Corollary (5.2), printed p. 156. DOI; Numdam.
Daniel S. Kubert, Universal bounds on the torsion of elliptic curves, Proceedings of the London Mathematical Society (3) 33 (1976), 193–237, especially Chapter IV's rational parametrizations, cited by Mazur's occurrence remark. DOI.
Vasily Ilin and contributors, MazurTheorem, public Lean formalization. Repository; campaign source snapshot 54d43d8dda8a6fcf069cc02a815f850d762c5c0c. Existing uncommitted changes are excluded from source-snapshot claims.

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Combinatorics Discrete Geometry Linear Optimization·Captain: mikedeng1

A Counterexample to the Hirsch Conjecture I: A 43-Dimensional Polytope with 86 Facets Whose Diameter Exceeds 43Research Paper

Motivation

The Hirsch conjecture, stated by Warren M. Hirsch in a 1957 letter to George Dantzig, asserts that the combinatorial diameter of a $d$ -dimensional polytope with $n$ facets is at most $n-d$ . The diameter is the largest number of edge steps needed to walk between two vertices of the polytope. The question matters to linear optimization: every edge-following pivot rule of the simplex method, starting at a vertex of the feasible region, needs at least as many pivots as the graph distance to an optimal vertex. A polynomial bound on polytope diameters is a necessary condition for a strongly polynomial simplex method, which is still unknown.

Timeline of the conjecture before the counterexample:

1967, Klee and Walkup: the conjecture fails for unbounded polyhedra, and for bounded polytopes it is equivalent to the $d$ -step conjecture (the case $n=2d$ ). They also proved it for $n-d\le 5$ .
1992, Kalai and Kleitman: a quasi-polynomial upper bound $n^{\log_2 d+2}$ on the diameter.
2010, Santos: a counterexample in dimension 43 with 86 facets (arXiv:1006.2814, published in Ann. of Math. 176 (2012) 383–412). It is the subject of this mission.
2012, Matschke, Santos and Weibel: smaller 5-dimensional spindles, giving counterexamples in dimension 20 (arXiv:1202.4701).

The polynomial Hirsch conjecture, which asks only for a bound polynomial in $n$ and $d$ , remains open.

Setting

A polytope in $\mathbb R^d$ is written as $\mathrm{Hpoly}(a,b)=\{x:\langle a_i,x\rangle\le b_i,\ i=1,\dots,n\}$ . A facet presentation is one in which $\mathrm{Hpoly}(a,b)$ has nonempty interior and no inequality is redundant. For a bounded polytope it then has dimension $d$ , and its $n$ inequalities correspond one to one to its facets. A step between two vertices crosses an edge (a one-dimensional face). The polytope is non-Hirsch when its diameter exceeds $n-d$ .

A spindle (Santos, Definition 1.4) is a polytope with two distinguished vertices $u,v$ such that every facet contains exactly one of them. Its length is the graph distance from $u$ to $v$ . The polar objects are prismatoids: polytopes with two parallel facets $Q^+,Q^-$ containing all vertices, whose width is the dual-graph distance between $Q^+$ and $Q^-$ .

Santos's explicit object is the 5-prismatoid $Q$ whose vertices are the 48 rows of Table 1 of the paper: $1^+,\dots,24^+$ with $x_5=1$ and $1^-,\dots,24^-$ with $x_5=-1$ . Its polar $Q^\Delta=\{x\in\mathbb R^5:\langle r,x\rangle\le1\ \text{for each row } r\}$ is a spindle with apices $e_5$ and $-e_5$ . Table 2 of the paper lists the 322 facets of $Q$ , which are the 322 vertices of $Q^\Delta$ : $\pm e_5$ and the points $\frac1{c_0}(\pm c_1,\pm c_2,\pm c_3,\pm c_4,c_5)$ of twenty types $B,B',\dots,K,K'$ with sixteen sign patterns each. The symmetry group $\Sigma$ of $Q$ has order 64; its index-two subgroup $\Sigma^+$ preserves $Q^+$ and $Q^-$ .

Formalization targets

Goal: Corollary 1.7

\exists\, a:\mathrm{Fin}\,86\to\mathbb R^{43},\ b\in\mathbb R^{86}:\quad \mathrm{Hpoly}(a,b)\ \text{nonempty, bounded, a facet presentation, and}\ \operatorname{diam}\mathrm{Hpoly}(a,b)>43.

The goal fixes the dimension (43) and the number of facets (86), as the paper does. It does not assert the exact diameter (44, remarked on p. 24).

Milestones

§3, first bullet (polar form): $Q^\Delta$ is a bounded 48-facet spindle with apices $\pm e_5$ , the rows tight at $e_5$ being $1^+,\dots,24^+$ .
Theorem 4.1(1) (polar form): the 322 points of Table 2 are distinct vertices of $Q^\Delta$ ; the letter classes are $\Sigma^+$ -orbits and the six $\Sigma$ -orbits are A ∪ L, B ∪ K, C ∪ J, D ∪ I, E ∪ H, F ∪ G.
Theorem 4.1(2) (polar form): $Q^\Delta$ has no other vertices.
Theorem 3.1 (polar form): the distance from $e_5$ to $-e_5$ in the graph of $Q^\Delta$ is exactly 6.
Theorem 1.6: a 5-spindle with 48 facets, 322 vertices and length exactly six exists.
Inductive step of Theorem 2.6 (polar form): a $d$ -spindle with $n>2d$ facets and length at least $l$ yields a $(d+1)$ -spindle with $n+1$ facets and length at least $l+1$ .
Theorem 1.5 (strong $d$ -step theorem): a $d$ -spindle with $n$ facets and length at least $l$ yields an $(n-d)$ -spindle with $2n-2d$ facets and length at least $l+n-2d$ , non-Hirsch when $l>d$ .

Significance

Corollary 1.7 settles the Hirsch conjecture in the negative. Theorem 1.5 is the reusable part: it turns any spindle whose length exceeds its dimension into a counterexample. That made the later search for small spindles (Matschke–Santos–Weibel) a finite, low-dimensional problem. Theorem 1.6 is the concrete certificate. Its claims about Tables 1 and 2 are finite statements about explicit rational data.

The result is proved, and several pieces are already formalized on Prove2Me. Hirsch.santos_counterexample (Proved) states that some polytope violates the bound $n-d$ , with no dimension or facet count. Hirsch.strong_dstep_spindle (Proved) is the "in particular" clause of Theorem 1.5, counting inequalities rather than facets. Lemmas 2.2 and 2.4 of the paper are Proved in polar form (Hirsch.spindle_inward_row_push_graph, Hirsch.spindle_wedge_vertex_graph_projection), as is the bound $n\ge2d$ for spindles (Hirsch.spindle_n_ge_two_d). This mission adds the dimension-43, 86-facet statement with facets counted exactly, the general length bound $l+n-2d$ with the spindle structure preserved, and a machine-checkable account of Santos's specific spindle.

Difficulty

The obvious first idea for the 5-dimensional milestones is brute computation. This is feasible in principle but heavy in Lean. Showing that 322 given points are extreme requires exhibiting, for each, five linearly independent tight rows among 48. Showing that no other vertex exists requires a complete enumeration argument rather than a membership check. Exact distance 6 requires the full adjacency structure between the 322 vertices, not only a path.

For the general theorems, the difficulty is that the construction is a one-point suspension followed by a perturbation. Edges and facets of the new polytope must be controlled, with the face lattice changing only in a prescribed way. Diameter is not monotone under arbitrary perturbations, and in particular a walk can become shorter. Keeping the facet count exact (not merely the inequality count) adds an irredundancy argument at each step.

Formalization scope

Polytopes are H-presentations Hirsch.Hpoly a b with a : Fin n → EuclideanSpace ℝ (Fin d), from the platform definition Hirsch_model. "A $d$ -polytope with $n$ facets" is boundedness plus IsFacetPresentation: nonempty interior and every row irredundant. Without this predicate, padding with redundant rows would change the bound $n-d$ , and the statement would collapse to the platform's dimension-free Hirsch.santos_counterexample. Diameter and length use the platform's padded walks (Hirsch.DiamLE, Hirsch.Reach). "Length at least $L$ " means no walk of fewer than $L$ steps, and "length exactly $L$ " adds a walk of $L$ steps. The junk-valued Hirsch.gdist is never used.

All prismatoid statements of the paper (§3, Theorems 2.6, 3.1, 4.1) are stated for the polar spindle, which the paper licenses (pp. 6, 8, 9). Vertices of $Q$ become rows, facets of $Q$ become vertices, the width becomes the apex distance, and $Q^\pm$ become $\pm e_5$ . Natural-number expressions $n-d$ , $2n-2d$ , $l+(n-2d)$ are faithful because spindles have $n\ge2d$ . Theorem 1.5's "length $l$ " is read as "length at least $l$ ", which is equivalent by monotonicity. Tables 1 and 2 are written entry by entry in Lean in the page's order.

Useful contributions include proofs of the finite Table 1/Table 2 facts by certified computation, reusable lemmas on one-point suspensions and on facet presentations under perturbation, and a proof of Corollary 1.7 from milestones 5 and 7.

Selected references

F. Santos, A counterexample to the Hirsch conjecture, Ann. of Math. 176 (2012) 383–412; cited version arXiv:1006.2814v3. https://arxiv.org/abs/1006.2814 , https://doi.org/10.4007/annals.2012.176.1.7
V. Klee and D. W. Walkup, The d-step conjecture for polyhedra of dimension d < 6, Acta Math. 117 (1967) 53–78. https://doi.org/10.1007/BF02395040
G. Kalai and D. J. Kleitman, A quasi-polynomial bound for the diameter of graphs of polyhedra, Bull. Amer. Math. Soc. 26 (1992) 315–316. https://arxiv.org/abs/math/9204233
B. Matschke, F. Santos and C. Weibel, The width of five-dimensional prismatoids, Proc. London Math. Soc. 110 (2015) 647–672. https://arxiv.org/abs/1202.4701

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Linear Optimization Operations Research Optimization·Captain: Shuze Chen

Disjunctive Programming XII: Intersection Cuts, Generalized Intersection Cuts, and Lift-and-Project CutsTextbook

Motivation

Intersection cuts (Balas, 1971) are the founding construction of cutting-plane theory for mixed 0-1 and mixed-integer programs: given a fractional LP solution $\bar x$ and a convex region $S$ around it known to contain no feasible integer point, the hyperplane through the points where $S$ 's boundary meets the extreme rays of the LP cone at $\bar x$ cuts off $\bar x$ without cutting off any feasible solution. What makes intersection cuts foundational rather than merely one technique among many is a completeness question: do intersection cuts, iterated over every choice of cutting region, exhaust the strongest possible cuts — the facets of the integer hull itself — or only some weaker subclass? Balas answered this affirmatively for standard intersection cuts (those derived from convex sets free of feasible integer points, as originally defined), while a narrower, more recently popular variant restricted to lattice-free sets provably falls short of this completeness (E. Balas, Intersection Cuts — A New Type of Cutting Planes for Integer Programming, Operations Research 19 (1971), 19–39, https://doi.org/10.1287/opre.19.1.19). This mission formalizes that completeness theorem, together with a companion pair of results (Balas and Kis, 2016) pinning down exactly when a lift-and-project cut — a strictly more general cutting-plane construction from an arbitrary disjunction — coincides with a standard intersection cut, and what happens when it provably does not (E. Balas and T. Kis, On the relationship between standard intersection cuts, lift-and-project cuts, and generalized intersection cuts, Mathematical Programming A 160 (2016), 85–114, https://doi.org/10.1007/s10107-015-0975-1).

Setting

Fix a finite index set $\iota$ (structural and surplus variables of an LP relaxation together), a basic index set $I \subseteq \iota$ and a nonbasic (cobasis) set $J$ , with optimal simplex-tableau coefficients $\bar a_{ij}$ for $i \in I$ , $j \in J$ . The extreme ray of the LP cone $C(J)$ at a basic solution $\bar x$ associated with $j \in J$ has direction $r^j$ with $r^j_i = -\bar a_{ij}$ for $i \in I$ , $r^j_j = 1$ , and $r^j_i = 0$ otherwise; $C(J)$ itself is the cone with apex $\bar x$ generated by these $n = |J|$ rays. A convex set $S$ is $P_I$ -free at $\bar x$ if $\bar x$ lies in $\mathrm{int}\, S$ and $\mathrm{int}\, S$ contains no point of the mixed-integer feasible set $P_I$ . The standard intersection cut (SIC) derived from such an $S$ is $\sum_{j\in J} \tfrac{1}{\lambda_j} x_j \ge 1$ , where $\lambda_j$ is the largest $t \ge 0$ with $\bar x - t r^j \in S$ .

The corner polyhedron $\mathrm{corner}(J)$ is the convex hull of the integer points contained in $C(J)$ ; it satisfies $C(J) \supset \mathrm{corner}(J) \supset \mathrm{conv}(P_I)$ . A set $F$ is a facet of a polyhedron $Q$ if it is a proper extreme subset of $Q$ of affine dimension exactly $\dim(Q) - 1$ .

For a lift-and-project cut, fix $P := \{x : \tilde A x \ge \tilde b\}$ and a family of inequalities $d^t x \ge d^t_0$ , $t \in T$ , presenting a $P_I$ -free polyhedron $S := \{x : d^t x \le d^t_0,\ t\in T\}$ . The associated cut-generating LP (CGLP) constraint set (11.6) is

\alpha - u^t \tilde A - u^t_0 d^t = 0, \qquad -\beta + u^t \tilde b + u^t_0 d^t_0 = 0 \ (t\in T), \qquad \textstyle\sum_{t\in T}(u^t e + u^t_0) = 1, \qquad u^t, u^t_0 \ge 0,

whose feasible solutions $(\alpha,\beta,\{u^t,u^t_0\})$ correspond to valid lift-and-project (L&P) cuts $\alpha x \ge \beta$ for the disjunction built from $P$ and the terms $d^t x \ge d^t_0$ . An inequality $\gamma^1 x \ge \gamma^1_0$ dominates $\gamma^2 x \ge \gamma^2_0$ on $P$ if every $x \in P$ satisfying the first also satisfies the second.

Formalization targets

Theorem 11.2 (goal). Every facet $F$ of $\mathrm{conv}(P_I)$ , defined by $\varphi x \ge \varphi_0$ and cutting off some vertex $v$ of $P$ (i.e. $\varphi v < \varphi_0$ ), is realized exactly by the standard intersection cut derived at $v$ from $T := \{x : \varphi x \le \varphi_0\}$ :

T \text{ is } P_I\text{-free at } v, \qquad \Big\{x : 1 \le \textstyle\sum_{j\in J} \tfrac{1}{\lambda_j} x_j\Big\} = \{x : \varphi_0 \le \varphi x\}.

Corollary 11.3. Every vertex of a corner polyhedron not already in $\mathrm{conv}(P_I)$ is cut off by some standard intersection cut — the same completeness claim restated at the level of individual excluded vertices rather than facets.

Theorem 11.9. A sufficient condition for an L&P cut to reduce to a standard intersection cut: if a basic feasible CGLP solution's multipliers $u^t$ are all supported on a single common nonsingular cobasis $\iota$ , then

\{x : \beta \le \alpha x\} = \{x : 1 \le \textstyle\sum_j \pi_j\, s_j(x)\}

for the intersection cut with coefficients $\pi_j := \max_{t} \pi^t_j$ , $\pi^t_j := d^t(-\bar a_j)/(d^t_0 - d^t \bar a_0)$ , expressed via the surplus values $s_j$ at $\iota$ 's rows.

Theorem 11.11. When Theorem 11.9's condition fails — even after every positive rescaling of the solution — no intersection cut from $S$ is equivalent to the L&P cut; and when the solution additionally uniquely minimizes the CGLP objective, the L&P cut is strictly better than, and dominated by none of, every intersection cut from $S$ .

The targets move from the completeness statement itself (11.2, its vertex-level restatement 11.3) to the mechanism explaining why completeness holds in general: a sufficient condition for literal coincidence (11.9), and a proof that failure of that condition is never fatal to completeness because the L&P cut remains at least as strong, in a precise domination sense (11.11).

Significance

Theorem 11.2 is the theoretical justification for standard intersection cuts as a complete cutting plane paradigm: no facet of the integer hull is out of reach of some choice of $P_I$ -free cutting region, in sharp contrast to the restricted (lattice-free) variant that dominates the modern multi-row cut literature but is provably incomplete in this sense. Theorems 11.9 and 11.11 locate lift-and-project cuts precisely relative to this complete family: L&P cuts specialize exactly to intersection cuts under an explicit, checkable structural condition on the CGLP solution, and strictly dominate the intersection-cut family whenever that condition cannot be met — which is what makes lift-and-project the strictly more general (and, on general non-split disjunctions, strictly more powerful) construction.

Both directions are proved in the source (Balas 1971 for Theorem 11.2; Balas and Kis 2016 for Theorems 11.9 and 11.11) but have no counterpart on this platform: nothing existing treats intersection cuts, corner polyhedra, cut-generating LPs, or the correspondence between these two cutting-plane families. This mission produces the first Lean statements of all four.

Difficulty

The obvious shortcut for Theorem 11.2 is to treat "cuts off a vertex" and "is $P_I$ -free" as producing merely some valid cut, and stop there — Theorem 1.1 already guarantees that much. The actual content is the equality: the specific intersection cut constructed from the halfspace $T$ does not just happen to be valid, it reconstructs $\varphi$ itself, coefficient for coefficient, because $T$ is a single hyperplane so every one of the LP cone's $n$ extreme rays exits it through the same boundary. Losing sight of this collapses the theorem into a restatement of Theorem 1.1 with no new content.

For Theorem 11.11, the difficulty is that "no intersection cut from $S$ is equivalent" must survive scaling: a naive argument might rule out one specific $(\alpha,\beta)$ -representative satisfying Theorem 11.9's condition while missing that a positive rescaling of the same cut could still satisfy it under a different multiplier vector. The theorem's hypothesis is deliberately built to close this gap by quantifying over every positive scalar and every feasible solution realizing the rescaled pair, not just the given one.

Formalization scope

The ambient space is a generic finite index type ι for Theorem 11.2/Corollary 11.3 (structural and surplus variables together, matching 01-intro-duality's own convention for the intersection- cut apparatus), and Fin n → ℝ for the CGLP-based Theorems 11.9/11.11, matching the series' default. P_I is left as an abstract parameter throughout (never expanded into an explicit integrality predicate on a specific coordinate subset for Theorem 11.2, matching 01-intro- duality's own treatment), except in the corner-polyhedron definitions, where it is made concrete via a coordinate set Nprime since the corollary's statement depends on it directly. "Facet" and "extreme ray" are restated from 02b-polarity's conventions (affine dimension via Module.finrank of vectorSpan; IsExtreme) rather than reinvented, since Chapter 2 already pins these down precisely for this series. "Basic feasible solution" to the CGLP in Theorem 11.9 is captured entirely by the theorem's own submatrix-support condition, not through a separate, independently-derived basicness predicate — the book's own proof uses no other property of basicness, so adding one would be unused decoration, not additional fidelity. Cut equivalence throughout is formalized as exact set equality of the two halfspaces, matching the series' established convention (e.g. 10-split-closure's Theorem 10.1) for what "equivalent cuts" means.

This mission depends on no other chunk's Lean definitions: the intersection-cut apparatus (extremeRay, PIFree) is restated from 01-intro-duality, the facet apparatus (PolyDim, IsFacet) from 02b-polarity, and the tableau apparatus (Ahat, Bhat, Abar, Abar0, SurplusM) from 08-cut-correspondence/09-simplex-tableau/10-split-closure, per the series convention against importing another draft mission's definitions while chunks are drafted concurrently. A trivializing formalization to rule out explicitly: collapsing Theorem 11.2 to "some intersection cut is valid and cuts off $v$ " (already implied by Theorem 1.1 alone) rather than the literal set-equality with the facet's own inequality, which is this theorem's actual content.

Selected references

E. Balas, Intersection Cuts — A New Type of Cutting Planes for Integer Programming, Operations Research 19 (1971), 19–39. https://doi.org/10.1287/opre.19.1.19
E. Balas and T. Kis, On the relationship between standard intersection cuts, lift-and-project cuts, and generalized intersection cuts, Mathematical Programming A 160 (2016), 85–114. https://doi.org/10.1007/s10107-015-0975-1
E. Balas and M. Perregaard, Generalized intersection cuts and a new cut generating paradigm, Mathematical Programming A 137 (2013), 19–35. https://doi.org/10.1007/s10107-011-0483-x
E. Balas, Disjunctive Programming, Springer, 2018, Chapter 11, §11.1–11.5. https://doi.org/10.1007/978-3-030-00148-3

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Linear Optimization Operations Research Optimization·Captain: Shuze Chen

Disjunctive Programming IX: The Correspondence Between Lift-and-Project Cuts and Simple Disjunctive CutsTextbook

Motivation

Chapter 6 built lift-and-project (L&P) cuts from a cut-generating LP, and Chapter 7 surveyed alternative nonlinear constructions reaching the same integer hull. This chapter asks a sharper question: how do L&P cuts relate, coefficient for coefficient, to older, more classical cutting planes — simple disjunctive cuts and mixed integer Gomory cuts, derived directly from a simplex tableau rather than from an auxiliary LP? The answer is an exact correspondence: every L&P cut from a basic solution of the cut-generating LP is equivalent to a specific simple disjunctive cut from a specific tableau basis, and conversely. This correspondence is not merely of theoretical interest — it converts a question about an infinite family of cuts into a finite, countable one (bases of a linear system), and it is what lets the chapter's capstone result, Theorem 8.7, establish a uniform rank bound of $p$ (the number of 0-1 variables) across four cut families at once, by proving it for one and transporting the proof to the other three.

Setting

$(CGLP)_k$ (eq. (8.1)) is the cut-generating LP for the disjunction $-x_k \ge 0 \lor x_k \ge 1$ , with an added normalization constraint $ue+u_0+ve+v_0=1$ that makes its feasible polytope bounded — so a "basic solution" can be identified with an extreme point of that polytope. Given a basic solution with $u_0,v_0>0$ and basic $u/v$ -components indexed by $M_1,M_2$ (Lemma 8.1-8.2), the $n\times n$ submatrix $\hat A$ of $\tilde A$ indexed by $J:=M_1\cup M_2$ is nonsingular, giving a simplex tableau in which $x_k$ is expressed as $x_k = \bar a_{k0} - \sum_{j\in J}\bar a_{kj}x_j$ (eq. (8.5)). The simple disjunctive cut from $x_k\le 0 \lor x_k\ge 1$ applied to this row has coefficients $\pi_j := \max\{\pi^1_j,\pi^2_j\}$ , $\pi_0 := \bar a_{k0}(1-\bar a_{k0})$ (eq. (8.7)-(8.8)).

Formalization targets

Theorem 8.7 (goal) — a uniform rank bound across four cut families

The rank of the LP relaxation $P$ with respect to (a) unstrengthened L&P cuts, (b) simple disjunctive cuts, (c) strengthened L&P cuts, (d) mixed integer Gomory cuts (equivalently, strengthened simple disjunctive cuts) is at most $p$ , the number of 0-1 variables.

The chain of results building toward it

Lemma 8.1 (basicness forces $u_0,v_0>0$ ), Lemma 8.2 (a basic solution's index sets give a nonsingular submatrix), Lemma 8.3 ( $0<\bar a_{k0}<1$ ), Theorem 8.4A (a basic L&P cut equals a simple disjunctive cut), Theorem 8.4B (the converse: every simple disjunctive cut from a valid basis equals some basic L&P cut), and Theorem 8.5 (the same correspondence, strengthened).

Significance

The results themselves. Theorems 8.4A/8.4B are, in the book's own words, an "exact correspondence between lift-and-project cuts for a mixed 0-1 program and earlier cuts from the literature" — placing L&P cuts, simple disjunctive cuts, and (via Theorem 8.5) mixed integer Gomory cuts on the same logical footing, all generated by choosing a basis of one underlying linear system. Theorem 8.7 is the payoff: a single uniform bound covering four cut families that the literature had previously bounded (if at all) by separate arguments, and by contrast to the unbounded rank of pure-integer fractional Gomory cuts, exhibiting a case where the mixed 0-1 structure yields much stronger guarantees.

Formalizing it. No object in this mission exists on the platform prior to it or in Mathlib. This mission restates 06-lift-project-cuts's $(CGLP)$ apparatus and Theorem 6.4's strengthened cut formula locally, per the series convention that a draft mission cannot import another draft mission's definitions, adapted throughout to this chapter's normalized $(CGLP)_k$ and its disjunction on a single fixed coordinate $k$ .

Difficulty

Formalizing "basic solution" required a genuine choice: unlike Chapter 6, $(CGLP)_k$ 's normalization constraint makes its feasible set a bounded polytope, so this mission identifies "basic solution" with an extreme point of that polytope (Set.extremePoints) for Lemma 8.1 (whose own statement has no reference to specific index sets), while Lemmas 8.2 onward take the basic index sets $M_1,M_2$ directly as hypothesis data, matching how those theorems are themselves phrased ("let the basic components... be indexed by $M_1$ and $M_2$ "). Theorem 8.7's rank bound required designing one generic HasRankAtMost predicate, parametrized by an abstract cut-closure operator, applicable uniformly to all four families — mirroring the book's own proof structure, which establishes the bound for one family and transports it to the other three via Theorems 8.4A/8.4B and 8.5, rather than arguing each part from scratch.

Formalization scope

The row/variable identification gap (see MODERATION_NOTES.md). The book's own eq. (8.4)- (8.5) identifies certain rows of the augmented, $m+p+n$ -row matrix $\tilde A$ (those that are bound constraints $x_j\ge 0$ ) with the variables they bound, so that a chosen nonbasic row set $J$ doubles as a set of "nonbasic variables." This mission's abstract row type does not track that identification (matching the abstraction already used throughout 06-lift-project-cuts and 07-higher-dim): Surplus instead defines the tableau row's nonbasic quantities directly as the slack expression $s_j := (\tilde Ax)_j - \tilde b_j$ , a genuine affine function of $x$ for every row, which reduces to $x_j$ itself exactly when row $j$ is that bound constraint — mathematically equivalent to the book's own substitution, stated without needing the row-to-variable lookup. Eq. (8.10)'s " $j\in J\cap N'$ " strengthening-eligibility test has the same gap; this mission takes the row-positions eligible for strengthening as an explicit Finset parameter rather than deriving membership from row identity.

Corollary 8.6 is out of scope for this mission — see HARD.md. Its facet-counting bound depends on the same row/variable identification (the printed bound is $\binom{m+p+n-1}{n}$ , excluding row $k$ specifically because it is $x_k$ 's own bound row) and would additionally require a general notion of "number of facets of a polyhedron" that this mission's abstraction, and Mathlib, do not provide; it does not feed Theorem 8.7's own proof, which cites only Theorems 8.4A/8.4B and 8.5.

Theorem 8.7 is stated via one generic HasRankAtMost predicate applied to SplitConvexify (part a) and three closure operators (SimpleDisjClosureOfSet, StrengthenedLPClosureOfSet, MIGClosureOfSet, parts b-d) defined by intersecting a represented polyhedron with every cut of the corresponding family, then lifted to bare sets by quantifying over every linear representation — since, unlike the split-convexification closure, these three cut families are defined via an explicit basis or CGLP solution and so genuinely need some concrete representation of the current polyhedron at each step of the recursion (the same representation-dependence Theorem 7.5's Lovász-Schrijver iteration required in 07-higher-dim).

Selected references

E. Balas, Disjunctive Programming, Springer, 2018. DOI: 10.1007/978-3-030-00148-3, Chapter 8.
E. Balas, M. Perregaard, A precise correspondence between lift-and-project cuts, simple disjunctive cuts, and mixed integer Gomory cuts for 0-1 programming, Mathematical Programming B 94 (2003), 221–245 (cited in the text as [33], the origin of Lemma 8.2 and Theorems 8.4A/8.4B).
E. Balas, M. Perregaard, Lift-and-project for mixed 0-1 programming: recent progress, Discrete Applied Mathematics 123 (2002), 129–154 (cited in the text as [32], the origin of Theorem 8.5's strengthened-cut coefficient identification).
F. Eisenbrand, A. Schulz, Bounds on the Chvátal rank of polytopes in the 0-1 cube, in Integer Programming and Combinatorial Optimization (IPCO 7), LNCS 1610 (1999), 137–150 (cited in the text as [73], the source of the unbounded pure-integer Gomory rank result this chapter's Theorem 8.7 contrasts with).

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Operations Research Probability Statistics+1·Captain: mikedeng1

Weak Convergence and Optimal Scaling of Random Walk Metropolis Algorithms: Langevin Diffusion Limit of the First CoordinateResearch Paper

Motivation

The random walk Metropolis algorithm is one of the most widely used Markov chain Monte Carlo methods for sampling from a density known up to a constant. Its one tuning parameter is the variance of the Gaussian proposal. If the variance is too small, the chain accepts almost every move but barely moves. If it is too large, it proposes long jumps that are almost always rejected. Practitioners need a rule for choosing it, and the rule has to work in high dimension, where both failure modes are severe.

Roberts, Gelman and Gilks (Ann. Appl. Probab. 7(1), 1997) gave the first rigorous answer for product targets. As the dimension grows, one coordinate of the suitably speeded-up chain converges to a Langevin diffusion. The speed of that diffusion is an explicit function of the proposal scale, and maximising it gives the rule "tune the proposal so that about 23% of proposals are accepted". This rule, and the 2.38/√I scaling behind it, is now standard advice in applied Bayesian statistics.

Setting

Let $f:\mathbb R\to\mathbb R$ be a target density: positive, $C^2$ , integrating to one, with $f'/f$ Lipschitz, and satisfying the moment conditions (A1) $\mathbb E_f[(f'/f)^8]<\infty$ and (A2) $\mathbb E_f[(f''/f)^4]<\infty$ . Here $\mathbb E_f[g(X)]=\int g(x)f(x)\,dx$ . In dimension $n\ge2$ the target is the product density $\pi_n(x)=\prod_{i=1}^n f(x_i)$ on $\mathbb R^n$ .

Fix a scale $l>0$ and set $\sigma_n^2=l^2/(n-1)$ . The random walk Metropolis chain $X^n=(X^n_0,X^n_1,\dots)$ moves as follows. From $X^n_{m-1}$ it proposes $Y\sim N(X^n_{m-1},\sigma_n^2I_n)$ . It sets $X^n_m=Y$ with probability $\alpha(X^n_{m-1},Y)=1\wedge\pi_n(Y)/\pi_n(X^n_{m-1})$ , and $X^n_m=X^n_{m-1}$ otherwise. The chain starts from $\pi_n$ , which is stationary for it. The speeded-up first coordinate is $U^n_t=X^n_{\lfloor nt\rfloor,1}$ for $t\ge0$ .

Let $\Phi$ be the standard normal distribution function, and define the roughness $I=\mathbb E_f[(f'(X)/f(X))^2]$ . The speed and the limiting acceptance rate are

h(l)=2l^2\,\Phi\!\Big(-\frac{l\sqrt I}{2}\Big),\qquad a(l)=2\,\Phi\!\Big(-\frac{l\sqrt I}{2}\Big).

The Langevin generator is $GV(x)=h(l)\big[\tfrac12V''(x)+\tfrac12(\log f)'(x)V'(x)\big]$ . It generates the Langevin diffusion $dU_t=h(l)^{1/2}dB_t+h(l)\frac{f'(U_t)}{2f(U_t)}dt$ .

Formalization targets

Goal: Theorem 1.1

As $n\to\infty$ ,

U^n\Rightarrow U,

where $\Rightarrow$ denotes weak convergence in the Skorokhod topology, $U_0$ has density $f$ , and $U$ is the Langevin diffusion with speed $h(l)$ . The limit is asserted to exist. No constants appear in the statement beyond those the model defines.

Milestones: the proof

Lemma 2.1. The stationary chain stays in the sets $F_n=\{|R_n-I|<n^{-1/8}\}\cap\{|S_n-I|<n^{-1/8}\}$ up to time $t$ with probability tending to one. Here $R_n$ and $S_n$ are the empirical averages of $((\log f)')^2$ and $-(\log f)''$ over coordinates $2,\dots,n$ .
Proposition 2.2. $|1\wedge e^x-1\wedge e^y|\le|x-y|$ .
Lemma 2.3. $\sup_{x\in F_n}\mathbb E|W_n|\to0$ , where $W_n$ is the second-order part of the log acceptance ratio.
Proposition 2.4. $\mathbb E[1\wedge e^A]=\Phi(\mu/\sigma)+e^{\mu+\sigma^2/2}\Phi(-\sigma-\mu/\sigma)$ for $A\sim N(\mu,\sigma^2)$ .
Lemma 2.5. $\limsup_n\sup_{x_1}n|\mathbb E[V(Y_1)-V(x_1)]|<\infty$ for $V\in C_c^\infty$ .
Lemma 2.6. The discrete generator $G_nV(x)=n\,\mathbb E[(V(Y)-V(x))\alpha(x,Y)]$ converges to $GV$ uniformly on $F_n$ , for $V\in C_c^\infty$ a function of the first coordinate (stated with bounded $(\log f)'''$ , the assumption its proof uses).

Milestones: the optimal-scaling corollary

Corollary 1.2 (i). $a_n(l)=\iint\pi_n(x)\alpha(x,y)q_n(x,y)\,dx\,dy\to a(l)$ .
Corollary 1.2 (ii). $h$ is maximised at $\hat l=2.38/\sqrt I$ , with $a(\hat l)=0.23$ and $h(\hat l)=1.3/I$ , to the printed precision.

Significance

Theorem 1.1 shows that, run for $n$ times as many steps, the chain in dimension $n$ looks like a fixed one-dimensional diffusion. The algorithm's cost therefore grows linearly in dimension, and its efficiency is measured by the single number $h(l)$ . Corollary 1.2 turns this into the 0.234 acceptance-rate heuristic and the $2.38/\sqrt I$ scaling. The same diffusion-limit method has since been applied to the Metropolis-adjusted Langevin algorithm, to Hamiltonian Monte Carlo and to non-product targets.

The theorem is proved on paper; it has no machine-checked proof. A formal development would give the first verified diffusion limit of an MCMC algorithm. It would also yield reusable components: the Metropolis chain on $\mathbb R^n$ as a measurable random mapping, a martingale-problem characterisation of one-dimensional diffusions, and a Gaussian computation (Proposition 2.4) that recurs throughout the optimal-scaling literature.

Difficulty

The obvious approach, a Taylor expansion of the log acceptance ratio, gives a sum of $n-1$ terms of size $1/n$ . That sum does not concentrate uniformly over the state space, since the coordinates $2,\dots,n$ are arbitrary. The expansion is controlled only on the sets $F_n$ , where the empirical averages $R_n$ and $S_n$ are close to $I$ . The limit therefore holds only after showing that the chain rarely leaves $F_n$ over a time horizon of $nt$ steps. A pointwise law of large numbers is not enough for that, because the bound has to survive a union over $nt$ steps. Passing from generator convergence on a set of high probability to weak convergence of processes requires the Ethier–Kurtz convergence theory: a core for the limit generator, and convergence of processes that are not themselves Markov. None of this theory is in Mathlib.

Formalization scope

All declarations live in the namespace Roberts1997.RWM. The following conventions are fixed.

Vectors are Fin n → ℝ, and the paper's first coordinate $x_1$ is index 0. Its coordinates $2,\dots,n$ are the indices i ≠ 0.
$\sigma_n^2=l^2/(n-1)$ is computed in $\mathbb R$ . All statements concern $n\ge2$ or large $n$ .
$l>0$ is assumed. The paper leaves it implicit, but $h(-l)\ne h(l)$ .
" $f$ is a density" is read as $\int f=1$ . The moment conditions are read as integrability of $(f'/f)^8f$ and $(f''/f)^4f$ . The standing assumption " $f'/f$ is Lipschitz" (p. 111) is carried by every statement.
The chain is built as a random mapping on an explicit probability space: $x_0\sim\pi_n$ , with i.i.d. standard normal innovations and uniform acceptance variables. Theorem 1.1's initial condition (components i.i.d. $f$ , shared across dimensions) is read as "the $n$ -th chain starts from $\pi_n$ ", since weak convergence depends only on the law of each $U^n$ .
" $U$ satisfies the Langevin SDE" is read as "the law of $U$ solves the martingale problem for $G$ on $C_c^\infty$ , with continuous paths and initial law $f(x)\,dx$ ". This is equivalent by Ethier–Kurtz (1986), Ch. 5, Prop. 3.1 and Thm 3.3, and follows the platform definition EthierKurtz_IsContinuousDiffusionLaw.
" $U^n\Rightarrow U$ " is read as the existence of an almost-sure coupling in which càdlàg copies of the $U^n$ converge to a continuous Langevin path uniformly on compact time intervals. For a continuous limit this is equivalent to weak convergence in $D_{\mathbb R}[0,\infty)$ , by Skorokhod's representation theorem and Ethier–Kurtz Ch. 3, Thm 1.8, Prop. 5.3 and Prop. 7.1. It follows the platform encoding of Ethier–Kurtz Theorem 7.4.1.
" $\sup\to0$ " and " $\limsup\sup<\infty$ " are stated as eventual uniform bounds. This avoids real suprema, whose value on an unbounded set is a default.
In Lemma 2.6, "as $d\to\infty$ " is a misprint for $n\to\infty$ , and " $2f(Ut)$ " in (1.2) is read as $2f(U_t)$ .
Corollary 1.2 (ii) is stated for an arbitrary constant $I>0$ . "To two decimal places" is read as explicit rounding intervals: $1.3$ is read to one decimal, and all maximisers over $l>0$ are covered.

The goal cannot be satisfied trivially. The limit law $Q$ must exist, and it must be a probability measure whose initial marginal is $f(x)\,dx$ , so the zero measure is excluded. The coupled copies must carry exactly the laws of the paths $U^n$ , not an arbitrary process with the same one-time marginals.

The statements carry the paper's hypotheses, with one exception. The printed proof of Lemma 2.6 bounds $\sup_z|(\log f)'''(z)|$ , which Theorem 1.1 does not assume, and under $C^2$ alone the uniform convergence over $F_n$ claimed by Lemma 2.6 fails (narrow spikes of $(\log f)''$ far out let a positive fraction of the coordinates shift the log acceptance ratio by a constant while $R_n$ and $S_n$ stay close to $I$ ). Lemma 2.6 is therefore stated with the proof's own assumption, $f\in C^3$ with $(\log f)'''$ bounded, named as an addition. Theorem 1.1 and the other results keep the paper's hypotheses.

A complete development needs several pieces not yet available: path spaces and the Skorokhod topology (or the coupling reading), the martingale problem and its well-posedness for Lipschitz drift, and the Ethier–Kurtz theorem on convergence of generators on sets of high probability. Proofs of the Gaussian milestones (Propositions 2.2 and 2.4, Lemma 2.5) and of Corollary 1.2 (ii) are independent of this infrastructure and are welcome contributions.

Selected references

G. O. Roberts, A. Gelman, W. R. Gilks, Weak convergence and optimal scaling of random walk Metropolis algorithms, Ann. Appl. Probab. 7(1), 110–120, 1997. https://doi.org/10.1214/aoap/1034625254
S. N. Ethier, T. G. Kurtz, Markov Processes: Characterization and Convergence, Wiley, 1986. https://doi.org/10.1002/9780470316658
A. Gelman, G. O. Roberts, W. R. Gilks, Efficient Metropolis jumping rules, Bayesian Statistics 5, Oxford University Press, 599–607, 1996.
G. O. Roberts, J. S. Rosenthal, Optimal scaling for various Metropolis–Hastings algorithms, Statistical Science 16(4), 351–367, 2001. https://doi.org/10.1214/ss/1015346320

15 thms4 active usersReviewed

Functional Analysis Probability·Captain: mikedeng1

Markov Processes: Characterization and Convergence 15: Diffusions in smooth bounded regionsTextbook

Why smooth-region boundary diffusions matter

Diffusion models in a bounded region need a rule for what happens when a path reaches the boundary. Two standard mechanisms are absorption, where the boundary kills the generator contribution, and oblique reflection, where a prescribed vector field pushes the process back into the region. Ethier and Kurtz treat these mechanisms as neighboring variants of the same uniformly elliptic model in Chapter 8, Section 1 of Markov Processes: Characterization and Convergence. The distinction is structural: the interior differential operator is shared, but its admissible generator graph changes with the boundary condition. This mission formalizes Theorems 1.4 and 1.5, retaining oblique reflection as the capstone and absorbed diffusion as the required related result.

The setting

Let $\Omega\subset\mathbb R^d$ be bounded, open, and connected, with $d\ge 2$ . Its boundary is locally represented in orthogonal coordinates as the graph of a scalar function. The formal predicate C²,μ boundary regularity requires one positive chart radius valid at every boundary point, a connected local boundary patch, and a graphing function whose second partial derivatives satisfy the source's componentwise Hölder oscillation condition. The exponent obeys $0<\mu\le 1$ .

The diffusion coefficients are a symmetric positive-semidefinite matrix field $a(x)$ and a drift field $b(x)$ . Their entries satisfy the same local componentwise Hölder convention. Uniform ellipticity means that one $\varepsilon>0$ satisfies

\varepsilon\le \sum_{i,j}\theta_i a_{ij}(x)\theta_j

for every $x\in\Omega$ and every unit vector $\theta$ . For smooth $f$ , the interior operator is

Gf(x)=\frac12\sum_{i,j}a_{ij}(x)\,\partial_{ij}f(x) +Df(x)[b(x)].

Functions live on the compact closure $\overline\Omega$ as bounded continuous functions. Their ambient extensions are differentiated only at interior points. The second coordinate of each graph is itself continuous on the closure, so it records the boundary trace of $Gf$ rather than assigning an arbitrary value after differentiation.

Formalization targets

Goal: obliquely reflected diffusion generation

Theorem 1.5 adds a reflection field $c$ whose components have C¹,μ boundary regularity. An outward unit normal $n(x)$ is oriented by a local defining function that is negative precisely inside $\Omega$ . Uniform obliqueness is the global lower bound

\varepsilon\le c(x)\cdot n(x),\qquad x\in\partial\Omega,

for one positive $\varepsilon$ . The reflected graph requires a continuous derivative trace $J$ that agrees with $Df$ in the interior and satisfies

J_x(c(x))=0,\qquad x\in\partial\Omega.

The target asserts that the uniform closure of this graph is single-valued and is exactly the full generator of a positive strongly continuous contraction semigroup preserving the constant function one. The generator condition is a biconditional derivative limit, so it identifies the complete domain rather than only a convenient operator restriction.

Related target: absorbed diffusion generation

Theorem 1.4 keeps the same smooth-region and ellipticity assumptions but uses the absorbed graph. Its continuous operator trace satisfies $Gf=0$ on the boundary. It has the same full generation conclusion: graph single-valuedness, a positive strongly continuous contraction semigroup, preservation of one, and exact identification of the generator. The absorbed zero trace is not imported into the reflected theorem, and the oblique derivative condition is not imposed on the absorbed graph.

Significance

These results connect a local elliptic expression and a geometric boundary condition to a global Markov evolution on continuous functions over the closed region. The conclusions contain more than existence of a semigroup: they determine the whole infinitesimal generator, ensure positivity and contraction, and retain the conservative convention used by Ethier and Kurtz. The paired statements make the effect of the boundary mechanism explicit while holding the interior model fixed.

The formal contribution is a machine-checkable statement layer, not a proof of the source theorems. It records the componentwise Hölder convention, uniform boundary charts, ellipticity, normal orientation, continuous derivative trace, the two distinct generator graphs, and every semigroup clause. These definitions can support later work on reflected processes, elliptic boundary problems, and martingale formulations without rebuilding the geometric conventions.

Where the difficulty lies

The main difficulty is simultaneous control of interior regularity, boundary geometry, and the closed generator domain. Pointwise obliqueness is insufficient: the source requires a uniform positive lower bound over the whole compact boundary. Likewise, merely writing $Df(c)=0$ for an arbitrary ambient extension would not supply a well-defined boundary derivative. The graph therefore carries a continuous derivative trace agreeing with the interior derivative. Replacing the biconditional generator equality by a one-way inclusion, or silently imposing the absorbed zero trace on the reflected graph, would weaken or change the source result.

Formalization scope and conventions

The state space is EuclideanSpace ℝ (Fin (n + 1)) with 2 ≤ n + 1. Connectedness supplies nonemptiness. Matrix positive semidefiniteness supplies symmetry and nonnegative quadratic forms; a separate hypothesis gives strict uniform ellipticity. Local oscillation bounds apply within connected components of small intersections and do not compare different components.

The absorbed and reflected graphs use bounded continuous functions on closure Ω. The arbitrary zero extension outside the closure is never differentiated at a boundary point. The reflected boundary equation uses a continuous field of linear derivative maps, while the absorbed equation uses the continuous second graph coordinate. No vacuous region, pointwise-only ellipticity or obliqueness, weakened generator inclusion, proof-only theorem dependency, or synthetic theorem combining the two boundary mechanisms is admitted.

Selected references

Stewart N. Ethier and Thomas G. Kurtz, Markov Processes: Characterization and Convergence, Wiley, 1986, Chapter 8, Section 1, Theorems 1.4–1.5 and equations (1.13)–(1.20). Wiley DOI
Stewart N. Ethier and Thomas G. Kurtz, same volume, Chapter 1 for strongly continuous contraction semigroups and generators, and Chapter 4 for the conservative Feller convention. Wiley DOI

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Machine Learning Probability Statistics+1·Captain: mikedeng1

Foundations of Machine Learning VIII: Multi-Class Classification and the Margin BoundTextbook

Motivation

Every generalization bound in chapters 2-5 is for binary classification. Most real-world classification problems have more than two classes, and the number of classes can itself be in the hundreds or thousands (topic classification, speech recognition). Chapter 9 extends the margin-based generalization theory of chapter 5 (SVMs) to this multi-class, mono-label setting, using the same Rademacher-complexity machinery as chunk 03-rademacher-vc, but with a new combinatorial ingredient — bounding the Rademacher complexity of a family built by taking a pointwise maximum over several hypothesis sets — needed because a multi-class prediction is itself an argmax over per-class scores.

Setting

A multi-class hypothesis is a scoring function $h:X\times Y\to\mathbb R$ with $Y=\{1,\dots,k\}$ (mono-label case); the predicted label is $\arg\max_y h(x,y)$ , and the margin $\rho_h(x,y)=h(x,y)-\max_{y'\ne y}h(x,y')$ (p. 215) is negative exactly when $h$ misclassifies $(x,y)$ . The empirical margin loss $\hat R_{S,\rho}(h)$ (Eq. 9.5) uses the same margin-loss function $\Phi_\rho$ (Definition 5.5) as chunk 05-svm, restated locally here. $\Pi_1(H) = \{x\mapsto h(x,y):y\in Y,h\in H\}$ (p. 217) projects a multi-class hypothesis set onto ordinary real-valued functions on $X$ — the object the chapter's Rademacher-complexity bound actually controls, since $H\subseteq\mathbb R^{X\times Y}$ has no norm of its own without such a projection. Lemma 9.1 is a purely combinatorial tool: the empirical Rademacher complexity of a family built by taking the pointwise max over $l$ hypothesis sets is bounded by the sum of their individual empirical Rademacher complexities — used to control the argmax structure of a multi-class prediction. Theorem 9.2 combines this with chunk 03's Rademacher-complexity generalization machinery (Theorem 3.3) to give the chapter's margin bound. Proposition 9.3 and Corollary 9.4 specialize this to kernel-based hypotheses, where each class has its own weight vector in a reproducing kernel Hilbert space and the $k$ weight vectors are jointly constrained by an $L^p$ -type group norm $\|W\|_{H,p}\le\Lambda$ .

Formalization targets

Lemma 9.1 (milestone). For $F_1,\dots,F_l$ hypothesis sets in $\mathbb R^X$ , $l\ge1$ , and $G=\{\max\{h_1,\dots,h_l\}:h_i\in F_i\}$ : $\hat R_S(G)\le\sum_{j=1}^l\hat R_S(F_j)$ .

Theorem 9.2 — the mission's goal. For $H\subseteq\mathbb R^{X\times Y}$ , $Y=\{1,\dots,k\}$ , fix $\rho>0$ . For any $\delta>0$ , with probability at least $1-\delta$ , for all $h\in H$ :

R(h) \le \hat R_{S,\rho}(h) + \tfrac{4k}\rho R_m(\Pi_1(H)) + \sqrt{\tfrac{\log(1/\delta)} {2m}}.

Proposition 9.3 (milestone). For a PDS kernel $K$ with feature map $\Phi$ and $K(x,x)\le r^2$ : $R_m(\Pi_1(H_{K,p})) \le \sqrt{r^2\Lambda^2/m}$ .

Corollary 9.4 (milestone). Under Proposition 9.3's hypotheses, fix $\rho>0$ . For any $\delta>0$ , with probability at least $1-\delta$ , for all $h\in H_{K,p}$ : $R(h) \le \hat R_{S,\rho}(h) + 4k\sqrt{r^2\Lambda^2/\rho^2/m} + \sqrt{\log(1/\delta)/(2m)}$ .

Significance

Theorem 9.2 is the multi-class generalization of chunk 05-svm's Theorem 5.8, and its proof is the chapter's genuine new technique rather than a restatement: it needs a $k$ -way application of Lemma 9.1 (once for the argmax structure of the margin, once summing over the $k$ possible labels), which is exactly where the $4k$ factor comes from. Corollary 9.4 is the direct theoretical basis for the multi-class SVM algorithm the chapter derives next (§9.3.1): the displayed dual optimization problem literally minimizes the right-hand side of the corollary's bound. No prior art exists on the platform: GET /theorems?q=multi-class%20classification returns zero hits, and chunk 03's Rademacher-complexity machinery (needed by the proof route) is a draft, not reusable, per the "drafts cannot import drafts" rule.

Difficulty

Lemma 9.1's proof is a genuine two-function argument (max as $\tfrac12(h_1+h_2+|h_1-h_2|)$ , Talagrand's lemma applied to $|\cdot|$ ) generalized to $l$ functions by induction, not a one-line consequence of chunk 03's single-hypothesis-set bound. Theorem 9.2's own proof (PDF pp. 234-236) is the chapter's most involved: it introduces an auxiliary margin function $\rho_{\theta,h}$ with a free parameter $\theta$ later fixed to $2\rho$ , splits the resulting Rademacher complexity into a "diagonal" term (bounded via a further one-hot decomposition across the $k$ classes, giving the first factor of $k$ ) and a "off-diagonal" term bounded via Lemma 9.1 (giving the second factor, folded into the same $4k$ constant). A formalization that stated Theorem 9.2 for $H$ itself rather than $\Pi_1(H)$ , or that treated $k$ as an unrelated free constant rather than the actual number of classes, would misstate the theorem — precisely the pitfall BRIEF.md names for this chapter. Proposition 9.3's proof is a clean Cauchy-Schwarz/Jensen argument in the RKHS but needs the $L^p$ -group-norm hypothesis class $H_{K,p}$ stated with its exact footnote definition (PDF p. 236), not a simplified $p=2$ special case.

Formalization scope

GeneralizationError, EmpiricalRademacherComplexity and RademacherComplexity are restated locally in this chunk's MultiClass namespace (the last two identical in content to chunk 03-rademacher-vc's own copies); MarginLossFunction restates chunk 05-svm's Definition 5.5 (the same function, needed here for this chapter's own EmpiricalMarginLoss); IsPDS restates chunk 06-kernels's PDS-kernel definition. All are duplicated rather than imported since a draft item cannot import another chunk's draft module, and none of 03, 05, 06 is listed as reusable in missions/README.md's "Published definitions" table at the time of this session. GeneralizationError is formalized via the book's own established equivalence " $h$ misclassifies $(x,y)$ iff $\rho_h(x,y)\le0$ " (the form Theorem 9.2's own proof displays and works with), rather than via an explicit argmax classifier construction — checked as faithful, not a weakening, since it is exactly the quantity the chapter's proof bounds. MarginFunction's ⨆_{y'≠y} is a real supremum rather than a Finset.sup', avoiding a nonempty-finset side proof at definition time; every consuming theorem supplies 2 ≤ k (Y = Fin k) to guard it against trap 5. MaxFamily's index type is Fintype+Nonempty rather than a Finset-cardinality parameter l, a harmless generalization matching "l ≥ 1 hypothesis sets" via Nonempty. IsPDS's feature map Φ and its defining property K(x,y) = ⟪Φ(x),Φ(y)⟫ are supplied as hypotheses to the two kernel theorems rather than as a separate "feature mapping associated to a kernel" definition — the book itself treats this as a given correspondence, not a construction. No numerical constant is altered: 4k/ρ and log(1/δ) in Theorem 9.2, r²Λ²/m in Proposition 9.3, and 4k and r²Λ²/ρ²/m in Corollary 9.4 are exactly as displayed.

Not formalized: §9.1's discussion of the multi-label case (Eq. 9.2/9.3, the Hamming-distance risk) and Eq. 9.4 (empirical Hamming error) — background for a case this chapter's own generalization-bound section (§9.2) does not cover (the mono-label case only); the multi-class SVM primal/dual optimization problems (§9.3.1, an algorithm derived from Corollary 9.4, not a generalization-theoretic theorem); AdaBoost.MH (§9.3.2, a boosting algorithm, analyzed via a convex-surrogate argument rather than the Rademacher-complexity route this mission formalizes); and the uniform-over- $\rho$ extension mentioned at the end of the Theorem 9.2 proof (an unnumbered remark referencing Theorem 5.9's technique from a different chapter, not restated here). Drafting only the algorithmic consequences (the multi-class SVM's optimization problem) in place of the generalization bounds themselves would be this chapter's trivializing formalization.

Selected references

M. Mohri, A. Rostamizadeh, A. Talwalkar, Foundations of Machine Learning, 2nd ed., MIT Press, 2018, Chapter 9.
V. Koltchinskii, D. Panchenko, "Empirical margin distributions and bounding the generalization error of combined classifiers," Annals of Statistics 30(1), 2002 (Lemma 9.1's technique).
K. Crammer, Y. Singer, "On the algorithmic implementation of multiclass kernel-based vector machines," JMLR 2, 2001 (the multi-class SVM algorithm §9.3.1 derives).

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Machine Learning Probability Random Matrix Theory+1·Captain: mikedeng1

High-Dimensional Probability IX: The Matrix Deviation InequalityTextbook

Motivation

Random matrices with independent rows are the workhorse of high-dimensional statistics and compressed sensing: sample covariance matrices, sub-sampled measurement operators, and randomized sketches are all of this form. A basic question about such a matrix $A$ is how close $\|Ax\|_2$ stays to its typical size $\mathbb E\|Ax\|_2\approx\sqrt m\|x\|_2$ — not just for one fixed $x$ , but simultaneously for every $x$ in some set $T$ of interest (a sphere, a cone, the difference set of a data cloud). A bound that holds only pointwise in $x$ is of limited use, since most applications need to reason about the worst case over an entire geometric set at once.

This chapter proves such a uniform bound — the matrix deviation inequality — for matrices with independent, isotropic, sub-gaussian rows, controlling the deviation by a single geometric parameter of $T$ , its Gaussian complexity. The result is a direct descendant of the chaining machinery of Chapter 8 (via Talagrand's comparison inequality, Chapter 8.6) and, in this book's own account, subsumes several results proved earlier by other methods — two-sided bounds on random matrices, the Johnson-Lindenstrauss lemma for infinite sets — while also yielding two new consequences central to high-dimensional convex geometry: the $M^*$ bound and the Escape theorem, both controlling how a random subspace intersects a fixed geometric set.

Setting

Fix a probability space $(\Omega,\mathcal F,P)$ . A random vector $X$ in $\mathbb R^n$ is isotropic if its covariance matrix is the identity, $\Sigma(X)=\mathbb E[XX^\top]=I_n$ — equivalently (the book's own Lemma 3.2.3), $\mathbb E\langle X,x\rangle^2=\|x\|_2^2$ for every $x\in\mathbb R^n$ . The sub-gaussian norm of a random vector $X$ is $\|X\|_{\psi_2}:=\sup_{x\in S^{n-1}}\|\langle X,x\rangle\|_{\psi_2}$ , the supremum over the unit sphere of the scalar sub-gaussian (Orlicz $\psi_2$ ) norm of its one-dimensional marginals; $X$ is sub-gaussian when this is finite.

Fix a standard Gaussian random vector $g\sim N(0,I_n)$ in $\mathbb R^n$ (a vector whose coordinates in any orthonormal basis are independent standard normal). For a subset $T\subseteq\mathbb R^n$ , the Gaussian width and Gaussian complexity of $T$ are

w(T) := \mathbb E\sup_{x\in T}\langle g,x\rangle, \qquad \gamma(T) := \mathbb E\sup_{x\in T}|\langle g,x\rangle|,

two closely related measures of the geometric size of $T$ — "cousins" that agree up to a factor of $2$ whenever $T$ contains the origin, and agree exactly when $T$ is origin-symmetric.

Formalization targets

Goal (Theorem 9.1.1, Matrix deviation inequality)

\exists\,C>0:\quad \mathbb E\sup_{x\in T}\bigl|\,\|Ax\|_2-\sqrt m\|x\|_2\,\bigr| \;\le\; CK^2\gamma(T)

for every $m\times n$ matrix $A$ whose rows $A_1,\dots,A_m$ are independent, isotropic, sub-gaussian random vectors with $K:=\max_i\|A_i\|_{\psi_2}$ , and every $T\subseteq\mathbb R^n$ (whenever $\gamma(T)$ is finite). $C$ is the book's own unnamed absolute constant, hard-coded to no numeral — the weakest stable form of the claim.

Milestone (Theorem 9.4.2, the $M^*$ bound)

\mathbb E\,\mathrm{diam}(T\cap\ker A) \;\le\; \frac{CK^2w(T)}{\sqrt m}

for the same class of matrices $A$ and any bounded $T\subseteq\mathbb R^n$ , where $\ker A$ is the (random) kernel of $A$ , a subspace of codimension at most $m$ . A direct one-paragraph consequence of the goal theorem (apply it to $T-T$ , then restrict to $\ker A$ , where $\|Ax-Ay\|_2$ vanishes).

Significance

The matrix deviation inequality converts a purely algebraic quantity — how close $\|Ax\|_2$ stays to $\sqrt m\|x\|_2$ — into a single geometric parameter of the index set $T$ , letting it subsume, via specializations of $T$ , results that were previously proved by separate ad hoc arguments: two-sided singular value bounds on random matrices ( $T$ a sphere), Johnson-Lindenstrauss-type embeddings for possibly infinite point sets ( $T$ a difference set), and covariance estimation. The $M^*$ bound is one of the two classical consequences the book develops fresh from the inequality (the other, the Escape theorem, is outside this mission's scope): it answers, quantitatively, how large a random affine section of a fixed convex body typically is, a question at the heart of the local theory of Banach spaces and of compressed sensing's recovery guarantees (Chapter 10 builds directly on this chapter's machinery). Both results have long-standing, well-understood classical proofs; this mission formalizes their statements, not open research.

Difficulty

The natural first idea — bound $\|Ax\|_2-\sqrt m\|x\|_2$ pointwise for a fixed $x$ using concentration of the norm of a sub-gaussian random vector, then take a union bound over $T$ — only works when $T$ is finite, and gives a bound that scales with $\log|T|$ rather than with the actual geometric size of $T$ . The book's actual route treats $X_x:=\|Ax\|_2-\sqrt m\|x\|_2$ , indexed by $x\in\mathbb R^n$ , as a genuine random process and shows it has sub-gaussian increments ( $\|X_x-X_y\|_{\psi_2}\le CK^2\|x-y\|_2$ ) — itself a nontrivial fact proved in stages (first for a single unit vector via concentration of the norm, Theorem 3.1.1; then for a pair of unit vectors via a squared-process argument; only then in full generality) — and then invokes Talagrand's comparison inequality (a consequence of the chaining machinery of Chapter 8) to pass from sub-gaussian increments directly to a bound in terms of Gaussian complexity, without ever performing a union bound over $T$ itself.

Formalization scope

A is represented by its rows, A : Fin m → Ω → EuclideanSpace ℝ (Fin n), with ‖Ax‖₂ recovered as Real.sqrt (∑ i, ⟨Aᵢ,x⟩²) rather than constructing A as a Matrix/LinearMap — this matches the book's own row-by-row hypotheses exactly and is what both theorems' own proofs use directly. IsIsotropic is formalized via the book's basis-free Lemma 3.2.3 characterization (E⟨X,x⟩² = ‖x‖² for every x) rather than the matrix equation Σ(X)=Iₙ, avoiding a fixed-basis covariance-matrix construction the rest of this chunk's definitions do not otherwise need. SubgaussianVectorNorm reuses the published scalar subgaussianNorm. GaussianWidth and GaussianComplexity realize the standard Gaussian vector g ∼ N(0,Iₙ) as the identity map on Mathlib's own standard Gaussian measure on a finite-dimensional inner product space (ProbabilityTheory.stdGaussian), and both, together with the goal's own left-hand side, use a locally-defined finite-marginal expected-supremum convention (ExpSup, EReal-valued) matching the book's own footnote-3 convention (Section 7.2), reused throughout the series. Both theorems' right-hand sides presuppose their respective geometric parameter ( $\gamma(T)$ or $w(T)$ ) is a finite real number; since both are EReal-valued in general, each theorem takes an explicit real witness together with a proof that it equals the true value — the same finiteness-disclosure pattern 07-chaining's Dudley inequality uses for its own right-hand integral, needed here for exactly the same reason (the book's own display does not spell out why the quantity is finite, true whenever $T$ is bounded, as in every application). $C$ (and, in the milestone, the same $C$ again — the two are not asserted equal, matching that the book states them as two separate "absolute constants") is existentially quantified before every type, instance and hypothesis it is uniform over. The $M^*$ bound milestone (m_star_bound) additionally carries the hypothesis $m > 0$ : its conclusion divides by $\sqrt m$ , and without this hypothesis Lean's real-division convention ( $x/0=0$ ) makes the right-hand side $0$ at $m=0$ regardless of $C, K, w(T)$ — false whenever $T$ has positive diameter, not merely a weaker or vacuous claim. The book's own proof ("Dividing by $\sqrt m$ yields …", p. 241) already implicitly assumes $m \ge 1$ , matching every other use of $m$ in the chapter as a positive count of measurement rows.

A trivializing formalization would fix $T$ to be a finite set, collapsing the goal to the elementary union-bound case the book explicitly contrasts its own more general statement against (Section 9.1's opening paragraph: "we may choose an arbitrary subset $T\subseteq\mathbb R^n$ "); this mission's goal quantifies over an arbitrary Set (EuclideanSpace ℝ (Fin n)) to rule that out.

This mission covers Theorem 9.1.1 and Theorem 9.4.2 only; Theorem 9.4.7 (the Escape theorem) and Theorem 9.2.4 (covariance estimation for lower-dimensional distributions), both named as candidate milestones, are left out for lack of session time given the substantial shared infrastructure this chapter needed from scratch. ExpSup, IsIsotropic, SubgaussianVectorNorm, GaussianWidth and GaussianComplexity are reusable by any later chapter needing an isotropic or sub-gaussian random vector, or a Gaussian-width-type quantity (Chapters 4, 10, 11 of this same book series all use one or more of these notions). Solvers' contributions are welcome on: Theorem 9.1.3 (the sub-gaussian increments of the deviation process, the technical heart of the goal's proof), Talagrand's comparison inequality itself (outside this mission, in 07-chaining's companion chapter), and the one-paragraph reduction from the goal to the $M^*$ bound.

Selected references

S. Mendelson, A. Pajor, N. Tomczak-Jaegermann, Reconstruction and subgaussian operators in asymptotic geometric analysis, Geometric and Functional Analysis 17 (2007), 1248–1282. https://doi.org/10.1007/s00039-007-0618-7
V. D. Milman, A new proof of A. Dvoretzky's theorem on cross-sections of convex bodies, Funkcional. Anal. i Priložen. 5 (1971), 28–37.
R. Vershynin, High-Dimensional Probability: An Introduction with Applications in Data Science, Cambridge University Press, 2018, Chapter 9. https://doi.org/10.1017/9781108231596

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Machine Learning Probability Statistics+1·Captain: mikedeng1

Foundations of Machine Learning IV: Support Vector Machines and the Margin BoundTextbook

Motivation

Support vector machines were, for two decades, the workhorse of applied classification, and the reason offered for their success was always geometric: SVMs maximize the margin between the two classes. Chapter 3's VC-dimension bound cannot explain why this should help — for linear hypotheses in $\mathbb R^N$ its bound depends on $N+1$ and is uninformative whenever the feature dimension is large relative to the sample size, exactly the regime (kernel-induced or high-dimensional features) where SVMs are most often used. Chapter 5 answers the question this leaves open: a generalization bound for a real-valued hypothesis, stated in terms of its margin on the training sample, that does not depend on the ambient dimension at all. This is also the template every later chapter's margin bound specializes (multi-class classification, ranking, and, indirectly, boosting all reuse the same Rademacher-complexity-of-a-Lipschitz-loss argument developed here).

Setting

A hypothesis here is a real-valued function $h:X\to\mathbb R$ , not (as in Chapters 2-3) a function into $\{-1,+1\}$ : for a labeled point $(x,y)$ with $y\in\{-1,+1\}$ , the sign of $h(x)$ gives the prediction and $|h(x)|$ is read as the classifier's confidence. The confidence margin of $h$ at $(x,y)$ is $y\,h(x)$ ; it is positive exactly when $h$ classifies $x$ correctly. For $\rho>0$ , the $\rho$ -margin loss $\Phi_\rho:\mathbb R\to\mathbb R$ (Definition 5.5) is

\Phi_\rho(x) = \min\Big(1,\max\Big(0,1-\frac x\rho\Big)\Big),

equal to $1$ when $x\le 0$ (misclassified), $0$ when $x\ge\rho$ (classified with confidence at least $\rho$ ), and interpolating linearly in between; it is $1/\rho$ -Lipschitz. The empirical margin loss on a sample $S=(x_1,\dots,x_m)$ with labels $y_1,\dots,y_m$ (Definition 5.6) is $\hat R_{S,\rho}(h) = \frac1m\sum_{i=1}^m\Phi_\rho(y_ih(x_i))$ — the fraction of training points misclassified or classified with confidence below $\rho$ , a strictly stronger requirement than plain misclassification. The (population) generalization error is $R(h)=\Pr_{(x,y)\sim D}[y\,h(x)\le 0]$ . Rademacher complexity, $\hat R_S(H)$ and $R_m(H)$ (Definitions 3.1-3.2, restated here since chunk 03-rademacher-vc's own copies are still drafts), measure how well a real-valued hypothesis class $H$ correlates with random sign noise on a sample, and are the vehicle through which the margin bound's complexity term is expressed.

Formalization targets

Lemma 5.7 (Talagrand's lemma, milestone). For $l$ -Lipschitz $\Phi_1,\dots,\Phi_m:\mathbb R\to\mathbb R$ and any hypothesis set $H$ of real-valued functions,

\frac1m\,\mathbb E_\sigma\Big[\sup_{h\in H}\sum_{i=1}^m\sigma_i(\Phi_i\circ h)(x_i)\Big] \le l\,\hat R_S(H).

Theorem 5.10 (Rademacher complexity of bounded-norm linear hypotheses, milestone). For $S\subseteq\{x:\|x\|\le r\}$ and $H=\{x\mapsto w\cdot x:\|w\|\le\Lambda\}$ ,

\hat R_S(H) \le \sqrt{r^2\Lambda^2/m}.

Corollary 5.11 (margin bound for linear hypotheses, milestone). For the same $H$ and $X\subseteq\{x:\|x\|\le r\}$ , fixing $\rho>0$ , with probability at least $1-\delta$ ,

R(h) \le \hat R_{S,\rho}(h) + 2\sqrt{\frac{r^2\Lambda^2/\rho^2}{m}} + \sqrt{\frac{\log(1/\delta)}{2m}} \quad\text{for all } h\in H.

Theorem 5.8 — the mission's goal. For any set $H$ of real-valued functions and $\rho>0$ , with probability at least $1-\delta$ , both

R(h) \le \hat R_{S,\rho}(h) + \frac2\rho R_m(H) + \sqrt{\frac{\log(1/\delta)}{2m}}

R(h) \le \hat R_{S,\rho}(h) + \frac2\rho \hat R_S(H) + 3\sqrt{\frac{\log(2/\delta)}{2m}}

hold simultaneously for all $h\in H$ .

Significance

Theorem 5.8 is genuinely dimension-free: unlike Chapter 3's VC-dimension bound (5.36 in the book, restated from Corollary 3.19), it holds regardless of the ambient feature dimension $N$ , depending instead only on the hypothesis class's Rademacher complexity and the chosen margin $\rho$ . Specialized to bounded-norm linear hypotheses (Corollary 5.11), this gives the theoretical justification most often cited for SVMs and every other margin-maximization algorithm: whenever the training data admits a large geometric margin, the empirical margin loss at that margin is small (often zero, in the separable case) and the bound is tight regardless of $N$ . Every later chapter's own margin bound (multi-class in Chapter 9, ranking in Chapter 10) is a direct structural descendant of Theorem 5.8's proof technique. No prior art on the Prove2Me platform is faithful: GET /theorems?q=support+vector+machine and q=margin+bound return no hits on this book's model (the two q=margin+bound hits found, both from the Aether Catalog, state a different comparison — a VC-type bound is eventually worse than a fixed Rademacher-type bound as a function of dimension — not Theorem 5.8 itself); q=Talagrand and q=contraction+principle return several hits (Ledoux/Talagrand convex-distance concentration, Rudin's Banach-space contraction-mapping theorem, a generic Rademacher-sign contraction lemma for quadratic sums) but every one states either a different mathematical object (metric-space fixed points, Talagrand's concentration inequality on product spaces) or a different idiom (squared vs. linear coordinate sums) from Lemma 5.7's function-composition contraction — none reused. All nine items are drafted fresh.

Not formalized here: Theorem 5.4 (the SVM sparsity/leave-one-out bound). It is listed as a candidate milestone in BRIEF.md, but its statement and proof depend on the primal/dual SVM optimization problem itself (the Lagrangian, KKT conditions, and the resulting definition of a "support vector" as a training point with nonzero dual coefficient) — a materially different, non-margin-based proof technique (leave-one-out stability of the trained hypothesis, via Lemma 5.3) that shares no definitions with the margin-bound family this mission's goal and other milestones are built on. Formalizing it faithfully would require standing up the SVM primal/dual formalism (Lagrangian, complementary slackness, the "support vector" predicate itself) from scratch, which is disproportionate to a single additional milestone within this mission's budget; per the captain brief's guidance to leave out, rather than approximate, a statement that cannot be made faithful in the time available, it is omitted.

Difficulty

The proof of Theorem 5.8 needs the empirical margin loss's zero-one-loss upper bound ( $\mathbb 1_{u\le 0}\le\Phi_\rho(u)$ ) applied before invoking Theorem 3.3's Rademacher generalization bound on the composed class $\tilde{\tilde H}=\{\Phi_\rho\circ f: f\in\tilde H\}$ , $\tilde H=\{(x,y)\mapsto y\,h(x):h\in H\}$ — reversing this order (bounding $R(h)$ by a Rademacher complexity computed on the zero-one loss directly) does not work, because the zero-one loss is not Lipschitz. Talagrand's lemma is exactly what lets the $1/\rho$ -Lipschitz surrogate $\Phi_\rho$ be pulled outside the Rademacher complexity, at the cost of a factor $1/\rho$ and no worse; its own proof is an induction removing one Rademacher variable at a time, using a two-point supremum argument (fixing $\epsilon>0$ , choosing near-optimal $h_1,h_2$ ) that does not simplify to anything less than genuine care with suprema of non-smooth objects — a formalization attempting to replace this with a naive linearity-of-expectation argument would be proving a false or vacuous statement, since $\sup$ does not commute with linear combinations. Theorem 5.10's bound needs the Cauchy-Schwarz and Jensen inequalities used in the particular order the book uses them (Cauchy-Schwarz on the empirical sup, then Jensen on the expectation of a norm, then the independence of the $\sigma_i$ s) — the bound $\hat R_S(H)\le r\Lambda/\sqrt m$ does not follow from either inequality alone.

Formalization scope

EmpiricalRademacherComplexity/RademacherComplexity are restated locally in SVM, byte-identical to chunk 03-rademacher-vc's own copies (a draft item cannot import another chunk's draft module); this duplication collapses once 03-rademacher-vc is uploaded and listed in missions/README.md's "Published definitions" table. MarginGeneralizationError is a new, real-valued-hypothesis specialization of Definition 2.1 (R(h) = P[y h(x) ≤ 0]), distinct from every earlier chunk's {-1,+1}-valued GeneralizationError, since no earlier chunk's own copy matches this chapter's real-valued convention. PhiRho/EmpiricalMarginLoss are new. The goal theorem (margin_bound_binary_classification) states H's own Rademacher complexity computed on the marginal X-distribution (D.map Prod.fst), matching the book's final displayed form — the proof's intermediate step (the lifted class $\tilde H=\{(x,y)\mapsto y h(x)\}$ having the same Rademacher complexity as $H$ itself, since $y\in\{-1,+1\}$ ) is not separately drafted, only the theorem's statement. Theorem 5.10/Corollary 5.11 generalize the book's ambient $\mathbb R^N$ to an arbitrary real inner-product space X ([NormedAddCommGroup X] [InnerProductSpace ℝ X]), a harmless generalization since the book's proof (Cauchy-Schwarz, Jensen, orthogonality of Rademacher signs) uses only the inner-product structure, never finite dimension; [BorelSpace X] is added to Corollary 5.11's statement to make the measurable structure under which MarginGeneralizationError is well-posed explicit, since every x ↦ ⟨w, x⟩ is automatically Borel-measurable — not a substantive restriction, the book never discusses measurability of linear functionals. No numerical constant in any of the four theorems is altered from the book's own displayed form. A trivializing formalization this mission avoids: stating Theorem 5.8 only for a Finset/finite H (which would make it a disguised instance of Chapter 2's finite-hypothesis bound rather than the chapter's genuinely new, complexity-based argument) — H : Set (X → ℝ) is left fully general, exactly as the book states it.

Selected references

M. Mohri, A. Rostamizadeh, A. Talwalkar, Foundations of Machine Learning, 2nd ed., MIT Press, 2018, Chapter 5.
C. Cortes, V. Vapnik, "Support-vector networks," Machine Learning 20(3), 1995, 273-297.
M. Talagrand, "Sharper bounds for Gaussian and empirical processes," The Annals of Probability 22(1), 1994, 28-76.
P. Bartlett, S. Mendelson, "Rademacher and Gaussian complexities: risk bounds and structural results," Journal of Machine Learning Research 3, 2002, 463-482.

12 thms4 active users

Machine Learning Probability Statistics+1·Captain: mikedeng1

Foundations of Machine Learning III: Structural Risk Minimization and Model SelectionTextbook

Motivation

Chapters 2 and 3 bound the estimation error of a hypothesis chosen from a fixed hypothesis set $H$ , but the choice of $H$ itself is left open: a richer $H$ lowers the approximation error (how close $H$ comes to the Bayes classifier) at the price of a looser generalization bound, and a poorer $H$ does the reverse. Chapter 4 is the book's answer to this trade-off. It first shows that Empirical Risk Minimization (ERM) alone cannot resolve it — ERM ignores the complexity of $H$ entirely — and then develops Structural Risk Minimization (SRM): decompose a rich hypothesis set into a nested countable union $H=\bigcup_{k\ge1}H_k$ of increasingly complex pieces, and let the learning algorithm balance empirical fit against a complexity penalty for each $H_k$ automatically. The chapter closes by showing how the same balance can be achieved computationally through convex surrogate losses, whose minimization is tractable where minimizing the zero-one loss directly is not.

Setting

For a hypothesis $h$ chosen from $H$ , the excess error $R(h)-R^*$ decomposes into an estimation term $R(h)-\inf_{h\in H}R(h)$ and an approximation term $\inf_{h\in H}R(h)-R^*$ (Eq. 4.1). Proposition 4.1 bounds ERM's estimation error by twice the uniform deviation $\sup_{h\in H}|R(h)-\hat R_S(h)|$ . For a nested family $(H_k)_{k\ge1}$ and $h\in H$ , $k(h)$ denotes the least index with $h\in H_{k(h)}$ ; SRM selects $h_S^{SRM}$ by minimizing $F_k(h)=\hat R_S(h)+R_m(H_k)+\sqrt{\log k/m}$ jointly over $k\ge1$ and $h\in H_k$ , where $R_m(H_k)$ is $H_k$ 's Rademacher complexity (Definitions 3.1/3.2, restated locally in this chunk's ModelSelection namespace). Theorem 4.2 is the resulting learning guarantee. Section 4.4 develops a competing model-selection procedure, cross-validation, and Theorem 4.4 directly compares its guarantee to SRM's on a held-out split of the sample. Section 4.7 turns to real-valued scoring functions $h:X\to\mathbb R$ with sign convention $f_h(x)=\mathrm{sign}(h(x))$ and a convex non-decreasing surrogate $\Phi$ of the zero-one loss; the Bayes scoring function $h^*(x)=\eta(x)-\tfrac12$ (Eq. 4.9) and the $\Phi$ -loss $L_\Phi$ (Eq. 4.10) let Theorem 4.7 bound the true excess error by a power of the surrogate's own excess loss.

Formalization targets

Proposition 4.1 (ERM bound, milestone). For any sample $S$ , $\Pr[R(h_S^{ERM}) - \inf_{h\in H}R(h) > \epsilon] \le \Pr[\sup_{h\in H}|R(h)-\hat R_S(h)| > \epsilon/2]$ .

Theorem 4.2 — the mission's goal. For a nested countable union $H=\bigcup_{k\ge1}H_k$ and $h_S^{SRM}$ minimizing $F_k(h)$ over the whole union, for any $\delta>0$ , with probability at least $1-\delta$ :

R(h_S^{SRM}) \le \inf_{h\in H}\Big[R(h)+2R_m(H_{k(h)})+\sqrt{\tfrac{\log k(h)}m}\Big] + \sqrt{\tfrac{2\log(3/\delta)}m}.

Theorem 4.4 (Cross-validation versus SRM, milestone). Splitting a sample of size $m$ into $S_1$ (size $(1-\alpha)m$ , training) and $S_2$ (size $\alpha m$ , validation), for any $\delta>0$ , with probability at least $1-\delta$ :

R(h_S^{CV}) - R(h_{S_1}^{SRM}) \le 2\sqrt{\tfrac{\log\max(k(h_S^{CV}),k(h_{S_1}^{SRM}))}{\alpha m}} + 2\sqrt{\tfrac{\log(4/\delta)}{2\alpha m}}.

Theorem 4.7 (Convex-surrogate excess-error bound, milestone). For $\Phi$ convex and non-decreasing with $s\ge1,c>0$ satisfying $|h^*(x)|^s \le c^s(L_\Phi(x,0)-L_\Phi(x,h^*_\Phi(x)))$ for all $x$ : $R(h)-R^* \le 2c(L_\Phi(h)-L^*_\Phi)^{1/s}$ .

Significance

Theorem 4.2 is the chapter's headline result and the theoretical justification for regularization-based learning: it shows that a single algorithm, without knowing in advance which $H_k$ contains a good hypothesis, achieves a guarantee that is — up to the $\sqrt{\log k(h)/m}$ penalty — as favorable as if an oracle had revealed the best-in-class index in advance (Eq. 4.6). It is also the chapter's genuine new content beyond chunk 03-rademacher-vc's single-hypothesis-set bound: the countable union bound (a 1/k²-weighted union over $k\ge1$ converging to $\pi^2/6$ , hence the $\log 3$ appearing in place of $\log 2$ ) is not a restatement of Theorem 3.3 but a distinct argument, and the goal's inf over the whole nested family is what makes SRM a model-selection method rather than a bound for one fixed $k$ . Theorem 4.4 is the chapter's only head-to-head comparison between two competing model-selection procedures, on two genuinely different samples. Theorem 4.7 is the bridge between the learning-theoretic guarantees of chapters 2-4 and the actually-implemented convex optimization problems of chapters 5 (SVM), 6 (kernels) and beyond, all of which minimize a convex surrogate rather than the zero-one loss directly. No prior art exists on the platform: GET /theorems?q=structural%20risk%20minimization and GET /theorems?q=model%20selection both return zero hits.

Difficulty

Theorem 4.2's proof genuinely uses the union bound over a countably infinite family indexed by $k\ge1$ with weight $1/k^2$ converging to $\pi^2/6 < 2$ (Eq. 4.5) — this is the chapter's distinct new technique, not an application of chunk 03's finite/VC-dimension machinery to a single $H_k$ ; a formalization that stated the bound only for one fixed $k$ , or dropped the inf over the whole union in favor of a single best-in-class $h^*$ , would be Theorem 4.2's named trivializing formalization (BRIEF.md's pitfall note) rather than the theorem itself. Theorem 4.4 requires keeping two distinct samples ( $S_1$ , $S_2$ , of different, precisely related sizes) and two distinct hypotheses ( $h_S^{CV}$ , $h_{S_1}^{SRM}$ ) apart throughout; conflating them collapses the comparison to a tautology. Theorem 4.7's difficulty is in its setup, not its statement: the Bayes scoring function, the $\Phi$ -loss, and the pointwise $\Phi$ -minimizer $h^*_\Phi$ (which the book allows to take the extended values $\pm\infty$ at the degenerate points $\eta(x)\in\{0,1\}$ ) all need care to state without silently altering the theorem's content.

Formalization scope

GeneralizationError, EmpiricalError, EmpiricalRademacherComplexity and RademacherComplexity are restated locally in this chunk's ModelSelection namespace (identical in content to chunk 03-rademacher-vc's own copies), since a draft item cannot import another chunk's draft module. LeastIndex H h (k(h)) is Nat.sInf {k | 1 ≤ k ∧ h ∈ H k}; every theorem using it carries the standing hypothesis that h lies in the relevant union, guarding against trap 5 (Nat.sInf of an empty set). Theorem 4.2's hSRM and Proposition 4.1's hERM are hypothesis-supplied functions satisfying the book's optimality property, not constructed via choice over an unconstrained H; H.Nonempty (Proposition 4.1) and (⋃ k ≥ 1, Hk k).Nonempty (Theorem 4.2) guard the outer sInf/inf terms against trap 5. Theorem 4.7's $h^*_\Phi$ is formalized as a real-valued function satisfying the pointwise minimization property for all $x$ ; the book's own extended-real convention ( $h^*_\Phi(x)=\pm\infty$ exactly where $\eta(x)\in\{0,1\}$ ) is outside this formalization — disclosed here and in MODERATION_NOTES.md — since no real number satisfies the minimizing property at those degenerate points, the theorem as stated applies precisely to the case a real-valued $h^*_\Phi$ can be supplied, which is the book's own generic case. No numerical constant is altered from the book in any of the four theorems: 2 and log(3/δ) in Theorem 4.2, 2 (twice) and log(4/δ) in Theorem 4.4, and 2c and the exponent 1/s in Theorem 4.7 are exactly as displayed.

Not formalized: the discussion of computing $k^*$ via binary search (a computational, not a statistical, result); $n$ -fold and leave-one-out cross-validation (Section 4.5, a practical variant of Theorem 4.4's two-sample cross-validation without its own numbered generalization bound); regularization-based algorithms (Section 4.6, the uncountable-union extension of SRM, which the book itself only sketches without a numbered theorem); Lemma 4.5 and Proposition 4.6 (intermediate results establishing that $h^*_\Phi$ induces the same classifier as $h^*$ , needed for Theorem 4.7's proof but not part of its statement); and the worked examples for the hinge, exponential and logistic losses (instantiations of Theorem 4.7's $s,c$ , not separate theorems). Drafting only these worked instantiations in place of Theorem 4.7's general statement would be a trivializing formalization for this chapter.

Selected references

M. Mohri, A. Rostamizadeh, A. Talwalkar, Foundations of Machine Learning, 2nd ed., MIT Press, 2018, Chapter 4.
V. Vapnik, Statistical Learning Theory, Wiley-Interscience, 1998 (structural risk minimization).
T. Zhang, "Statistical behavior and consistency of classification methods based on convex risk minimization," Annals of Statistics 32(1), 2003 (Theorem 4.7's origin).

16 thms4 active users

Analysis Dynamical Systems Topology·Captain: Lucas

Local Connectivity of the Mandelbrot Set (MLC)Open Problem

The set

For a complex parameter $c$ , iterate the quadratic map

f_c(z) = z^2 + c

starting at the critical point $z = 0$ . The Mandelbrot set is the set of parameters for which this orbit stays bounded:

M = \{\, c \in \mathbb{C} \ : \ \sup_{k \in \mathbb{N}} \left| f_c^{\,k}(0) \right| < \infty \,\}.

Equivalently -- and this is the first milestone of the mission -- $c \in M$ if and only if $|f_c^{\,k}(0)| \le 2$ for every $k$ , which exhibits $M$ as a compact subset of the plane.

$M$ is the parameter-space picture of the simplest non-trivial family in complex dynamics, and it acts as a dictionary: the shape of $M$ near a parameter $c$ encodes the dynamics of $f_c$ on its Julia set, so structural questions about $M$ are questions about the whole quadratic family at once. Douady and Hubbard proved in 1982 that $M$ is connected, by exhibiting a conformal isomorphism

\Phi : \mathbb{C} \setminus M \longrightarrow \mathbb{C} \setminus \overline{\mathbb{D}}

between the complement of $M$ and the exterior of the closed unit disk.

The question

MLC conjecture. $M$ is locally connected: every point of $M$ has a neighbourhood basis, in the subspace topology, consisting of connected sets.

By Caratheodory's theorem, MLC is equivalent to the statement that $\Phi^{-1}$ extends continuously to the unit circle. That extension would deliver a complete combinatorial description of $M$ -- the pinched disk model of Douady and Thurston -- in which every boundary point is labelled by the external rays landing on it. Two headline consequences follow: the density of hyperbolicity in the quadratic family (Fatou's conjecture: every quadratic polynomial can be perturbed to one with an attracting cycle), and zero area for $\partial M$ .

MLC has been open since the early 1980s and is regarded as the central problem of one-dimensional complex dynamics.

Timeline

1982 -- Douady and Hubbard prove that $M$ is connected, via the Boettcher uniformisation of its complement, and formulate MLC.
1984/85 -- The Orsay notes develop the combinatorics of external rays and the pinched-disk model, and show that MLC implies the density of hyperbolicity in the quadratic family.
1990 -- Yoccoz proves MLC at every finitely renormalizable parameter without an indifferent periodic point, introducing the Yoccoz puzzle and the rigidity techniques that dominate later work.
1997 -- Lyubich extends local connectivity to infinitely renormalizable parameters of bounded type, using complex bounds for quadratic-like renormalization.
1997 -- Graczyk-Swiatek and Lyubich prove density of hyperbolicity in the real quadratic family.
1998 -- Shishikura proves that $\partial M$ has Hausdorff dimension $2$ , by parabolic implosion. Whether $\partial M$ has positive area remains open.
2005 -- Buff and Cheritat construct quadratic Julia sets of positive area, showing that the analogous area question in the dynamical plane has a negative answer.
Today -- MLC is known at large classes of parameters, but the general case, and with it the density of hyperbolicity, remain open.

What this mission asks for

The goal theorem is MLC itself, in the form "the Mandelbrot set, as a topological subspace of $\mathbb{C}$ , is a locally connected space".

The milestones are of three kinds, and are ordered accordingly:

Foundations provable today -- the escape criterion (in the quadratic and the general unicritical degree) and compactness. These make the filter-theoretic definition usable and are the natural entry point for a solver new to the mission.
Known theorems from the literature -- connectedness of $M$ (Douady-Hubbard), the implication MLC $\Rightarrow$ density of hyperbolicity (Douady-Hubbard), and $\dim_H(\partial M) = 2$ (Shishikura). These are hard but settled, and formalizing them builds the infrastructure -- Boettcher coordinates, external rays, parabolic implosion -- that any attack on the goal will need.
The open companions -- density of hyperbolicity in the quadratic and unicritical families, zero area of $\partial M$ , and MLC for all Multibrot sets $M_n$ , the parameter sets of $z \mapsto z^n + c$ .

All statements are phrased against a single shared definition file, so a solver can move between milestones without re-fixing conventions.

24 thms4 active usersReviewed

Algebra·Captain: mysticflounder

Equational Magmas: E677 → E255 (finite case)Open Problem

Motivation

An equation for a magma constrains a binary operation without assuming that it is associative, commutative, or has an identity. Which laws imply which others, over all magmas or over finite magmas only, is the question the Equational Theories Project studies at scale. Restricting the underlying set to be finite can change the answer: a structural argument may depend on the fact that a surjective self-map of a finite set is injective.

The project's December 2025 paper reports the finite implication from E677 to E255 as unresolved, while reporting a counterexample to the implication when infinite magmas are allowed. The paper also tentatively conjectures that a finite counterexample exists. This mission makes the affirmative implication its formal target and also accepts a rigorous refutation of the complete finite statement.

This mission treats the universal target as open. Supporting structural facts and conditional reductions are separately identified, so that progress on one does not assert completion of the target.

Setting

A magma here is a type $\alpha$ with a total binary operation $\diamond:\alpha\times\alpha\to\alpha$ . Parentheses specify the order of evaluation throughout; no reassociation is permitted. The condition E677 means

\forall x,y\in\alpha,\quad x=y\diamond\bigl(x\diamond((y\diamond x)\diamond y)\bigr).

The condition E255 means

\forall x\in\alpha,\quad x=((x\diamond x)\diamond x)\diamond x.

These are the two laws used in Chapter 13 of the project blueprint. For a fixed element $y$ , the left translation (left multiplication) map is $L_y(x)=y\diamond x$ . A fixer of $x$ is an element $y$ with $y\diamond x=x$ ; it acts on $x$ from the left. This definition concerns one element $x$ ; it does not require $y$ to act as an identity on every element. The left orbit of $x$ is the forward orbit $x,L_x(x),L_x^2(x),\ldots$ of $x$ under its own left translation.

Formalization targets

The supporting targets expose the relevant distinction between a constraint on a possible fixer and the existence of a fixer. For every finite $\alpha$ satisfying E677, the first supporting statement is

\forall y\in\alpha,\quad L_y\text{ is bijective}.

The second supporting statement determines any fixer:

\forall x,y\in\alpha,\quad y\diamond x=x\ \Longrightarrow\ y=(x\diamond x)\diamond x.

The third supporting statement is the backward recurrence

\forall x,y\in\alpha,\quad x=(y\diamond x)\diamond\bigl((y\diamond(y\diamond x))\diamond y\bigr).

These three statements are blueprint Lemma 13.1(i)–(iii) and are Proved on this platform. The following fixer-existence assertion is the same obligation as E255 under E677 and finiteness (blueprint Lemma 13.2); it is a reduction node, not a milestone:

\forall x\in\alpha,\quad\exists y\in\alpha,\quad y\diamond x=x.

The mission goal is

\forall\text{ finite magmas }\alpha,\quad \operatorname{E677}(\alpha)\ \Longrightarrow\ \operatorname{E255}(\alpha).

For finite E677 magmas, E255 at $x$ supplies the fixer $(x\diamond x)\diamond x$ ; from any fixer $y$ of $x$ , Lemma 13.1(ii) gives $y=(x\diamond x)\diamond x$ and hence E255 at $x$ . Each direction is one line, so fixer existence is not a strictly weaker milestone.

Three open sub-goals come from the case split in the accepted reduction sketches for the fixer-existence node. Sub-goal 1 is orbit-local: for a fixed $x$ , if two elements of the left orbit of $x$ have equal right products by $x$ , then they are equal or $x$ has a fixer. Sub-goal 2 concerns the case of exactly one element $A$ outside the left orbit of $x$ , with $x$ having no fixer: a right-translation collision $c\diamond x=A\diamond x$ between an orbit point $c$ and $A$ must propagate to the orbit successor $x\diamond c$ . Sub-goal 3 assumes $x\diamond x\ne x$ and at least two elements outside the left orbit: if right translation by $x$ sends an orbit point outside the orbit, then $x$ has a fixer. Sub-goal 3 is not a smaller case of the goal. If it holds for every finite E677 magma, the mission goal follows by a product construction (milestone "Sub-goal 3 support (a)"), and the mission goal gives Sub-goal 3 because E255 at $x$ supplies the fixer $(x\diamond x)\diamond x$ . So a proof of Sub-goal 3 alone proves the goal, and a route through Sub-goals 1 and 2 still needs Sub-goal 3. The D4 milestones specialize Sub-goal 1 to a left orbit of exact period four; D4.0, D4.1, D4.2 and D4.4a–e are Proved, and the series is open at D4.3 and its five branch closeouts D4.3a–e. One conditional result (E677 together with conjugacy-closed left translations implies E255; Padilla, 2026) is Proved on this platform. The milestone "Alternative resolution — a finite E677 counterexample to E255" (theorem finite_counterexample) is the disproof target. The twelve milestones "Counterexample structure (a)–(l)" are constraints on a counterexample and rules for a counterexample search, all Proved; (h)–(j) take small-order facts as hypotheses. None of the twelve is a disproof. None of the displayed statements restricts the cardinality to a tested range.

In the authors' repository, Sub-goal 1 is called Piece 1, the unique-outsider case containing Sub-goal 2 is called Gap 1, and the case of Sub-goal 3 is called Gap ≥ 2; those names appear in some theorem sources and file names.

Significance

A resolution determines whether this particular law gains E255 as a consequence upon restriction to finite carriers. An affirmative proof must cover every finite cardinality, every operation on each carrier, and every assignment of the universally quantified elements. A finite counterexample must supply an operation that satisfies every instance of E677 while failing E255 at some element.

The formal package provides small, reusable statements of the two laws, the left translation property, and the fixer constraint. Keeping these statements separate allows their precise hypotheses and conclusions to be checked individually. In particular, the second supporting result says what a fixer must be when one exists; the fixer-existence formulation records the additional mathematical content needed to ensure existence.

Difficulty

The left translation conclusion concerns maps with the left input fixed. The fixer-existence formulation instead asks about the image of the map $y\mapsto y\diamond x$ , with its right input fixed. No assumption in the formal goal makes these two maps interchangeable. Bijectivity of every left translation map alone does not state that a fixer exists.

Because the implication fails for infinite magmas, E255 is not an equational consequence of E677, so any proof must use finiteness in an essential way. The blueprint's linear and linear-extension constructions cannot produce a finite counterexample (blueprint Lemmas 13.3 and 13.4); community members have formalized on this platform the scalar-coefficient case of the linear lemma, the Type I and Type II affine families, and the linear-extension lemma (see the discussion thread).

Likewise, checking a collection of finite operation tables does not quantify over arbitrary finite cardinalities. Such computation does not discharge the goal submitted here. Any proof must justify every use of finiteness and retain the displayed parenthesization of the laws.

Formalization scope

The representation uses an arbitrary universe-polymorphic type, an explicit binary operation, and a Fintype instance for finite targets. Passing the operation explicitly avoids importing a separate magma package or imposing algebraic typeclass laws. The predicates E677 and E255 themselves do not assume finiteness; each theorem states its own finite-carrier hypothesis.

Empty carriers are included. Both laws hold vacuously on them; the pointwise fixer statement is also vacuous because there is no element $x$ . Consequently an empty carrier cannot refute the main goal. Nonempty carriers of every finite size are included without further assumptions. There is no associativity, commutativity, idempotence, identity element, or cancellation hypothesis hidden in the representation.

The definition bundle FiniteMagmaE677 provides E677, E255, HasFixerAt (existence of a fixer of $x$ ), InLeftOrbit (membership in the left orbit of $x$ ) and OrbitRightCollisionOrFixer. Three further bundles, FiniteMagmaE677_seventh_element_growth, FiniteMagmaE677_d4_first_return and FiniteMagmaE677_sswap_renewal, hold the period-four packet types used by the D4 milestones.

Status (2026-10-22 UTC). Proved on this platform: Lemma 13.1(i)–(iii), the Sub-goal 1, Sub-goal 2 and Sub-goal 3 support lemmas, D4.0, D4.1, D4.2, D4.4a–e, the conditional conjugacy-closure theorem, Counterexample structure (a)–(l), and three further conditional theorems by omegaestable (existential row transport, parametrized row transport, left-parameter transport; see the discussion thread). Open: Sub-goals 1–3, D4.3 with D4.3a–e, and the counterexample target. Community members have also formalized the blueprint's 496-element non-right-cancellative E677 magma (see the discussion thread).

Welcome contributions: a proof of any open sub-goal or D4.3 branch (a proof of Sub-goal 3 proves the goal; see the milestone "Sub-goal 3 support (a)"); a proof sketch that reduces an open leaf to new lemmas; a concrete operation table on Fin n proving finite_counterexample, which would make the main theorem false.

Results in the captain's private repository that are not on this platform

Results that are Proved on this platform are not repeated here. Readers cannot open the captain's private repository, so each statement is given in full. PROVEN means a complete proof, in the form stated. BOUNDED means a SAT-solver result for one encoded finite structure: each UNSAT result used here has a DRAT proof that drat-trim accepted (unverified C code, not a kernel-checked proof). EMPIRICAL means a computation over finite data, with no certificate.

Not ported

A port of these results to this platform is a long job.

1. The finite facts that Counterexample structure (h)–(j) take as hypotheses. BOUNDED. No E677 magma has order 2, 3, 4, 6 or 8, and every E677 magma of order 7 or 9 has an idempotent. Counterexample structure (h) and (j) give the earlier sources and the captain's check. These facts, except those for orders 2 and 3, are hypotheses of theorems that are Proved on this platform, so a port of them would make (h)–(j) unconditional; it needs kernel-checked proofs made from the DRAT certificates. Source: the captain's private repository, revision dd2ee3e5, experiments/analysis/idempotent-free-small-orders-2026-10-21/ and, for order 8, experiments/analysis/ti-groups-2026-10-19/ (section "Order 8").

2. Period-four refinements of D4.2. PROVEN (Lean; axioms propext, Classical.choice, Quot.sound). A repository theorem refines the five branches of the D4.2 first-return packet and keeps every case. Write $q=c_2\diamond x$ . In the R-through-one branch (the repository's name), each outcome is a fixer of $x$ , a strict decrease of a finite contact budget, a new $L_x$ -cycle disjoint from the elements already accounted for (with its exact size), or a closed trace of $L_q^{-1}$ that is reduced to two phase-contact families. In the two unswapped S branches, each outcome is a fixer of $x$ , a decrease by one of the number of unmarked phases on the $L_x$ -cycle of $q$ , or a new disjoint $L_x$ -cycle with its exact size. These results do not close D4.3 or any of D4.3a–e, because the outcomes are not yet combined into one state with a strictly decreasing well-founded measure. A port needs the repository's library of period-four packets. Source: the captain's private repository, revision 4100763b, lean/E677/Spine/Piece1D4RefinedFirstReturn.lean, e677_d4_q_refinedFirstReturnPacket_of_firstReturnPacket.

Not formalized

These results have only solver or computation evidence, and no Lean proof, and no milestone on this platform uses them as hypotheses.

3. A counterexample has at least 11 elements. BOUNDED. Every E677 magma with at most 10 elements satisfies E255. By item 1, no E677 magma has order 2, 3, 4, 6 or 8. Every E677 magma of order 5, 7 or 9 is right-cancellative; then each right translation $y\mapsto y\diamond x$ is a bijection, so $x$ has a fixer, and Lemma 13.1(ii) gives E255 at $x$ . At order 10, a point $x$ at which E255 fails has no fixer, so $x$ is not in its own column $\{y\diamond x\}$ ; a search for such a table, split into 67 cases by the cycle type of $L_x$ (the length $l\ge 2$ of the left orbit of $x$ , and the cycle lengths of $L_x$ on the other $10-l$ elements), is UNSAT in each case. Earlier reports: the sources listed in Counterexample structure (h)–(j) for the E677 magmas of orders 2–4 and 6–9, among them a Lean proof for order 6 by J. Hyun. Without certificates (ETP Zulip thread "FINITE: 677 -> 255"): J. Brox (Mace4) reports no counterexample of order 9 (2024-12-10), none of order at most 10 (2025-01-10), no E677 magma of order 10 (2025-01-12) and no counterexample of order 11 (2025-03-12); B. Le Floch (2024-12-22, Mace4) reports none of order at most 9; J. Hyun (2025-01-09, CaDiCaL) none of order 5 to 8; S. Chuks-Onah (2026-05-01, Z3) none of order 5 or 6; R. Schneider (2026-05-15, own search program) and B. Subercaseaux (2026-05-15, SAT) none of order at most 11. The new content here is the DRAT certificates. Source: the captain's private repository, revision dd2ee3e5, experiments/analysis/rc-small-orders-2026-10-19/ and experiments/analysis/rc-cubes-2026-10-19/.

4. Order 35 over the quotient $x\diamond y=4x+3y$ on $\mathbb{F}_7$ . With the order-7 fact of item 1, Counterexample structure (i) gives: if a minimum counterexample $M$ has order 35 and a non-universal congruence $\theta$ that relates $z$ and $q(z)$ , then $M/\theta$ has 7 elements and an idempotent, and the classes have 5 elements. By the classification of J. Brox (ETP Zulip thread "FINITE: 677 -> 255", 2025-03-26, Mace4, no certificate; not checked by the captain), $M/\theta$ is then isomorphic to $4x+3y$ or to $4x+y$ on $\mathbb{F}_7$ . Take $Q$ : $x\diamond y=4x+3y$ on $\mathbb{F}_7$ (0 is its only idempotent), write the elements of $M$ as pairs $(a,s)$ with $a\in\mathbb{F}_7$ and $s\in\mathbb{F}_5$ , and call the map $(s,t)\mapsto$ (second coordinate of $(a,s)\diamond(b,t)$ ) the block over $(a,b)$ . Results, each for an E677 magma of this form:

If at most five blocks other than the block over $(0,0)$ do not have the form $(s,t)\mapsto s+f_{ab}(t-s)$ with a permutation $f_{ab}$ of $\mathbb{F}_5$ , then E255 holds. BOUNDED (two short paper lemmas reduce it to 48 patterns of non-cyclic blocks, each UNSAT). This includes the case where every block has this form.
If every block has the form $(s,t)\mapsto A\,t+B$ with $A\in\{1,4\}$ (with $A$ and $B$ chosen for each block and each $s$ ), then E255 holds. BOUNDED.
With all blocks free, the solver runs did not finish, so there is no result. The quotient $4x+y$ was not tested.

The only known E677 magma of order 35 with this quotient and classes of 5 elements is isomorphic to $Q\times M_5$ , where $M_5$ is $s\diamond t=2s-t$ on $\mathbb{F}_5$ (EMPIRICAL). T. Tao (ETP Zulip thread "FINITE: 677 -> 255", 2024-12-13) proposed the extensions $(x,s)\diamond(y,t)=(4x+3y,\ s\diamond_{x,y}t)$ , and (same thread, 2024-12-22) the fibre operations $s+f_{x,y}(t-s)$ of the first result. The second result is a case of the form $(x\diamond y,\ b_{x,y,s}t+c_{x,y,s})$ for which B. Reinke (same thread, 2026-06-11) expected E255 to lift, without a proof. Source: the captain's private repository, revision dd2ee3e5, docs/current/e677-coverage-matrix-2026-10-19.md (the rows for this quotient) and experiments/analysis/pro-counterexample-audit-2026-10-20/.

5. Extensions and row changes of the order-31 magma $x\diamond y=5x-4y+1$ on $\mathbb{F}_{31}$ . Call it $Q_{31}$ . It has no idempotent, and it is the smallest known E677 magma without an idempotent; every known E677 magma without an idempotent has order divisible by 31 (EMPIRICAL). So $Q_{31}$ is the smallest known candidate quotient in Counterexample structure (j). BOUNDED results:

No E677 magma has a congruence with quotient $Q_{31}$ and classes of 2 or 3 elements.
No E677 magma with a congruence with quotient $Q_{31}$ and classes of 4 elements fails E255.
In the family $(u,s)\diamond(w,t)=(5u-4w+1,\ s+h_{w-u}(t-s))$ on $\mathbb{F}_{31}\times\mathbb{F}_5$ , with permutations $h_d$ of $\mathbb{F}_5$ , no E677 member fails E255.
An E677 operation on $\mathbb{F}_{31}$ that fails E255 differs from $Q_{31}$ in at least 20 rows and at least 40 cells.

B. Le Floch (ETP Zulip thread "FINITE: 677 -> 255", 2024-12-13) found $5x-4y+c$ mod 31 as the one linear E677 model in which $c$ cannot be removed, and R. Schneider (same thread, 2025-11-21) noted that $Q_{31}$ has no idempotent. T. Tao (same thread, 2024-12-13) proposed the extensions $(x,s)\diamond(y,t)=(x\diamond y,\ s\diamond_{y-x}t)$ of $Q_{31}$ . B. Reinke (same thread, 2025-11-07) encoded them for z3 and wrote that he thought he could exclude classes of fewer than 4 elements (no certificate). The first result above covers all extensions with classes of 2 or 3 elements, not only these; the third result is the case of Tao's extensions with fibre operations $s+h(t-s)$ . Source: the captain's private repository, revision dd2ee3e5, experiments/analysis/gap2-monolith-m3-2026-10-17/ (§2, §6), experiments/analysis/pro-counterexample-audit-2026-10-20/ (cyclic-section-19375/, construction-19350/) and experiments/analysis/idempotent-free-2026-10-19/.

6. Tables with an automorphism of prime order, orders 12–21. BOUNDED, incomplete. Let $\alpha$ be an automorphism of prime order $p$ of an E677 magma of order $n$ , $12\le n\le 21$ , with $f$ fixed points, and let $z$ be a point with no fixer. The fixed points of $\alpha$ form a submagma, so an E677 magma of order $f$ ; by item 1, $f\notin\{2,3,4,6,8\}$ . The search covers $f\in\{0,1,5,7,9,11\}$ . If $\alpha$ fixes $z$ , the fixed points form an E677 magma of order $f$ that fails E255 at $z$ ; this is excluded for $f\le 9$ by item 3, and for $f=11$ 4 shapes $(n,p,f)$ are UNSAT and 7 were not run. If $z$ is in a cycle of $\alpha$ , 26 shapes are UNSAT (all 9 shapes with $p\ge 11$ , and 17 shapes with $p\le 7$ ); 14 shapes are UNKNOWN after 3600 s, and 40 were not run. Not covered: the shapes with $f=0$ and one cycle ( $n=p\in\{13,17,19\}$ , translation-invariant tables on $\mathbb{Z}_n$ ), fixed-point count 10, and fixed-point counts of at least 12. Earlier searches without certificates (ETP Zulip thread "FINITE: 677 -> 255"): R. Schneider (2025-11-20 and 2025-11-21) searched the tables with the automorphism $(0)(1\,2\,\dots\,n-1)$ up to $n=31$ and found only right-cancellative ones (the shapes with $f=1$ and one cycle); B. Reinke (2025-11-21) noted that the fixed points form a submagma and found no table with one cycle and 5 fixed points of order at most 25 other than order 5 (the shapes with $f=5$ and one cycle), and (2025-11-26) noted that in a minimum counterexample only the identity automorphism fixes the failing point. Source: the captain's private repository, revision dd2ee3e5, experiments/analysis/symmetric-simple-2026-10-20/.

7. A refuted intermediate statement. EMPIRICAL (exhaustive check of one explicit table, run twice with independent encodings; not kernel-checked). Let $S(x)$ be the closure of the left orbit of $x$ under $z\mapsto z\diamond x$ . The repository's gate CP asserted: in a finite E677 magma, if the hypotheses of Sub-goal 3 hold at $x$ and $z\mapsto z\diamond x$ is not injective on $S(x)$ , then the left orbit of $x$ has minimal period $7$ . Let $K$ be the 7-element E677 magma of Sub-goal 3 support (a) (file db/7/1 of the public collection memoryleak47/eq677), and let $M$ be the E677 magma of order $77$ in file db/77/65 of the same collection. In $M\times K$ (order $539$ ; E677 checked on all $539^2$ pairs), the point $(0,0)$ satisfies these hypotheses, and its left orbit has minimal period $\operatorname{lcm}(7,6)=42$ . The point $(0,0)$ has the fixer $(76,4)$ , so this is not a counterexample to E255, and it refutes CP only in the form without a no-fixer hypothesis. Source: the captain's private repository, revision 4100763b, docs/current/dead-ends/gap-ge-2.md, entry D7-6.

8. $\Phi(x)=(x\diamond x)\diamond x$ is not injective in general. EMPIRICAL. In the 2026-09-27 snapshot of the community database (4091 tables), every table satisfies E255, and $\Phi$ is not injective in 1107 of them; the smallest such order is $7$ . In one table of order 7, row 6 is the identity row, so $6$ is a fixer of every $x$ and Lemma 13.1(ii) gives $\Phi(x)=6$ for every $x$ . So global injectivity of $\Phi$ is not a usable intermediate target. Source: the captain's private repository, revision 4100763b, docs/professor-phi-injectivity.md, correction of 2026-10-14.

Selected references

Matthew Bolan et al., The Equational Theories Project: Advancing Collaborative Mathematical Research at Scale, arXiv:2512.07087v2 (December 16, 2025), paper.
The Equational Theories Project contributors, Equational Theories, online proof blueprint, Chapter 13, equations (1)–(2) and Lemmas 13.1–13.4, chapter, accessed September 7, 2026 and October 13, 2026.
J. I. Padilla (GitHub: omegaestable), Prove E677 implies E255 under left translation conjugacy closure, Equational Theories Project pull request #1518, 2026 (closed without merge by its author on 2026-10-13; the theorem is Proved on this platform), pull request.
Lean Zulip, channel "Equational", thread "FINITE: 677 -> 255", thread.

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Computational Geometry Discrete Geometry Graph Theory·Captain: hao jia

Uniform Obstacle Bounds for Planar Graphs (OPG-37357)Open Problem

Motivation

An obstacle representation turns a graph into a visibility system: vertices are points in the plane, and nonedges are blocked by polygonal obstacles. The obstacle number asks for the minimum number of obstacles needed. OPG-37357 records two different questions for planar graphs. The first asks whether one obstacle can ever be insufficient. The second asks whether some universal constant bounds the ordinary obstacle number of every planar graph.

The status of the two parts is different. Berman, Chappell, Faudree, Gimbel, Hartman, and Williams proved in 2017 that explicit planar graphs, including the icosahedron and their graphs $X_4$ and $X_6$ , have ordinary obstacle number two. Thus the first question has a published positive answer. The universal-constant question remains the research target here. A separate invariant called planar or plane obstacle number requires a crossing-free visibility drawing; results for that invariant must not be substituted for the ordinary obstacle number used by this mission.

Setting

A finite simple graph $G$ has a $k$ -obstacle drawing when its vertices are placed injectively as points in $\mathbb R^2$ and there are $k$ pairwise disjoint closed connected polygonal obstacles such that

uv\in E(G) \quad\Longleftrightarrow\quad [p(u),p(v)]\text{ meets no obstacle}.

Graph vertices lie outside every obstacle. The ordinary obstacle number $\operatorname{obs}(G)$ is the least such $k$ . The drawing itself may contain crossings between visible graph edges; planarity is a property of the abstract input graph, not an extra constraint on the obstacle drawing.

The Lean model represents a polygonal obstacle as a connected finite union of closed filled triangles. This gives a compact polygonal region with exact real-coordinate segment incidence. Straight-line planarity of the abstract graph is represented separately.

Formalization targets

The two-part OPG record

The source records both

\exists\text{ finite planar }G,\ \operatorname{obs}(G)>1

and

\exists k\in\mathbb N\ \forall\text{ finite planar }H, \ \operatorname{obs}(H)\le k.

The first assertion is known in the literature and appears as a published-result milestone. The second is open and is therefore the mission's main theorem. Together they preserve the two-part source without presenting the whole record as unresolved.

Published first part

A milestone formalizes the stronger published statement

\exists\text{ finite planar }G, \qquad \operatorname{obs}(G)\le2 \quad\text{and}\quad \operatorname{obs}(G)\not\le1.

This captures ordinary obstacle number exactly two without hard-coding one graph before its adjacency data and lower-bound certificate are formalized.

Universal bound

The open milestone asks for a single natural number $k$ , chosen before the graph, that works for every finite planar graph. The number of obstacle corners is not bounded by this theorem; only the number of connected polygonal obstacles is.

Significance

The published first part establishes that planarity alone does not force a one-obstacle representation. The second part asks whether planar graphs nevertheless have uniformly bounded visibility complexity. A positive answer would produce a common finite obstacle budget independent of graph order; a negative answer would require a family of planar graphs with unbounded ordinary obstacle number.

Formalization is especially useful because several nearby notions differ by one word but have different known bounds: ordinary versus plane obstacle number, arbitrary polygonal versus convex obstacles, and fixed-placement versus freely chosen drawings. The mission's definitions make those choices explicit and provide reusable segment-obstacle semantics for later geometric graph formalizations.

Difficulty

A finite combinatorial graph does not come with a canonical visibility drawing. Even when one starts with an arbitrary connected blocking set, replacing it by one bounded simple polygon requires compactness, component, incidence, and polygonal-neighborhood arguments. Conversely, lower bounds must quantify over every possible placement and obstacle, not merely refute a selected coordinate drawing.

Counting results for unrestricted graphs do not automatically preserve planarity. Bounds for planar obstacle number impose a crossing-free drawing and therefore answer a different question. The known two-obstacle examples close only the existential first part and give no universal $k$ .

Formalization scope

All graph vertex types are finite. Obstacles are closed connected polygonal regions represented by finite triangle unions; they are pairwise disjoint and avoid graph vertices. Visibility uses the full closed segment, so tangency or boundary contact blocks a nonedge. The planarity witness is independent of the obstacle drawing. Empty and one-vertex graphs remain in the universal quantifier and should be handled without division or nonemptiness assumptions.

The repository's fixed-placement polygonization argument and finite arrangement code are candidate_only. They may motivate supporting lemmas, but they neither prove the unrestricted obstacle-drawing completeness theorem nor settle the universal bound. Contributions are welcome on exact geometry primitives, the published two-obstacle construction and lower bound, conversions between connected blockers and polygonal obstacles, and the universal root. A proof for the plane invariant, convex invariant, one fixed drawing, or a finite order cutoff must be labeled at that narrower scope.

Selected references

L. W. Berman, G. G. Chappell, J. R. Faudree, J. Gimbel, C. Hartman, and G. I. Williams, Graphs with Obstacle Number Greater than One, JGAA 21(6), 2017. https://doi.org/10.7155/jgaa.00452
J. Gimbel, P. Ossona de Mendez, and P. Valtr, Obstacle Numbers of Planar Graphs, Graph Drawing 2017. https://arxiv.org/abs/1706.06992
M. Balko, S. Chaplick, R. Ganian, S. Gupta, M. Hoffmann, P. Valtr, and A. Wolff, Bounding and Computing Obstacle Numbers of Graphs, SIAM Journal on Discrete Mathematics 38(2), 2024. https://arxiv.org/abs/2206.15414
Open Problem Garden / UnsolvedMath, OPG-37357. https://www.unsolvedmath.com/problems/OPG-37357

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Combinatorics Graph Theory Optimization·Captain: hao jia

Clique Partitions of Chordal Graphs (Erdos Problem 81)Open Problem

Motivation

An edge partition into cliques compresses the adjacency structure of a graph into complete pieces without allowing any edge to be counted twice. Erdős Problem 81 asks for the asymptotically sharp upper bound on the number of pieces needed when the graph is chordal. Chordal graphs have strong elimination structure, but that structure does not make the partition parameter additive under arbitrary edge deletion, and obtaining a linear error term remains substantially stronger than identifying the leading quadratic coefficient.

Erdős, Ordman, and Zalcstein studied clique partitions of chordal graphs in 1993. Their examples already exhibit the $n^2/6$ scale, while their general upper estimate had a larger quadratic coefficient. Later dense-packing results of Haxell–Rödl and Yuster compare fractional and integer triangle packings with an $o(n^2)$ gap. The project candidate combines that interface with chordal elimination arguments to formulate a uniform $n^2/6+o(n^2)$ milestone. It does not supply the $O(n)$ remainder asked for by the root.

Setting

A finite simple graph is chordal when it has no induced cycle of length greater than three. The Lean definition uses the equivalent perfect-elimination form: vertices admit an injective ranking such that the later neighbors of every vertex form a clique.

An edge partition into cliques is a finite family $\mathcal P$ of complete vertex sets such that every edge of $G$ belongs to exactly one member of $\mathcal P$ . Members may share vertices but may not share edges. Write $\operatorname{cp}(G)$ for the minimum possible number of pieces.

The asymptotic notation

\frac{n^2}{6}+O(n)

means that there are constants $C>0$ and $n_0\ge1$ , chosen independently of $G$ and $n$ , such that every chordal $n$ -vertex graph with $n\ge n_0$ has a clique partition with at most $n^2/6+Cn$ pieces.

Formalization targets

Erdős Problem 81

The root theorem is

\exists C>0\ \exists n_0\ge1\ \forall n\ge n_0\ \forall G\text{ chordal on }n\text{ vertices}, \qquad \operatorname{cp}(G)\le \frac{n^2}{6}+Cn.

The quantifier order is essential: $C$ and $n_0$ are universal and cannot depend on the graph.

Leading-coefficient milestone

The supporting target records the weaker uniform statement

\forall\varepsilon>0\ \exists n_0\ \forall n\ge n_0\ \forall G\text{ chordal on }n\text{ vertices}, \qquad \operatorname{cp}(G)\le \left(\frac16+\varepsilon\right)n^2.

This is the precise $n^2/6+o(n^2)$ form. It is not equivalent to the root: choosing $\varepsilon=1/n$ is invalid because the cutoff may depend on the fixed value of $\varepsilon$ .

Significance

The root would determine the clique-partition extremum for chordal graphs up to a linear remainder, matching the scale of the complete-split examples that motivate the coefficient $1/6$ . It would refine a leading-order asymptotic theorem into a uniform estimate strong enough to distinguish second-order behavior.

Formalization creates a clean interface among perfect elimination orderings, exact edge partitions, fractional edge-and-triangle decompositions, and integer triangle packings. It also forces the proof to distinguish a partition from a cover and original graph order from the order of any auxiliary hypergraph. These definitions can support other decomposition problems on chordal and split graphs.

Difficulty

Perfect elimination does not by itself give the sharp partition count. Greedily taking maximal cliques may overlap in edges or accumulate too many singleton pieces. Similarly, a fractional edge-and-triangle partition can achieve the right leading coefficient while integer rounding loses $o(n^2)$ pieces; the root requires that loss to be only $O(n)$ .

The dense-packing theorem has quantifiers of the form “for every fixed $\varepsilon>0$ there exists $N(\varepsilon)$ .” It therefore yields a uniform subquadratic error but no linear error. Any proof of the root must add a chordal-specific rounding or extremal reduction rather than treating the general packing theorem as if its $\varepsilon$ could vary with $n$ .

Formalization scope

Graphs are finite and simple. Chordality is encoded by existence of a perfect-elimination ranking, including disconnected and edgeless graphs. A clique piece is a finite vertex set that spans a complete subgraph. Exactness means every actual edge occurs in exactly one piece; no nonedge can occur inside a piece. Bounds are compared in $\mathbb R$ so the displayed asymptotic expressions retain their conventional form, while the number of parts remains a natural number.

The candidate derivation of the leading coefficient imports finite linear-programming duality and the Haxell–Rödl/Yuster fixed-triangle packing approximation. It is candidate_only, not an admitted result or kernel proof. Contributions may formalize the perfect-elimination lemmas, the fractional compression, the uniform packing interface, complete-split lower examples, or the root linear rounding theorem. A result for edge-and-triangle pieces only, a fractional partition, or one fixed order must not be presented as the unrestricted integer clique-partition theorem.

Selected references

P. Erdős, E. T. Ordman, and Y. Zalcstein, Clique Partitions of Chordal Graphs, Combinatorics, Probability and Computing 2(4), 1993. https://doi.org/10.1017/S0963548300000808
P. E. Haxell and V. Rödl, Integer and Fractional Packings in Dense Graphs, Combinatorica 21, 2001. https://doi.org/10.1007/s004930170003
R. Yuster, Integer and fractional packing of families of graphs, 2003. https://arxiv.org/abs/math/0305350
Erdős Problems, Problem 81. https://www.erdosproblems.com/81

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