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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.

≥ 2.131121719190Formalized record≥ 2.15Open frontier

9 provers on it18 of 19 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 $2$ .

≥ 2.131121719190Formalized record≥ 2.15Open frontier

9 provers on it18 of 19 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

Operations Research Probability Statistics·Captain: mikedeng1

Achieving Efficiency in Black-Box Simulation of Distribution Tails with Self-Structuring Importance Samplers: The Self-Structuring Importance Sampler Is Logarithmically EfficientResearch Paper

Motivation

Many operational risk questions ask for a small probability $p_u=P(L(X)\ge u)$ : the chance that the cost $L$ of a decision, evaluated at a random scenario $X\in\mathbb R^d$ , exceeds a large threshold $u$ . Examples are large delays in project networks or vehicle routing, large portfolio credit losses, and failures in engineered systems. Plain Monte Carlo needs on the order of $p_u^{-1}$ samples to estimate $p_u$ to fixed relative accuracy, which is prohibitive when $p_u$ is $10^{-6}$ or smaller. Importance sampling (IS) draws samples from a different distribution and reweights them by a likelihood ratio, but classical efficient IS schemes are built for one model at a time: they need the distribution of $X$ and the structure of $L$ (e.g. linearity or convexity) to solve for a good sampling distribution.

Deo and Murthy (arXiv:2102.07060v3, published in Operations Research, 2025) propose a self-structuring sampler that needs neither: it applies a fixed multiplicative stretching to ordinary samples of $X$ and uses only black-box evaluations of $L$ . Their main result is that this sampler is asymptotically optimal in the standard sense of logarithmic efficiency (Asmussen and Glynn 2007, Ch. 6) over a nonparametric class of light- and Weibull-type-tailed distributions and nonconvex losses.

Setting

Write $\mathbb R^d_+=\{x:x\ge0\}$ and $\mathbb R^d_{++}=\{x:x>0\}$ (componentwise), and $\|x\|=\max_i|x_i|$ . The random vector $X=(X_1,\dots,X_d)$ has density $f_X$ . Its marginal cumulative hazards are $\Lambda_i(x)=-\log P(X_i>x)$ , $\Lambda(x)=(\Lambda_1(x_1),\dots,\Lambda_d(x_d))$ , and $Y=\Lambda(X)$ is the standardized vector. Let $q_i=\Lambda_i^{\leftarrow}$ be the left-continuous inverse, $q(y)=(q_1(y_1),\dots,q_d(y_d))$ , $\hat q(t)=q(t\mathbf 1)/\|q(t\mathbf 1)\|$ and $\Lambda_{\min}(u)=\min_i\Lambda_i(u)$ .

The standing assumptions are:

Assumption 2.1 on the loss: risk sets $\{x\in\operatorname{supp}X: L(x)\ge u\}$ lie in $\mathbb R^d_+$ for large $u$ , and $L(nx_n)/n^\rho\to L^*(x)$ for every $x_n\to x$ in $\mathbb R^d_+$ , with $\rho>0$ and $L^*>0$ somewhere on $\mathbb R^d_+$ .
Assumption 3.1 on the marginals: each $\Lambda_i$ is continuous, eventually strictly increasing, and regularly varying with index $\alpha_i>0$ ( $\Lambda_i(sx)/\Lambda_i(s)\to x^{\alpha_i}$ ).
Assumption 3.2 on the dependence: $Y$ has density $p(y)e^{-\varphi(y)}$ with $n^{-1}\varphi(ny_n)\to I(y)$ and $n^{-\varepsilon}\log p(ny_n)\to0$ along every $y_n\to y\ne0$ in $\mathbb R^d_+$ .
Assumption 5.1: each $\Lambda_i$ is eventually convex or concave.
The limit $q^*=\lim_{t\to\infty}\hat q(t)$ exists.

The sampler (Algorithm 1) fixes a hyper-parameter $l=l(u)$ and maps a sample $x$ to $T(x)=x\,(u/l)^{\kappa(x)}$ , with either exponent

\kappa^{(1)}(x)=\frac1\rho\frac{\log(1+|x|)}{\|\log(1+|x|)\|_\infty}\qquad\text{or}\qquad \kappa^{(2)}(x)=\frac{\log(1+|x|)}{\log l}.

With $J$ the closed-form Jacobian of Table 1 and $Z=T(X)$ , the single-sample estimator is

\zeta(u)=\frac{J(X)f_X(Z)}{f_X(X)}\,\mathbb I\big(L(Z)\ge u\big),\qquad M_{2,u}=E[\zeta(u)^2].

Formalization targets

Goal: Theorem 5.1 (logarithmic efficiency)

Under Assumptions 2.1, 3.1, 3.2, 5.1, existence of $q^*$ , $L^*(q^*x)>0$ for some $x\in\mathbb R^d_+$ , and any slowly varying $l(u)\to\infty$ , for either exponent,

M_{2,u}<\infty\ \text{for large }u,\qquad \lim_{u\to\infty}\frac{\log M_{2,u}}{\log p_u^2}=1.

The goal fixes neither the exponent, nor the hyper-parameter $l$ , nor the loss or the distribution beyond the stated assumptions.

Milestones

The milestones follow the paper's proof chain (Appendix A.3 for $\kappa^{(1)}$ , Appendix E for $\kappa^{(2)}$ ):

Proposition 2.1: $\zeta(u)$ is unbiased and $T$ is invertible.
Lemma 3.1: the coordinates of $Y$ are standard exponential.
Theorem 3.1: $\{Y/t\}$ satisfies a large deviations principle with rate $I$ .
Lemma 3.2: $I$ is continuous and homogeneous, and $\inf_{y_i>c}I=c$ .
Lemma A.1: the rare event in standardized coordinates.
Theorem 4.1: the tail asymptotic

\log p_u=-\Lambda_{\min}(u^{1/\rho})\big[I^*+o(1)\big],\qquad I^*=\inf\{I(y):L^*(q^*y^{1/\alpha})\ge1,\ y\ge0\}.

Lemmas A.4 and A.5: bounds on $T^{-1}$ , and $\psi_u=\Lambda\circ T^{-1}\circ q$ is $o(t(u))$ on compacts.
Lemma A.6: $M_{2,u}$ as a Laplace-type functional of $Y_u=Y/t(u)$ .
Lemma A.7: the density exponent $a_u\ge I+o(1)$ .
Lemma A.9: the rare set stays a fixed distance from the origin.
Display (85): the $\kappa^{(2)}$ bound on $T^{-1}$ .

Significance

Theorem 4.1 gives the exponential decay rate of $p_u$ for a broad class of nonconvex losses (piecewise affine and quadratic functions, values of mixed integer programs, neural-network feature maps) under nonparametric, copula-style dependence. Previously such asymptotics were known only for specific losses and distribution families. Theorem 5.1 shows that one simple, model-oblivious transformation attains the best possible exponential rate for the second moment, so the relative error grows more slowly than any power $p_u^{-\varepsilon}$ . This supports using the sampler for value-at-risk, conditional value-at-risk and chance-constrained optimization.

The results are proved in the paper's appendices (Appendices A, D.1 and E). None of them has a machine-checked proof. Formalizing them would require regular variation and Potter bounds, a tail large deviations principle in $\mathbb R^d$ , and a Varadhan-type integral lemma along a real speed. These are general tools that are missing from Mathlib and are reusable well beyond this paper.

Difficulty

The obvious argument is to bound $M_{2,u}$ by computing the likelihood ratio on the rare set. This fails because the rare set, after standardization, is described only asymptotically: $L_u(x)=u^{-1}L(q(t(u)x))$ converges to $f_{LD}(x)=L^*(q^*x^{1/\alpha})$ only continuously on $\mathbb R^d_{++}$ , not uniformly. The exponent of the likelihood ratio involves the density of $Y$ at two different scales ( $t(u)p$ and $\psi_u(t(u)p)$ ) and the hazard rates at points pulled back by $T^{-1}$ . A lower bound on it must hold uniformly enough to apply a Varadhan-type lemma with a varying family of functions, while the transformed sample stays inside a region where the asymptotic density assumptions apply. The upper and lower large-deviations bounds also rely on homogeneity of $I$ and $f_{LD}$ , which hold only for the heaviest-tailed coordinates of $q^*$ .

Formalization scope

Points of $\mathbb R^d$ are Fin d → ℝ with the sup norm, and components are 0-based. $X$ is modelled by its law $\mu$ together with a density $f_X$ . $M_{2,u}$ is a lower Lebesgue integral and $p_u=\mu\{L\ge u\}$ ; both are extended nonnegative reals, so neither $+\infty$ nor $0$ is collapsed to a junk value. Logarithms of probabilities in large-deviations statements are taken in EReal, where $\log0=-\infty$ . $I^*$ is an EReal infimum, equal to $+\infty$ over an empty set. Regular variation is the real-variable limit with an eventually positive denominator, and reuses the published definition HeavyTailNV.Tail.IsRegularlyVarying. The Jacobian is Table 1's closed form. Statements that evaluate $q$ at $t(u)p$ take $p\in\mathbb R^d_{++}$ .

The following hypotheses are added relative to the printed statements, each disclosed on its item:

$l(u)\to\infty$ in Theorem 5.1 and Lemma A.7 (the paper states it in §2.2 and uses it in Lemma A.5 and Appendix D.1).
" $L^*(q^*x)$ not identically zero" is read as positivity somewhere.
Measurability of $L$ .
$0<l\le u$ , and $l>1$ for $\kappa^{(2)}$ , together with an IS support condition in Proposition 2.1 and Lemma A.6. Proposition 2.1 is false as printed without them.
$p(ny_n)>0$ eventually inside (13).
$d\ge1$ where junk values at $d=0$ would otherwise matter.
In Theorem 4.1, continuity of the rate $I$ on $\mathbb R^d_+\setminus\{0\}$ , which the theorem's hypothesis "the tail LDP in Theorem 3.1" carries through Lemma 3.2(a) and which the proof uses.

The following printed statements are corrected:

Lemma A.6 is stated without its spurious factor $t(u)^{2d}$ .
Lemma A.9 is restricted to points with $q(t(u)p)\in\operatorname{supp}X$ . As printed it fails when $L$ is large off the support.
Lemma 3.2's " $I(0)=0$ " becomes a limit at $0$ , since (13) does not determine $I(0)$ .
Theorem 3.1 is stated with real speed $t$ , which implies the printed sequence form.

Lemmas A.8 and A.10 and Corollary A.1 are not posed. Their uniform claims over all of $\mathbb R^d_+$ depend on pointwise versions of the hazard rates and on the values of $L$ off the support and off $\mathbb R^d_+$ , which the hypotheses do not control; Corollary A.1 fails as printed when $X$ has negative components.

A trivializing formalization is ruled out explicitly. The goal is the limit (23) for the estimator $\zeta(u)$ of §5.2 with Table 1's Jacobian, for every admissible $l$ and both exponents. It is not the bound $M_{2,u}\ge p_u^2$ , not a statement about an unspecified IS density, and not a statement for one fixed $l$ . The finiteness conjunct prevents Lean's $\log 0=0$ convention from satisfying it.

Contributions are welcome at every level. The general tools (Potter bounds, the tail LDP, a Varadhan lemma for continuously converging functions) are useful as separate developments.

Selected references

A. Deo, K. Murthy, Achieving Efficiency in Black-Box Simulation of Distribution Tails with Self-Structuring Importance Samplers, Operations Research, 2025; preprint arXiv:2102.07060v3, 2023. https://arxiv.org/abs/2102.07060
S. Asmussen, P. W. Glynn, Stochastic Simulation: Algorithms and Analysis, Springer, 2007. https://doi.org/10.1007/978-0-387-69033-9
L. de Haan, A. Ferreira, Extreme Value Theory: An Introduction, Springer, 2006. https://doi.org/10.1007/0-387-34471-3
A. Dembo, O. Zeitouni, Large Deviations Techniques and Applications, 2nd ed., Springer, 1998. https://doi.org/10.1007/978-1-4612-5320-4

17 thms2 active usersReviewed

Algorithmic Game Theory Mechanism Design·Captain: mikedeng1

Round-Robin Beyond Additive Agents: Existence and Fairness of Approximate Equilibria 1: For Cancelable Valuations the Bluff Profile Is an Exact Pure Nash Equilibrium of Round-RobinResearch Paper

Round-Robin as a game

Round-Robin is the simplest allocation procedure for indivisible goods: agents take turns, in a fixed order, and at each turn the active agent takes her favourite good among those still available. Run on truthful preferences with additive valuations it returns an allocation that is envy-free up to one good (EF1), which is why it is the default baseline of discrete fair division. The procedure, however, only asks each agent for a ranking of the goods, and an agent may gain by misreporting: taking a good that others want early, and a good nobody else wants later, can improve her bundle. Round-Robin is therefore a game whose strategies are rankings, and the natural questions are whether it has pure Nash equilibria and whether the allocations at equilibrium are still fair.

For additive valuations the answer to the first question is known: an exact equilibrium exists, given by a profile now called the bluff profile, with a proof fragmented over Aziz, Goldberg and Walsh (2017), Bouveret and Lemaître (2016) and Amanatidis et al. (2021), as recalled on p. 7 of the paper below. Amanatidis, Birmpas, Lazos, Leonardi and Reiffenhäuser (arXiv:2301.13652, Mathematics of Operations Research, 2025) move beyond additive agents, to valuations for which the goods already held change the value of the remaining ones. This mission formalizes their first result: for cancelable valuations, the bluff profile is an exact pure Nash equilibrium.

Setting

There are $n \ge 1$ agents $N = [n]$ and a finite set $M$ of $m$ goods. Agent $i$ has a valuation function $v_i : 2^M \to \mathbb R$ that is normalized ( $v_i(\emptyset) = 0$ ) and non-decreasing. A valuation $v$ is cancelable if for all $S, T \subseteq M$ and every good $g \notin S \cup T$ ,

v(S \cup \{g\}) > v(T \cup \{g\}) \;\Longrightarrow\; v(S) > v(T).

Cancelability says that adding a common good never reverses a strict comparison. Additive valuations are cancelable; the paper notes (p. 3) that cancelable functions also generalize budget-additive, unit-demand and multiplicative valuations, and (p. 5) that neither the cancelable nor the submodular class contains the other.

A ranking $\succ_i$ is a total order on $M$ . Given a profile $\boldsymbol\succ = (\succ_1, \dots, \succ_n)$ , Round-Robin (Mechanism 1) runs $m/n$ rounds; in each round agents $1, \dots, n$ in turn receive $\mathrm{top}(\succ_i, S)$ , the $\succ_i$ -largest good of the set $S$ of goods still available. The bundle of agent $i$ is $A_i(\boldsymbol\succ)$ . The profile $\boldsymbol\succ$ is an exact pure Nash equilibrium (PNE) if for every agent $i$ and every ranking $\succ_i'$ ,

v_i\big(A_i(\succ_i', \boldsymbol\succ_{-i})\big) \le v_i\big(A_i(\boldsymbol\succ)\big).

A strict truthful profile $\boldsymbol\succ^*$ has every agent rank the goods by non-increasing singleton value $v_i(\{g\})$ , with ties broken in any fixed way. Let $h_1, \dots, h_m$ be the goods in the order Round-Robin $(\boldsymbol\succ^*)$ allocates them. The bluff profile $\boldsymbol\succ^{\mathrm b}$ has every agent report the same ranking $h_1 \succ h_2 \succ \cdots \succ h_m$ .

Formalization targets

Goal: Theorem 3.2

For cancelable valuations and every strict truthful profile $\boldsymbol\succ^*$ , the bluff profile is an exact PNE of Round-Robin:

\forall i \in N,\ \forall \succ_i' :\quad v_i\big(A_i(\succ_i', \boldsymbol\succ^{\mathrm b}_{-i})\big) \le v_i\big(A_i(\boldsymbol\succ^{\mathrm b})\big).

Milestones

Lemma 2.2. If $v$ is cancelable and $R \cap (S \cup T) = \emptyset$ , then $v(S \cup R) > v(T \cup R) \Rightarrow v(S) > v(T)$ , $v(S) \ge v(T) \Rightarrow v(S \cup R) \ge v(T \cup R)$ , and $v(S) = v(T) \Rightarrow v(S \cup R) = v(T \cup R)$ .
Lemma 3.3. For a cancelable valuation and $k$ -element sets $X = \{x_1, \dots, x_k\}$ , $Y = \{y_1, \dots, y_k\}$ : if $v(\{x_j\}) \ge v(\{y_j\})$ for every $j$ , then $v(X) \ge v(Y)$ .
Fact 3.1. Round-Robin $(\boldsymbol\succ^{\mathrm b})$ and Round-Robin $(\boldsymbol\succ^*)$ return the same allocation.
The ordering $Y^*$ (proof of Theorem 3.2). If agent $i$ deviates to any ranking and receives $A_i$ , then $A_i$ has an ordering $(y^*_1, \dots, y^*_k)$ with $v_i(\{y^*_r\}) \le v_i(\{h_{(r-1)n+i}\})$ for every round $r$ .

A companion item, Proposition 2.5, states that Round-Robin $(\boldsymbol\succ^*)$ is EF1 for cancelable valuations; with Fact 3.1 and Theorem 3.2 it exhibits an exact PNE whose allocation is EF1.

Significance

The theorem gives an explicit equilibrium of Round-Robin, computable by one run of the mechanism on truthful singleton rankings, for a valuation class strictly larger than the additive one. Combined with Proposition 2.5 it shows that Round-Robin with cancelable agents always has an exact equilibrium that is EF1, so strategic behaviour need not cost fairness. The same paper shows that this fails for submodular valuations, where the best one can guarantee is a $1/2$ -approximate equilibrium (the second mission of this series), and that every approximate equilibrium remains approximately EF1 (the third and fourth missions).

The result is proved in the paper; nothing about its truth is open. As far as is known, no part of it, and no part of the Round-Robin mechanism, the EF1 notion or the class of cancelable valuations, has been machine-checked. This mission formalizes the known proof and, on the way, builds a reusable model of Round-Robin as a function of reported rankings.

Difficulty

The proof compares two executions of Round-Robin: one in which everybody reports the bluff order, and one in which agent $i$ reports a different ranking. A deviation changes which goods are available at every later step, and so changes what every other agent takes; the obvious attempt, comparing agent $i$ 's $r$ -th good in the two runs directly, fails because her $r$ -th good under the deviation may be a good that was already gone in round $r$ of the bluff run. The argument needs a reordering of the deviation bundle under which the round-by-round comparison holds, and then a set-level inequality that cancelable (not additive) valuations still support. In Lean, the main work is in the bookkeeping of the two executions: which goods are allocated before a given step, and how a change of one report propagates.

Formalization scope

Agents are Fin n with $n \ge 1$ , numbered from $0$ ; goods form a finite type $M$ with decidable equality. Valuations are real-valued functions on Finset M, normalized and non-decreasing, hence nonnegative. A ranking is a list of all goods without repetition, most preferred first. Round-Robin is defined by recursion on steps: step $t$ (from $0$ ) belongs to agent $t \bmod n$ , and agent $i$ 's bundle collects the goods picked at her steps among the first $m$ ; the paper's step $(r-1)n + i$ is step $(r-1)n + (i-1)$ . The bluff profile is computed from the strict truthful profile, and the theorem quantifies over every strict truthful profile, i.e. every tie-breaking. Deviations range over rankings only, as in Definition 2.4.

The paper assumes $m = kn$ "for simplicity", noting that dummy goods reduce the general case to it. Every statement that uses the round structure carries the hypothesis $|M| = kn$ ; the dummy-goods reduction is not formalized. Lemma 2.2 is stated for any real-valued cancelable function, which the page's proof covers. A bluff profile defined as an arbitrary equilibrium, deviations over arbitrary lists, or an EF1 computed from reports instead of true valuations would each trivialize or change the statement; none is used.

Definitions needed: valuations, cancelability, rankings and top, Round-Robin with its step bookkeeping, approximate PNE and EF1, strict truthful profiles and the bluff profile. All are in one definitions file, written to be shared with the other missions of the series. Proofs of the milestones, and lemmas about Round-Robin executions (one good per step, bundles of size $k$ , the effect of changing one report), are welcome contributions.

Selected references

G. Amanatidis, G. Birmpas, P. Lazos, S. Leonardi, R. Reiffenhäuser, Round-Robin Beyond Additive Agents: Existence and Fairness of Approximate Equilibria, arXiv:2301.13652v1, 2023; Mathematics of Operations Research, 2025. https://arxiv.org/abs/2301.13652
G. Amanatidis, G. Birmpas, F. Fusco, P. Lazos, S. Leonardi, R. Reiffenhäuser, Allocating Indivisible Goods to Strategic Agents: Pure Nash Equilibria and Fairness, WINE 2021, LNCS 13112, pp. 149–166.
H. Aziz, P. Goldberg, T. Walsh, Equilibria in Sequential Allocation, ADT 2017, LNCS 10576, pp. 270–283.
S. Bouveret, M. Lemaître, Characterizing Conflicts in Fair Division of Indivisible Goods Using a Scale of Criteria, Autonomous Agents and Multi-Agent Systems 30(2):259–290, 2016.
R. J. Lipton, E. Markakis, E. Mossel, A. Saberi, On Approximately Fair Allocations of Indivisible Goods, EC 2004, pp. 125–131.

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Operations Research Probability Stochastic Systems·Captain: mikedeng1

Learning and Balancing Unknown Loads in Large-Scale Systems I: The Refined Learning Scheme Settles the Threshold at mΔ and Balances the Load on (m, Δ)-Bounded IntervalsResearch Paper

Motivation

In adaptive video streaming and online gaming, tasks are dispatched to one of $n$ identical server pools; all tasks in a pool are served in parallel, so a task's duration does not depend on the load, but its quality of service does. The total number of tasks then does not depend on how tasks are dispatched (the pools together form an infinite-server system), and the relevant objective is a balanced distribution of tasks over the pools: every Schur-concave utility of the occupancy vector, including the $\alpha$ -fair utilities, rewards balance (Goldsztajn, Borst & van Leeuwaarden 2024, §1). Join the Shortest Queue is optimal for this purpose but needs the exact occupancy of every pool.

A threshold dispatching rule needs much less. With an integer $\Delta\ge1$ and a threshold $\ell_n$ , it keeps every pool between $\ell_n$ and $h_n:=\ell_n+\Delta$ tasks whenever it can. Choosing $\ell_n$ well requires the offered load $\rho=\lambda/\mu$ , which in practice is unknown and varies in time. The refined learning scheme of Goldsztajn, Borst and van Leeuwaarden adjusts $\ell_n$ online, in steps of $\Delta$ , using the same coarse state information as the dispatcher. This mission formalizes their main result on that scheme (Theorem 2 of arXiv:2012.10142v3): along time intervals where the load is suitably bounded, the threshold settles at the right value and the load becomes balanced.

Timeline. The threshold rule with $\Delta=1$ and the fixed threshold $\lfloor\rho\rfloor$ was shown to have the same fluid and diffusion limits as Join the Shortest Queue for constant arrival rates and exponential service (Goldsztajn et al., INFORMS Journal on Computing, 2022). The present paper (arXiv 2020, revised 2024; Mathematics of Operations Research, 2025) removes the knowledge of $\rho$ , allows a time-varying arrival rate, and proves almost-sure statements about the learned threshold.

Setting

Fix a horizon $T>0$ , a measurable arrival-rate function $\lambda:[0,T]\to[0,C]$ , a service rate $\mu>0$ and an integer $\Delta\ge1$ . The system with $n$ pools receives tasks at rate $n\lambda(t)$ ; service times are exponential with rate $\mu$ . The occupancy state $q_n(t)=(q_n(t,i))_{i\ge0}$ records the fraction $q_n(t,i)$ of pools with at least $i$ tasks, so it lies in

Q=\{q\in[0,1]^{\mathbb N}: q(i+1)\le q(i)\le q(0)=1\}.

The threshold $\ell_n(t)$ is a nonnegative multiple of $\Delta$ . An arriving task goes to a pool, chosen uniformly at random, with fewer than $\ell_n$ tasks if there is one; otherwise to one with fewer than $h_n$ tasks if there is one; otherwise to any pool. The refined learning scheme with parameters $\alpha_n\in(0,1)$ acts at arrivals: if, right before the arrival, the fraction of pools with at least $\ell_n$ tasks is at most $\alpha_n$ , the threshold drops by $\Delta$ ; if all pools have at least $\ell_n$ tasks and at least $n-1$ have at least $h_n$ , it rises by $\Delta$ .

All systems are built on one probability space from independent unit-rate Poisson processes $\mathcal N_i$ , i.i.d. uniform selection variables $U_j$ , and random initial conditions with $q_n(0)\to q(0)$ almost surely, $q_n(0,B+1)=0$ and $\ell_n(0)\le B$ . The paths $(q_n,\ell_n)$ are the solutions of the implicit equations (9a)–(9b): arrivals follow $\mathcal N^\lambda_n(t)=\mathcal N_0(n\int_0^t\lambda)$ and are routed by the $U_j$ ; departures from pools with $i$ tasks follow $\mathcal N_i$ run at the clock $n\int_0^t\mu i[q_n(s,i)-q_n(s,i+1)]ds$ .

The fluid total mass is $u(t)=u(0)e^{-\mu t}+\int_0^t\rho(s)\mu e^{-\mu(t-s)}ds$ with $u(0)=\sum_{i\ge1}q(0,i)$ . On an interval $[a,b]\subset[0,T]$ let $\rho_{\min},\rho_{\max}$ be the infimum and supremum of $\rho$ . The load is $(m,\Delta)$ -bounded on $[a,b]$ (Definition 1) if $m\Delta<\rho_{\min}\le\rho_{\max}<(m+1)\Delta$ and $b-a$ exceeds an explicit constant $\sigma(a,b,m,\Delta)$ built from $\log\frac{\rho_{\min}}{\rho_{\min}-m\Delta}$ and $\big[\log\frac{u(a)-\rho_{\max}}{(m+1)\Delta-\rho_{\max}}\big]^+$ .

Formalization targets

Goal: Theorem 2

Assume $\alpha_n\to1$ and $\liminf_n n^{\gamma_0}(1-\alpha_n)>0$ for some $\gamma_0\in(0,1/2)$ , the load is $(m,\Delta)$ -bounded on $[a,b]$ , and $\sigma(a,b,m,\Delta)<\sigma<b-a$ . Then almost surely

\lim_{n\to\infty}\sup_{t\in[a+\sigma,b]}|\ell_n(t)-m\Delta|=0,\qquad \lim_{n\to\infty}\sup_{t\in[a+\sigma,b]}n^{\gamma}|1-q_n(t,i)|=0\ \ (i\le m\Delta,\ \gamma\in[0,\tfrac12)),

\limsup_{n\to\infty}\sup_{t\in[a+\sigma,b]}\sum_{i>(m+1)\Delta}q_n(t,i)\,e^{\mu[t-(a+\sigma)]}\le u(a+\sigma).

Milestones

In the order of the proof: well-posedness of (9a)–(9b) (§5.1.1); Lemmas 21 and 22 and Proposition 4 (relative compactness); Lemmas 5 and 6 and Proposition 7 (the set $\Omega_T$ of nice sample paths); Proposition 8 ( $u_n\to u$ uniformly); Proposition 9 (decay of tail masses below the threshold); Proposition 10 (the threshold eventually stays at most $m\Delta$ after $a+\sigma_{\mathrm{bd}}$ ); Corollary 11 (the bound (6c)); Lemmas 12 and 13 (the threshold locks at $m\Delta$ ).

Significance

The result. Theorem 2 shows that a scheme using only two bits of state per arrival finds the balancing threshold $m\Delta$ without knowing $\rho$ , within the explicit time $\sigma(a,b,m,\Delta)$ , which is tight for an initially empty system with constant load. It quantifies balance: the fraction of pools with fewer than $m\Delta$ tasks is $o(n^{-\gamma})$ for every $\gamma<1/2$ , and the mass beyond column $(m+1)\Delta$ decays exponentially. Conditions (5) are sharp in kind: $\alpha_n\to1$ prevents settling too high, and the rate $n^{-\gamma_0}$ stops $O(\sqrt n)$ fluctuations from pushing the threshold down.

Formalizing it. The result is proved in the paper; nothing here is open mathematically. Nothing of it is machine-checked. A formal proof would add an almost-sure coupled construction of a many-server Markov process on Poisson primitives, a rate- $n^{-\gamma}$ functional strong law for the Poisson process, and a pathwise relative-compactness argument in the uniform topology. None of these is in Mathlib.

Difficulty

The usual fluid-limit route fails here. The learning scheme reacts to excursions of the occupancy process of vanishing size: a threshold increase needs $n q_n(h_n)\ge n-1$ , a decrease $q_n(\ell_n)\le\alpha_n$ with $\alpha_n\to1$ . So the threshold is not a continuous function of the fluid state, and limits of $(q_n,\ell_n)$ need not satisfy any closed ODE. The natural first idea, deriving a limiting ODE for $(q,\ell)$ jointly and studying its equilibria, fails for exactly this reason: whether the threshold moves is decided by fluctuations of order $n^{-1/2}$ around the fluid path, which a law of large numbers does not see. The statements must therefore hold almost surely and uniformly in time for the coupled sequence, not in law.

Formalization scope

Model. UnknownLoads.Refined.LoadData holds $T,\lambda,C,\mu,\Delta$ (with $\lambda$ measurable, an assumption the paper uses tacitly). Primitives holds the Poisson processes (the published BesbesZeevi.Parametric.PoissonProcess, independent as a family), the uniform $U_j$ , the initial conditions, and the independence of the three families. Any probability space carrying them is allowed; completion is immaterial for almost-sure statements.
Initial conditions are random and satisfy, almost surely, $q_n(0)\in Q$ , $nq_n(0,i)\in\mathbb N$ , $\ell_n(0)\in\Delta\mathbb N$ , $q_n(0,B+1)=0$ , $\ell_n(0)\le B$ , and $q_n(0)\to q(0)$ coordinatewise (the form of (2) for exponential service).
The processes are given in System with the hypothesis that, almost surely, for every $n\ge1$ , $(q_n,\ell_n)$ is a càdlàg, $Q$ -valued solution of (9a)–(9b) on $[0,T]$ . Left limits are Function.leftLim, and $\tau_{n,k}=\inf\{t\ge0:\mathcal N^\lambda_n(t)\ge k\}$ . The well-posedness milestone states that such solutions exist and are unique almost surely, so System is not vacuous.
$\Omega_T$ is the predicate IsNicePath B_T ω. It holds the equations at $\omega$ , simple events (unit jumps, no coincidences) and the initial conditions at $\omega$ , Proposition 4's property, (11) and (12a)–(12c). Proposition 7 asserts it almost surely; "for each $\omega\in\Omega_T$ " becomes "for every $B_T,\omega$ with IsNicePath B_T ω".
Conventions. Time is real, with conditions on $[0,T]$ . $\lim_n\sup_t(\cdot)=0$ is written in $\varepsilon$ – $n$ form. Relative compactness in $(D_{\mathbb R^{\mathbb N}}[0,T],\varrho)$ is coordinatewise uniform convergence along sub-subsequences. Subsequences are StrictMono maps. Tail sums in inequalities ((6c), Corollary 11) are in $[0,\infty]$ ; $u_n$ and $v_n$ in limit statements are real series, finite sums for large $n$ on $\Omega_T$ . $\liminf_n n^{\gamma_0}(1-\alpha_n)>0$ is "eventually $\ge\kappa$ for some $\kappa>0$ ". The constant in (6c) is fixed to $u(a+\sigma)$ .
Ruled out. A real tsum in (6c) would make the bound vacuous for non-summable tails, and a real supremum would make (6a) vacuous for unbounded thresholds. Neither appears. (6a) says that $\ell_n$ equals $m\Delta$ on $[a+\sigma,b]$ for all large $n$ , a property of the learned threshold that no fixed threshold could satisfy for all admissible loads.
One added hypothesis. Proposition 9 assumes $u(t_0)>0$ : for an empty system at $t_0$ the paper's strict inequality reads $0<0$ .
Welcome contributions: the Poisson functional strong law with rate (Lemma 6), Bramson's approximation lemma (Lemma 21), and the construction's well-posedness. All three are reusable for other many-server models.

Selected references

D. Goldsztajn, S. C. Borst, J. S. H. van Leeuwaarden, Learning and balancing unknown loads in large-scale systems, arXiv:2012.10142v3, 2024 (Mathematics of Operations Research, 2025). https://arxiv.org/abs/2012.10142
D. Goldsztajn, S. C. Borst, J. S. H. van Leeuwaarden, D. Mukherjee, P. A. Whiting, Self-learning threshold-based load balancing, INFORMS Journal on Computing 34(1), 39–54, 2022.
M. Bramson, State space collapse with application to heavy traffic limits for multiclass queueing networks, Queueing Systems 30, 89–140, 1998. https://doi.org/10.1023/A:1019160803783
P. D. Sparaggis, D. Towsley, C. Cassandras, Extremal properties of the shortest/longest non-full queue policies in finite-capacity systems with state-dependent service rates, Journal of Applied Probability 30, 223–236, 1993.

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Linear Optimization Operations Research Optimization·Captain: mikedeng1

On the Optimality of Affine Policies for Budgeted Uncertainty Sets 2: For Disjoint Constrained Budgeted Sets, Affine Policies Approximate the Adjustable Robust Problem within O(log² n / log log n)Research Paper

Motivation

Two-stage adjustable robust optimization models decisions taken in two steps around an uncertain demand: a first-stage decision is fixed now, the demand is revealed, and a second-stage (recourse) decision reacts to it. The objective is the worst-case total cost over a set of possible demands. Computing the optimal adjustable solution is intractable in general, and even for a single budget constraint it cannot be approximated within $\Omega(\log n/\log\log n)$ under a standard complexity assumption (Feige, Jain, Mahdian, Mirrokni, IPCO 2007), where $n$ is the number of decision variables.

Affine policies, in which the recourse is restricted to an affine function of the revealed demand (Ben-Tal et al. 2004), are computable by one linear program and are widely used in practice. The question is how much they lose in the worst case.

Uncertainty sets in applications are rarely a single budget: demand is usually grouped (by region, product family or time period) with a separate budget per group. When the groups are disjoint the set is a Cartesian product of budget sets, a class studied by Gupta et al. and by Feige et al. in the robust covering literature.

Timeline. Bertsimas and Goyal (2012) proved that affine policies are within $O(\sqrt m)$ of the adjustable optimum for general uncertainty sets and that this is tight for some sets. El Housni and Goyal (arXiv:1807.00163, 2018; Mathematics of Operations Research 2021) proved $O(\log n/\log\log n)$ for a single budget set (Theorem 3.1, the first mission of this series) and $O(\log^2 n/\log\log n)$ for disjoint constrained budgeted sets (Theorem 4.1, the goal of this mission).

Setting

Let $A\in\mathbb R^{m\times n}$ , a non-negative recourse matrix $B\in\mathbb R^{m\times n}_+$ , costs $c,d\in\mathbb R^n_+$ and a polyhedral set $\mathcal X\subseteq\mathbb R^n$ be given. For an uncertainty set $\mathcal U\subseteq\mathbb R^m$ of right-hand sides, problem (1.1) is

z_{AR}(\mathcal U)=\min_{x\in\mathcal X}\ c^Tx+\max_{h\in\mathcal U}\ \min_{y(h)\ge0}\big\{d^Ty(h)\ :\ Ax+By(h)\ge h\big\}.

A first-stage $x$ and a recourse rule $y(\cdot)$ form a feasible solution when $x\in\mathcal X$ and $y(h)\ge0$ , $Ax+By(h)\ge h$ for every $h\in\mathcal U$ . The affine problem (2.1) restricts the rule to $y(h)=Ph+q$ (still requiring $Ph+q\ge0$ on $\mathcal U$ ); its value is $z_{Aff}(\mathcal U)\ge z_{AR}(\mathcal U)$ .

Let $S_1,\dots,S_L$ be a partition of $[m]$ and $w_{\ell i}\in[0,1]$ weights. The disjoint constrained budgeted set (4.1) is

\mathcal U=\Big\{h\in[0,1]^m\ :\ \sum_{i\in S_\ell}w_{\ell i}h_i\le1\ \ \forall\ell\in[L]\Big\}.

With $L=1$ it is the budget of uncertainty set of the first mission.

The fractional covering value (3.2) of a requirement $h$ is $z(h)=\min_{y\ge0}\{d^Ty: By\ge h\}$ . In an online fractional covering problem the rows to be covered arrive in batches and a solution may only be increased; an online algorithm $\mathcal A$ is $\rho$ -competitive when its cost on every sequence is at most $\rho$ times the offline optimum $\Theta$ . Throughout, $\log$ is the natural logarithm and $\beta=8\log n/\log\log n$ .

Formalization targets

Goal: Theorem 4.1

z_{Aff}(\mathcal U)=O\Big(\frac{\log^2n}{\log\log n}\Big)\cdot z_{AR}(\mathcal U)

for every instance with $\mathcal U$ a disjoint constrained budgeted set: there are $C>0$ and $N$ such that $z_{Aff}\le C\frac{\log^2n}{\log\log n}z_{AR}$ for all $n\ge N$ , all $m$ , every number $L$ of blocks and all data. The constant is unfixed, and the bound must not depend on $L$ .

Milestones (the paper's proof, in order)

Lemma 3.3 (structural result), restated from the first mission: if covering each row $i\in\mathcal J$ costs more than $4\gamma\frac{\log n}{\log\log n}w_i$ while every scenario $\mathcal W\subseteq\mathcal J$ with $\sum_{\mathcal W}w_i\le1$ costs at most $\gamma$ , then $z(\mathcal J)\le4\gamma\frac{\log n}{\log\log n}$ (for $n$ large).
Lemma 4.1 — for a greedy scenario $a_1,\dots,a_L$ chosen by Algorithm 1, the online cost $\nu_L$ is at most $\rho\cdot\mathrm{OPT}$ .
Lemma 4.2 — the affine solution (4.7), a linear part on the thresholded components plus a static part, is feasible.
Lemma 4.3 — the linear part costs at most $\beta\rho\cdot\mathrm{OPT}$ for every $h\in\mathcal U$ .
Lemma 4.4 — the static part costs at most $(1+2\rho+\beta\rho)\cdot\mathrm{OPT}$ .

With $\rho=O(\log n)$ , Lemmas 4.3 and 4.4 give the goal.

Significance

The result. For disjoint budgets the bound is within a $\log n$ factor of the hardness bound for a single budget, and it does not depend on the number of groups $L$ . It shows that affine policies stay near-optimal beyond the single-budget case, for a class that models grouped demand. The paper notes that it is open whether the extra $\log n$ factor is necessary. Gupta et al. give an $O(\log n)$ -approximation for a special case with 0/1 matrices and equal weights, but their algorithm is not an affine policy.

Formalizing it. The theorem is proved on paper; no machine-checked proof is known. The formal development makes explicit what the paper uses about the online covering algorithm (monotone, covering, competitive), handles the "without loss of generality" relabelling of the blocks in the greedy scenario, and records two gaps in the proof of Lemma 4.4 (zero weights, and a zero increment $\nu_\ell-\nu_{\ell-1}$ , for which Lemma 3.3 is stated with $\gamma>0$ ).

Difficulty

The thresholds that split cheap from expensive components cannot use the single quantity $\mathrm{OPT}$ as in the single-budget case: a scenario may activate one budget in every block at once, and summing per-block bounds gives a factor $L$ . That is the $O(L\log n/\log\log n)$ bound of the paper's Section 5. Removing the dependence on $L$ requires per-block thresholds that add up to $O(\log n)\cdot\mathrm{OPT}$ , and the paper obtains them from the increments of an online covering algorithm along a greedily chosen worst sequence of scenarios. A proof of the goal therefore needs an $O(\log n)$ -competitive online fractional covering algorithm, Buchbinder and Naor's Theorem 4.1, posed on this platform as BNCovPack.Covering.theorem_4_1 (b57de23d-5953-424f-a729-5a018aa74b37, open). That theorem requires positive costs, so zero-cost columns, which this paper allows, need separate handling. The static part also needs Lemma 3.3, the core of the first mission.

Formalization scope

Data. Vectors are functions on Fin m, Fin n with the componentwise order; matrices are Matrix (Fin m) (Fin n) ℝ; Lean indices and blocks are 0-based. First stage and recourse share the dimension $n$ , as on p. 1. The partition is S : Fin L → Finset (Fin m), pairwise disjoint and covering Fin m (empty blocks allowed).
Values. $z_{AR}$ , $z_{Aff}$ , $z(h)$ , $\Theta$ and $z_{Sta}$ are infima in the extended reals, $+\infty$ when infeasible. A formalization with real infima (equal to $0$ on infeasible problems) or without $Ph+q\ge0$ would trivialize the goal and is excluded, as is a constant $C$ that depends on the instance or on $L$ .
Standing assumptions of p. 1: $B\ge0$ , $c,d\ge0$ , $\mathcal X$ polyhedral, $w_{\ell i}\in[0,1]$ , $S$ a partition. " $\mathcal U\subseteq[0,1]^m$ compact convex with $e_i\in\mathcal U$ " holds for the set (4.1) and is not added.
Added hypotheses and corrections. (i) $c^Tx\ge0$ for $x\in\mathcal X$ : Theorem 4.1 is false without it (already for $L=1$ ), and it holds whenever $\mathcal X\subseteq\mathbb R^n_+$ . (ii) $n\ge3$ where $\beta\ge0$ is needed, and a threshold $N_0$ in Lemma 3.3, which as printed fails at $n=2$ . (iii) Covering solutions $v_i$ are required only for $i\in\mathcal I$ .
The online algorithm is an abstract structure OnlineCover with ratio $\rho$ : starting at $0$ , non-negative, monotone, covering every revealed coverable row, and $\rho$ -competitive on disjoint batches. The milestones hold for every such algorithm, with the paper's $O(\log n)$ written as $\rho$ . The goal does not mention the algorithm. Algorithm 1 is a predicate on a run (a bijection of blocks and the chosen scenarios) with its maximality clause, so ties are arbitrary.
Thresholds are in multiplied form $\alpha_iz(e_i)\le w\beta(\nu_\ell-\nu_{\ell-1})$ ; no division by a weight that may be $0$ .
Infrastructure. Lemma 3.3 is shared with the first mission and needs LP duality and randomized rounding. Buchbinder–Naor's theorem, or any $O(\log n)$ -competitive online fractional covering algorithm, is needed for the goal. Proofs of any milestone are welcome.

The source is arXiv:1807.00163v2; its printed page numbers equal the PDF page numbers.

Selected references

O. El Housni, V. Goyal, On the Optimality of Affine Policies for Budgeted Uncertainty Sets, arXiv:1807.00163v2, 2019; Mathematics of Operations Research 46(2), 2021. https://arxiv.org/abs/1807.00163
N. Buchbinder, J. Naor, Online primal-dual algorithms for covering and packing, Mathematics of Operations Research 34(2), 2009. https://doi.org/10.1287/moor.1080.0363
D. Bertsimas, V. Goyal, On the power and limitations of affine policies in two-stage adaptive optimization, Mathematical Programming 134(2), 2012. https://doi.org/10.1007/s10107-011-0444-4
U. Feige, K. Jain, M. Mahdian, V. Mirrokni, Robust combinatorial optimization with exponential scenarios, IPCO 2007, LNCS 4513.
A. Ben-Tal, A. Goryashko, E. Guslitzer, A. Nemirovski, Adjustable robust solutions of uncertain linear programs, Mathematical Programming 99(2), 2004. https://doi.org/10.1007/s10107-003-0454-y

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Dynamic Programming Operations Research Optimization+1·Captain: mikedeng1

Coordinating Inventory Control and Pricing Strategies for Perishable Products 2: With Zero Lead Time and Minimal Disposal, Two Myopic Solutions Bound the Optimal Order-Up-To LevelResearch Paper

Why bound the order-up-to level of a perishable product

Food, blood products, pharmaceuticals and many retail goods are perishable: a unit is usable for a fixed number of periods and is then discarded. A seller of such a product decides every period how much to order, what price to charge, and how much stock to deplete or dispose of. Because units of different ages coexist, the state of the system is a vector of inventory levels by remaining lifetime, and the exact dynamic program suffers from the curse of dimensionality in that vector. Nahmias's survey (Nahmias 1982) records this obstacle for the pure ordering problem; adding pricing makes it worse.

Chen, Pang and Pan (Operations Research 62(2), 2014) analyse the joint ordering, pricing and disposal problem with the tools of L♮-concavity (Murota's discrete convex analysis, extended to real vectors). Their §4 shows that the optimal profit-to-go is L♮-concave, which yields monotone policies with bounded sensitivity. Their §6.1 then shows that two one-dimensional, myopic problems bracket the optimal order-up-to level, which is the starting point of the heuristics of §6.2. This mission formalizes the backlogging case of that bracketing result, Theorem 3.

Setting

A single product has lifetime $l\ge 2$ periods; the replenishment lead time is zero, and unmet demand is backlogged. The unit costs are $c$ (ordering), $h^+$ (holding), $h^-$ (backlogging) and $\theta$ (disposal); $\gamma$ is the discount factor. The seller chooses the expected demand $d$ in an interval $\mathcal D=[\underline d,\bar d]$ , which is equivalent to choosing the price $P(d)$ , where $P$ is strictly decreasing. The expected revenue $P(d)\,d$ is continuous and concave on $\mathcal D$ (Assumption 1). Realized demand is $d_t=d+\varepsilon_t$ , with independent, identically distributed, zero-mean noise supported in $[A,B]$ , $A\le 0\le B$ , and $\underline d+A\ge 0$ .

The state $s=(s_1,\dots,s_{l-1})$ lies in $\mathcal S=\{s_1\le\dots\le s_{l-1}\}$ : $s_i$ is the inventory with residual lifetime at most $i$ periods (negative values are backorders), and $s_{l-1}$ is the inventory position. Each period the seller picks an order-up-to level $s_l\ge s_{l-1}$ , the demand level $d$ , and, after demand is realized, a depletion $a$ with $s_1\vee d_t\le a\le s_l\vee d_t$ . The next state is $\tilde s=(s_2-a,\dots,s_l-a)$ . The one-period objective is

\varphi_t(s,s_l,d,a\mid\varepsilon_t)=\gamma c\,\tilde s_{l-1}-cs_l-\theta(a-d_t)-h^+(s_l-a)^+-h^-(a-s_l)^++\gamma f_{t+1}(\tilde s),

$g_t$ maximizes it over $a$ , $G_t(s,s_l,d)=P(d)d+\mathbb E[g_t]$ , and the profit-to-go is $f_t(s)=\max_{s_l\ge s_{l-1},\,d\in\mathcal D}G_t(s,s_l,d)$ with $f_{T+1}=0$ .

Under $\theta\ge h^+/(1-\gamma)$ only expiring stock is disposed of (Proposition 1). With $\tilde\theta=\theta+\gamma c-h^+$ and $v_t(s_l,d_t)=-(1-\gamma)cs_l-\gamma cd_t-h^+(s_l-d_t)^+-h^-(d_t-s_l)^+$ , the two myopic objectives are

\Pi_t(s_l,d)=P(d)d+\mathbb E[v_t(s_l,d_t)],\qquad \tilde\Pi_t(s_l,d)=\Pi_t(s_l,d)-\gamma cs_l-K\,\mathbb E[(s_l-d_t)^+],

with $K=\gamma h^++\dots+\gamma^{l-2}h^++\gamma^{l-1}\tilde\theta$ . $\bar S_t(d)$ and $\underline S_t(d)$ are their largest maximizers over $s_l\ge0$ .

Formalization targets

Goal: Theorem 3 (Bounds), backlogging case

For every period, every state $s\in\mathcal S$ and every $d\in\mathcal D$ , the largest optimal order-up-to level $s_{lt}(s,d)$ satisfies

\underline S_t(d)\vee s_{l-1}\ \le\ s_{lt}(s,d)\ \le\ \bar S_t(d)\vee s_{l-1}.

Milestones, in attack order

Lemma 6 (p. 290): $f_t$ is nonincreasing in each coordinate of the state.
Proposition 1 (p. 292): if $\theta\ge h^+/(1-\gamma)$ , depleting $d_t\vee s_1$ is optimal.
The p. 294 decomposition: $\varphi_t$ at $a=d_t\vee s_1$ equals $v_t(s_l,d_t)-\tilde\theta(s_1-d_t)^++\gamma f_{t+1}(\tilde s)$ .
Theorem 1, first sentence (p. 291), at zero lead time: $f_t$ , $g_t$ , $\varphi_t$ are L♮-concave.
Concavity in $s_l$ (p. 295): $\Pi_t(\cdot,d)$ and $G_t(s,\cdot,d)$ are concave.
Upper step (p. 295): $G_t(s,s'_l,d)-G_t(s,s_l,d)\le\Pi_t(s'_l,d)-\Pi_t(s_l,d)$ for $s'_l>s_l\ge s_{l-1}$ .

The proof's lower-bound display (p. 295) is not a milestone: as printed, for arbitrary $s'_l>s_l\ge s_{l-1}$ , it is false when $h^+>\gamma c$ (see Formalization scope).

Significance

The result. The bounds reduce an $(l-1)$ -dimensional optimization to two scalar newsvendor-type problems whose solutions do not depend on the period (Remark 6), and they give interpretable brackets: $\bar S_t(d)$ ignores the future cost of excess stock, while $\underline S_t(d)$ charges every excess unit holding until expiry and then disposal. They justify the one-dimensional heuristics of §6.2 and give a check on any computed policy.

Formalizing it. The paper's proof is short but compresses three steps that need care: the structural results of §4 at zero lead time, the minimal-disposal property, and a coupling argument for the lower bound whose printed display is not correct as stated (see below). A machine-checked development would make the whole chain from the recursion to the bounds explicit, including the existence conventions for optimal decisions. To the knowledge of this mission, none of these results has been formalized in any proof assistant.

Difficulty

The central obstacle is the concavity of $G_t(s,\cdot,d)$ in the order-up-to level. It is not visible from the one-period terms: it rests on L♮-concavity of the whole recursion (Theorem 1), and the depletion constraint $s_1\vee d_t\le a\le s_l\vee d_t$ is not an L♮-convex set, so the standard preservation lemma does not apply to (5) as written.

The lower bound needs a quantitative estimate of how much an extra unit of fresh stock can cost in the future. The printed estimate charges the expiring unit $\tilde\theta$ where an unused unit actually costs $\theta$ , so it cannot hold for every pair of order-up-to levels; the lower bound must use that the comparison is only needed below $\underline S_t(d)$ , where excess stock is not pushed past the next period's optimal order-up-to level.

Formalization scope

The Lean development lives in the namespace PerishablePricing.Bounds. Its conventions are:

Carrier $\mathbb R$ . The paper treats real and integer vectors at once; only the real case is formalized. L♮-convexity is defined on an explicit domain with real values, the paper's convention for "L♮-convex on a set".
Indices. States are Fin L → ℝ with $L=l-1\ge1$ , 0-based.
Time. Time counts periods to go: f 0 = 0 is $f_{T+1}$ , and $G_t$ , $g_t$ , $\varphi_t$ use the continuation f n.
Maxima and expectations. The maxima in the recursion are suprema over nonempty sets whose values are bounded above, and expectations are Bochner integrals against the noise law.
Optimal decisions. Every optimal decision is a largest maximizer, taken as a hypothesis at the points where it is used. With an arbitrary maximizer the upper bound fails when $\Pi_t(\cdot,d)$ is flat.

The standing assumptions are the hypotheses of every statement:

nonnegative unit costs and $0\le\gamma\le1$ ;
the noise conditions and the strictly decreasing inverse demand;
Assumption 1;
$(1-\gamma)c\le h^-$ , the §4 assumption $c\le h^-/(1-\gamma)$ ;
$h^+\le(1-\gamma)\theta$ with $\gamma<1$ , the §6.1 assumption $\theta\ge h^+/(1-\gamma)$ , which divides by $1-\gamma$ .

Deviations from the page, all disclosed in the items:

Only the backlogging case of Theorem 3 is stated. The lost-sales case rests on Theorem 2, which is not posed.
The hypothesis "the starting inventory level of the planning horizon is zero" constrains no object of the statement, and the conclusion is stated for every state.
$\tilde\Pi_t$ 's printed term $-\gamma c\,\tilde s_{l-1}$ depends on the state and the noise. It is read as $-\gamma c\,s_l$ , which leaves the maximizers unchanged.
The objective missing from the printed definition of $\bar S_t(d)$ is $\Pi_t$ .
The upper step's " $G_t=\Pi_t+\gamma\mathbb E[f_{t+1}]$ " omits a term constant in $s_l$ .
The lower-bound display of the proof of Theorem 3 is not stated. Its exponents $t+l-3$ , $t+l-2$ should read $l-3$ , $l-2$ , and even then, for arbitrary $s'_l>s_l\ge s_{l-1}$ , it fails when $h^+>\gamma c$ . Example: $l=2$ , deterministic demand, $c=1$ , $\gamma=\tfrac12$ , $h^+=h^-=1$ , $\theta=2$ , one period after the current one, $s_1=0$ , $d=\tfrac12$ , $s_l=100$ , $s'_l=101$ : the left side is $-3$ and the right side is $-2.75$ . In that case an unused unit is disposed of at cost $\theta$ , not $\tilde\theta$ . Theorem 3 itself is stated as printed, with $\underline S_t(d)$ defined through $\tilde\Pi_t$ .

The goal must not be trivialized. Its continuation, the concavity of $G_t$ and the monotonicity of $f_{t+1}$ are properties of the model's own recursion, never hypotheses on abstract functions. Likewise $s_{lt}(s,d)$ is defined through $G_t$ , never through $\Pi_t$ or $\tilde\Pi_t$ .

A complete development needs:

L♮-concavity preservation under partial maximization over an L♮-convex set and under expectation;
concavity of expectations of piecewise-linear functions of the noise;
measurability and integrability of the stage value $g_t$ in the noise;
the coupling arguments for Proposition 1 and the lower bound.

The L♮ preservation lemmas are reusable for other discrete-convexity models in inventory theory. Contributions toward any milestone, or toward that infrastructure, are welcome.

The source is the published Operations Research article. In the held scan, printed page = PDF page + 282.

Selected references

X. Chen, Z. Pang, L. Pan, Coordinating Inventory Control and Pricing Strategies for Perishable Products, Operations Research 62(2):284–300, 2014. https://doi.org/10.1287/opre.2014.1261
K. Murota, Discrete Convex Analysis, SIAM, 2003. https://doi.org/10.1137/1.9780898718508
P. Zipkin, On the Structure of Lost-Sales Inventory Models, Operations Research 56(4):937–944, 2008. https://doi.org/10.1287/opre.1070.0482
S. Nahmias, Perishable Inventory Theory: A Review, Operations Research 30(4):680–708, 1982. https://doi.org/10.1287/opre.30.4.680

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Combinatorics Operations Research Optimization·Captain: mikedeng1

The Continuous-Time Joint Replenishment Problem: ϵ-Optimal Policies via Pairwise Alignment: Ψ-Pairwise Alignment Yields a Replenishment Policy Within Factor 1 + 5ϵ of Any PolicyResearch Paper

Motivation

The joint replenishment problem is a basic model of coordinated inventory control. A firm stocks $n$ commodities, each of which follows its own Economic Order Quantity (EOQ) model, but every order also incurs a joint ordering cost $K_0$ , regardless of which commodities it contains. Ordering commodities together therefore saves money, and the question is how to synchronize their replenishment cycles.

Since the mid-1980s the best known guarantee for the continuous-time problem came from power-of-2 policies (Roundy 1985, 1986; Jackson, Maxwell and Muckstadt 1985; Muckstadt and Roundy 1987). Their cost is within $\sqrt{9/8}\approx1.06$ of optimal for a fixed base period, and within $1/(\sqrt2\ln2)\approx1.02$ when the base is optimized. On the hardness side, Schulz and Telha (2022) and Tuisov and Yedidsion (2020) showed the problem to be intractable; the latter proved it strongly NP-hard, which rules out a fully polynomial-time approximation scheme.

Segev (arXiv:2302.09941, published in Management Science, 2025) gave the first improvement over power-of-2 policies. For every $\epsilon\in(0,\tfrac12)$ the problem can be approximated within $1+\epsilon$ of optimal in time $O(2^{\tilde O(1/\epsilon^3)}\cdot n^{O(1)})$ . This mission formalizes the approximation guarantee behind that theorem. The source is the arXiv v1 preprint, and all theorem numbers refer to it.

Setting

Commodity $i\in[n]$ has ordering cost $K_i>0$ and holding cost $H_i>0$ . Ordering it every $T_i>0$ time units costs per unit time

C_i(T_i)=\frac{K_i}{T_i}+H_iT_i .

A policy is a vector $T=(T_1,\dots,T_n)$ of positive intervals. For $t>0$ and $\Delta\ge0$ , let $\mathcal M_{t,\Delta}=\{0,t,2t,\dots,\lfloor\Delta/t\rfloor t\}$ be the multiples of $t$ in $[0,\Delta]$ . A policy places

N(T,\Delta)=\Big|\bigcup_{i\in[n]}\mathcal M_{T_i,\Delta}\Big|

joint orders in $[0,\Delta]$ . Its joint ordering cost and total cost are

J(T)=K_0\cdot\lim_{\Delta\to\infty}\frac{N(T,\Delta)}{\Delta},\qquad F(T)=J(T)+\sum_{i\in[n]}C_i(T_i).

For $\mathcal N\subseteq[n]$ , $M_{\mathcal N}$ is the least common multiple of $\{T_i\}_{i\in\mathcal N}$ , and $M_{\mathcal N}=\infty$ when the intervals have no common multiple.

The algorithm works against a reference policy $T^*$ with $T^*_{\min}=\min_iT^*_i$ and an estimate $F(T^*)\le\widetilde{\mathrm{OPT}}\le2F(T^*)$ .

Easy regime. When $T^*_{\min}\ge\frac n\epsilon\cdot\frac{K_0}{\widetilde{\mathrm{OPT}}}$ , it uses the easy policy $\hat T_i=\sqrt{(K_0+K_i)/H_i}$ .
Difficult regime. Otherwise it guesses $\tilde T_{\min}$ with $(1-\epsilon/2)T^*_{\min}\le\tilde T_{\min}\le T^*_{\min}$ (2). It cuts $[\tilde T_{\min},\tilde T_{\min}/\epsilon]$ into segments $S_\ell=[(1+\epsilon)^{\ell-1}\tilde T_{\min},(1+\epsilon)^\ell\tilde T_{\min})$ , $\ell=1,\dots,L$ . It keeps the active segments $\mathcal A^*$ , those containing some $T^*_i$ , and picks one representative $R^*_\ell$ among the $T^*_i$ in each.

Two representatives are $\Psi$ -aligned when $aR^*_{\ell_1}=bR^*_{\ell_2}$ for positive integers $a,b\le\Psi=2L^2 2^L/\epsilon$ . The alignment graph $G^*_\Psi$ joins $\Psi$ -aligned active segments. The linear system (LP) asks for $R_\ell\in\bar S_\ell$ satisfying the same alignment relations as $R^*$ . A solution is misaligned across components when representatives in different components of $G^*_\Psi$ are not $\Psi$ -aligned. The final policy gives commodity $i$ the $C_i$ -cheapest of the $R_\ell$ and of the single large interval $\lceil\max\{\tilde T_{\min}/\epsilon,\sqrt{K_i/H_i}\}\rceil^{(R_1)}$ , rounded up to a multiple of $R_1$ .

Formalization targets

Goal: the approximation half of Theorem 1.3

Fix $n\ge1$ , $K_0,K_i,H_i>0$ , $\epsilon\in(0,\tfrac12)$ , a policy $T^*$ with all $T^*_i>0$ , and $\widetilde{\mathrm{OPT}}$ with $F(T^*)\le\widetilde{\mathrm{OPT}}\le2F(T^*)$ .

(a) In the easy regime,

F(\hat T)\le(1+2\epsilon)F(T^*).

(b) In the difficult regime, the grid of powers of $1+\epsilon/2$ contains a guess satisfying (2). For every such guess and every choice of representatives, a feasible, cross-misaligned solution of (LP) exists, and every final policy $\hat T$ built from such a solution satisfies

F(\hat T)\le(1+5\epsilon)\,F(T^*).

Milestones

The goal's proof uses the following results, in this order:

Claim 1.1 (EOQ facts) and Lemma 1.2 (the density exists and equals $\sum_{\mathcal N}(-1)^{|\mathcal N|+1}/M_{\mathcal N}$ ).
Lemma 2.1 (the easy regime) and Observations 2.2 and 2.3.
Lemma 2.4, Lemma 2.5 and Corollary 2.6 (the structure of (LP)).
Lemmas 3.2 and 3.3 and Claims 3.4 and 3.5 (crossing and uncrossing terms).
Lemma 3.1 and inequality (3) (density), and Lemma 3.6 (assignability).

Significance

The theorem gives the first approximation scheme for continuous-time joint replenishment. It settles the problem's approximability up to the form of the exponential dependence on $\epsilon$ , after nearly four decades in which power-of-2 policies were the best available. Its central device, synchronizing pairs of representatives through bounded integer multiples and controlling everything else by inclusion–exclusion over the alignment graph, is proposed in the paper as a template for other multi-item lot-sizing models.

The result is proved on paper, and no machine-checked version is known. A formalization adds three things.

Lemma 1.2. A verified proof that the joint ordering cost of every positive policy is well defined. The paper notes that this is not obvious from (1).
Bookkeeping. A check of the paper's constants and signs across a long inclusion–exclusion argument.
Generality. Making explicit that the guarantee holds against any positive reference policy, not only an optimal one, whose existence the paper never establishes.

Difficulty

The obvious argument runs as follows: round every $T^*_i$ to a nearby point of a small grid and charge joint orders to the grid. It fails because the cost of the joint orders depends on the least common multiples of the intervals, not on their sizes. Two nearby reals can have a small lcm, or none at all, and an arbitrarily small perturbation changes $N(T,\Delta)$ by a constant factor. The proof must therefore keep the arithmetic relations among the representatives exactly where they matter: within components of $G^*_\Psi$ , through (LP). It must also show that relations across components contribute only lower-order terms. This requires two nested inclusion–exclusion expansions (Lemma 3.3 and Claim 3.5) whose error terms, of order $|\mathcal A^*|^2 2^{|\mathcal A^*|}$ , are absorbed by the choice of $\Psi$ .

Lemma 1.2 is a squeeze argument, but it rests on showing that the common multiples of finitely many positive reals form the set $M\cdot\mathbb N$ .

Formalization scope

All declarations live in the namespace ContJRP.Align.

Representation.

The model is defined on Mathlib with no external dependency.
Commodities are Fin n with $n\ge1$ .
$N(\cdot,\Delta)$ counts a finite union of Finset ℝ, and $0$ is a member of every $\mathcal M_{t,\Delta}$ .
The density is a limUnder.
$1/M_{\mathcal N}$ is 0 when no common multiple exists.
Segments are indexed from $0$ in Lean, so the paper's $S_\ell$ and $R_1$ are Lean's index $\ell-1$ and R 0.

Committed conventions.

$K_0,K_i,H_i>0$ and $\epsilon\in(0,\tfrac12)$ everywhere.
$L$ is the least natural number with $(1+\epsilon)^L>1/\epsilon$ , which the page leaves as $O(\frac1\epsilon\log\frac1\epsilon)$ .
Every alignment multiple and the $\beta^\pm$ of Lemma 2.4 are positive integers.
Inclusion–exclusion sums range over nonempty index sets.
The reference policy $T^*$ is any positive policy whose cost the estimate brackets; it need not be optimal.
Lemmas in §2–§3 do not assume the difficult-regime inequality, which their proofs do not use.

What is left out. The running-time half of Theorem 1.3, the time bounds of Lemmas 2.4 and 2.5, and the final selection of the cheapest candidate (§3.3) are not formalized. The goal carries the constants the analysis proves, $1+2\epsilon$ and $1+5\epsilon$ , rather than the rescaled $1+\epsilon$ .

Trivializing formalizations ruled out.

(LP) cannot be empty, because the goal asserts that a feasible, cross-misaligned solution exists.
limUnder cannot return a junk value, because Lemma 1.2 and (3) are stated as Tendsto limits.
The final policy is not a free parameter: each $\hat T_i$ must be one of its options and $C_i$ -minimal among them.
Misalignment quantifies over positive multiples only, so it is satisfiable.

Infrastructure and contributions. A complete development needs three pieces:

the lattice of common multiples of positive reals, and the floor-count $|\bigcap_{i}\mathcal M_{T_i,\Delta}|=\lfloor\Delta/M\rfloor+1$ ;
inclusion–exclusion for finite unions (Finset.inclusion_exclusion_card_biUnion in Mathlib);
elementary calculus for the EOQ cost.

The density results (Lemma 1.2) are reusable for any periodic-scheduling model. Proofs of any milestone are welcome, as are alternative statements of the selection step.

Selected references

D. Segev, The Continuous-Time Joint Replenishment Problem: ϵ-Optimal Policies via Pairwise Alignment, arXiv:2302.09941v1, 2023; Management Science, 2025. https://arxiv.org/abs/2302.09941
R. Roundy, 98%-effective integer-ratio lot-sizing for one-warehouse multi-retailer systems, Management Science 31(11), 1985. https://doi.org/10.1287/mnsc.31.11.1416
P. Jackson, W. Maxwell, J. Muckstadt, The joint replenishment problem with a powers-of-two restriction, IIE Transactions 17(1), 1985. https://doi.org/10.1080/07408178508975270
C.-P. Teo, D. Bertsimas, Multistage lot sizing problems via randomized rounding, Operations Research 49(4), 2001. https://doi.org/10.1287/opre.49.4.599.11225
A. S. Schulz, C. Telha, Integer factorization: Why two-item joint replenishment is hard, Operations Research, 2022 (forthcoming at the time of the preprint).
A. Tuisov, L. Yedidsion, The continuous joint replenishment problem is strongly NP-hard, arXiv:2006.05310, 2020. https://arxiv.org/abs/2006.05310

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Economics Operations Research Optimization·Captain: mikedeng1

Advance Selling and Upgrading in Priority Queues: With Only Online Customers, Upgrading Raises the Optimal Revenue Exactly When Bₛ − Bₐ > α(p(λₕ) − p(λₗ))Research Paper

Motivation

Amusement parks and similar attractions sell two kinds of ticket for the same ride: a regular ticket, which joins a first-come-first-served queue, and a fast-track ticket, which joins a separate queue served with non-preemptive priority. Increasingly these tickets are sold online in advance, before the customer knows whether the park will be crowded on the day, and some parks also let a customer who bought a regular ticket online upgrade to fast track on arrival for a fee. Whether such an upgrading option earns the seller anything beyond what advance selling already earns is a pricing question that combines two literatures: the economics of priority queues, going back to Naor's toll model (Naor 1969) and equilibrium threshold strategies in priority queues (Hassin and Haviv 1997), and advance selling under demand uncertainty (Xie and Shugan 2001), where upgrades had been studied for inventory but not for queues (Cui, Duenyas and Sahin 2018).

Wang, Cao, Xie and Zhan (MSOM 2025) answer it in a stylised M/M/1 model. This mission formalizes their main result for the case in which every customer buys online, Theorem 1, together with the propositions its proof rests on.

Setting

A single server has exponential service times with rate $\mu$ . The day is crowded (arrival rate $\lambda_h$ ) or uncrowded (rate $\lambda_l$ ), with $0<\lambda_l<\lambda_h<\mu$ ; an arriving customer sees a crowded day with probability $\alpha\in(0,1)$ . Each customer has a cost $c$ per unit of waiting time, uniform on $[0,\bar c]$ with $\bar c>0$ . Regular tickets cost $B_a$ in advance and $B_s$ on the spot; these prices are exogenous and may be any real numbers. The seller chooses the fast-track prices $P_a$ (advance) and $P_s$ (spot), and in the upgrading scheme also the fee $K$ .

Customers sort themselves by two delay-sensitivity cutoffs $0<c_r\le c_f\le\bar c$ . If on a day of rate $\lambda$ the customers with cost above a cutoff $c$ use the fast track, the expected waits are

W_F(c)=\frac{\lambda}{\mu\bigl(\mu-(1-c/\bar c)\lambda\bigr)},\qquad W_R(c)=\frac{\mu W_F(c)}{\mu-\lambda},

Cobham's formulas for a two-class priority queue. The crowded-day waits $W^h$ are evaluated at $(\lambda_h,c_r)$ and the uncrowded-day waits $W^l$ at $(\lambda_l,c_f)$ .

The paper reduces the seller's problem to two explicit programs. Under AS (advance and spot selling) the seller maximises

\Pi^{AS}=P_a\Bigl(1-\frac{c_f}{\bar c}\Bigr)+\bigl(\alpha P_s+(1-\alpha)B_s\bigr)\frac{c_f-c_r}{\bar c}+\min\{B_a,B_s\}\frac{c_r}{\bar c}

over $(c_r,c_f,P_a,P_s)$ subject to $P_a\le P_s$ , $0<c_r\le c_f\le\bar c$ , and three indifference conditions that apply only when $c_r<c_f$ , when $c_f<\bar c$ , and when $c_f=\bar c$ respectively (program (7)). Under AUS (advance and spot selling with upgrading), the term $\alpha P_s+(1-\alpha)B_s$ becomes $\min\{\alpha P_s+(1-\alpha)B_s,\ B_a+\alpha K\}$ and $P_a\le B_a+K$ is added (program (8)). Write $\Pi^{AS\star}$ and $\Pi^{AUS\star}$ for their optimal values, and

p(\lambda)=\frac{\bigl(\mu-\sqrt{\mu(\mu-\lambda)}\bigr)\lambda\bar c}{(\mu-\lambda)\mu^2},\quad c(\lambda)=\frac{\bigl(\sqrt{\mu(\mu-\lambda)}-(\mu-\lambda)\bigr)\bar c}{\lambda},\quad \pi(\lambda)=\frac{(\sqrt\mu-\sqrt{\mu-\lambda})^2\bar c}{\mu(\mu-\lambda)}.

These are the optimal price premium, cutoff and revenue gain when the rate $\lambda$ is known.

Formalization targets

Goal: Theorem 1

\Pi^{AS\star}=\Pi^{AUS\star}\ \text{ if } B_s-B_a\le\alpha\bigl(p(\lambda_h)-p(\lambda_l)\bigr),\qquad \Pi^{AUS\star}>\Pi^{AS\star}\ \text{ if } B_s-B_a>\alpha\bigl(p(\lambda_h)-p(\lambda_l)\bigr).

The goal compares the two optima and does not mention their common value. It asserts that the AS supremum exists, that the AUS maximum is attained, and the comparison.

Milestones

The ordering $0<c(\lambda_h)<c(\lambda_l)<\bar c$ (§5.1, p. 15).
Lemma 1: with all customers buying on the spot, the equilibrium cutoffs are the closed form (5).
The single-rate cutoff problem from the proof of Proposition 3: $\psi(\hat c)=\hat c(W_R(\hat c)-W_F(\hat c))$ is strictly increasing, and the revenue as a function of the cutoff is maximised at $c(\lambda)$ with value $\pi(\lambda)$ .
Proposition 3: with a known rate $\lambda$ , the optimal spot price is $B_s+p(\lambda)$ and the optimal revenue $\pi(\lambda)+B_s$ .
Proposition 5, parts 1)–2): if $B_s-B_a<\alpha(p(\lambda_h)-p(\lambda_l))$ , the AS optimum is $\alpha\pi(\lambda_h)+(1-\alpha)\pi(\lambda_l)+\min\{B_a,B_s\}$ , attained only at cutoffs $(c(\lambda_h),c(\lambda_l))$ and explicit prices.
Proposition 5, last sentence: if $B_s-B_a>\alpha(p(\lambda_h)-p(\lambda_l))$ , the AS supremum is strictly below $\alpha\pi(\lambda_h)+(1-\alpha)\pi(\lambda_l)+B_a$ .
Proposition 6: the AUS optimum is always $\alpha\pi(\lambda_h)+(1-\alpha)\pi(\lambda_l)+\min\{B_a,B_s\}$ , with explicit optimal prices and fees in three regimes.

Significance

The value $\alpha\pi(\lambda_h)+(1-\alpha)\pi(\lambda_l)+B$ is what a seller would earn by charging a different fast-track price on crowded and uncrowded days. Theorem 1 says that the upgrading option always recovers this state-dependent benchmark, whereas advance selling alone recovers it only while the spot regular premium $B_s-B_a$ stays below the threshold $\alpha(p(\lambda_h)-p(\lambda_l))$ . The threshold gives a concrete pricing rule: offer upgrades exactly when the spot regular ticket is much dearer than the advance one. The paper builds its welfare comparison (§5.4) and its extensions with balking and extra on-site waiting (§6) on the same programs.

The result is proved in the paper, with proofs in its Online Appendix. Nothing in this model has a machine-checked proof. A formalization would check the case analysis of the conditional constraints, which the printed statement gets wrong at the boundary (see Formalization scope), and would provide a Lean model of the waits, the single-rate optimum and the two programs that later missions on this paper can reuse.

Difficulty

Each single-rate step is one-variable calculus with square roots. The difficulty lies in the two programs. Their constraints switch on and off with the cutoffs. Treating them as equalities throughout, or as a disjunction of cases, changes the feasible set. The obvious argument shows that every feasible value is at most the benchmark, by maximising the separable revenue in each cutoff. That is enough for Proposition 6 and for the equality case of Theorem 1, but not for the strict inequality. There the benchmark's maximiser violates $P_a\le P_s$ , the supremum of (7) need not be attained, and the AS value must be shown to stay a positive distance below the benchmark over a feasible set that is not closed.

Formalization scope

All objects are explicit real expressions in a definitions file, AdvSellUpgrade.Online.Setting. The parameters $(\mu,\lambda_h,\lambda_l,\alpha,\bar c,B_a,B_s)$ are bundled in a structure. The standing assumptions are $0<\lambda_l<\lambda_h<\mu$ , $\bar c>0$ and $0<\alpha<1$ ; no sign is assumed on any price. The conventions are:

The mission is stated on the reduced programs (7) and (8) as printed. The queueing game they come from (Definition 1, Propositions 1–2 and the revenue (3)) is not formalized; the waits are taken in Cobham's closed form.
$\alpha$ is a free parameter in $(0,1)$ . The paper defines it as $\beta\lambda_h/\bar\lambda$ , and every value in $(0,1)$ arises from exactly one $\beta$ . Theorem 1 fails at $\alpha=1$ .
The "if" constraints of (7) and (8) are implications.
The fourth constraint of (8) is printed with $W^h$ . It is read with $W^l$ , as in (7). It binds only at $c_f=\bar c$ and does not affect any statement of the mission.
Optimal values are stated as a least upper bound (for AS) or a greatest element (for AUS) of the set of attainable objective values, never as a bare supremum, which Lean sets to $0$ on an empty or unbounded set. This rules out the trivializing reading of Theorem 1 as $0=0$ . The goal does not contain the closed-form optimum.
Boundary repairs. Theorem 1 prints " $<$ " for the equality case and "Otherwise" for the strict one. At $B_s-B_a=\alpha(p(\lambda_h)-p(\lambda_l))$ the AS program attains the AUS optimum, so the boundary point belongs to the equality case, as in the paper's own proof (Online Appendix p. 13). The same repair applies to the last sentence of Proposition 5.
Proposition 5 is formalized in parts 1)–2) and its last sentence. Parts 3)–4) use a threshold and a root defined only implicitly in the appendix.
Proposition 3 and the single-rate milestones are stated for every rate $\lambda\in(0,\mu)$ , which covers both states.

Useful contributions include the one-variable lemmas on $p$ , $c$ , $\pi$ and the waits, and the reduction of (7) and (8) to the separable revenue in the cutoffs.

Selected references

Y. Wang, P. Cao, J. Xie, D. Zhan, Advance Selling and Upgrading in Priority Queues, Manufacturing & Service Operations Management, 2025. https://doi.org/10.1287/msom.2023.0410
A. Cobham, Priority assignment in waiting line problems, Journal of the Operations Research Society of America 2(1), 1954. https://doi.org/10.1287/opre.2.1.70
P. Naor, The regulation of queue size by levying tolls, Econometrica 37(1), 1969. https://doi.org/10.2307/1909200
R. Hassin, M. Haviv, Equilibrium threshold strategies: the case of queues with priorities, Operations Research 45(6), 1997. https://doi.org/10.1287/opre.45.6.966
J. Xie, S. M. Shugan, Electronic tickets, smart cards, and online prepayments: when and how to advance sell, Marketing Science 20(3), 2001. https://doi.org/10.1287/mksc.20.3.219.9765
Y. Cui, I. Duenyas, Ö. Şahin, Pricing of conditional upgrades in the presence of strategic consumers, Management Science 64(7), 2018. https://doi.org/10.1287/mnsc.2017.2783

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Economics Operations Research Optimization·Captain: mikedeng1

A Model of Shoppertainment Live Streaming: Zero Entertainment Bandwidth Is Never Optimal, the Optimal Bandwidth Lies Beyond the Traffic Peak, and Is Full Entertainment When vₑ > v(1+q)/4Research Paper

Motivation

In shoppertainment live streaming a single streamer sells products from many manufacturers during live sessions that mix product presentation with entertainment. Some of the most successful streamers spend much of their air time on content unrelated to the products; the paper cites Dong Yuhui, who devotes his sessions to English lessons. Traditional TV shopping (QVC) does the opposite and spends all air time on product features. This raises a question for operations and marketing: how should a streamer split a session's limited bandwidth between entertainment and product information?

Gu, Zhao and Wu (Management Science, 2024) answer it in a three-stage game between a streamer, a pool of manufacturers and a population of consumers. Entertainment attracts consumers directly, but it also leaves them less certain whether a product fits, and that uncertainty changes how manufacturers price. The paper's headline result, Proposition 3, says that zero entertainment is never optimal. This mission formalizes Proposition 3 together with the chain of lemmas and propositions of Section 4 on which it rests.

Setting

A product has quality $q \in [0,1]$ , the probability that it fits a consumer. A good fit is worth $v > 0$ and a bad fit $0$ . The qualities of the contracted manufacturers are uniform on $[\underline q, 1]$ , where $\underline q \in [0,1)$ is the streamer's quality threshold; write $dF(q) = dq/(1-\underline q)$ . The streamer keeps a commission rate $\alpha \in (0,1)$ of sales revenue.

The streamer chooses an entertainment bandwidth $t_e \in [0,1]$ , the share of a session spent on entertainment. Each attending consumer receives a fit signal whose accuracy is $1 - t_e$ . After a good signal the posterior fit probability is $1 - t_e + t_e q$ , after a bad one it is $t_e q$ . The two willingness-to-pay levels are therefore

p_H = (1 - t_e + t_e q)\,v, \qquad p_L = t_e q v .

With $T$ attending consumers, the demand at price $p$ is $T$ if $p \le p_L$ (everyone buys), $qT$ if $p_L < p \le p_H$ (only good-signal consumers buy), and $0$ above $p_H$ . A manufacturer earns $(1-\alpha)\,p\,D(p)$ .

Write $\hat t_e = 1/(2-\underline q)$ and $\hat q = 2 - 1/t_e$ . In equilibrium a manufacturer of quality $q$ prices low ( $p_L$ ) exactly when $t_e > 1/(2-q)$ , that is, when $q < \hat q$ . A consumer's expected shopping surplus $Eu_p(t_e)$ is the average, over manufacturers, of what the consumer gains at the equilibrium price. Attending costs are uniform on $[0,K]$ , and the streamer's entertaining capability $v_e \ge 0$ gives the entertainment utility $v_e t_e$ , so the attending traffic is

T(t_e) = \frac{v_e t_e + Eu_p(t_e)}{K}.

The individual profitability $\pi(t_e)$ is the streamer's expected commission from one attending consumer, and the streamer's payoff is $\Pi(t_e) = T(t_e)\,\pi(t_e)$ .

Formalization targets

Goal: Proposition 3 (p. 33)

For $0 < \alpha < 1$ , $v > 0$ , $K > 0$ , $0 \le \underline q < 1$ and $v_e \ge 0$ :

$t_e = 0$ is not a maximizer of $\Pi$ on $[0,1]$ , and $\Pi$ attains its maximum on $[0,1]$ ;
if $v_e \le v(1+\underline q)/4$ , every maximizer $t^*_e$ of $\Pi$ satisfies $t^*_e \ge t^T_e$ for every maximizer $t^T_e$ of $T$ ;
if $v_e > v(1+\underline q)/4$ , then

\Pi(t_e) < \Pi(1) \qquad \text{for all } t_e \in [0,1).

Milestones

Lemma 2 (p. 24): a manufacturer of quality $q > 0$ maximizes profit at $p_L$ (uniquely) when $t_e > 1/(2-q)$ and at $p_H$ when $t_e \le 1/(2-q)$ .
Proposition 1 (p. 25): the proportion of low-pricing manufacturers is $\psi(t_e) = (\hat q - \underline q)/(1-\underline q)$ for $t_e > \hat t_e$ and $0$ otherwise; it increases in $t_e$ and decreases in $\underline q$ .
Lemma 3 (p. 27): $Eu_p = \int_{\underline q}^{\hat q}(1-t_e)\,q v\,dF(q)$ for $t_e > \hat t_e$ , and $0$ otherwise.
Display (14) (p. 31): $\pi(t_e) = \alpha\big(\int_{\underline q}^{\hat q} p_L\,dF + \int_{\hat q}^1 p_H q\,dF\big)$ for $t_e > \hat t_e$ , and $\alpha \int_{\underline q}^1 p_H q\,dF$ otherwise.
Proposition 2 (p. 27): $T$ is strictly increasing on $[0,1]$ if $v_e > v(1+\underline q)/2$ ; otherwise it strictly increases up to a peak $t^T_e \in [\hat t_e, 1]$ and strictly decreases after it.
Lemma 4 (p. 32): $\pi$ strictly decreases and then strictly increases on $[0,1]$ , is maximal at both $t_e = 0$ and $t_e = 1$ , and is strictly increasing in $\underline q$ .

Significance

Proposition 3 is the paper's managerial conclusion. Spending all bandwidth on product information, as TV shopping does, is never optimal. A streamer should allocate at least the traffic-maximizing share to entertainment, and should go to full entertainment once its entertaining capability exceeds $v(1+\underline q)/4$ . Corollary 2 and the extensions of Section 5 of the paper (informing capability, in-session adjustment, competing streamers) build on this result and on the Section 4 chain.

The results are proved in the paper and its Online Supplementary, and none is machine-checked. The supplement's argument for the regime $v_e > v(1+\underline q)/4$ shows only that $d\Pi/dt_e > 0$ at $t_e = 1$ , a local statement. The goal here asks for the global claim that $t_e = 1$ is the unique maximizer, so a complete formalization closes a gap in the published argument. The milestones make the paper's equilibrium derivation (manufacturer pricing, consumer surplus, traffic, profitability) checkable step by step.

Difficulty

Lemma 2, Proposition 1, Lemma 3 and display (14) are elementary: case analysis of a price and integrals of piecewise polynomials whose switch point $\hat q(t_e)$ depends on $t_e$ . Proposition 2 and Lemma 4 need sign analysis of explicit rational functions of $t_e$ on $[\hat t_e, 1]$ , uniformly in $\underline q \in [0,1)$ . The traffic is not concave there: its surplus component is convex on part of the interval and concave on the rest.

The goal is the hard part. $\Pi = T\pi$ is the product of a function that rises and then falls (Proposition 2) and a function that falls and then rises (Lemma 4), so the shape of either factor does not settle where the product peaks. Comparing maximizers of $\Pi$ with maximizers of $T$ , and ruling out an interior maximum of $\Pi$ above the threshold $v(1+\underline q)/4$ for every $\underline q$ and $v_e$ , requires quantitative control of the product, not just the monotonicity pattern of its factors. Checking the sign of $d\Pi/dt_e$ at $t_e = 1$ alone is not enough.

Formalization scope

All parameters are real numbers and every function of the model is total; the statements carry the standing ranges as hypotheses. The bandwidth domain is Set.Icc 0 1, and "optimal" means IsMaxOn on that set. The single definitions file Shoppertainment.OptimalBandwidth.Model contains $p_H$ , $p_L$ , the demand (8) extended by $0$ above $p_H$ , the manufacturer's profit, $\hat t_e$ , $\hat q$ , the equilibrium pricing rule, $\psi$ (as a Lebesgue measure), the consumer's ex-ante surplus, $Eu_p$ , $T$ , the revenue per attending consumer, $\pi$ and $\Pi$ . $Eu_p$ and $\pi$ are integrals over the uniform law of quantities evaluated at each manufacturer's equilibrium price. They are not defined by the closed forms (12) and (14). This keeps Lemma 3 and (14) genuine statements and ties the goal to the manufacturers' equilibrium.

Conventions and pinned hypotheses:

$\underline q < 1$ , so that the uniform law on $[\underline q,1]$ exists (the paper allows $\underline q \le 1$ ).
$\alpha < 1$ , the standing range of p. 13.
Lemma 2 pins $q > 0$ and $T > 0$ . Without them every positive price earns $0$ .
Lemma 2 uses the manufacturer's own threshold $1/(2-q)$ , as in the Online Supplementary A.2 and Figure 4(b). The printed display (10) reads $1/(2-\underline q)$ , which would contradict Proposition 1.
A tie $t_e = 1/(2-q)$ is priced high, as in (10). It is a null set for every integral.
Proposition 2 assumes $v_e > 0$ (Table 1), and the goal assumes $v_e \ge 0$ (as printed in Proposition 3).
The peaks $t^T_e$ and $t^*_e$ are stated as maximizers, never through the first-order conditions of the supplement.
Proposition 1's decrease in $\underline q$ is strict for $t_e < 1$ and weak at $t_e = 1$ , where $\psi \equiv 1$ .

Two formalizations would trivialize the mission, and both are excluded. Without the existence clause, a "for every maximizer" statement could hold vacuously, so the goal asserts existence. Defining $Eu_p$ and $\pi$ by (12) and (14) would make two milestones true by definition.

Lemma 1, Corollaries 1–2, Remark 1 and the Section 5 extensions are not posed. A complete development needs only Mathlib's interval integrals, Lebesgue measure on $\mathbb R$ and elementary real analysis. Proofs of the closed forms of the Online Supplementary ( $\Gamma$ in A.4, (A1) in A.6) are welcome as auxiliary lemmas.

Selected references

Z. Gu, X. Zhao, D. J. Wu, A Model of Shoppertainment Live Streaming, Management Science, 2024; authors' accepted manuscript, October 2024. https://doi.org/10.1287/mnsc.2023.01724
R. Lal, M. Sarvary, When and How Is the Internet Likely to Decrease Price Competition?, Marketing Science 18(4), 1999. https://doi.org/10.1287/mksc.18.4.485

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Linear Optimization Operations Research Optimization+1·Captain: mikedeng1

On the Optimality of Affine Policies for Budgeted Uncertainty Sets 1: For Budget of Uncertainty Sets, Affine Policies Approximate the Two-Stage Adjustable Robust Problem within O(log n / log log n)Research Paper

Motivation

Two-stage adjustable robust optimization models decisions taken in two steps around an uncertain demand: a first-stage decision is fixed now, the demand is revealed, and a second-stage (recourse) decision reacts to it. The objective is the worst-case total cost over a set of possible demands. The model covers set cover, capacity planning and network design under uncertain demand (Bertsimas, Goyal 2012; Gupta, Nagarajan, Ravi 2014). Computing the optimal adjustable solution is intractable in general: Feige, Jain, Mahdian and Mirrokni (2007) showed that, for the budget of uncertainty sets below, the problem cannot be approximated within a factor better than $\Omega(\log n/\log\log n)$ under a standard complexity assumption, where $n$ is the number of decision variables.

Affine policies, in which the recourse is restricted to an affine function of the revealed demand (Ben-Tal et al. 2004), are computable by a single linear program and perform well empirically. The question is how much they can lose in the worst case.

Timeline. Bertsimas and Goyal (2012) proved that affine policies are within $O(\sqrt m)$ of the adjustable optimum for general uncertainty sets ( $m$ = number of uncertain parameters) and that this is tight for some sets. El Housni and Goyal (arXiv:1807.00163, 2018; Mathematics of Operations Research 2021) proved that for budget of uncertainty sets the factor is $O(\log n/\log\log n)$ , matching the hardness bound. That result is the goal of this mission.

Setting

z_{AR}(\mathcal U)=\min_{x\in\mathcal X}\ c^Tx+\max_{h\in\mathcal U}\ \min_{y(h)\ge0}\big\{d^Ty(h)\ :\ Ax+By(h)\ge h\big\}.

A first-stage $x$ and a recourse rule $y(\cdot)$ form a feasible solution when $x\in\mathcal X$ and $y(h)\ge0$ , $Ax+By(h)\ge h$ for every $h\in\mathcal U$ . The affine problem (2.1) restricts the rule to $y(h)=Ph+q$ with $P\in\mathbb R^{n\times m}$ , $q\in\mathbb R^n$ (still requiring $Ph+q\ge0$ on $\mathcal U$ ); its value is $z_{Aff}(\mathcal U)\ge z_{AR}(\mathcal U)$ .

The budget of uncertainty set (1.2) with weights $w_i\in[0,1]$ is

\mathcal U=\Big\{h\in[0,1]^m\ :\ \sum_{i=1}^mw_ih_i\le1\Big\}.

The fractional covering value (3.2) of a requirement $h$ is $z(h)=\min_{y\ge0}\{d^Ty: By\ge h\}$ , and $z(\mathcal W)=z(\mathbb 1(\mathcal W))$ for $\mathcal W\subseteq[m]$ . Throughout, $\log$ is the natural logarithm and $\beta=4\log n/\log\log n$ .

Formalization targets

Goal: Theorem 3.1

z_{Aff}(\mathcal U)=O\Big(\frac{\log n}{\log\log n}\Big)\cdot z_{AR}(\mathcal U)

for every instance with $\mathcal U$ a budget of uncertainty set: there are $C>0$ and $N$ such that $z_{Aff}\le C\frac{\log n}{\log\log n}z_{AR}$ for all $n\ge N$ , all $m$ and all data. The constant is left unfixed, so the goal survives any improvement of the constant.

Milestones (the paper's proof, in order)

Lemma 3.1 — the explicit affine solution (3.5), a linear part $y_{Lin}(h)=\sum_{i\in\mathcal I}\alpha_ih_iv_i$ plus a static part from (3.4), is feasible.
Lemma 3.2 — the linear part costs at most $\beta\cdot\mathrm{OPT}$ for every $h\in\mathcal U$ .
Lemmas B.1, B.2 — two Chernoff bounds for weighted sums of independent Bernoulli variables.
Claims 3.1–3.3 — rescaling of the covering problem, and two probability bounds for a randomized rounding of a dual (packing) solution.
Lemma 3.3 (structural result) — if covering each single row $i\in\mathcal J$ costs more than $4\gamma\frac{\log n}{\log\log n}w_i$ , while every scenario $\mathcal W\subseteq\mathcal J$ with $\sum_{\mathcal W}w_i\le1$ costs at most $\gamma$ , then $z(\mathcal J)\le4\gamma\frac{\log n}{\log\log n}$ .
Lemma 3.4 — the static part costs $O(\beta)\cdot\mathrm{OPT}$ .

Significance

The result. The bound shows that affine policies are an optimal approximation for this class, up to a constant factor: no polynomial-time algorithm can beat them by more than a constant in the worst case (under the hardness assumption). It improves the earlier $O(\sqrt m)$ bound for budget sets exponentially, and it is the base case from which the paper derives bounds for intersections of budget constraints (Sections 4–5). Lemma 3.3 is a statement about fractional covering problems alone; it generalizes a set-cover result of Gupta, Nagarajan and Ravi (2014) to arbitrary non-negative matrices and weights, and improves its bound from $O(\log n)$ to $O(\log n/\log\log n)$ .

Formalizing it. The theorem is proved on paper; no machine-checked proof of it or of the structural lemma is known. A formal development would also settle the paper's slips: Claim 3.2's $1-O(1/n)$ does not follow from its proof (stated here in the explicit form the proof gives), Lemma B.2's strict inequality fails when $\mu=0$ , and Lemma 3.3 needs $n$ large.

Difficulty

Feasibility and the cost of the linear part follow directly from the construction. The difficulty is the static part, that is, Lemma 3.3. The obvious argument covers $\mathcal J$ by a union of cheap scenarios, and it fails: each scenario covers rows of total weight at most $1$ , there can be many disjoint scenarios, and adding up their costs gives a bound proportional to $\sum_{i\in\mathcal J}w_i$ , which may be as large as $m$ . The lemma asserts a bound that depends on $n$ alone, at the rate $\log n/\log\log n$ of the hardness result. Any argument therefore has to exploit condition 1 (every single row is expensive) together with condition 2 across all $2^{|\mathcal J|}$ scenarios at once. As printed, the lemma holds only for $n$ large; for $n=2$ it is false.

Formalization scope

Data. Vectors are functions on Fin m, Fin n with the componentwise order; matrices are Matrix (Fin m) (Fin n) ℝ; Lean index $i$ is the paper's $i+1$ . First stage and recourse share the same dimension $n$ , as on p. 1.
Values. $z_{AR}$ , $z_{Aff}$ , $z(h)$ and the static value $z_{Sta}$ are infima in the extended reals, $+\infty$ when infeasible. The recourse is a function of $h$ , which gives the same value as the paper's inner minimum. A formalization with real infima (equal to $0$ on infeasible problems) or without $Ph+q\ge0$ would trivialize the goal and is excluded.
Standing assumptions of p. 1: $B\ge0$ , $c,d\ge0$ , $\mathcal X$ polyhedral, $w_i\in[0,1]$ . The assumption " $\mathcal U\subseteq[0,1]^m$ compact convex with $e_i\in\mathcal U$ " holds for the budget set and is not added.
Added hypotheses and corrections. (i) $c^Tx\ge0$ for $x\in\mathcal X$ : Theorem 3.1 is false without it, and it holds whenever $\mathcal X\subseteq\mathbb R^n_+$ . (ii) $n\ge3$ where $\beta>0$ is needed, and a threshold $N_0$ in Lemma 3.3. (iii) Lemma B.2 in non-strict form. (iv) Claim 3.2 with the explicit bound $1-n\,e^\delta/(1+\delta)^{1+\delta}$ , $\delta=\eta/2-2$ . (v) Covering solutions $v_i$ are required only for $i\in\mathcal I$ .
Quantifiers. In the goal and in Lemmas 3.3, 3.4 the constants $C$ , $N$ , $N_0$ are chosen before $m$ and the instance; a constant allowed to depend on the instance would make the goal trivial.
Randomness is finite: independent Bernoulli selection with probabilities and expectations as finite sums, reusing the published definition SpectralSparsify.Sampling.BernoulliSubset. No measure theory is needed.
Infrastructure. LP strong duality (LinearOptimization.lp_strong_duality, proved on the platform) is a tool for Lemma 3.3. The Chernoff bounds and the structural lemma are reusable beyond this mission. Proofs of any milestone, and alternative arguments for Lemma 3.3, are welcome.

The source is arXiv:1807.00163v2; its printed page numbers equal the PDF page numbers.

Selected references

O. El Housni, V. Goyal, On the Optimality of Affine Policies for Budgeted Uncertainty Sets, arXiv:1807.00163v2, 2019; Mathematics of Operations Research 46(2), 2021. https://arxiv.org/abs/1807.00163
D. Bertsimas, V. Goyal, On the power and limitations of affine policies in two-stage adaptive optimization, Mathematical Programming 134(2), 2012. https://doi.org/10.1007/s10107-011-0444-4
U. Feige, K. Jain, M. Mahdian, V. Mirrokni, Robust combinatorial optimization with exponential scenarios, IPCO 2007, LNCS 4513 (cited as [19] in the source).
A. Gupta, V. Nagarajan, R. Ravi, Thresholded covering algorithms for robust and max–min optimization, Mathematical Programming 146(1–2), 2014 (cited as [21] in the source).
A. Ben-Tal, A. Goryashko, E. Guslitzer, A. Nemirovski, Adjustable robust solutions of uncertain linear programs, Mathematical Programming 99(2), 2004. https://doi.org/10.1007/s10107-003-0454-y
P. Raghavan, Probabilistic construction of deterministic algorithms: approximating packing integer programs, FOCS 1986 (cited as [26] in the source).

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Dynamic Programming Operations Research Optimization+1·Captain: mikedeng1

Coordinating Inventory Control and Pricing Strategies for Perishable Products 1: With Backlogging, the Profit-to-Go Is L♮-Concave and the Optimal Policy Is Monotone with Bounded SensitivityResearch Paper

Motivation

Food, blood and pharmaceuticals are perishable: a unit not sold within its lifetime is thrown away. A seller decides each period how much to order, which price to charge, and whether to dispose of stock early. A large stock of old units argues for a lower price and a smaller order.

Optimal policies for fixed-lifetime inventory are hard to characterize: the state is a vector of inventory levels by remaining lifetime (Fries 1975; Nahmias 1982), and later work turned largely to heuristics.

Chen, Pang and Pan (Oper. Res. 62(2), 2014) obtained structural results for the joint ordering, pricing and disposal problem with a positive replenishment lead time. They used L♮-concavity, a notion from discrete convex analysis (Murota 2003) that Zipkin (2008) used to analyze lost-sales inventory systems with lead times. This mission formalizes the paper's result for the case where unmet demand is backlogged.

Setting

A product with a lifetime of exactly $l \ge 2$ periods is reviewed periodically over a finite horizon of $T$ periods. Orders arrive after a lead time of $k$ periods, $0 \le k < l$ . Write $L = l - 1$ .

Vectors and L♮-convexity. For vectors $u, v \in \mathbb{R}^n$ , $u \wedge v$ and $u \vee v$ are the componentwise minimum and maximum, and $e$ is the all-ones vector. A set $\mathcal{V}$ is L♮-convex if $(u + \alpha e) \wedge v$ and $u \vee (v - \alpha e)$ lie in $\mathcal{V}$ for all $u, v \in \mathcal{V}$ and $\alpha \ge 0$ . A function $f$ is L♮-convex on $\mathcal{V}$ if moreover

f(u) + f(v) \ge f\big((u + \alpha e) \wedge v\big) + f\big(u \vee (v - \alpha e)\big),

and L♮-concave if $-f$ is L♮-convex. The ordered sets are $\mathcal{V}_{n,k} = \{s \in \mathbb{R}^n : s_1 \le \dots \le s_k\}$ .

Data. The unit costs are $c$ (ordering), $h^+$ (holding), $h^-$ (backlogging) and $\theta$ (disposal), all nonnegative, and $\gamma \in [0, 1]$ is the discount factor. The seller chooses the expected demand $d \in \mathcal{D} = [\underline d, \bar d]$ in place of the price. The price is $P(d)$ , strictly decreasing, and the expected revenue $P(d)d$ is continuous and concave on $\mathcal{D}$ . The realized demand is $d_t = d + \varepsilon_t$ , where the noise terms are i.i.d. with mean zero and support in $[A, B]$ , $A \le 0 \le B$ , and $\underline d + A \ge 0$ .

State and decisions. The state $s = (s_1, \dots, s_{l-1})$ satisfies $s_1 \le \dots \le s_{l-1}$ , so the state space is $\mathcal{S} = \mathcal{V}_{l-1,l-1}$ . The coordinate $s_i$ is the inventory, net of backorders, with residual lifetime at most $i$ , including orders in transit. Each period the seller chooses:

an order-up-to level $s_l \ge s_{l-1}$ ;
a demand level $d \in \mathcal{D}$ ;
after demand is realized, a depletion $a$ with $s_1 \vee d_t \le a \le s_{l-k} \vee d_t$ , so $a - d_t$ units are disposed of.

The next state is $\tilde s = (s_2 - a, \dots, s_l - a)$ .

Recursion. The modified profit-to-go satisfies $f_{T+1} = 0$ and

f_t(s) = \max_{s_l \ge s_{l-1},\ d \in \mathcal{D}} \Big\{ P(d)d + \mathbb{E}\big[g_t(s, s_l, d \mid \varepsilon_t)\big] \Big\}, \qquad g_t = \max_{s_1 \vee d_t \le a \le s_{l-k} \vee d_t} \phi_t,

\phi_t(s, s_l, d, a \mid \varepsilon_t) = \gamma c\,\tilde s_{l-1} - c s_l - \theta(a - d_t) - h^+(s_{l-k} - a)^+ - h^-(a - s_{l-k})^+ + \gamma f_{t+1}(\tilde s).

In §4 the paper also assumes $c \le h^-/(1 - \gamma)$ .

Formalization targets

Goal: Theorem 1 (p. 291)

For $t = 1, \dots, T+1$ , $f_t$ , $g_t$ and $\phi_t$ are L♮-concave in $s$ , $(s, s_l, d)$ and $(s, s_l, d, a)$ . The largest optimal order-up-to level $s_{lt}(s)$ and demand level $d_t(s)$ are nondecreasing in $s$ , the optimal price is nonincreasing, and

s_{lt}(s + \omega e) \le s_{lt}(s) + \omega, \qquad d_t(s + \omega e) \le d_t(s) + \omega \qquad (\omega \ge 0).

The largest optimal depletion $a_t(s, s_l, d \mid \varepsilon)$ is nondecreasing, and $a_t(s + \omega e, s_l + \omega, d + \omega \mid \varepsilon) \le a_t(s, s_l, d \mid \varepsilon) + \omega$ .

Milestones

The ordered sets $\mathcal{V}_{n,k}$ , $\mathcal{V}^+_{n,k}$ are L♮-convex (p. 287).
Lemma 1: $g(x, \xi) = f(x - \xi e)$ is L♮-convex when $f$ is (p. 287).
Lemma 2: partial minimization over an L♮-convex set preserves L♮-convexity (p. 287).
The set $\{a \ge s_1 \vee d_t\}$ is L♮-convex (p. 291).
The bound $f_{t+1}(\tilde s') - f_{t+1}(\tilde s) \le (\theta + c)\delta$ for the next states generated by $a' > a \ge s_{l-k}\vee d_t$ (p. 291).
(9): $\phi_t$ is nonincreasing in $a$ beyond $s_{l-k} \vee d_t$ , so the upper bound on $a$ can be dropped (p. 291).
Lemma 3: the largest minimizer of an L♮-convex function is monotone, with $\xi(x + \omega e) \le \xi(x) + \omega$ (p. 287).
The first sentence of Theorem 1, the L♮-concavity of $f_t$ , $g_t$ , $\phi_t$ .
Corollary 1: monotone sensitivity of the order quantity and the demand level in each age class (p. 292).
Proposition 1: if $\theta \ge h^+/(1-\gamma)$ , depleting only $d_t \vee s_1$ is optimal (p. 292).

Significance

The result. Theorem 1 gives a qualitative description of the optimal policy in a model where no closed form is available. Fresher inventory leads to ordering less and charging more. One more unit of inventory of any age raises the order-up-to level and the demand level by at most one unit. Corollary 1 refines this by age: the order quantity is most sensitive to young inventory, and the price discount is most sensitive to inventory close to expiry. These properties underlie the bounds and heuristics of §6 of the paper.

The formalization. The result is proved in the paper, but parts of the argument are informal: the $(\theta + c)\delta$ bound is a coupling argument, existence of optimal solutions is "straightforward to check and thus omitted", and Lemmas 1–3 are stated without proof. A formalization makes the induction checkable and yields a reusable library of L♮-convexity on real vectors with domains, used by the wider family of inventory models built on the same technique. None of these results is known to be machine-checked.

Difficulty

The obvious induction, that each step of the recursion preserves L♮-concavity, fails at the depletion problem (5): its constraint set is not L♮-convex, because of the upper bound $a \le s_{l-k} \vee d_t$ , so Lemma 2 does not apply. The bound must first be shown inactive, which needs a quantitative estimate of how the profit-to-go changes when backorders are added, a property of the value function itself.

Lemma 3 handles a scalar decision, while Theorem 1 has a pair $(s_l, d)$ and needs the componentwise-largest maximizer; every supremum and expectation must also be shown to be finite.

Formalization scope

Carrier. The paper treats $\mathbb{F} \in \{\mathbb{R}, \mathbb{Z}\}$ at once. The mission formalizes the real case only.
L♮-convexity. Functions are real-valued on an explicit domain set, which is the paper's own convention for "L♮-convex on a set". The submodularity characterization is not used as the definition.
Indexing. States are Fin L → ℝ with $L = l - 1 \ge 1$ , indexed from 0, and the lead time satisfies $k \le L$ . The paper's $s_{l-k}$ is coordinate $L - k$ of $(s_1, \dots, s_l)$ : at $k = 0$ it is the decision $s_l$ , and at $k = L$ it is $s_1$ .
Time. Time is counted in periods to go, which is faithful because the data are stationary. f 0 = 0 is $f_{T+1}$ . The parts of Theorem 1 about $g_t$ , $\phi_t$ and decisions cover the decision periods $t \le T$ .
Suprema and expectations. Every maximum is sSup of the image of a nonempty feasible set that is bounded above. Expectations are Bochner integrals against the noise law $\mu$ .
Optimal decisions. Largest optimal decisions are hypotheses at the points compared; at $\gamma = 1$ , $k \ge 1$ no largest order-up-to level exists in the last period.
Standing assumptions. All the §3 assumptions (nonnegative costs, $\gamma \in [0,1]$ , strictly decreasing $P$ , continuous concave revenue, zero-mean noise with support in $[A, B]$ , $\underline d + A \ge 0$ ) are fields of the model structure. The §4 assumption $c \le h^-/(1-\gamma)$ is a hypothesis $(1-\gamma)c \le h^-$ of every §4 statement.
Corrected slips.
- The paper sets $f_t = +\infty$ off $\mathcal{S}$ , a slip for $-\infty$ . The Lean evaluates $f_t$ on $\mathcal{S}$ only.
- In Lemma 3, " $\xi(x) + \omega \in \mathrm{dom}(f)$ " reads $x + \omega e \in \mathrm{dom}(f)$ .
- The final " $< 0$ " of (9) is stated as " $\le 0$ ". It is an equality when $h^- = 0$ and either $\gamma = 1$ or $\theta = 0$ .
- Proposition 1 needs $\gamma < 1$ for its quotient, and it asserts that $d_t \vee s_1$ is optimal, not that it is unique.
Ruled-out formalizations. Any of the following would trivialize the mission:
- defining $f_t$ as an arbitrary L♮-concave function, or taking the continuation in $\phi_t$ as a hypothesis-bound function;
- dropping the upper bound $a \le s_{l-k} \vee d_t$ from the depletion interval;
- restricting to $l = 2$ .
The value functions here are the model's own recursion, for every $l \ge 2$ .
What is needed. L♮-convexity on $\mathbb{R}^n$ with domains (shifts, sums, expectations, partial maximization, comparative statics for largest maximizers) and integrability of $g_t$ in the noise; all reusable. Proofs of single milestones are welcome.
Source. The source is the published Operations Research article; in the copy held, printed page = PDF page + 282.

Selected references

X. Chen, Z. Pang, L. Pan, Coordinating Inventory Control and Pricing Strategies for Perishable Products, Operations Research 62(2):284–300, 2014. https://doi.org/10.1287/opre.2014.1261
P. Zipkin, On the Structure of Lost-Sales Inventory Models, Operations Research 56(4):937–944, 2008. https://doi.org/10.1287/opre.1070.0482
K. Murota, Discrete Convex Analysis, SIAM Monographs on Discrete Mathematics and Applications, 2003. https://doi.org/10.1137/1.9780898718508
B. Fries, Optimal Ordering Policy for a Perishable Commodity with Fixed Lifetime, Operations Research 23(1):46–61, 1975. https://doi.org/10.1287/opre.23.1.46
S. Nahmias, Perishable Inventory Theory: A Review, Operations Research 30(4):680–708, 1982. https://doi.org/10.1287/opre.30.4.680

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Mathematical Physics·Captain: Lucas

Giddings 1999: Boundary S matrix and the AdS/CFT dictionary — free-field spectrum on global AdSResearch Paper

Motivation

The AdS/CFT correspondence (Maldacena 1997) relates quantum gravity on anti-de Sitter (AdS) space to a conformal field theory (CFT) living on its timelike boundary. Gubser–Klebanov–Polyakov and Witten gave the CFT-from-AdS direction of the dictionary: CFT correlators are computed from bulk Green functions. Giddings (PRL 83, 2707, 1999) supplies an entry in the reverse direction. Because AdS confines particles and has no ordinary asymptotic states, a flat-space S matrix does not exist; the paper defines a boundary S matrix from in/out states built out of wave packets incident from the boundary, derives an LSZ-type reduction formula (eq. 42), and concludes that the boundary S matrix equals the boundary CFT correlator (eqs. 44–45).

Every step of that argument rests on the classical theory of the free Klein–Gordon field on global AdS: the parameters $h_\pm,\nu$ (eq. 4), the discrete normalizable spectrum $\omega_{nl}=2h_++2n+l$ with modes behaving like $(\cos r)^{2h_+}$ at the boundary (eqs. 5–7), the non-normalizable modes behaving like $(\cos r)^{2h_-}$ (eqs. 8–9), and the boundary limit (eqs. 19–20) that fixes the normalization $2\nu$ of the bulk–boundary propagator.

Setting

Fix an integer $d\ge 2$ , an AdS radius $R>0$ and a scalar mass $m\in\mathbb R$ . Global $\mathrm{AdS}_{d+1}$ has metric (eq. 2)

ds^2=R^2\sec^2 r\,\bigl(-dt^2+dr^2+\sin^2 r\,d\Omega_{d-1}^2\bigr),\qquad 0\le r<\tfrac\pi2 .

Define (eq. 4)

\nu=\tfrac12\sqrt{d^2+4m^2R^2},\qquad 2h_\pm=\tfrac d2\pm\nu .

A mode $\phi=e^{-i\omega t}Y_{l\vec m}(\Omega)\chi(r)$ , with $Y_{l\vec m}$ a spherical harmonic on $S^{d-1}$ , solves the free equation $(\Box-m^2)\phi=0$ (eq. 3) exactly when the radial profile $\chi$ satisfies, on $0<r<\pi/2$ ,

\chi''+\frac{d-1}{\sin r\cos r}\chi'+\omega^2\chi-\frac{l(l+d-2)}{\sin^2 r}\chi-\frac{m^2R^2}{\cos^2 r}\chi=0 .

A normalizable mode is a nonzero $C^2$ solution that is bounded as $r\to0$ (regular at the centre of AdS) and has finite Klein–Gordon norm, i.e. $\int_0^{\pi/2}\tan^{d-1}r\,\chi(r)^2\,dr<\infty$ . The explicit modes of eq. (5) are

\chi_{nl}(r)=\sin^l r\,(\cos r)^{2h_+}\,P_n^{(l+d/2-1,\;\nu)}(\cos 2r),

with $P_n^{(\alpha,\beta)}$ the Jacobi polynomial, and the eigenfrequencies of eq. (7) are $\omega_{nl}=2h_++2n+l$ .

Formalization targets

Goal: the normalizable spectrum (eqs. 3–7)

For every $\omega\in\mathbb R$ , $l\in\mathbb N$ and radial profile $\chi$ ,

\chi \text{ is a normalizable mode at }(\omega,l)\iff \exists n\in\mathbb N:\ |\omega|=\omega_{nl}\ \text{and}\ \chi=c\,\chi_{nl}\text{ on }(0,\tfrac\pi2)\text{ for some }c\neq0 .

Milestones

Eq. (4): $2h_\pm$ are the two roots of $\Delta(\Delta-d)=m^2R^2$ , and $2h_+-2h_-=2\nu$ .
Eqs. (5), (7): $\chi_{nl}$ solves the radial equation at $\omega=\omega_{nl}$ .
Eq. (6): $\chi_{nl}(r)/(\cos r)^{2h_+}\to k_{nl}\neq0$ as $r\to\pi/2$ .
Eq. (5): $\chi_{nl}$ is a normalizable mode at $(\omega_{nl},l)$ .
Eqs. (8)–(9): for $|\omega|$ outside the spectrum there is a solution regular at $r=0$ with $\chi(r)/(\cos r)^{2h_-}\to1$ .
Eqs. (19)–(20): $\displaystyle\lim_{r\to\pi/2}(\tan r)^{d-1}\bigl[(\cos r)^{2h_-}\overleftrightarrow{\partial_r}(\cos r)^{2h_+}\bigr]=-2\nu$ .
Eqs. (12), (33): $\Delta_b>\Delta\Rightarrow(\cos r)^{\Delta_b-\Delta}\to0$ .
Eqs. (23), (42)–(44): given Witten's formula (43), the LSZ expression (42) built from the wave packets (23) equals $\int\prod_a[db_a\,f_a(b_a)/Z_a]\,\langle T\prod_a\mathcal O(b_a)\rangle$ .

Significance

The goal is the precise mathematical content behind the paper's statement that free solutions "exist at arbitrary frequency but are only normalizable for a discrete set of frequencies". This spectrum is what makes the boundary operators $a_{\omega l\vec m}$ of eq. (25) act as creation and annihilation operators (eqs. 27–29), and the boundary exponents $2h_\pm$ and the constant $2\nu$ determine the normalizations in eqs. (20), (21), (28) and (29). The physics derivation of eqs. (28)–(42) is not rigorous mathematics; milestone 8 isolates the final step from (42)–(43) to (44), which is an exact identity about integral kernels.

The radial results are classical (the paper attributes the explicit solutions to Balasubramanian–Kraus–Lawrence). This mission asks for machine-checked proofs of these results, together with the reusable Jacobi-polynomial and boundary-asymptotics infrastructure they require.

Difficulty

Verifying that $\chi_{nl}$ solves the equation is a computation. The goal's converse direction is the hard part: one must show that a solution regular at $r=0$ can have finite Klein–Gordon norm only when $|\omega|$ equals some $\omega_{nl}$ , and that it is then unique up to a scalar. This requires controlling both regular singular points $r=0$ and $r=\pi/2$ of the radial equation, including the resonant cases where $2\nu$ or $l+d/2-1$ is an integer and logarithms appear. Since $d\ge2$ forces $\nu\ge1$ , the $(\cos r)^{2h_-}$ branch is never square-integrable, so the second quantization available for $0<\nu<1$ does not arise.

Formalization scope

$d$ is a natural number with $d\ge2$ ; $m,R$ are real with $R>0$ ; $l,n$ are natural numbers; $\omega$ is real and only $|\omega|$ matters.
Radial profiles are real-valued functions $\mathbb R\to\mathbb R$ ; only their values on $(0,\pi/2)$ enter any definition. Complex modes reduce to their real and imaginary parts.
Regularity at the origin is encoded as boundedness of $|\chi|$ on $(0,\pi/4]$ . Finite Klein–Gordon norm is integrability of $\tan^{d-1}r\,\chi(r)^2$ on $(0,\pi/2)$ .
The Jacobi polynomial is defined by the explicit finite sum with generalized binomial coefficients. Powers of $\cos r$ with real exponents use Real.rpow; $\cos r>0$ on the relevant range.
Nontriviality ( $\chi\neq0$ somewhere) is part of the definition of a normalizable mode, so the zero function never counts.
Milestone 8 takes the LSZ right-hand side (42) as the definition of the boundary S matrix and Witten's relation (43) as a hypothesis, with joint integrability of the integrand. In- and out-packets are listed together, and the plane-wave limit (45) is not formalized.

Selected references

S. B. Giddings, Boundary S matrix and the anti–de Sitter space to conformal field theory dictionary, Phys. Rev. Lett. 83, 2707 (1999). https://doi.org/10.1103/PhysRevLett.83.2707
V. Balasubramanian, P. Kraus, A. Lawrence, Bulk vs. boundary dynamics in anti-de Sitter spacetime, Phys. Rev. D 59, 046003 (1999). https://arxiv.org/abs/hep-th/9805171
E. Witten, Anti de Sitter space and holography, Adv. Theor. Math. Phys. 2, 253 (1998). https://arxiv.org/abs/hep-th/9802150
J. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2, 231 (1998). https://arxiv.org/abs/hep-th/9711200
S. S. Gubser, I. Klebanov, A. Polyakov, Gauge theory correlators from non-critical string theory, Phys. Lett. B 428, 105 (1998). https://arxiv.org/abs/hep-th/9802109

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Quantum Error Correction Quantum Information·Captain: Lucas

Quantum Threshold Theorem: Polylogarithmic-Overhead Fault-Tolerant Simulation Below a Constant Fault RateTextbook

Motivation

A quantum computer cannot perform gate operations perfectly, so a small constant error per operation is unavoidable. A priori this could mean that a noisy quantum computer can apply only a bounded number of gates before the computation is destroyed by noise, which would rule out long quantum computations in practice. The threshold theorem (or quantum fault-tolerance theorem) says that this does not happen: if the error rate of the physical components is below a constant threshold, quantum error correction can suppress the logical error rate to arbitrarily low levels at a modest cost in size. It is the quantum analogue of von Neumann's threshold theorem for classical computation with unreliable components (von Neumann 1956).

Timeline.

1956: von Neumann shows that reliable classical computation is possible with components that fail with a small constant probability.
1996: Shor gives the first fault-tolerant quantum computation scheme, proving a weaker version of the threshold theorem (Shor 1996).
1997-1999: the threshold theorem is proved independently, for various error models, by Aharonov and Ben-Or (arXiv:quant-ph/9906129); Knill, Laflamme and Zurek (arXiv:quant-ph/9702058); and Kitaev (arXiv:quant-ph/9707021).
2000: Nielsen and Chuang give the general framework stated below in their textbook.

Setting

A register of qubits is labelled by a finite set $Q$ , and its states are density matrices indexed by bit strings $x:Q\to\{0,1\}$ . A location applies one elementary operation to an ordered tuple of distinct qubits. The elementary operations are the gates $H$ , $T=\mathrm{diag}(1,e^{i\pi/4})$ and $\mathrm{CNOT}$ , the preparation of a qubit in $|0\rangle$ , and a single-qubit wait. A layer is a set of locations whose supports partition the register, so idle qubits carry explicit waits. A circuit $C$ is a sequence of layers started in $|0\cdots0\rangle$ , and its size $|C|$ is its total number of locations. Measuring every qubit at the end gives the outcome distribution $P_{\mathrm{ideal}}(C)$ .

The hardware follows an independent stochastic noise model with fault rate at most $p$ : each location, independently, is faulty with some probability $q\le p$ , and a faulty location applies an arbitrary, adversarially chosen quantum channel to its own qubits instead of its ideal operation. Running a physical circuit $C'$ on such hardware gives the outcome distribution $P_{\mathrm{noisy}}(C')$ . A physical circuit simulating $n$ logical qubits acts on $n$ blocks of $b$ qubits; at the end each block's measured bits are decoded to one logical bit by a fixed map $\mathrm{dec}:\{0,1\}^b\to\{0,1\}$ .

Formalization targets

Goal: threshold theorem

The source states the theorem as follows: a quantum circuit on $n$ qubits containing $p(n)$ gates may be simulated with probability of error at most $\varepsilon$ using $O(\log^{c}(p(n)/\varepsilon)\,p(n))$ gates, for some constant $c$ , on hardware whose components fail with probability at most $p$ , provided $p$ is below some constant threshold $p_{\mathrm{th}}$ , and given reasonable assumptions about the noise in the underlying hardware. Formally: there are $p_{\mathrm{th}}>0$ and $c\in\mathbb N$ such that for each $0\le p<p_{\mathrm{th}}$ there is $A$ such that for all $n$ , $D$ and $0<\varepsilon<1$ there are $b$ and $\mathrm{dec}$ such that every depth- $D$ circuit $C$ on $n$ qubits has a simulation $C'$ with

|C'|\le A\Big(\log\tfrac{|C|}{\varepsilon}+1\Big)^{c}|C|,\qquad d_{\mathrm{TV}}\big(\mathrm{dec}_*P_{\mathrm{noisy}}(C'),P_{\mathrm{ideal}}(C)\big)\le\varepsilon

for every noise model with fault rate at most $p$ . The goal leaves the threshold, the exponent and the constants unspecified; it asserts only the shape of the result.

Milestones

Errors accumulate at most linearly: without encoding, $d_{\mathrm{TV}}\le p\,|C|$ .
Better gates: one level of encoding gives $d_{\mathrm{TV}}\le c\,p^2|C|$ with $|C'|\le A|C|$ .
Recursion: $e_{k+1}\le c\,e_k^2$ and $e_0\le p$ imply $e_k\le (cp)^{2^k}/c$ .
Concatenation: for every $k$ , $d_{\mathrm{TV}}\le \frac{(cp)^{2^k}}{c}|C|$ with $|C'|\le A d^k|C|$ .
Overhead: choosing $k$ to make the total error at most $\varepsilon$ costs $d^k=O\big((\log(g/\varepsilon)+1)^{\lceil\log_2 d\rceil}\big)$ .

Significance

The theorem shows that noise is not a fundamental obstacle to scalable quantum computation: any efficient ideal quantum algorithm remains efficient on noisy hardware once the physical error rate is below a constant. Current estimates put the threshold for the surface code on the order of $1\%$ (Fowler, Stephens, Groszkowski 2009), and the theorem underlies the resource estimates that guide current hardware development (Campbell, Terhal, Vuillot 2017).

The theorem has been proved on paper in several forms. A machine-checked proof requires an explicit model of noisy circuits, the construction of fault-tolerant gadgets for a universal gate set, and the probabilistic bookkeeping that is usually left informal. This mission provides the model and an attack path that follows the standard proof strategy.

Difficulty

Encoding a gate in an error-correcting code makes it larger, and therefore exposes it to more faults. The whole difficulty is to show that errors are corrected faster than they are created: every encoded gadget, including error correction itself and the non-Clifford $T$ gate, must be designed so that a single fault cannot cause a logical failure, and this must hold uniformly at every level of concatenation. Simply repeating each operation and taking a majority vote does not work for quantum states, which cannot be copied, and measuring a qubit directly destroys the encoded information.

Formalization scope

All statements use one definition file. Qubits of the physical register are labelled by Fin n × Fin b; states are complex matrices indexed by Q → Fin 2; channels are lists of Kraus operators; noise is the independent stochastic model above, with per-location fault probabilities in $[0,p]$ and arbitrary fault channels. No mid-circuit measurement is available, but qubits can be re-prepared in $|0\rangle$ at any time. Sizes count every location, idle steps included. The block size and the decoder may depend on $n$ , the depth, $\varepsilon$ and $p$ , but never on the circuit being simulated; this rules out a decoder that simply writes down the answer, and the size bound rules out classical brute-force simulation inside the physical circuit.

A complete development needs: trace-distance and contractivity facts for channels, a quantum error-correcting code with fault-tolerant gadgets for $H$ , $T$ , $\mathrm{CNOT}$ , preparation and error correction, a fault-counting argument for one level of encoding, and its iteration under concatenation. The channel and trace-distance infrastructure is reusable beyond this mission.

Selected references

J. von Neumann, Probabilistic Logics and the Synthesis of Reliable Organisms From Unreliable Components, Automata Studies, Princeton University Press, 1956. https://doi.org/10.1515/9781400882618-003
P. W. Shor, Fault-tolerant quantum computation, Proc. 37th FOCS, 1996. https://doi.org/10.1109/SFCS.1996.548464
D. Aharonov and M. Ben-Or, Fault-Tolerant Quantum Computation with Constant Error Rate, SIAM J. Comput. 38(4), 2008. https://arxiv.org/abs/quant-ph/9906129
E. Knill, Resilient Quantum Computation, Science 279, 1998. https://arxiv.org/abs/quant-ph/9702058
A. Yu. Kitaev, Fault-tolerant quantum computation by anyons, Annals of Physics 303, 2003. https://arxiv.org/abs/quant-ph/9707021
M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary ed., Cambridge University Press, 2010.
Wikipedia, Threshold theorem. https://en.wikipedia.org/w/index.php?title=Threshold_theorem&oldid=1360742213

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Quantum Error Correction Quantum Information·Captain: Goku

Stabilizer Codes and Quantum Error Correction 4: The standard-form encoding network maps |c⟩ to the encoded basis state (4.8)Textbook

Motivation

This mission develops Chapter 4 of Stabilizer Codes and Quantum Error Correction by Daniel Gottesman. The chapter's selected results describe the mathematical guarantees below.

Setting and conventions

The linked theorem statements and shared definitions specify the register types, oracle access, normalization, and resource conventions. Their source locators and independent readbacks accompany each target.

Chapter goal

Let a stabilizer on $n=m+k$ qubits ( $m=n-k$ generators) be given in the standard form (4.3) of §4.1,

\left(\begin{array}{ccc|ccc} I & A_1 & A_2 & B & C_1 & C_2\\ 0 & 0 & 0 & D & I & E\end{array}\right),

with $r$ rows in the first block, and assume its rows commute: $H_xH_z^{\mathsf T}+H_zH_x^{\mathsf T}=0$ . For a binary row $(a\,|\,b)$ write $P(a|b)=\bigotimes_q \sigma_z^{b_q}\sigma_x^{a_q}$ (the real form, sign $+1$ ). Let $M_i=P(\text{row } i)$ be the generators and let $\bar X_1,\dots,\bar X_k$ be the logical $\sigma_x$ operators in the standard form of p.40: $\bar X_i=P(0\,u_2\,e_i\,|\,v_1\,0\,0)$ with $u_2$ the $i$ -th row of $E^{\mathsf T}$ and $v_1$ the $i$ -th row of $E^{\mathsf T}C_1^{\mathsf T}+C_2^{\mathsf T}$ .

Let $U_{\mathrm{enc}}$ be the encoding network of §4.2 (with the author's errata): first, for each $i$ , the controlled-NOTs that apply $\bar X_i$ without its final $\sigma_x$ , controlled on qubit $n-k+i$ ; then, for $i=1,\dots,r$ in turn, the Hadamard $R$ on qubit $i$ , a $\sigma_z$ on qubit $i$ if $B_{ii}=1$ , and $M_i$ without its qubit- $i$ factor, controlled on qubit $i$ (controlled- $\sigma_z^{b}\sigma_x^{a}$ gates). Then for every $c=c_1\dots c_k\in\{0,1\}^k$ ,

U_{\mathrm{enc}}\,|0\dots0\rangle|c_1\dots c_k\rangle=\frac{(I+M_1)\cdots(I+M_{n-k})\,\bar X_1^{c_1}\cdots\bar X_k^{c_k}|0\dots0\rangle}{\big\|(I+M_1)\cdots(I+M_{n-k})\,\bar X_1^{c_1}\cdots\bar X_k^{c_k}|0\dots0\rangle\big\|},

and in particular the vector on the right-hand side is nonzero.

This is the correctness of Gottesman's general encoding circuit for stabilizer codes: the network prepares exactly the encoded basis states, with no leftover global phase.

Formalization Note The errata are built in: the Hadamard on qubit $i$ comes just before that qubit's control operations, controlled- $Y$ gates are controlled- $\sigma_z\sigma_x$ gates in the real form, and every generator has sign $+1$ . The page writes encoded states unnormalized; the unitary network yields the normalized vector. Since the real form $\sigma_z\sigma_x=i\sigma_y$ , a generator with an odd number of $(1|1)$ positions is $i$ times a Hermitian Pauli operator; the identity is the one the page's network computes in this convention.

Significance

These are known mathematical results. Their Lean statements have been checked and reviewed against the cited source; formal proofs remain open for contributors.

Supporting targets

§4.1: Every stabilizer code can be put into the standard form (4.3).
§4.2: The standard-form encoding network maps |c⟩ to the encoded basis state (4.8).

Formalization scope

Prove the existing statements under their stated hypotheses and definitions. Any correction to a printed source claim remains explicit in its target description. All listed target proofs are currently sorry; no proof is claimed. The mission goal is the designated chapter capstone; the remaining targets are separate milestones. Parts a, b, c and d are merged into their original chapter.

Selected references

Daniel Gottesman, Stabilizer Codes and Quantum Error Correction, 1997 thesis, downloaded arXiv v1 PDF; PDF title page now prints 2024. https://arxiv.org/pdf/quant-ph/9705052v1.

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Optimization Quantum Information·Captain: Goku

Quantum algorithms for optimizers 9: QAOA with enough layers reaches a maximum eigenstate of H_FTextbook

Motivation

This mission develops Chapter 9 of Quantum algorithms for optimizers by Giacomo Nannicini. The chapter's selected results describe the mathematical guarantees below.

Setting and conventions

Chapter goal

Let $n \ge 0$ and let $f : \{0,1\}^n \to \mathbb{R}$ be the objective of the maximization problem (9.38). Define the $n$ -qubit Hamiltonians

H_F = \sum_{x \in \{0,1\}^n} f(x)\,|x\rangle\langle x|, \qquad H_I = \sum_{j=1}^n \sigma_j^X,

where $\sigma_j^X$ is the Pauli $X$ on qubit $j$ and the identity elsewhere. For $p \ge 1$ and $\beta, \theta \in [0, 2\pi]^p$ let

U_{\mathrm{QAOA}}(p, \beta, \theta) = e^{-i\beta_p H_I} e^{-i\theta_p H_F} \cdots e^{-i\beta_1 H_I} e^{-i\theta_1 H_F}.

Then for every $\delta > 0$ there are a unit vector $|\varphi\rangle$ with $H_F|\varphi\rangle = (\max_x f(x))\,|\varphi\rangle$ , that is an eigenstate of $H_F$ with maximum eigenvalue, and an integer $P \ge 1$ such that for every $p \ge P$ there are angles $\beta^*, \theta^* \in [0, 2\pi]^p$ with

\bigl\| U_{\mathrm{QAOA}}(p, \beta^*, \theta^*)\,(H|0\rangle)^{\otimes n} - |\varphi\rangle \bigr\| \le \delta .

Here $(H|0\rangle)^{\otimes n}$ is the uniform superposition, the maximum eigenstate of $H_I$ . The proposition says that with enough layers, the QAOA ansatz can reproduce adiabatic evolution along $H(s) = (1-s)H_I + sH_F$ and so prepares a state close to one encoding an optimal solution of (9.38).

Formalization Note The source's limit over $p$ , with angles that depend on $p$ , is stated as "for all sufficiently large $p$ some angles achieve distance at most $\delta$ ", with the target eigenvector fixed before $p$ . The global phase allowed by Theorem 9.11 is absorbed into the choice of the eigenvector, since a phase multiple of a maximum eigenvector is again one. The informal claim that $U_{\mathrm{QAOA}}$ "approximates an adiabatic evolution" is represented by its stated "in particular" consequence.

Significance

These are known mathematical results. Their Lean statements have been checked and reviewed against the cited source; formal proofs remain open for contributors.

Supporting targets

Proposition 9.4: The Ising Hamiltonian is diagonal with the QUBO objective values.
Theorem 9.11: Adiabatic theorem with T ≥ (10⁴/δ²)(‖H′‖³/γ⁴ + ‖H′‖‖H″‖/γ³).
Theorem 9.17: Adiabatic theorem when the tracked eigenvalue is identically 0.
Lemma 9.18: Derivative bounds for the phase-fixed instantaneous zero eigenvector.
Lemma 9.20: Replacing U_k ⋯ U_j by powers of U_1 in the discretized adiabatic evolution.
Corollary 9.21: The T-bound of Theorem 9.11 suffices (derived from Theorem 9.17).
Theorem 9.23: Adiabatic theorem with the integrated 1/γ³ gap dependence.
Corollary 9.24: Adiabatic theorem for the linear interpolation (1 − s)H_I + sH_F.
Proposition 9.28: QAOA with enough layers reaches a maximum eigenstate of H_F.

Formalization scope

Selected references

Giacomo Nannicini, Quantum algorithms for optimizers, 2025, v5; notes dated 9 July 2025. https://arxiv.org/pdf/2408.07086v5.

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Machine Learning Quantum Information·Captain: Goku

Quantum Algorithms and Learning Theory — Part II 6: Shadow tomography with poly(log m, log d, 1/ε, log(1/δ)) copiesTextbook

Motivation

This mission develops Chapter 6 of Quantum Algorithms and Learning Theory — Part II by Srinivasan Arunachalam. The chapter's selected results describe the mathematical guarantees below.

Setting and conventions

Chapter goal

This is Aaronson's shadow tomography theorem.

A two-outcome measurement on $\mathbb C^d$ is a POVM $\{E,\mathbb 1-E\}$ with $0\preceq E\preceq\mathbb 1$ . A learner using $k$ copies of an unknown $d$ -dimensional density matrix $\rho$ performs a measurement (a POVM $\{N_o\}_{o\in O}$ with finite outcome set $O$ ) on $\rho^{\otimes k}$ and outputs numbers $b(o)=(b_1(o),\dots,b_m(o))\in[0,1]^m$ .

There is a polynomial $p$ in four variables such that for all $\varepsilon,\delta\in(0,1]$ , all integers $m,d>0$ and all effects $E_1,\dots,E_m$ on $\mathbb C^d$ there exist $k\le p(\log m,\log d,1/\varepsilon,\log(1/\delta))$ and such a learner (depending on $m,d,\varepsilon,\delta$ and $E_1,\dots,E_m$ , but not on $\rho$ ) with

\Pr_{o\sim\{\operatorname{Tr}(N_o\rho^{\otimes k})\}}\Big[\ \forall i\in[m]:\ |\operatorname{Tr}(E_i\rho)-b_i(o)|\le\varepsilon\ \Big]\ \ge\ 1-\delta

for every $d$ -dimensional density matrix $\rho$ .

The number of copies is exponentially smaller in $m$ and $d$ than for full state tomography or for estimating each $\operatorname{Tr}(E_i\rho)$ separately.

Formalization Note The source writes $\varepsilon,\delta\in[0,1]$ ; the endpoints $0$ are excluded because there $1/\varepsilon$ or $\log(1/\delta)$ is undefined. Internal randomness and post-processing of the learner are absorbed into the POVM; $\rho^{\otimes k}$ is the $k$ -fold Kronecker power and logarithms are natural (the base is absorbed by the polynomial).

Significance

These are known mathematical results. Their Lean statements have been checked and reviewed against the cited source; formal proofs remain open for contributors.

Supporting targets

Lemma 6.4.8: A class of size at most M needing Ω(√((N−k+1)k)) quantum membership queries.
Fact 6.4.9: An admissible weight k of order log M/(log(N/log M)+1).
Theorem 6.6.2: Shadow tomography with poly(log m, log d, 1/ε, log(1/δ)) copies.

Formalization scope

Selected references

Srinivasan Arunachalam, Quantum Algorithms and Learning Theory — Part II, 2018 thesis, ILLC DS-2018-08. https://ir.cwi.nl/pub/27536/Thesis_S_Arunachalam.

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Quantum Information·Captain: Goku

Quantum Computing: Lecture Notes 15: 2-query locally decodable codes have exponential lengthTextbook

Motivation

This mission develops Chapter 15 of Quantum Computing: Lecture Notes by Ronald de Wolf. The chapter's selected results describe the mathematical guarantees below.

Setting and conventions

Chapter goal

This is the exponential lower bound on the length of 2-query locally decodable codes.

A map $C : \{0,1\}^n \to \{0,1\}^N$ is a $(2,\delta,\varepsilon)$ -locally decodable code if for every $i \in [n]$ a classical randomized algorithm making at most $2$ queries to an $N$ -bit string $y$ outputs $x_i$ with probability at least $\frac12 + \varepsilon$ , for every $x \in \{0,1\}^n$ and every $y$ within Hamming distance $\delta N$ of $C(x)$ .

There are constants $c > 0$ and $n_0 \in \mathbb N$ such that for all $n \ge n_0$ , all $N$ , all $\delta > 0$ , $\varepsilon > 0$ , and every $(2,\delta,\varepsilon)$ -locally decodable code $C : \{0,1\}^n \to \{0,1\}^N$ ,

N \ge 2^{c\,\delta^2 \varepsilon^4 n}.

Hence the Hadamard code, of length $2^n$ , is essentially optimal among 2-query LDCs.

Formalization Note The asymptotic $2^{\Omega(\delta^2\varepsilon^4 n)}$ is read with absolute constants $c$ and $n_0$ , quantified before $N, \delta, \varepsilon$ and $C$ . The threshold $n_0$ is needed: the identity code with $n = N = 1$ is a $(2, \delta, 1/2)$ -LDC for every $\delta < 1$ , while $1 \ge 2^{c\delta^2\varepsilon^4}$ fails for every $c > 0$ .

Significance

These are known mathematical results. Their Lean statements have been checked and reviewed against the cited source; formal proofs remain open for contributors.

Supporting targets

§15.2, Theorem 4: Nayak: quantum random access codes need (1 - H(p))n qubits.
§15.3, Fact 1: Katz–Trevisan normal form for decoders of locally decodable codes.
§15.3, Theorem 5: 2-query locally decodable codes have exponential length.

Formalization scope

Selected references

Ronald de Wolf, Quantum Computing: Lecture Notes, 27 August 2026. https://homepages.cwi.nl/~rdewolf/qcnotes.pdf.

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Optimization Quantum Information·Captain: Goku

Quantum algorithms for optimizers 8: End-to-end complexity of the quantum MaxCut SDP solver in the QRAM model (corrected statement)Textbook

Motivation

This mission develops Chapter 8 of Quantum algorithms for optimizers by Giacomo Nannicini. The chapter's selected results describe the mathematical guarantees below.

Setting and conventions

Chapter goal

This is Proposition 8.34 of Nannicini, the end-to-end cost of the quantum mirror-descent solver for the MaxCut SDP relaxation, stated in an explicit hybrid classical–quantum QRAM cost model.

Let $n=2^q$ . For a nonzero real symmetric $C\in\mathbb R^{n\times n}$ put $\hat C=C/\|C\|_F$ . Call a density matrix $\rho\in\mathbb C^{n\times n}$ an $\epsilon$ -solution of (MaxCutSDP) if

\sum_{j=1}^n\Big|\rho_{jj}-\frac1n\Big|\le\epsilon\qquad\text{and}\qquad \operatorname{Tr}(\hat C\rho)\ \ge\ \operatorname{Tr}(\hat C\sigma)-\epsilon\ \text{ for every density matrix }\sigma\text{ with }\operatorname{diag}(\sigma)=\tfrac1n\mathbb 1.

A hybrid algorithm consists of (i) a classical preprocessing program, a straight-line program of real arithmetic operations ( $+,-,\times,\div,\sqrt\cdot,\arccos,\arg$ , rational constants), which is the only part that reads the entries of $C$ and which writes the input half of a QRAM ( $U_{\mathrm{QRAM}}|j\rangle|y\rangle=|j\rangle|y\oplus M_j\rangle$ , Def. 5.39); (ii) $R$ quantum runs, each applying a circuit of two-qubit gates and QRAM calls to $|0\cdots0\rangle$ and measuring all qubits, where the circuit of each run and the contents of the work half of the QRAM are chosen classically from the outcomes of the earlier runs; (iii) an output, computed from all outcomes, in the form of a list of pairs $(y^{(i)},d^{(i)})\in\mathbb R^2\times\mathbb R^n$ , which stands for the matrix

\bar\rho=\frac1{L}\sum_{i=1}^{L}\frac{\exp\big(y^{(i)}_1\hat C+y^{(i)}_2\operatorname{diag}(d^{(i)})\big)}{\operatorname{Tr}\exp\big(y^{(i)}_1\hat C+y^{(i)}_2\operatorname{diag}(d^{(i)})\big)}.

The theorem: there are absolute constants $K>0$ and $c$ such that for all $q$ , all $\epsilon\in(0,1]$ and all $\delta\in(0,1)$ , with $\Lambda=\log(2+n)+\log(2+1/\epsilon)+\log(2+1/\delta)$ , there is a hybrid algorithm, fixed before $C$ , such that

the QRAM has size at most $K n^2\Lambda^c$ , with words of at most $K\Lambda^c$ bits;
the preprocessing uses at most $K n^2\Lambda^c$ classical operations;
along every measurement record the total number of QRAM accesses and additional gates is at most $K\,\dfrac{n^{1.5}}{\epsilon^5}\Lambda^c$ ;
for every nonzero real symmetric $C$ , with probability at least $1-\delta$ the output $\bar\rho$ is an $\epsilon$ -solution of (MaxCutSDP).

The proposition is the payoff of the chapter's MaxCut analysis: the subgradient estimation of Prop. 8.28, the iteration count of Thm. 8.30 and the Hamiltonian structure of Lem. 8.33 combine into a quantum solver whose quantum cost is sublinear in the $n^2$ entries of $C$ .

Formalization Note The printed proposition has no cost model, no success probability, and no precise meaning for the tolerance over the binary search on $\gamma$ ; these are fixed as above. The classical bookkeeping between quantum runs (tallying samples, forming the subgradients, updating $y$ and $D$ ) is not counted, matching the proof, which takes the classical cost from Prop. 7.29 alone; that bookkeeping never sees $C$ . The output is the source's own representation of the iterate (Rem. 8.27, Lem. 8.33): with an arbitrary decoder, the optimum could be read off $\hat C$ for free.

Significance

These are known mathematical results. Their Lean statements have been checked and reviewed against the cited source; formal proofs remain open for contributors.

Supporting targets

Proposition 8.7: Entropic mirror descent iterates are the Gibbs states of the accumulated subgradients.
Theorem 8.10: Regret bound of the matrix multiplicative weights update.
Lemma 8.12: Emptiness of the PIC polytope certifies primal feasibility after rescaling.
Theorem 8.15: Algorithm 7 (MMWU for SDP) returns a primal or a dual certificate.
Proposition 8.19: The generalized PIC polytope built from θ-accurate traces is sandwiched between P_0 and P_{4Rrθ}.
Proposition 8.21: Computing the dual vector with Õ(√m) calls to a trace-estimate oracle (corrected statement).
Proposition 8.28: The estimated MaxCut subgradient is an ϵ/2-subgradient (corrected zero branch).
Lemma 8.29: A subgradient of the active piece at a nearby point is an approximate subgradient of the max.
Theorem 8.30: Convergence of inexact entropic mirror descent for the MaxCut SDP feasibility problem.
Lemma 8.33: The MaxCut mirror-descent Hamiltonians are y₁Ĉ + y₂D with ‖y‖₁ ≤ (4/ϵ) log n.
Proposition 8.34: End-to-end complexity of the quantum MaxCut SDP solver in the QRAM model (corrected statement).

Formalization scope

Selected references

Giacomo Nannicini, Quantum algorithms for optimizers, 2025, v5; notes dated 9 July 2025. https://arxiv.org/pdf/2408.07086v5.

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Quantum Error Correction Quantum Information·Captain: Goku

An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation 2: Quantum Gilbert–Varshamov, quantum Hamming, and asymptotic rate boundsTextbook

Motivation

This mission develops Chapter 2 of An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation by Daniel Gottesman. The chapter's selected results describe the mathematical guarantees below.

Setting and conventions

Chapter goal

An $[[n,k,d]]$ code is a subspace $C$ of $(\mathbb C^2)^{\otimes n}$ of dimension $2^k$ such that every operator $E$ acting on at most $d-1$ qubits satisfies $\langle\psi|E|\psi\rangle=c'(E)$ for all unit vectors $|\psi\rangle\in C$ , with $c'(E)$ independent of $|\psi\rangle$ (eq. (34)); that is, $C$ has distance at least $d$ . Put $t=\lfloor (d-1)/2\rfloor$ . The code is nondegenerate if, for every unit vector $|\psi\rangle\in C$ , the Gram matrix $\big(\langle\psi|P_a^\dagger P_b|\psi\rangle\big)_{a,b}$ over the phase-free Pauli operators $P_a$ of weight at most $t$ is nonsingular. Let $H(x)=-x\log_2x-(1-x)\log_2(1-x)$ .

(Quantum Gilbert–Varshamov bound.) If $d\ge1$ and

\Big(\sum_{j=0}^{d-1}3^j\binom nj\Big)2^k\le 2^n,

then an $[[n,k,d]]$ code exists. 2. (Quantum Hamming bound.) Every nondegenerate $[[n,k,d]]$ code satisfies

\Big(\sum_{j=0}^{t}3^j\binom nj\Big)2^k\le 2^n.

(Asymptotic rates.) Let $0<p<1/2$ $0 < p < 1/2$ and $\varepsilon>0$ $ε > 0$ . There is $N$ $N$ such that for every $n\ge N$ $n \geq N$ :
- every nondegenerate $[[n,k,d]]$ code with $d\ge 2pn$ has $k/n\le 1-p\log_2 3-H(p)+\varepsilon$ ;
- if $1-2p\log_2 3-H(2p)>0$ , there is a nondegenerate $[[n,k,d]]$ code with $d\ge 2pn$ and $k/n\ge 1-2p\log_2 3-H(2p)-\varepsilon$ .

The first two bounds are counting arguments. The third gives the rate window, eq. (37), between which the best nondegenerate codes lie.

Formalization Note The source's 'distance $d$ ' is read as distance at least $d$ , so existence is meaningful also for $k=0$ . The hypothesis $d\ge1$ excludes $d=0$ , where (36) is an empty sum and the claim would demand a $2^k$ -dimensional code for every $k$ . The asymptotic clause is read as an $\varepsilon$ – $N$ statement. The lower rate bound is asserted only when it is positive.

Significance

These are known mathematical results. Their Lean statements have been checked and reviewed against the cited source; formal proofs remain open for contributors.

Supporting targets

Theorem 1: No-Cloning: no quantum operation copies every state.
Theorem 2: Correctable errors are closed under linear combinations; weight-t Pauli errors give all t-qubit errors.
Theorem 3: Knill–Laflamme condition for quantum error correction.
Theorem 4: Quantum Gilbert–Varshamov, quantum Hamming, and asymptotic rate bounds.

Formalization scope

Selected references

Daniel Gottesman, An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation, 2009, v1. https://arxiv.org/pdf/0904.2557v1.

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Optimization Quantum Information·Captain: Goku

Quantum algorithms for optimizers 7: A (‖A‖_F, Õ(q), ξ)-block-encoding of a classical matrix from QRAM, with an explicit cost modelTextbook

Motivation

This mission develops Chapter 7 of Quantum algorithms for optimizers by Giacomo Nannicini. The chapter's selected results describe the mathematical guarantees below.

Setting and conventions

Chapter goal

Fix constants $C_0 \in \mathbb{R}$ and $c_0 \in \mathbb{N}$ (those of the assumption $p = \widetilde{O}(1)$ ). There are constants $C > 0$ and $c \in \mathbb{N}$ such that the following holds. Let $q, p \in \mathbb{N}$ , $d = 2^q$ , $\xi > 0$ , write $L = \log(2+d) + \log(2+1/\xi)$ , and assume $p \le C_0 L^{c_0}$ . Then there exist

a QRAM (Def. 5.39) with $2^b$ addresses and words of $r$ bits, of size $2^b r \le C d^2 (p+1) L^c$ ;
a straight-line real-arithmetic program of at most $C d^2 L^c$ operations ( $+, -, \times, \div, \sqrt{\cdot}, \arccos$ , complex argument, rational constants) and an addressing map, which compute the QRAM contents from the entries of the matrix;
a quantum circuit $\mathcal{V}$ on $w + q$ qubits with $w \le C (q+1) L^c$ , making at most $C L^c$ uncontrolled calls to the QRAM (or its inverse) and using at most $C L^c$ two-qubit gates,

all independent of the matrix, such that for every $A \in \mathbb{C}^{d\times d}$ whose entries satisfy $|\operatorname{Re} a_{ij}|, |\operatorname{Im} a_{ij}| \le 2^p$ , the unitary $\mathcal{V}(U_{\mathrm{QRAM}}(A))$ obtained with the QRAM loaded by the program is an $(\|A\|_F, w, \xi)$ -block-encoding of $A$ :

\bigl\| \|A\|_F\,(\langle 0|_w \otimes I)\,\mathcal{V}\,(|0\rangle_w \otimes I) - A \bigr\| \le \xi .

This is the data-structure route to block-encodings of classically stored dense matrices, with subnormalization the Frobenius norm and only polylogarithmic quantum cost.

Formalization Note Corrected statement. The source gives no cost model for classical arithmetic, does not say in which parameters $p$ is polylogarithmic, and does not say how the costs depend on $\xi$ . Here classical cost is the length of a straight-line real-arithmetic program (the model of the Cor. 5.45 chunk), $p = \widetilde{O}(1)$ means $p \le C_0 L^{c_0}$ for arbitrary $C_0, c_0$ , all $\widetilde{O}$ bounds are polylogarithmic in $d$ and $1/\xi$ , and the QRAM size is $\widetilde{O}(d^2 p)$ rather than $O(d^2 p)$ because the word size must grow with $\log(1/\xi)$ . A $p$ -bit entry is modelled by the magnitude bound $2^p$ on its real and imaginary parts.

Significance

These are known mathematical results. Their Lean statements have been checked and reviewed against the cited source; formal proofs remain open for contributors.

Supporting targets

Proposition 7.9: Iterative refinement reaches residual ϵ in O(log 1/ϵ) iterations.
Proposition 7.14: Product of block-encodings (error term corrected).
Proposition 7.18: Linear combination of block-encodings (error term corrected for α < 1).
Lemma 7.20: Block-encoding of the diagonal matrix of inner products ⟨ϕ_j|ψ_j⟩.
Proposition 7.21: Block-encoding from sparse-access oracles (rotation cost and entry encoding made explicit).
Lemma 7.37: A bounded polynomial of degree Õ(β) approximating exp(2βx)/4 on [−1, 0].
Lemma 7.39: Block-encoding a density matrix from a unitary preparing its purification.
Proposition 7.41: One-copy trace estimator: a random variable with bias ≤ θ/4 for Tr(Aρ) and constant variance.
Proposition 7.38: Gibbs-state preparation from a block-encoding of H, for the polynomial of Lemma 7.37 made explicit.
Proposition 7.29: A (‖A‖_F, Õ(q), ξ)-block-encoding of a classical matrix from QRAM, with an explicit cost model.

Formalization scope

Selected references

Giacomo Nannicini, Quantum algorithms for optimizers, 2025, v5; notes dated 9 July 2025. https://arxiv.org/pdf/2408.07086v5.

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Optimization Quantum Information·Captain: Goku

Quantum algorithms for optimizers 5: Separation oracle from polylogarithmically many calls to a quantum membership oracle (corrected)Textbook

Motivation

This mission develops Chapter 5 of Quantum algorithms for optimizers by Giacomo Nannicini. The chapter's selected results describe the mathematical guarantees below.

Setting and conventions

Chapter goal

This is a corrected form of the quantum construction of a separation oracle from a membership oracle.

Quantum membership oracle. Points of $\mathbb{R}^d$ are written in an $m$ -qubit register through an encoding $p \mapsto \mathrm{enc}(p)$ . A quantum oracle implementing $\mathrm{MEM}_{0,\epsilon'}(K)$ is a unitary $O$ acting on the point register, an answer qubit and $k$ workspace qubits. On every basis point it acts as

O\,|p\rangle|0\rangle|0^k\rangle = |p\rangle|\varphi_p\rangle ,

where measuring the answer qubit of $\varphi_p$ returns the correct membership bit $[\mathrm{enc}(p) \in K]$ with probability at least $1-\epsilon'$ .

Separation oracle. $\mathrm{SEP}_{\epsilon,\delta}(x)$ (Definition 5.22) must, with probability at least $1-\delta$ , either assert $x \in B_2(K,\epsilon)$ or return a unit vector $g$ with $\langle g, y\rangle \le \langle g, x\rangle + \epsilon$ for all $y \in B_2(K,-\epsilon) = \{y : B_2(y,\epsilon) \subseteq K\}$ .

Theorem. There are absolute constants $C > 0$ and $c \in \mathbb{N}$ with the following property. Let $d \in \mathbb{N}$ and $r, R, \epsilon, \delta \in \mathbb{R}$ with $0 < r < R$ , $0 < \epsilon < R$ and $0 < \delta < 1/3$ . Then there is $\epsilon' > 0$ such that for every $x \in \mathbb{R}^d$ there exist an encoding $\mathrm{enc}$ and, for every workspace size $k$ , a quantum circuit with measurement decoding. The circuit makes at most

C\,\bigl(\log(2+d) + \log(2+R) + \log(2+1/r) + \log(2+1/\epsilon) + \log(2+1/\delta)\bigr)^{c}

uncontrolled calls to the oracle or its inverse. It correctly implements $\mathrm{SEP}_{\epsilon,\delta}(x)$ for every convex $K$ with $B_2(0,r) \subseteq K \subseteq B_2(0,R)$ and $x \notin B_2(K,-\epsilon)$ , and for every unitary implementing $\mathrm{MEM}_{0,\epsilon'}(K)$ .

Classically, separation from membership needs $\tilde O(d)$ membership calls. This result gives an exponential improvement in the dimension for the query complexity of convex optimization with a membership oracle.

Formalization Note The source leaves several objects open: the unitary implementing $\mathrm{MEM}$ , the value of $\epsilon'$ , and the parameters hidden in $\tilde O(1)$ . The statement makes each of them explicit. The unit-norm requirement on $g$ is added because Definition 5.22 as printed accepts $g = 0$ , which would make the separation oracle trivial.

Significance

These are known mathematical results. Their Lean statements have been checked and reviewed against the cited source; formal proofs remain open for contributors.

Supporting targets

Proposition 5.3: Jordan's gradient circuit recovers the gradient of an exactly representable linear function.
Proposition 5.6: The gradient algorithm for approximately linear functions.
Proposition 5.16: A Hadamard-test circuit is a probability oracle for (1 + Re⟨ψ|ϕ_x⟩)/2.
Corollary 5.18: Pure-state tomography with Euclidean error using Õ(d/ϵ) uses of U.
Proposition 5.25: Properties of the height function: convexity, Lipschitz continuity, and separation from an ϵ̄-subgradient (corrected).
Lemma 5.28: The central-difference gradient is within (η/2)Δ^(η)f(x) of every subgradient.
Lemma 5.30: The Laplace approximation is at most dL/r on average over a cube.
Lemma 5.32: Deviation from the finite-difference linearization on the ℓ1 ball is at most (η²/2)Δ^(η)f(z).
Proposition 5.31: Finite differences of an approximate function at a random point give an explicit ϵ′-subgradient at the origin (corrected).
Proposition 5.36: Algorithm 4 (binary tree of controlled R_Y rotations) prepares the amplitude encoding of a real unit vector.
Corollary 5.37: Amplitude encoding of a complex vector to error ϵ with Õ(d) gates from {H, T, CX}.
Corollary 5.45: Amplitude encoding with O(log d) QRAM queries and O(log d (log d + log 1/ϵ)) two-qubit gates after Õ(d) classical preprocessing (corrected).
Theorem 5.33: Separation oracle from polylogarithmically many calls to a quantum membership oracle (corrected).

Formalization scope

Selected references

Giacomo Nannicini, Quantum algorithms for optimizers, 2025, v5; notes dated 9 July 2025. https://arxiv.org/pdf/2408.07086v5.

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Machine Learning Quantum Information·Captain: Goku

A Survey of Quantum Learning Theory 4: Quantum states are PAC-learnable from n·poly(1/ε, 1/γ, log(1/δ)) measurement resultsTextbook

Motivation

This mission develops Chapter 4 of A Survey of Quantum Learning Theory by Srinivasan Arunachalam and Ronald de Wolf. The chapter's selected results describe the mathematical guarantees below.

Setting and conventions

Chapter goal

This is Aaronson's theorem on "pretty good tomography": an $n$ -qubit state can be PAC-learned from linearly many (in $n$ ) measurement results.

An $n$ -qubit state is a density matrix $\rho\in\mathbb C^{2^n\times 2^n}$ (positive semidefinite, trace $1$ ). A two-outcome measurement is an effect $E$ , i.e. $0\preceq E\preceq \mathrm{Id}$ ; its first outcome has probability $\mathrm{Tr}(E\rho)$ . Let $D$ be a probability distribution (Borel) on the set of effects.

There is a polynomial $p$ in three variables with nonnegative coefficients such that for every $n$ and all $\varepsilon,\gamma>0$ , $\delta\in(0,1)$ there are a number

T\ \le\ n\cdot p\big(1/\varepsilon,\ 1/\gamma,\ \log(1/\delta)\big)

and a (measurable) learner $L$ , mapping $T$ measurement results to an $n$ -qubit density matrix, with the following property: for every $n$ -qubit state $\rho$ and every distribution $D$ , if $E_1,\dots,E_T$ are drawn i.i.d. from $D$ and, independently given the $E_i$ , each $b_i$ is a bit with $\Pr[b_i=1]=\mathrm{Tr}(E_i\rho)$ , then with probability at least $1-\delta$ the output $\sigma=L((E_1,b_1),\dots,(E_T,b_T))$ satisfies

\Pr_{E\sim D}\big[\,|\mathrm{Tr}(E\sigma)-\mathrm{Tr}(E\rho)|>\gamma\,\big]\ \le\ \varepsilon .

The learner knows neither $\rho$ nor $D$ . Only the number of measurement results is bounded, not the running time.

Formalization Note " $T=n\cdot\mathrm{poly}$ " is read as an upper bound on $T$ (a learner may ignore extra results). The condition $\delta<1$ is added: for $\delta\ge 1$ the claim is empty and $\log(1/\delta)\le 0$ . The learner is deterministic and measurable; the probability of success is written out as an integral over $D^{T}$ of the probabilities of the bit strings.

Significance

These are known mathematical results. Their Lean statements have been checked and reviewed against the cited source; formal proofs remain open for contributors.

Supporting targets

§4.1.1, Theorem 4.3: Servedio–Gortler: quantum exact learning needs Ω(max{1/√γ(C), log|C|/n}) queries.
§4.1.1, Corollary 4.4: Servedio–Gortler: D(C) = O(n·Q(C)^3).
§4.1.2, Theorem 4.5: Classical (N,M)-query complexity of exact learning is Θ(min{M,N}).
§4.1.2, Theorem 4.6: Kothari: optimal quantum (N,M)-query complexity of exact learning.
§4.1.2, Theorem 4.8: Kothari: quantum exact learning with O(√((1/γ)/log(1/γ))·log|C|) queries.
Theorem 4.12: Quantum PAC learning needs Ω(d/ε + log(1/δ)/ε) quantum examples.
Theorem 4.13: Diagonal of the square root of an XOR-invariant Gram matrix (corrected: with the factor 2^{-k/2}).
Theorem 4.15: Quantum agnostic learning needs Ω(d/ε² + log(1/δ)/ε²) quantum examples.
Theorem 4.16: Quantum states are PAC-learnable from n·poly(1/ε, 1/γ, log(1/δ)) measurement results.

Formalization scope

Selected references

Srinivasan Arunachalam and Ronald de Wolf, A Survey of Quantum Learning Theory, 2017, v3. https://arxiv.org/pdf/1701.06806v3.

11 thms2 active usersReviewed

Quantum Error Correction Quantum Information·Captain: Goku

An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation 5: Threshold theorem: below p_T, concatenated circuits simulate C to distance ε with polylog overheadTextbook

Motivation

This mission develops Chapter 5 of An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation by Daniel Gottesman. The chapter's selected results describe the mathematical guarantees below.

Setting and conventions

Chapter goal

There exist an $[[n,1,2t+1]]$ code with fault-tolerant gadgets of a common depth (Definitions 2–5) and a threshold $p_T>0$ with the following property. For every $p$ with $0\le p<p_T$ there are constants $K>0$ and $c\in\mathbb N$ such that for every ideal circuit $C$ (Definition 6: preparation, $H$ , $R_{\pi/8}$ , CNOT, wait and measurement locations, no input) and every $\varepsilon>0$ there is a level $L$ for which the $L$ -fold concatenated fault-tolerant simulation $C'=C^{(L)}$ satisfies

$\#\mathrm{qubits}(C')\le K\,\big(\log_2(2+|C|/\varepsilon)\big)^{c}\,\#\mathrm{qubits}(C)$ and $\#\mathrm{steps}(C')\le K\,\big(\log_2(2+|C|/\varepsilon)\big)^{c}\,\#\mathrm{steps}(C)$ , where $|C|$ is the number of locations of $C$ ;
for every local stochastic error model on $C'$ (Definition 12) with error bounds $p_i\le p$ , $p_i<1$ ,

d_{\mathrm{TV}}\big(\text{decoded output of } C',\ \text{output of } C\big)\le\varepsilon.

The output of $C'$ is decoded classically level by level. The threshold theorem shows that arbitrarily long quantum computations can be performed reliably with polylogarithmic overhead once the physical error rate is below a constant.

Formalization Note The fault-tolerant circuit is pinned to the concatenated circuit of the proof (p.42); an unconstrained circuit with free classical computation could sample the output of $C$ classically. $\mathrm{polylog}(x)$ is encoded as $K(\log_2(2+x))^c$ , which is positive also for $x\le 1$ .

Significance

These are known mathematical results. Their Lean statements have been checked and reviewed against the cited source; formal proofs remain open for contributors.

Supporting targets

§5.1, Theorem 6: Good implies correct: an extended rectangle with at most t faults is correct.
§5.2, Theorem 7: A fault-tolerant circuit with only good ExRecs has the ideal output distribution.
§5.2, Lemma 2: *Correctness for the ideal decoder implies correctness for a -decoder.
§5.2, Lemma 3: *A good truncated ExRec is correct for ideal decoders and -decoders.
Theorem 8: A fault-tolerant circuit simulates the ideal circuit with errors at the bad ExRecs.
Theorem 9: Level reduction: a local stochastic model on C′ induces one on C with bounds Σ_{|R|=t+1} Π p_j.
Theorem 10: Threshold theorem: below p_T, concatenated circuits simulate C to distance ε with polylog overhead.

Formalization scope

Selected references

Daniel Gottesman, An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation, 2009, v1. https://arxiv.org/pdf/0904.2557v1.

8 thms2 active usersReviewed

Convex Optimization Operations Research Optimization+1·Captain: mikedeng1

A Soft Robust Model for Optimization Under Ambiguity: For Every δ > 0, Soft Robust Feasibility Is a Convex Risk Measure Constraint, and Every Normalized Convex Risk Measure Arises This WayResearch Paper

Motivation

Optimization under uncertainty usually fixes one probability distribution $\mathbb P$ for the uncertain data and optimizes an expected payoff, or fixes a set $\mathcal Q$ of distributions and protects against the worst one (robust optimization). The first is sensitive to a misspecified $\mathbb P$ ; the second can be very conservative, because it treats a barely plausible distribution exactly like the nominal one. Ben-Tal, Bertsimas and Brown (Oper. Res. 58(4), 2010) propose a graded alternative, the soft robust model: the decision maker accepts a loss of at most $\varepsilon$ in expected payoff under every distribution of a set $\mathcal Q(\varepsilon)$ , simultaneously for all levels $\varepsilon$ up to a budget $\delta$ . Larger sets of distributions are allowed a larger shortfall.

The paper's central result, Theorem 2.1, shows that this model is exactly a constraint on a convex risk measure in the sense of Föllmer and Schied (Finance Stoch. 6, 2002), and that every normalized convex risk measure arises from some soft robust model. This connects distributionally robust optimization with the theory of monetary risk measures, and gives a tractable convex reformulation of soft robust constraints. Related earlier work connected coherent risk measures with standard robust optimization (Bertsimas and Brown, Oper. Res. 57(6), 2009).

Setting

Let $(\Omega,\mathcal F)$ be a measurable space and $\mathcal P$ the set of all probability measures on it. A decision $x$ ranges over a nonempty, closed, convex set $X\subseteq\mathbb R^n$ ; a random variable $u:\Omega\to\mathbb R^m$ carries the uncertain parameters, and $f(x,u(\omega))$ is the payoff, concave in $x\in X$ for every $\omega$ . For each $x\in X$ the random variable $f(x)=f(x,u(\cdot))$ is assumed bounded and measurable.

A family of measures is $\mathbf Q=\{\mathcal Q(\varepsilon)\subseteq\mathcal P:\varepsilon\ge0\}$ . It is nondecreasing if $\mathcal Q(\varepsilon_1)\subseteq\mathcal Q(\varepsilon_2)$ for $\varepsilon_1\le\varepsilon_2$ , and convex if $\mathbb Q_i\in\mathcal Q(\varepsilon_i)$ implies $\lambda\mathbb Q_1+(1-\lambda)\mathbb Q_2\in\mathcal Q(\lambda\varepsilon_1+(1-\lambda)\varepsilon_2)$ for $\lambda\in[0,1]$ . The standing assumptions are that $\mathbf Q$ is nondecreasing and convex and that $\mathcal Q(0)\neq\emptyset$ . The soft robust feasible set at level $\delta\ge0$ is

X^Q(\delta)=\Big\{x\in X:\ \inf_{\mathbb Q\in\mathcal Q(\varepsilon)}\mathbb E_{\mathbb Q}[f(x)]\ \ge\ -\varepsilon\quad\forall\,\varepsilon\in[0,\delta]\Big\}.\tag{5}

A risk measure, on the space $\mathcal Y$ of bounded payoffs, is a map $\rho:\mathcal Y\to\mathbb R$ with $Y_1\ge Y_2\Rightarrow\rho(Y_1)\le\rho(Y_2)$ and $\rho(Y+c)=\rho(Y)-c$ for $c\in\mathbb R$ ; it is convex if $\rho(\lambda Y_1+(1-\lambda)Y_2)\le\lambda\rho(Y_1)+(1-\lambda)\rho(Y_2)$ . It is generated by a convex penalty $\alpha:\mathcal P\to[0,\infty]$ if $\rho(Y)=\sup_{\mathbb Q\in\mathcal P}\{-\mathbb E_{\mathbb Q}[Y]-\alpha(\mathbb Q)\}$ , and normalized if in addition $\alpha(\mathbb Q)=0$ for some $\mathbb Q$ . A family and a penalty determine each other:

\alpha^Q(\mathbb Q)=\inf\{\varepsilon\ge0:\mathbb Q\in\mathcal Q(\varepsilon)\},\qquad \mathcal Q^\alpha(\varepsilon)=\{\mathbb Q\in\mathcal P:\alpha(\mathbb Q)\le\varepsilon\},

and $\rho^Q$ denotes the risk measure generated by $\alpha^Q$ .

Formalization targets

Goal: Theorem 2.1 for $\delta>0$

For every nondecreasing, convex family $\mathbf Q$ with $\mathcal Q(0)\ne\emptyset$ , $\rho^Q$ is a convex risk measure generated by the normalized convex penalty $\alpha^Q$ , the minimum below is attained for every $x\in X$ , and

X^Q(\delta)=\Big\{x\in X:\ \min_{\lambda\ge0}\Big\{\lambda\delta+(\lambda+1)\,\rho^Q\Big(\frac{f(x)}{\lambda+1}\Big)\Big\}\le0\Big\}.\tag{6}

Conversely, for every convex risk measure $\rho$ generated by a normalized convex penalty $\alpha$ , the family $\mathbf Q^\alpha$ is nondecreasing and convex with $\mathcal Q^\alpha(0)\ne\emptyset$ , and (6) holds with $\rho$ on the right and $\mathbf Q^\alpha$ on the left.

Milestones: the steps of the proof (p. 1224)

The soft robust infimum $\inf\{\mathbb E_{\mathbb Q}[f(x)]+\varepsilon:\varepsilon\in[0,\delta],\ \mathbb Q\in\mathcal Q(\varepsilon)\}$ equals $\inf\{\mathbb E_{\mathbb Q}[f(x)]+\alpha^Q(\mathbb Q):\alpha^Q(\mathbb Q)\le\delta\}$ .
Strong duality with an attained multiplier: that infimum equals $\max_{\lambda\ge0}\{-\lambda\delta+\inf_{\mathbb Q}\{\mathbb E_{\mathbb Q}[f(x)]+(\lambda+1)\alpha^Q(\mathbb Q)\}\}$ .
The scaling identity $\inf_{\mathbb Q}\{\mathbb E_{\mathbb Q}[Y]+(\lambda+1)\alpha^Q(\mathbb Q)\}=-(\lambda+1)\rho^Q(Y/(\lambda+1))$ .
$\rho^Q$ is a convex risk measure generated by the convex, normalized penalty $\alpha^Q$ .
For a convex normalized $\alpha$ , $\mathbf Q^\alpha$ is nondecreasing and convex with $\mathcal Q^\alpha(0)\ne\emptyset$ .
$\alpha^{Q^\alpha}=\alpha$ .

Significance

Theorem 2.1 makes the soft robust constraint usable: the right side of (6) is a single constraint in $(\lambda,x)$ , and the function $(\lambda+1)\rho(f(x)/(\lambda+1))$ is jointly convex in $(\lambda,x)$ for $\lambda\ge0$ , so soft robust problems are convex programs whenever the risk measure can be evaluated. Read in the other direction, it gives every normalized convex risk measure an interpretation as a graded robustness requirement over the level sets of its penalty, which is how the paper derives soft robust counterparts of optimized certainty equivalents, CVaR and entropic risk, and its later bounds on the probability of a loss.

The theorem is proved in the paper; to our knowledge it has no machine-checked proof. Formalizing it produces a Lean development of convex risk measures generated by penalties on an arbitrary measurable space, with extended-valued penalties and mixtures of probability measures, and a one-constraint Lagrange duality theorem over a convex set of probability measures. Mathlib does not currently provide Lagrange duality in this form.

Difficulty

The equalities of the proof read as routine, and two of them are not. The first equality is described as "the definition of $\alpha^Q$ ", but a measure with $\alpha^Q(\mathbb Q)=\delta$ need not belong to $\mathcal Q(\delta)$ , because the infimum defining $\alpha^Q$ need not be attained; the equality relies on the convexity of the family, the nonemptiness of $\mathcal Q(0)$ and the boundedness of $f(x)$ . The second equality is Lagrange duality with dual attainment for a convex program whose variable is a probability measure and whose objective and constraint are extended-valued; the classical theorem applies to convex subsets of vector spaces, and transferring it to mixtures of probability measures with $[0,\infty]$ -valued penalties is the main formal work. Attainment of the minimum in (6), which the paper asserts by writing "min", is exactly dual attainment and does not follow from the inequality alone.

Formalization scope

The probability measures are ProbabilityMeasure Ω on a measurable space; a family is a map from levels $\varepsilon\in\mathbb R_{\ge0}$ to sets of probability measures, and mixtures are built from the measures $t\,\mathbb Q_1+(1-t)\,\mathbb Q_2$ . Penalties, including $\alpha^Q$ , take values in $[0,\infty]$ , so that a measure in no $\mathcal Q(\varepsilon)$ has penalty $+\infty$ ; a real-valued $\alpha^Q$ would give such measures penalty $0$ through the junk value of an empty infimum, and is excluded. Risk measures are real functions whose axioms are imposed on bounded measurable payoffs, "generated by" includes the least-upper-bound property of the supremum, and "min" in (6) is attainment on $[0,\infty)$ together with a nonpositive minimum. The decision set and payoff carry the paper's assumptions; concavity of the payoff and closedness and convexity of $X$ are kept although Theorem 2.1 does not use them. Infima over measures in the milestones are taken in the extended reals.

The case $\delta=0$ is excluded. The paper states Theorem 2.1 for $\delta\ge0$ , but at $\delta=0$ its proof interchanges a limit in $\lambda$ with a supremum over measures without justification, and (6) fails: on $\Omega=\{a,b\}$ with $\mathcal Q(0)$ the measures giving $a$ mass at most $1/4$ and $\mathcal Q(\varepsilon)$ those giving it mass at most $1/2$ for $\varepsilon>0$ , the payoff $f(a)=-2$ , $f(b)=1$ is soft robust at level $0$ while the right side of (6) is empty. The goal and the milestones therefore fix $\delta>0$ , the case the paper's main argument treats.

A complete development needs: convex risk measures and their generation by penalties, mixtures of probability measures, Lagrange duality with Slater's condition over convex sets of measures, and the scaling identity for $\rho^Q$ . The duality theorem and the risk-measure layer are reusable beyond this mission. Contributions of any milestone are welcome, as are general convex-duality lemmas that the strong-duality step can import.

Selected references

A. Ben-Tal, D. Bertsimas, D. B. Brown, A Soft Robust Model for Optimization Under Ambiguity, Operations Research 58(4, Part 2):1220–1234, 2010. https://doi.org/10.1287/opre.1100.0821
H. Föllmer, A. Schied, Convex measures of risk and trading constraints, Finance and Stochastics 6(4):429–447, 2002. https://doi.org/10.1007/s007800200072
H. Föllmer, A. Schied, Stochastic Finance: An Introduction in Discrete Time, 2nd ed., de Gruyter, 2004. https://doi.org/10.1515/9783110212075
D. G. Luenberger, Optimization by Vector Space Methods, Wiley, 1969, Chapter 8.
D. Bertsimas, D. B. Brown, Constructing Uncertainty Sets for Robust Linear Optimization, Operations Research 57(6):1483–1495, 2009. https://doi.org/10.1287/opre.1080.0695

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Mathematical Logic Model Theory·Captain: Lucas

Continuum Hypothesis: Independence from ZFCTextbook

Motivation

The continuum hypothesis (CH) asks how large the set of real numbers is. Cantor proved that $|\mathbb Z|=|\mathbb Q|<|\mathbb R|$ , and in 1878 he conjectured that no set has cardinality strictly between that of the integers and that of the reals. Settling CH was the first of Hilbert's problems at the 1900 International Congress of Mathematicians. The question was answered by showing that it cannot be answered from the standard axioms: CH is independent of Zermelo–Fraenkel set theory with choice (ZFC).

Timeline (as recorded in the source):

1878 — Cantor formulates the (weak) continuum hypothesis; in 1882 he states CH in its modern form.
1900 — Hilbert lists CH as the first of his problems.
1906 — Kőnig proves a theorem that, combined with Hausdorff's notion of cofinality (1908), shows $2^{\aleph_0}\neq\aleph_\alpha$ whenever $\alpha$ has cofinality $\omega$ .
1930 — Patai: if $2^\kappa$ is the $\gamma$ -th successor of $\kappa$ for every infinite $\kappa$ , then $\gamma$ is finite.
1938/1940 — Gödel proves that AC and GCH hold in the constructible universe $L$ , so $\mathrm{Con}(\mathrm{ZF})\Rightarrow\mathrm{Con}(\mathrm{ZFC}+\mathrm{GCH})$ .
1947 — Sierpiński proves that ZF + GCH implies AC.
1963 — Cohen invents forcing and proves $\mathrm{Con}(\mathrm{ZFC})\Rightarrow\mathrm{Con}(\mathrm{ZFC}+\neg\mathrm{CH})$ .
1986 — Freiling shows that $\neg$ CH is equivalent to his axiom of symmetry.

Setting

A cardinal measures the size of a set; $|A|\le|B|$ means there is an injection $A\to B$ , and $|A|=|B|$ a bijection. $\aleph_0=|\mathbb N|$ , $2^\kappa=|\mathcal P(X)|$ for $|X|=\kappa$ , so $|\mathbb R|=2^{\aleph_0}$ . $\aleph_\alpha$ is the $\alpha$ -th infinite cardinal, $\beth_\alpha$ is defined by $\beth_0=\aleph_0$ , $\beth_{\alpha+1}=2^{\beth_\alpha}$ and suprema at limits, and $\mathrm{cf}$ denotes cofinality.

CH: there is no $S\subseteq\mathbb R$ with $\aleph_0<|S|<2^{\aleph_0}$ (CH).
GCH: for every infinite cardinal $\kappa$ there is no cardinal strictly between $\kappa$ and $2^\kappa$ (GCH).

Independence is a statement about a first-order theory. The language of set theory has one binary relation symbol $\in$ (setLang). ZF (ZF) consists of extensionality, foundation, pairing, union, power set, infinity, and all instances of the separation and replacement schemas; ZFC (ZFC) adds the axiom of choice (acSentence). CH and GCH are also written as first-order sentences (chSentence, gchSentence): CH says that every subset of $\mathcal P(\omega)$ injects into $\omega$ or receives an injection from $\mathcal P(\omega)$ . A theory is consistent if it has a model, and $T\models\sigma$ means $\sigma$ holds in every model of $T$ ; by Gödel's completeness theorem these coincide with the proof-theoretic notions.

Formalization targets

Goal: independence of CH from ZFC

\mathrm{Con}(\mathrm{ZFC})\ \Longrightarrow\ \mathrm{ZFC}\not\models\mathrm{CH}\ \text{ and }\ \mathrm{ZFC}\not\models\neg\mathrm{CH}.

Milestones

$|\mathbb Z|=|\mathbb Q|<|\mathbb R|$ .
$\mathrm{CH}\iff 2^{\aleph_0}=\aleph_1$ .
(Gödel) $\mathrm{Con}(\mathrm{ZF})\Rightarrow\mathrm{Con}(\mathrm{ZFC}+\mathrm{GCH})$ .
$\mathrm{ZFC}+\mathrm{GCH}\models\mathrm{CH}$ .
(Cohen) $\mathrm{Con}(\mathrm{ZFC})\Rightarrow\mathrm{Con}(\mathrm{ZFC}+\neg\mathrm{CH})$ .
(Kőnig) $2^{\aleph_0}\neq\aleph_\alpha$ when $\mathrm{cf}(\alpha)=\omega$ ; and $\mathrm{cf}(2^{\aleph_0})>\aleph_0$ .
(Sierpiński) $\mathrm{ZF}+\mathrm{GCH}\models\mathrm{AC}$ .
$\mathrm{GCH}\iff\forall\alpha,\ 2^{\aleph_\alpha}=\aleph_{\alpha+1}\iff\forall\alpha,\ \aleph_\alpha=\beth_\alpha$ .
Under GCH: $A<B\Rightarrow 2^A<2^B$ for infinite $A$ , and the three-case formula for $\aleph_\alpha^{\aleph_\beta}$ .
(Patai) $\forall\alpha,\ 2^{\aleph_\alpha}=\aleph_{\alpha+\gamma}$ implies $\gamma<\omega$ .
(Freiling) the axiom of symmetry is equivalent to $\neg$ CH.

Significance

The independence theorem shows that the ZFC axioms do not determine the size of the continuum, and it was among the first independence results for a mathematical statement. Its two halves introduced the two main tools of modern set theory: inner models (Gödel's $L$ ) and forcing (Cohen). Many statements in analysis, topology and measure theory were subsequently shown to be independent by the same methods.

For formalization, the goal requires a model theory of ZF inside Lean: interpreting the first-order theory, building $L$ inside an arbitrary model of ZF, and building forcing extensions of an arbitrary model of ZFC. The independence of CH was formalized in Lean 3 by the Flypitch project (Han and van Doorn, 2019–2020); no Lean 4 / Mathlib formalization is known to this mission. The cardinal-arithmetic milestones (2, 6, 8–11) are theorems of ZFC that can be proved directly in Lean's cardinal library.

Difficulty

Neither half follows from cardinal arithmetic in Lean: Lean's foundations prove that models of ZFC exist, but (assuming they are consistent) they prove neither CH nor its negation about Lean's own Cardinal, so an argument inside Lean's cardinal arithmetic alone cannot produce the required models. Gödel's half needs relativization of formulas to a definable inner class and absoluteness of the construction of $L$ ; Cohen's half needs a generic extension in which a new family of $\aleph_2$ reals is added while cardinals are preserved (the countable chain condition). Both require a theory of definability inside arbitrary, possibly non-well-founded, models of a first-order theory.

Formalization scope

Syntax and semantics are Mathlib's first-order logic (FirstOrder.Language, Theory.IsSatisfiable, Theory.ModelsBoundedFormula written ⊨ᵇ). Formulas name their variables by natural numbers; close φ is the universal closure. Equality is the built-in equality of first-order logic.
Consistency is satisfiability and provability is semantic consequence over models with universe in Type. By the completeness theorem these agree with the syntactic notions.
The first-order CH uses $\mathcal P(\omega)$ in place of $\mathbb R$ (they are equinumerous in ZF). The first-order GCH is the choice-free form " $|S|\le|T|\le|\mathcal P(S)|\Rightarrow|T|=|S|$ or $|T|=|\mathcal P(S)|$ , for $S$ infinite".
The consistency results carry their hypotheses $\mathrm{Con}(\mathrm{ZF})$ or $\mathrm{Con}(\mathrm{ZFC})$ explicitly, even though these are provable in Lean. A solution may not replace a model of ZF by Lean's own universe in a way that assumes CH or its negation.
Results that need large-cardinal hypotheses or class forcing (Easton, Solovay, Foreman–Woodin, Merimovich) are not part of this mission.

Reusable infrastructure: the language of set theory with standard abbreviations (pairs, functions, $\omega$ , power set), satisfaction lemmas for the abbreviations, relativization, absoluteness, the constructible hierarchy, and forcing over models of ZFC.

Selected references

G. Cantor, Ein Beitrag zur Mannigfaltigkeitslehre, J. Reine Angew. Math. 84 (1878) 242–258. https://doi.org/10.1515/crll.1878.84.242
K. Gödel, The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis, PNAS 24 (1938) 556–557. https://doi.org/10.1073/pnas.24.12.556
K. Gödel, The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory, Annals of Math. Studies 3, Princeton, 1940.
P. J. Cohen, The independence of the Continuum Hypothesis I, II, PNAS 50 (1963) 1143–1148; 51 (1964) 105–110. https://doi.org/10.1073/pnas.50.6.1143, https://doi.org/10.1073/pnas.51.1.105
C. Freiling, Axioms of symmetry: throwing darts at the real number line, J. Symbolic Logic 51 (1986) 190–200. https://doi.org/10.2307/2273955
L. Gillman, Two classical surprises concerning the Axiom of Choice and the Continuum Hypothesis, Amer. Math. Monthly 109 (2002) 544–553. https://doi.org/10.2307/2695444
K. Kunen, Set Theory: An Introduction to Independence Proofs, North-Holland, 1980.
Wikipedia, Continuum hypothesis (revision 1376315189). https://en.wikipedia.org/w/index.php?title=Continuum_hypothesis&oldid=1376315189

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Analysis Functional Analysis·Captain: Lucas

Invariant Subspace ProblemOpen Problem

Motivation

The invariant subspace problem asks whether every bounded linear operator on an infinite-dimensional separable complex Hilbert space leaves some non-trivial closed subspace invariant. It is listed as a high-importance open problem in analysis in the Open Problem Garden (entry) and is widely regarded as one of the most famous open problems in functional analysis. Invariant subspaces are the infinite-dimensional replacement for eigenvectors: in finite dimensions every complex matrix has an eigenvector, and the structure theory of matrices (triangular and Jordan forms) is built from chains of invariant subspaces. Whether any such structure is available for an arbitrary bounded operator on Hilbert space is exactly what the problem asks.

Timeline

1930s (von Neumann, unpublished). Compact operators on Hilbert space have non-trivial closed invariant subspaces.
1954 (Aronszajn–Smith). The compact case on arbitrary Banach spaces (doi:10.2307/1969637).
1966 (Bernstein–Robinson; Halmos). Polynomially compact operators on Hilbert space.
1973 (Lomonosov). Every non-zero compact operator has a non-trivial closed hyperinvariant subspace; consequently every operator commuting with a non-zero compact operator has a non-trivial closed invariant subspace (doi:10.1007/BF01080698).
1975 announced / 1987 published (Enflo). A bounded operator on a separable Banach space with no non-trivial closed invariant subspace (doi:10.1007/BF02392260).
1985 (Read). Such an operator on the classical space $\ell^1$ (doi:10.1112/blms/17.4.305).
2011 (Argyros–Haydon). An infinite-dimensional Banach space on which every operator is a scalar plus a compact operator, so that every operator on it has a non-trivial closed invariant subspace (arXiv:0903.3921).

The Hilbert-space question remains open.

Setting

Let $\mathbb{K}$ be $\mathbb{R}$ or $\mathbb{C}$ , let $E$ be a normed space over $\mathbb{K}$ , and let $T\colon E\to E$ be a bounded linear operator. A linear subspace $S\subseteq E$ is invariant under $T$ if $T(S)\subseteq S$ . It is a non-trivial closed invariant subspace if moreover $S$ is closed in the norm topology and $\{0\}\neq S\neq E$ . A closed subspace $S$ with $\{0\}\neq S\neq E$ is a non-trivial closed hyperinvariant subspace of $T$ if $A(S)\subseteq S$ for every bounded operator $A$ with $AT=TA$ .

A Hilbert space $H$ is a complete complex inner product space; it is separable if it has a countable dense subset. An operator $K$ is compact if it maps bounded sets to relatively compact sets, and $T$ is normal if $T^*T=TT^*$ . In the Lean development these notions are IsInvariantSubspace, IsNontrivialClosedInvariantSubspace, HasNontrivialClosedInvariantSubspace and HasNontrivialClosedHyperinvariantSubspace (namespace InvariantSubspaceProblem), together with Mathlib's IsCompactOperator and IsStarNormal.

Formalization targets

Goal: the invariant subspace problem

\forall\, H \text{ separable, } \dim H=\infty,\ \forall\, T\in\mathcal{B}(H):\quad \exists\, S\subseteq H \text{ closed},\ \{0\}\neq S\neq H,\ T(S)\subseteq S .

The source poses a yes/no question; the goal states the affirmative answer, and a formal proof of its negation equally settles the problem.

Milestones (known results)

Finite-dimensional case: every operator on a complex space of dimension $\ge 2$ has a non-trivial invariant subspace.
Non-separable case: every operator on a non-separable complex normed space has a non-trivial closed invariant subspace.
Normal operators on a complex Hilbert space of dimension $\ge 2$ .
Aronszajn–Smith: compact operators on infinite-dimensional real or complex Banach spaces.
Lomonosov: non-zero compact operators have non-trivial closed hyperinvariant subspaces.
Lomonosov: operators commuting with a non-zero compact operator.
Enflo: a separable infinite-dimensional complex Banach space with an operator having no non-trivial closed invariant subspace.
Read: such an operator on $\ell^1$ .
Argyros–Haydon: a separable infinite-dimensional real Banach space on which every operator is $\lambda I+K$ with $K$ compact.
Argyros–Haydon: a separable infinite-dimensional real Banach space on which every operator has a non-trivial closed invariant subspace.

Significance

The result itself. A positive answer would give every bounded operator on Hilbert space a non-trivial piece of structure and would be the first step towards triangular-form type theorems for general operators. A negative answer would, after Enflo and Read, show that even the Hilbert-space geometry does not force invariant subspaces. Either answer resolves a question that has organised much of operator theory since the 1930s.

Formalizing it. The goal is open; no proof, formal or informal, is known. The milestones are proved results whose formal proofs would build reusable infrastructure absent from Mathlib: the Aronszajn–Smith and Lomonosov theorems (the latter through Schauder's fixed point theorem), invariant subspaces from the spectral theorem for normal operators, and the large explicit constructions of Enflo, Read and Argyros–Haydon. None of these results is formalized in Mathlib at the time of writing.

Difficulty

The obvious approach, taking the closed linear span of an orbit $\{x, Tx, T^2x,\dots\}$ , fails exactly when every non-zero vector is cyclic, which is the situation that has to be ruled out; for separable spaces nothing prevents this a priori. Spectral methods fail because an operator can have a one-point spectrum and no eigenvectors (quasinilpotent operators). Methods that work on Hilbert space must use the inner-product structure in an essential way, since Enflo's and Read's examples show that the statement is false on some Banach spaces. Lomonosov's technique needs a non-zero compact operator in the commutant, which a general operator need not have.

Formalization scope

Hilbert spaces are Mathlib types H with [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]; separability is TopologicalSpace.SeparableSpace H and infinite dimension is ¬ FiniteDimensional ℂ H. Operators are continuous linear maps H →L[ℂ] H. Scalars are complex in the goal. Subspaces are Submodules, and closedness is required explicitly: without it the statement is trivially true in infinite dimensions (the algebraic span of an orbit is a proper invariant subspace), and a published platform statement that omits closedness should not be confused with this problem. Existential milestones quantify over spaces in Type.

Useful infrastructure includes: Schauder's fixed point theorem, Riesz theory of compact operators, the spectral theorem for normal operators, and $\ell^1$ machinery for Read's construction. Contributions of reusable lemmas, partial results (for instance further classes of operators known to have invariant subspaces), and proof sketches for the milestones are welcome.

Selected references

Open Problem Garden, Invariant subspace problem, posted by T. Chow, 2009. http://www.openproblemgarden.org/op/invariant_subspace_problem
N. Aronszajn, K. T. Smith, Invariant subspaces of completely continuous operators, Ann. of Math. 60 (1954) 345–350. https://doi.org/10.2307/1969637
V. I. Lomonosov, Invariant subspaces for the family of operators which commute with a completely continuous operator, Funct. Anal. Appl. 7 (1973) 213–214. https://doi.org/10.1007/BF01080698
P. Enflo, On the invariant subspace problem for Banach spaces, Acta Math. 158 (1987) 213–313. https://doi.org/10.1007/BF02392260
C. J. Read, A solution to the invariant subspace problem on the space $\ell_1$ , Bull. London Math. Soc. 17 (1985) 305–317. https://doi.org/10.1112/blms/17.4.305
S. A. Argyros, R. G. Haydon, A hereditarily indecomposable $\mathcal{L}_\infty$ -space that solves the scalar-plus-compact problem, Acta Math. 206 (2011) 1–54. https://arxiv.org/abs/0903.3921
H. Radjavi, P. Rosenthal, The invariant subspace problem, Math. Intelligencer 4 (1982) 33–37.
B. S. Yadav, The present state and heritages of the invariant subspace problem, Milan J. Math. 73 (2005) 289–316.

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