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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

Completed

Algorithmic Game Theory Economics Operations Research·Captain: mikedeng1

Coopetition in Platform-Based Retailing: On the Platform’s Entry I: The Seller Prefers Entry Exactly Above Thresholds in Spillover and Referral Rate and Outside a Band of Competition IntensityResearch Paper

Motivation

Large online marketplaces such as Amazon, JD.com and Walmart.com act both as a sales channel for third-party sellers and as a retailer of their own. A platform watches which third-party products sell well and may start selling the same product itself, a move called platform entry. Over ten months of 2013–2014, Amazon entered about 3% of the products first offered only by third-party sellers, and its entry lowered the affected products' sales rankings, that is, raised their total sales (Zhu & Liu, Strategic Management Journal 2018). The conventional view is that a seller should deter entry, since the platform becomes a competitor that also collects a referral fee on every sale the seller makes.

L. Qi, H. Song and W. Xiao (Management Science 2025) build a two-stage model with three features: the platform charges a referral rate on the seller's revenue, the platform's marketing effort spills over to the seller's product, and the two parties compete in quantities with differentiated products. Their Proposition 2 identifies exactly when the seller is better off if the platform enters. This mission formalizes that result together with the equilibrium analysis it rests on.

Setting

There are five parameters: the degree of spillover $\delta \in [0,1]$ , the referral rate $f \in [0,1)$ , the competition intensity $\beta \in [0,1]$ , the seller's effort-cost coefficient $c_S$ and the platform's entry cost $K$ . The product has base value $a$ . The platform's effort-cost coefficient is normalized to $1/2$ .

Stage 1. The seller chooses an effort $e_S$ at cost $c_S e_S^2$ , which raises the product value to $a + e_S$ , and a quantity $q_1$ . The price is $p_1 = a + e_S - q_1$ , and the seller earns $q_1 p_1 (1-f) - c_S e_S^2$ after paying the referral fee $f q_1 p_1$ .

Stage 2 without entry. The seller alone sells $q_S$ at price $p_S = a + e_S - q_S$ and earns $q_S p_S(1-f)$ .

Stage 2 with entry. The platform chooses an effort $e_P \ge 0$ at cost $e_P^2/2$ . Its own product's value becomes $a + e_S + e_P$ , and the seller's product's value becomes $a + e_S + \delta e_P$ . The two then choose quantities simultaneously, with prices

p_S = a + e_S + \delta e_P - q_S - \beta q_P, \qquad p_P = a + e_S + e_P - q_P - \beta q_S .

The seller earns $q_S p_S (1-f)$ . The platform earns $q_S p_S f + q_P p_P - e_P^2/2 - K$ : the referral fee on the seller's revenue plus its own margin.

Backward induction gives closed forms. Write

N_5 = 4(1+\delta) - 2\beta(2-\delta+\delta^2) + \beta^3(1+f) - \beta^2(2+f+\delta f)

and let $D$ be the quartic denominator of the platform's optimal effort (Lemma 2). Condition (5) is $N_5 > 0$ : under it the seller stays in the market after entry. The paper assumes (5) from that point on. Let

H = \begin{cases} N_5^2/D^2 & \text{if (5) holds},\\ 0 & \text{otherwise,}\end{cases}

so that the seller's two-stage profits as functions of the effort $e_S$ are

\Pi^S_N(e_S) = \tfrac12 (1-f)(a+e_S)^2 - c_S e_S^2, \qquad \Pi^S_E(e_S) = \big(\tfrac14 + H\big)(1-f)(a+e_S)^2 - c_S e_S^2 .

The receptive regime is $\Omega = \{(\delta,f,\beta) \mid H > 1/4\}$ .

Formalization targets

Goal: Proposition 2

\Pi^S_E(e_S) > \Pi^S_N(e_S) \iff (\delta,f,\beta) \in \Omega \qquad (a + e_S \ne 0),

and the shape of $\Omega$ on the box $[0,1]\times[0,1)\times[0,1]$ :

for fixed $(f,\beta)$ there is $\tilde\delta(f,\beta)$ with $(\delta,f,\beta)\in\Omega \iff \delta > \tilde\delta(f,\beta)$ ;
for fixed $(\delta,\beta)$ there is $\tilde f(\delta,\beta)$ with $(\delta,f,\beta)\in\Omega \iff f > \tilde f(\delta,\beta)$ ;
for fixed $(\delta,f)$ there are $\beta_1(\delta,f) \le \beta_2(\delta,f)$ with $(\delta,f,\beta)\in\Omega \iff \beta < \beta_1$ or $\beta > \beta_2$ .

The thresholds are existentials, as the paper defines them only inside its proof.

Milestones

Lemma 1: the stage-2 quantity game after entry has a unique equilibrium, given in closed form.
Lemma 2 under (5): the platform's optimal effort $e^*_P$ in closed form, and the seller stays ( $q^*_S > 0$ ). The positivity of $e^*_P$ and of its denominator is a separate milestone.
Equations (6)–(8): the seller's equilibrium two-stage profits are $\Pi^S_N$ and $\Pi^S_E$ above.
The identity $4N_5^2 - D^2 = P_{11}\cdot S_2$ from the proof of Proposition 2, where $P_{11}$ is the function (11).
Under (5) the second factor $S_2$ is positive.
Lemma 3, in three parts: the sign pattern of (11) in $\delta$ , in $f$ and in $\beta$ .

Significance

The result. Proposition 2 says that the seller's preference does not depend on the seller's own effort or on the base value, only on $(\delta, f, \beta)$ . It shows that the conventional advice to deter entry is wrong on a nonempty region: a strong enough spillover, or a high enough referral rate (which gives the platform a stake in the seller's sales and so a reason to invest more effort), makes entry welcome. In the competition intensity the regime is not monotone. The seller welcomes entry when products are well differentiated, opposes it at intermediate $\beta$ , and can welcome it again near perfect substitutes. The paper's later results on the seller's optimal effort and on entry deterrence (Proposition 3 onward) are stated separately inside and outside $\Omega$ , so this classification is their starting point.

Formalizing it. The result is proved in the paper, and nothing about it is machine-checked yet. Its proof leans on "one can verify" claims about quartic polynomials on a box: the positivity of $D$ , of the platform's effort and of $S_2$ under (5), and the sign patterns of (11). A formal proof checks each of them. The mission also ties the closed forms (6)–(7) to an explicit game, with equilibria defined as best responses, so the comparison is between equilibrium profits and not between formulas taken on trust.

Difficulty

The first part of the goal is immediate once (6)–(7) are known, since $\Pi^S_E - \Pi^S_N = (H - \tfrac14)(1-f)(a+e_S)^2$ . The content lies elsewhere.

The equilibrium analysis. The platform's payoff includes the referral fee on the seller's revenue, so its best-response slope is $\beta(1+f)/2$ , not the textbook $\beta/2$ . The seller's quantity has a corner at zero. The platform's effort problem must be solved against every equilibrium after every deviation, and it reduces to a single variable only after uniqueness in Lemma 1.
Polynomial sign conditions. Positivity of $D$ and $S_2$ under (5), and the threshold structure of (11) in each variable, are statements about polynomials of degree up to four on a box with the semialgebraic side condition (5).
A step the paper omits. $\Omega$ is defined through $H$ , and $H = 0$ off (5). Moving Lemma 3's thresholds from (11) to $\Omega$ therefore also needs "(11) $> 0$ implies (5)", which the paper does not state. It holds numerically: on a $201^3$ grid, wherever (11) is positive the left side of (5) is above $1.01$ . A complete proof of the goal must supply it.

Formalization scope

All quantities are real numbers. The parameters are explicit arguments, and the standing ranges $\delta\in[0,1]$ , $f\in[0,1)$ , $\beta\in[0,1]$ are a hypothesis InBox δ f β. Every polynomial ( $N_5$ , $D$ , the numerator of $e^*_P$ , (11), $S_2$ ) is written term for term as printed. $H$ keeps the case split of (8), so $\Omega$ is false wherever (5) fails. Conventions:

Defined from the game: the quantity equilibrium (a pure Nash equilibrium over nonnegative quantities, through the published general two-player definition CompetingChains.Cournot.IsPureNash), the entry equilibrium (the platform's effort $e_P \ge 0$ is a global maximizer against every equilibrium after every deviation), and the stage-1 and no-entry problems.
The paper's closed forms: $H$ , $\Omega$ , $\Pi^S_N$ and $\Pi^S_E$ . Milestone 3 proves that the closed forms equal the equilibrium profits.
"Optimal" means a global maximizer, ties allowed.
The platform's effort is nonnegative. The seller's effort $e_S$ is any real number, as in the paper.
Lemmas 1–2 assume $a + e_S > 0$ (a positive product value), and Lemma 2 assumes (5), the paper's standing assumption.
Only the branch of Lemma 2 under (5) is formalized.
The goal's first part assumes $a + e_S \ne 0$ . At $a + e_S = 0$ both profits equal $-c_S e_S^2$ and no strict preference exists.
Condition (5) is not a hypothesis of the goal.

The first part of the goal is near-definitional given (6)–(7). The mission's content is the threshold structure of $\Omega$ and milestones 1–3, which make (6)–(7) the equilibrium profits. A formalization in which the thresholds may depend on the varying parameter, or in which $\Omega$ is taken as $\{(11) > 0\}$ by definition, would not be this theorem.

A complete development needs real polynomial inequalities on boxes (nlinarith/polyrith-style certificates or interval arguments) and elementary concave maximization. A uniqueness lemma for two-player games whose best responses have slopes with product below one would be reusable for other differentiated Cournot models. Contributions are welcome for any milestone, for the step "(11) $> 0$ implies (5)", and for proofs of the sign lemmas by certificates.

Selected references

L. Qi, H. Song, W. Xiao, Coopetition in Platform-Based Retailing: On the Platform's Entry, Management Science, 2025. https://doi.org/10.1287/mnsc.2023.00260 (formalized from the authors' manuscript with Online Supplement, 2024-08-03).
F. Zhu, Q. Liu, Competing with complementors: An empirical look at Amazon.com, Strategic Management Journal 39(10), 2018. https://doi.org/10.1002/smj.2932
N. Singh, X. Vives, Price and quantity competition in a differentiated duopoly, RAND Journal of Economics 15(4), 1984. https://doi.org/10.2307/2555525

12 thms3 active usersReviewed

Completed

Economics Operations Research Optimization·Captain: mikedeng1

Capturing the Benefits of Autonomous Vehicles in Ride-Hailing: The Role of Market Configuration 1: A Pure Human-Driver Market Has a Unique Maximal Stable Equilibrium, Increasing in Market SizeResearch Paper

Motivation

Ride-hailing platforms match riders with drivers who choose for themselves whether to work. Autonomous vehicles (AVs) would replace part of this self-scheduling labor with capital whose fleet size the platform chooses. Lian and van Ryzin study when, and under which market configuration, the cost savings of AVs reach riders. Their comparison needs a benchmark: the market served by human drivers (HVs) alone. This mission formalizes that benchmark, Proposition 1 of the paper, together with the lemmas on minimal supply and utilization that the rest of the paper uses.

Two economic forces drive the benchmark. The first is the density effect: in a denser market an idle car is closer to the next rider, so pickups are shorter and each driver spends more of the hour on paid trips. The second is the supply effect of labor: a larger fleet requires higher hourly earnings to attract the marginal driver. Proposition 1 describes how the equilibrium demand, fleet size, utilization and price respond to market size when both forces are present.

The source is the authors' 2023 SSRN working paper (abstract 3716491, version of 23 August 2023); all theorem, lemma and equation numbers refer to that version, not to the 2025 Management Science article.

Setting

A market has an ETA function $t_1(s) = a s^{-r}$ with $a > 0$ and $0 \le r < 1$ , the expected pickup time when $s$ cars are open (idle); $r$ is the ETA elasticity, and $r > 0$ means a density effect. A trip lasts $t_2 > 0$ hours. Rider values for a ride hour are uniform on $[0, V]$ , $V > 0$ . HVs have the supply curve $w(n) = w_0 + \alpha n$ , $w_0 > 0$ , $\alpha \ge 0$ : a fleet of $n$ drivers participates when hourly earnings are $w(n)$ .

There are $I$ demand scenarios with potential demand masses $0 < m_1 < \dots < m_I$ . In scenario $i$ , the inverse demand at demand rate $d \in [0, m_i]$ is $p_i(d) = V(1 - d/m_i)$ , and the market revenue per hour is $r_i(d) = d\,p_i(d)\,t_2$ .

To serve demand $d$ with $s$ open cars, Little's law requires $s + d(t_1(s) + t_2)$ cars. The minimal supply level is

\underline{n}(d) = \tilde a\, d^{\frac{1}{r+1}} + t_2 d, \qquad \tilde a = r^{\frac{1}{r+1}} a^{\frac{1}{r+1}} + r^{-\frac{r}{r+1}} a^{\frac{1}{r+1}},

and the maximal utilization is $\bar u(d) = d t_2/\underline{n}(d)$ . A demand rate $d \in [0, m_i]$ is an equilibrium when revenue covers exactly the minimal driver cost,

r_i(d) = w(\underline{n}(d))\,\underline{n}(d). \qquad (13)

It is stable (Definition 1) if there is no $\epsilon > 0$ with $r_i(d-\epsilon) < w(\underline{n}(d-\epsilon))\underline{n}(d-\epsilon)$ and $r_i(d+\epsilon) > w(\underline{n}(d+\epsilon))\underline{n}(d+\epsilon)$ . The maximal stable equilibrium $d^*_{HV,i}$ is the greatest stable equilibrium; then $n^*_{HV,i} = \underline{n}(d^*_{HV,i})$ , $u^*_{HV,i} = \bar u(d^*_{HV,i})$ and $p^*_{HV,i} = p_i(d^*_{HV,i})$ . The loose labor market condition of scenario $i$ is

r_i(m_i/2) > w(\underline{n}(m_i/2))\,\underline{n}(m_i/2), \qquad (33)

the paper's restatement of its Assumption 2: the revenue curve meets the cost curve on its decreasing branch.

In Lean the parameters form the structure MarketParams, and the objects above are aTilde, nMin, uBar, priceFn, revenue, wage, hvCost, IsEquilibrium, IsStable, IsMaxStableEq and LooseMarket, in the namespace AVRideHail.PureHV.

Formalization targets

Goal: Proposition 1 (p. 14)

Under the standing parameter assumptions and (33) in every scenario, each scenario has a unique maximal stable equilibrium, and for the family $d^*_{HV,i}$ :

d^*_{HV,i} > 0,\quad r_i(d^*_{HV,i}) = w(n^*_{HV,i})\, n^*_{HV,i}, \quad p^*_{HV,i} = \frac{w(n^*_{HV,i})}{u^*_{HV,i}},

d^*_{HV,i}\ \text{and}\ n^*_{HV,i}\ \text{strictly increasing in } i, \qquad u^*_{HV,i}\ \begin{cases}\text{strictly increasing in } i, & r > 0,\\ \text{constant in } i, & r = 0.\end{cases}

Milestones

Lemma 1 Item 1 (p. 38, with (26)): $\underline{n}(d)$ is the minimum of $s + d(as^{-r} + t_2)$ over $s > 0$ , attained only at $s = (ra)^{1/(r+1)} d^{1/(r+1)}$ .
Lemma 1 Item 2 (p. 38): $\underline{n}$ is a strictly increasing bijection of $[0,\infty)$ , so its inverse $\bar d(n)$ exists.
Lemma 2 (p. 39): $\bar u$ is strictly increasing and strictly concave, $\bar u(0) = 0$ and $\bar u(d) \to 1$ .
App. B.1.1 (p. 42): under (33), (13) has exactly one root in $[m_i/2, m_i]$ , and it is the maximal stable equilibrium.
App. B.1.3 (p. 43): $d^*_{HV,i}$ is strictly increasing in $m_i$ .
App. B.1.6 (pp. 45–46): $p^*_{HV,i}$ is strictly increasing in $m_i$ when $r = 0$ , $\alpha > 0$ , and strictly decreasing when $r > 0$ , $\alpha = 0$ .

Significance

Proposition 1 is the reference point for every comparison in the paper: the AV markets of Sections 5–6 (common platform, independent platforms, monopoly AV supplier) are judged by how far their prices and served demand move from $p^*_{HV,i}$ and $d^*_{HV,i}$ . Its Item 3 isolates the density effect as the source of rising utilization, and the special cases of Item 5 show that the HV price can rise or fall with market size, so neither direction can be assumed in the AV analysis. Lemmas 1 and 2 are used throughout the appendices.

The paper proves these results by hand; to our knowledge none of them has a machine-checked proof. The mission produces a checked version of the benchmark and of the two technical lemmas, with every boundary case ( $r = 0$ , $\alpha = 0$ , $d = 0$ ) stated explicitly.

Difficulty

The equation (13) always has the root $d = 0$ , and it can have a second, unstable positive root below the stable one; the naive reading "the equilibrium is the root of (13)" therefore picks the wrong point. The work is to show that the greatest stable root exists, lies on the decreasing branch of the revenue curve, and moves monotonically with $m_i$ . This requires strict monotonicity of the cost curve $w(\underline{n}(d))\underline{n}(d)$ , which combines a real power $d^{1/(r+1)}$ with the linear supply curve, and continuity on $[m_i/2, m_i]$ . Lemma 2 needs strict concavity of a quotient of real powers on a closed half-line including $0$ , where the function is defined by its limit.

Formalization scope

All quantities are real numbers; fleet sizes and demands are continuous, as in the paper. Powers are Real.rpow. Scenarios are indexed by Fin I, 0-based. The maximal stable equilibrium is never chosen by Classical.choose: statements take a family dstar with IsMaxStableEq in each scenario.

Committed conventions and pinned hypotheses:

$a, t_2, V, w_0 > 0$ and $0 \le r < 1$ , $\alpha \ge 0$ are explicit binders. Lemma 1 Item 1 and Lemma 2 add $r > 0$ , which the page leaves implicit and without which they fail.
The loose-market condition (33) is a hypothesis of the goal in every scenario. Proposition 1's printed statement omits it, but its proof uses it, and without it a thin scenario has $d^*_{HV,i} = 0$ .
In Definition 1, $\epsilon$ ranges over $0 < \epsilon \le d$ with $d + \epsilon \le m_i$ , keeping $d \pm \epsilon$ in the domain $[0, m_i]$ of $p_i$ .
At $r = 0$ , Real.rpow gives $\tilde a = a$ and $\underline{n}(d) = (a + t_2)d$ ; $\bar u(0) = 0/0 = 0$ agrees with Lemma 2.
"Increasing" means strictly increasing.
The uniqueness in "unique maximal stable equilibrium" is automatic for a greatest element; the content of the first conjunct is existence.

Encoding the equilibrium as "the root of (13) in $[m_i/2, m_i]$ " would make the goal follow from its own hypotheses; the goal instead quantifies over greatest stable equilibria, and that root is the conclusion of a milestone.

A complete development needs the intermediate value theorem on $[m_i/2, m_i]$ , strict monotonicity and concavity facts for real powers, and limits at infinity. The minimal-supply and utilization lemmas are reusable in any model with a pickup-time function $a s^{-r}$ . Proofs of any milestone are welcome, as are proofs of the remark of Item 5 in other parameter regimes.

Selected references

Z. Lian and G. van Ryzin, Capturing the Benefits of Autonomous Vehicles in Ride-Hailing: The Role of Market Configuration, SSRN working paper 3716491, version of 23 August 2023. https://ssrn.com/abstract=3716491
Z. Lian and G. van Ryzin, Capturing the Benefits of Autonomous Vehicles in Ride-Hailing: The Role of Market Configuration, Management Science, 2025. https://doi.org/10.1287/mnsc.2020.03112

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Economics Mechanism Design Operations Research+1·Captain: mikedeng1

Selling Data to Marketers 1: In Selling Marketing Lists, the Low-Cost Plan Is Efficient and the High-Cost Plan Carries More Information Than the Benchmark If φ < 1/2 and Less If φ > 1/2Research Paper

Why sell information to marketers

Data brokers such as Acxiom, Nielsen and Oracle, and data generators such as Google and Facebook, sell processed consumer data to firms that use it to decide whom to target, whom to lend to, or which market to enter. The U.S. Federal Trade Commission's 2014 report on data brokers distinguishes two products: a marketing list, which carries contact information without which the buyer cannot act at all, and a data append, which enriches data the buyer already holds. Liang Guo's Selling Data to Marketers (Management Science, doi:10.1287/mnsc.2023.01350) asks how a seller should design a menu of data products of different informativeness when buyers differ in a privately known cost of acting.

The question belongs to the theory of optimal screening (Mussa and Rosen 1978; Maskin and Riley 1984), but with a twist: the buyer's marginal value of information is not a primitive. It is derived from the buyer's decision problem, so the direction of the single-crossing condition is itself an outcome of the model. This mission formalizes the paper's analysis of marketing lists in the base model with a binary value.

Setting

A marketer can take an action (target a consumer) at cost $c>0$ . The action's value $u$ is $\bar u$ with prior probability $\phi\in(0,1)$ and $\underline u<\bar u$ otherwise. The seller can provide a binary signal $w\in\{w_h,w_l\}$ with $\Pr(w_h\mid\bar u)=\Pr(w_l\mid\underline u)=\alpha$ , where the informativeness $\alpha$ lies in an interval $[\underline\alpha,\bar\alpha]\subset(1/2,1)$ . The probability of the optimistic signal is $\sigma_h(\alpha)=\phi\alpha+(1-\phi)(1-\alpha)$ , the posterior probability of $\bar u$ after it is $\phi_h(\alpha)=\phi\alpha/\sigma_h(\alpha)$ , and the posterior mean is $v_h(\alpha)=\phi_h(\alpha)\bar u+[1-\phi_h(\alpha)]\underline u$ ; $\sigma_l,\phi_l,v_l$ are defined symmetrically. The paper assumes $v_l(\underline\alpha)<c<v_h(\underline\alpha)$ , so a marketer acts exactly after $w_h$ , and a marketer with cost $c$ who pays $T$ for a signal of informativeness $\alpha$ gets

U=-T+\sigma_h(\alpha)[v_h(\alpha)-c].

Serving a marketer at informativeness $\alpha$ costs the seller $k(\alpha)$ , with $k(\underline\alpha)=0$ and $k$ increasing.

There are two cost types, $c_H>c_L>0$ , with $\Pr(H)=\theta\in(0,1)$ ; write $\theta_H=\theta$ , $\theta_L=1-\theta$ . Under the benchmark with observable cost, the seller extracts all surplus and chooses $\alpha$ to maximize

\Pi_i(\alpha)=\theta_i\big[\sigma_h(\alpha)[v_h(\alpha)-c_i]-k(\alpha)\big],

whose maximizers are the efficient levels $\alpha^e_i$ . With private costs the seller offers a menu $(T_H,\alpha_H),(T_L,\alpha_L)$ with $T_j\ge0$ and maximizes $\Pi=\sum_i\theta_i[T_i-k(\alpha_i)]$ subject to individual rationality $U_{ii}\ge0$ and incentive compatibility $U_{ii}\ge U_{ii'}$ , where $U_{ij}=-T_j+\sigma_h(\alpha_j)[v_h(\alpha_j)-c_i]$ . This is problem (6); its solutions are the optimal menus, with informativeness levels $\alpha^*_H,\alpha^*_L$ .

Formalization targets

Goal: Proposition 2

Under the regularity of $k$ described below,

\alpha^*_L=\alpha^e_L,\qquad \alpha^*_H>\alpha^e_H\ \text{ if }\phi<\tfrac12,\qquad \alpha^*_H<\alpha^e_H\ \text{ if }\phi>\tfrac12.

The first claim is an equality of sets of maximizers; the inequalities compare every optimal menu with every efficient level.

Milestones

The derivative of the benchmark profit: $\partial\Pi_i/\partial\alpha_i=\theta_i[\phi(\bar u-c_i)-(1-\phi)(\underline u-c_i)-k'(\alpha_i)]$ .
The cross derivative $\partial^2\Pi_i/\partial\alpha_i\partial c_i=\theta_i(1-2\phi)$ , stated as a mixed second difference.
Proposition 1: $\alpha^e_i$ increases with $c_i$ if $\phi<1/2$ and decreases if $\phi>1/2$ .
The reduction of (6) to the problem in $(\alpha_H,\alpha_L)$ alone,

\max_{\alpha_H,\alpha_L}\ \theta\big[\sigma_h(\alpha_H)[v_h(\alpha_H)-c_H]-k(\alpha_H)\big]+(1-\theta)\big[\sigma_h(\alpha_L)[v_h(\alpha_L)-c_L]-\sigma_h(\alpha_H)(c_H-c_L)-k(\alpha_L)\big],\qquad(7)

obtained by making IR $_H$ and IC $_{L,H}$ bind. 5. The derivative of (7) in $\alpha_H$ : $\theta[\phi(\bar u-c_H)-(1-\phi)(\underline u-c_H)-k'(\alpha_H)]+(1-\theta)(1-2\phi)(c_H-c_L)$ .

Significance

In textbook screening the low type's allocation is distorted downward to reduce the high type's rent. Here the low type of the screening problem is the high-cost marketer, and its information can be distorted in either direction: the rent $\sigma_h(\alpha_H)(c_H-c_L)$ conceded to low-cost marketers falls with $\alpha_H$ when $\phi<1/2$ and rises when $\phi>1/2$ . Proposition 2 therefore says that a seller of marketing lists may supply socially excessive information to some buyers, a possibility absent from the data-appends literature the paper compares against, where provision can only be insufficient. Proposition 1 isolates the mechanism: the prior decides whether information mainly prevents wasteful targeting or restores profitable targeting, and the cost of acting amplifies one or the other.

The result is proved in the paper; nothing in it has been machine-checked. The mission produces a checked version of the two-type screening reduction (including the verification that the constraints dropped in (7) hold at its solutions, which the paper asserts as standard) and of the comparative statics with set-valued optima, since $k$ is not assumed convex and optima need not be unique.

Difficulty

The derivative computations are routine once $\sigma_h(\alpha)v_h(\alpha)$ is simplified to $\phi\alpha\bar u+(1-\phi)(1-\alpha)\underline u$ . The work lies elsewhere. First, the reduction: "IR binds for the high-cost type and IC binds for the low-cost type" is the standard heuristic, but at a solution of (7) one still has to check the dropped constraint IC $_{H,L}$ and the nonnegativity $T_L\ge0$ ; both depend on the ordering of $\alpha_H$ and $\alpha_L$ at the optimum, which in turn depends on the sign of $1-2\phi$ . Second, strictness: comparative statics through increasing differences give only weak inequalities between maximizer sets. Upgrading $\alpha^*_H\ge\alpha^e_H$ to $\alpha^*_H>\alpha^e_H$ requires ruling out a common maximizer, which needs the optimum to be interior and $k$ differentiable there; the naive argument by first-order conditions fails at the corners $\underline\alpha,\bar\alpha$ . Third, $\alpha^*_L=\alpha^e_L$ as sets requires existence of optimal menus, hence attainment of the maximum of (7).

Formalization scope

Everything is over $\mathbb R$ . The model is a structure Model (values, prior, informativeness interval, selling cost) extended by Screening (type probability and the two costs); the standing assumptions $\underline u<\bar u$ , $0<\phi<1$ , $1/2<\underline\alpha<\bar\alpha<1$ , $k(\underline\alpha)=0$ , $k$ increasing on $[\underline\alpha,\bar\alpha]$ (weakly), $0<\theta<1$ , $0<c_L<c_H$ , and $v_l(\underline\alpha)<c_i<v_h(\underline\alpha)$ for both types are fields. The posteriors are defined with the divisions printed in (2). Efficient levels and optimal menus are sets of maximizers, never chosen by Classical.choose or a first-order condition. Menus include $T_j\ge0$ .

The goal additionally assumes the part of footnote 5 (p. 7) that makes optima interior: $k$ continuous on $[\underline\alpha,\bar\alpha]$ , differentiable on $(\underline\alpha,\bar\alpha)$ , with right derivative $k'(\underline\alpha)=0$ and $k'(\alpha)\to+\infty$ as $\alpha\to\bar\alpha^-$ . Convexity and twice differentiability, also in the footnote, are not assumed. The milestones carry no regularity beyond differentiability at the point considered (milestones 1, 5) and continuity for the optimality characterization in milestone 4.

The optimal menu is a maximizer of (6) under all four constraints, not a maximizer of (7): defining it through (7) would make half of the goal immediate and skip the reduction.

The development needs the algebra of the binary posterior, the two-type screening reduction, and a strict version of monotone comparative statics; Topkis's theorem on strictly increasing differences (Theorem 2.8.4) is already proved on the platform and referenced. Proofs of any milestone, and alternative arguments for the strict inequalities, are welcome.

Selected references

L. Guo, Selling Data to Marketers, Management Science, 2025 (author's manuscript). https://doi.org/10.1287/mnsc.2023.01350
D. M. Topkis, Supermodularity and Complementarity, Princeton University Press, 1998. https://doi.org/10.1515/9781400822539
M. Mussa and S. Rosen, Monopoly and Product Quality, Journal of Economic Theory 18(2), 1978. https://doi.org/10.1016/0022-0531(78)90085-6
E. Maskin and J. Riley, Monopoly with Incomplete Information, RAND Journal of Economics 15(2), 1984. https://doi.org/10.2307/2555674
Federal Trade Commission, Data Brokers: A Call for Transparency and Accountability, 2014. https://www.ftc.gov/reports/data-brokers-call-transparency-accountability-report-federal-trade-commission-may-2014

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Algorithmic Game Theory Economics Operations Research·Captain: mikedeng1

Revenue Sharing at Music Streaming Platforms I: Homogeneity and Additivity Characterize Weighted Indices; Adding Equal Individual or Global Impact of Users Gives Pro-Rata or User-CentricResearch Paper

Motivation

Music streaming platforms collect a subscription fee from each user and pay part of the revenue to the artists whose content was played. The industry standard is the pro-rata method: the revenue pool is divided in proportion to each artist's total number of streams across all users. A competing method, the user-centric method, divides each user's own payment among the artists that user played, in proportion to that user's streams. SoundCloud adopted it in 2021 and Deezer has campaigned for it; the debate between the two methods is documented in the literature the paper cites, among others Dimont (2017), Haampland et al. (2022), Page and Safir (2018) and Meyn et al. (2023), listed below.

Bergantiños and Moreno-Ternero (arXiv:2310.11861, published in Management Science, 2025) put the debate on an axiomatic footing. Their Theorem 1 identifies the two basic principles every reasonable method in a large family satisfies, and isolates pro-rata and user-centric inside that family by one further principle each. This mission formalizes that theorem and the steps of its proof.

Setting

There are $n$ artists $N$ and a finite set $M$ of users, $m=|M|$ . A streaming problem $(N,M,t)$ records, for every artist $i$ and user $j$ , the number $t_{ij}\in\mathbb N$ of times $j$ streamed content of $i$ . Each user pays $1$ , so the amount to share is $m$ . Write

$T_i=\sum_{j\in M}t_{ij}$ , the streams of artist $i$ ;
$T^j=\sum_{i\in N}t_{ij}$ , the streams of user $j$ ;
$L^j=\{i\in N : t_{ij}>0\}$ , the artists user $j$ played, assumed nonempty for every user;
$t_{\cdot j}=(t_{ij})_{i\in N}$ , the profile of user $j$ , and $t_{-j}$ the table without user $j$ .

A popularity index $I$ assigns to every problem a vector $I(N,M,t)\in\mathbb R^N_{\ge 0}$ with positive sum, and artist $i$ is paid the reward

R^I_i(N,M,t)=\frac{I_i(N,M,t)}{\sum_{i'\in N}I_{i'}(N,M,t)}\,m .

Since $R^{cI}=R^I$ for every $c>0$ , an index is identified with its positive multiples. The pro-rata index is $P_i=T_i$ and the user-centric index is $U_i=\sum_{j\in M}t_{ij}/T^j$ . A weight system assigns a weight $\omega(j,x)>0$ to every user $j$ and every profile $x\in\mathbb N^N$ ; its weighted index is $I^\omega_i=\sum_{j\in M}\omega(j,t_{\cdot j})\,t_{ij}$ . Pro-rata is $\omega\equiv 1$ and user-centric is $\omega(j,x)=1/\sum_i x_i$ .

The axioms are:

Homogeneity: if $t_{ij}=\lambda t_{i'j}$ for all $j\in M$ , with $\lambda\ge 0$ , then $I_i=\lambda I_{i'}$ .
Additivity: if $M$ splits into disjoint $M^1,M^2$ , then $I(N,M,t)=I(N,M^1,t^1)+I(N,M^2,t^2)$ .
Equal individual impact of similar users: if $t_{ij}=t_{ij'}$ , then $I_i(N,M\setminus\{j\},t_{-j})=I_i(N,M\setminus\{j'\},t_{-j'})$ .
Equal global impact of users: $\sum_i I_i(N,M\setminus\{j\},t_{-j})=\sum_i I_i(N,M\setminus\{j'\},t_{-j'})$ for all $j,j'\in M$ .

Two further axioms of §3 compare the methods: reasonable lower bound (the artists played by a group $C$ of users receive at least $|C|$ ) and click-fraud-proofness (changing one user's streams moves no artist's reward by more than $1$ ).

Formalization targets

Goal: Theorem 1 (p. 12)

For every popularity index $I$ :

\text{(a)}\quad I \text{ satisfies homogeneity and additivity} \iff \exists\,\omega>0:\ I=I^\omega ;

\text{(b)}\quad (n\ge 2)\quad \text{homogeneity, additivity, equal individual impact} \iff \exists\,c>0:\ I=c\,P ;

\text{(c)}\quad \text{homogeneity, additivity, equal global impact} \iff \exists\,c>0:\ I=c\,U .

Milestones (the proof, Appendix pp. 18–20)

Every weighted index satisfies homogeneity and additivity (p. 18).
An additive index decomposes as $I_i(N,M,t)=\sum_{j\in M}I_i(N,\{j\},t_{\cdot j})$ (p. 18).
On a one-user problem a homogeneous index is $I_i(N,\{j\},x)=\frac{\sum_{i'}I_{i'}(N,\{j\},x)}{T^j}\,x_i$ (pp. 18–19).
Theorem 1(a) as a statement on its own (p. 12).
Pro-rata satisfies the three axioms of (b) (p. 19).
Under the axioms of (b), any representing weight $\omega(j,x)$ is the same at every nonzero profile (pp. 19–20).
$\sum_i U_i(N,M\setminus\{j\},t_{-j})=m-1$ , and user-centric satisfies the axioms of (c) (p. 20).
Under equal global impact, $\sum_i I_i(N,\{j\},x)$ is the same for every user and nonzero profile (p. 20).

Companions (§3, p. 13; Appendix, pp. 21–22): user-centric satisfies reasonable lower bound and click-fraud-proofness; with at least two artists, pro-rata satisfies neither.

Significance

Theorem 1 says that the choice between the two methods used in practice is a choice between two principles, and nothing else, once homogeneity and additivity are accepted: equal individual impact of similar users ("all streams count the same") gives pro-rata, and equal global impact of users ("all users count the same") gives user-centric. Part (a) describes the whole family of methods that respect the two basic axioms, which includes capped and hybrid schemes proposed in the industry (p. 10). The companion results show that the further axioms of §3 separate the two methods in favour of user-centric. The paper's further results, the core of the associated cooperative game (Theorem 2) and the core-selection characterization (Theorem 3), are separate missions of this series.

The result is proved in the paper; its proofs are in the appendix (pp. 18–22). No machine-checked version is known. A formalization fixes the model precisely, including the conventions the paper leaves implicit (the disjointness in additivity, nonempty user lists, the identification of indices up to a positive factor), and records exactly where two artists are needed.

Difficulty

The "if" directions are direct computations. The "only if" directions are a functional equation on indices: additivity reduces an index to its values on one-user problems, and homogeneity forces those values to be proportional to the profile, so the remaining freedom is one weight per user and profile. The work in (b) and (c) is to show that this weight is constant, using the extra axiom only on small auxiliary problems that contain users outside the given problem. The constant must be one number for all problems: a constant depending on the problem would make (b) and (c) much weaker, and is not what the paper proves.

Formalization scope

Artists are Fin n, indexed from $0$ . Users are natural numbers, an infinite universe, and a problem is a finitely supported table ℕ →₀ (Fin n → ℕ) whose support is the user set $M$ ; every user of a problem therefore has a nonzero profile, which is the paper's standing assumption $L^j\neq\varnothing$ , and an index can only read the columns of users in $M$ . Streams are natural numbers. An index is a function from problems to Fin n → ℝ; it is a popularity index when it is nonnegative with positive sum on every problem with at least one user. The empty problem has nothing to share, and no positivity is demanded there; additivity forces the index to vanish on it. Additivity is stated for disjoint user sets: the printed axiom only says $M=M^1\cup M^2$ , which with $M^1=M^2=M$ would force $I=2I$ and leave no index at all. "Is the pro-rata (user-centric) index" means "is a positive multiple of it, with one constant for all problems". Part (b) assumes at least two artists: with one artist the user-centric index satisfies its axioms and is not a multiple of pro-rata (every index then pays the whole $m$ to the single artist, so (b) holds at the level of rewards). Weights at the zero profile never enter a weighted index and are only required to be positive.

A formalization in which the positivity requirement applies to the empty problem, additivity allows overlapping user sets, or the constant in (b) or (c) depends on the problem is trivial or empty and is ruled out.

All objects are defined in one module, MusicStreaming.Axioms.Setting, on top of Mathlib's Finsupp and finite sums; nothing else is needed. Proofs of the milestones, alternative proofs of Theorem 1, and the verification that the definitions compute the paper's Example 1 are welcome.

Selected references

G. Bergantiños and J. D. Moreno-Ternero, Revenue sharing at music streaming platforms, Management Science, 2025; preprint arXiv:2310.11861v1, 2023. https://arxiv.org/abs/2310.11861 (the source of every statement in this mission; Theorem 1 on p. 12, proofs on pp. 18–22).
J. Meyn, M. Kandziora, S. Albers, M. Clement, Consequences of platforms' remuneration models for digital content: initial evidence and a research agenda for streaming services, Journal of the Academy of Marketing Science 51, 114–131, 2023 (reference [39] of the paper).
J. Dimont, Royalty inequity: Why music streaming services should switch to a per-subscriber model, Hastings Law Journal 69, 675, 2017 (reference [22] of the paper).
W. Page and D. Safir, Money in, money out: Lessons from CMOs in allocating and distributing licensing revenue, technical report, 2018 (reference [45] of the paper).
O. Haampland, R. Johannessen, F. Moreau, P. Wikström, Fairness and royalty payment systems on music streaming platforms, mimeo, 2022 (reference [29] of the paper).

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Functional Analysis Partial Differential Equations Symplectic Geometry·Captain: Mazecto

Fredholm Theory and the Index Formula for Cauchy–Riemann Operators on the CylinderTextbook

Motivation

Linear Cauchy–Riemann operators are the linearizations of the nonlinear Cauchy–Riemann equation for pseudoholomorphic curves. On the cylinder $\mathbb{R}\times S^1$ with the trivial bundle they take the form

D_S=\partial_s+J_0\,\partial_t+S(s,t):W^{1,p}(\mathbb{R}\times S^1,\mathbb{R}^{2n})\to L^p(\mathbb{R}\times S^1,\mathbb{R}^{2n}),

where $S(s,\cdot)$ converges to loops $S^\pm$ of symmetric matrices as $s\to\pm\infty$ . These are the operators of Floer homology, of symplectic field theory and of contact homology. The dimension of their moduli spaces is the Fredholm index of $D_S$ , and the Riemann–Roch formula with punctures computes it from the Conley–Zehnder indices of the asymptotic operators $A^\pm=-J_0\partial_t-S^\pm(t)$ :

\operatorname{ind}D_S=\mu_{CZ}(A^+)-\mu_{CZ}(A^-).

This mission formalizes this formula, together with the Fredholm property of $D_S$ , following C. Wendl, Lectures on Symplectic Field Theory, Lectures 3–5, in the case of the cylinder (the sphere with one positive and one negative puncture). Mathlib has the notion of a Fredholm operator but no Sobolev spaces on cylinders, no Cauchy–Riemann operators and no Conley–Zehnder index.

Setting

Cylinder. $S^1=\mathbb{R}/\mathbb{Z}$ . Maps on $\mathbb{R}\times S^1$ are maps $u(s,t)$ on $\mathbb{R}^2$ with $u(s,t+1)=u(s,t)$ , and $L^p(\mathbb{R}\times S^1)$ is $L^p$ of the strip $\mathbb{R}\times[0,1)$ with Lebesgue measure. $\mathbb{R}^{2n}=\mathbb{C}^n$ with $J_0=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$ , multiplication by $i$ .
Sobolev space. $W^{1,p}(\mathbb{R}\times S^1,\mathbb{R}^{2n})$ is the closure of the smooth maps that vanish for large $|s|$ under the norm $\|u\|_p+\|\partial_su\|_p+\|\partial_tu\|_p$ .
Asymptotic operators. For a loop $S_\infty$ of symmetric matrices, $A=-J_0\partial_t-S_\infty(t)$ . Its fundamental solution $\Psi$ solves $\Psi'=J_0S_\infty\Psi$ , $\Psi(0)=\mathrm{Id}$ . $A$ is nondegenerate if $1$ is not an eigenvalue of $\Psi(1)$ , and $\mu_{CZ}(A)$ is the Conley–Zehnder index of the symplectic path $\Psi|_{[0,1]}$ .
Cauchy–Riemann operators. $S:\mathbb{R}\times S^1\to\mathbb{R}^{2n\times2n}$ is continuous and $S(s,\cdot)\to S^\pm$ uniformly as $s\to\pm\infty$ . $D_S$ acts on $W^{1,p}$ ; its index is $\dim\ker D_S-\dim\operatorname{coker}D_S$ .

Formalization targets

Goal: the index formula on the cylinder

If $1<p<\infty$ and the asymptotic operators $A^\pm$ are nondegenerate, then $D_S:W^{1,p}\to L^p$ is Fredholm and $\operatorname{ind}D_S=\mu_{CZ}(A^+)-\mu_{CZ}(A^-)$ (Wendl, Theorem 5.4 with Theorem 4.3, for the cylinder).

Milestones

Translation-invariant operators. For a nondegenerate loop $S_\infty$ , $\partial_s+J_0\partial_t+S_\infty(t):W^{1,p}\to L^p$ is an isomorphism (Theorem 4.11).
Global regularity. An $L^p$ weak solution of $D_S\eta=f\in L^p$ lies in $W^{1,p}$ (Proposition 4.6).
Local estimate. $\|\eta\|_{W^{1,p}}\le c\|D_S\eta\|_p+c\|\eta\|_p$ for $\eta$ supported in $[-R,R]\times S^1$ (Proposition 4.8).
Estimate on the ends. $\|\eta\|_{W^{1,p}}\le c\|D_S\eta\|_p$ for $\eta$ supported far out in an end with nondegenerate asymptotic operator (Proposition 4.9).
An abstract estimate. If $\|x\|\le c\|Tx\|+c\|Kx\|$ with $K$ compact, then $\ker T$ is finite-dimensional and $T$ has closed range (Lemma 4.12).
Global estimate. $\|\eta\|_{W^{1,p}}\le c\|D_S\eta\|_p+c\|\eta\|_{L^p([-R,R]\times S^1)}$ (Lemma 4.13).
Compactness on finite cylinders. Bounded sequences in $W^{1,p}$ have subsequences converging in $L^p([-R,R]\times S^1)$ (Wendl, p. 74).
Semi-Fredholm property of $D_S$ (Corollary 4.14).
Cokernel and formal adjoint. $\dim\operatorname{coker}D_S$ is finite and equals the kernel dimension of the Cauchy–Riemann operator $D_{\bar S}$ on $W^{1,q}$ , $\bar S=-CS^{\mathsf T}C$ , $1/p+1/q=1$ , with $C$ complex conjugation (Lemma 4.17, Proposition 4.18, Theorem 4.20).
Fredholm property of $D_S$ (Theorem 4.3).
Independence of $p$ of the kernel dimension and of the index (Theorem 4.3).
Classification of nondegenerate asymptotic operators. Two nondegenerate loops have the same Conley–Zehnder index if and only if they are joined by a path of nondegenerate loops (Theorem 3.34).
Conjugation. $\mu_{CZ}(-J_0\partial_t+CS_\infty C)=-\mu_{CZ}(-J_0\partial_t-S_\infty)$ (Lemma 5.8).
Model operators. $\eta\mapsto-J_0\dot\eta-\pi k\eta-\sigma e^{2\pi ikt}\bar\eta$ on loops in $\mathbb{C}$ is nondegenerate with Conley–Zehnder index $k$ , for $k\in\mathbb{Z}$ and $\sigma>0$ (Lemma 5.29).
A plane equation without $L^2$ solutions. $\bar\partial f+\bar z\bar f=0$ has no nonzero smooth solution in $L^2(\mathbb{C})$ (Proposition 5.20).
A plane equation with Gaussian solutions. The smooth $L^2$ solutions of $\bar\partial f+z\bar f=0$ are the real multiples of $e^{-|z|^2/2}$ (Proposition 5.21).
Large antilinear perturbations. For a function $\beta$ on the cylinder with normalized zeros, $D_S+\sigma\beta\bar{\ \cdot\ }$ is onto with kernel of dimension $\#Z^+$ when all zeros are positive, and injective with cokernel of dimension $\#Z^-$ when all zeros are negative, for all large $\sigma$ (Proposition 5.27).
Antilinear deformations of asymptotic operators. If each asymptotic operator has the Conley–Zehnder index of an operator $-J_0\partial_t-S'^\pm$ whose deformations $-J_0\partial_t-S'^\pm-\sigma\beta_\pm\bar{\ \cdot\ }$ , $\sigma\ge0$ , are all nondegenerate, then $\operatorname{ind}D_S=\operatorname{wind}\beta_+-\operatorname{wind}\beta_-$ (Lemma 5.28).

Significance

The result. The index formula is the dimension formula for moduli spaces of pseudoholomorphic cylinders: Floer trajectories in Hamiltonian Floer homology and holomorphic cylinders in symplectizations. It fixes the grading of these theories, and its sign conventions are those of the Conley–Zehnder index of the asymptotic orbits.

Formalizing it. The mission builds Sobolev spaces on the cylinder, the Fredholm theory of Cauchy–Riemann operators with cylindrical ends, and the Conley–Zehnder index of asymptotic operators. These are the analytic foundations of Floer theory and symplectic field theory and are reusable for transversality arguments and for the indices of other operators. Milestone 5 is a general statement about Banach spaces.

Difficulty

The operators act between infinite-dimensional spaces of functions on a non-compact domain, so finiteness of kernel and cokernel is a global statement that sees the behaviour at both ends. The Sobolev space is defined as a closure, and the operator acts on its elements through their weak derivatives. The Conley–Zehnder index is defined through a continuous argument of a normalized complex determinant along a path of symplectic matrices, extended to a normal form; the conventions (sign of $J_0$ , orientation of the cylinder, which end is positive) all enter the final formula. Several milestones are uniform estimates in which the constant is chosen before the function.

Formalization scope

Namespace CauchyRiemannIndex. The definition module CauchyRiemannIndex_Setting defines the cylinder measure, W1p, crKernel, crRange, crIndex, IsFredholmCR, asymptotic operators through IsFundamentalSolution, IsNondegenerate and asymptoticCZIndex, and the Conley–Zehnder index czIndex of a symplectic path. The mission is standalone; czIndex and its auxiliary definitions repeat those of the ConleyZehnder mission word for word.
Spaces. $\mathbb{R}^{2n}$ is EuclideanSpace ℝ (Fin n ⊕ Fin n), matrices are Matrix (Fin n ⊕ Fin n) (Fin n ⊕ Fin n) ℝ, and $J_0$ is Matrix.J. Maps on the cylinder are maps ℝ × ℝ → ℝ²ⁿ, periodic in the second variable, and $L^p$ is taken on the strip $\mathbb{R}\times[0,1)$ . $W^{1,p}$ is a closed subspace of Lp × Lp × Lp, consisting of triples $(u,\partial_su,\partial_tu)$ .
Exponents. $p$ is in ℝ≥0∞ with $1<p$ and $p\ne\infty$ . Norms of smooth maps are eLpNorm in ℝ≥0∞; constants are in ℝ≥0.
Indices. The Fredholm index is finrank ker - finrank (Lp ⧸ range) in ℤ; statements that use it also assert that both are finite-dimensional.
Smoothness. Smoothness is ContDiff ℝ ∞ ( $C^\infty$ , not analytic). The coefficient $S$ is only assumed continuous, with uniform convergence at the ends.

Contributions welcome: general lemmas on Sobolev spaces of periodic functions, on Fredholm operators between Banach spaces, and on the Conley–Zehnder index.

Selected references

C. Wendl, Lectures on Symplectic Field Theory, arXiv:1612.01009, Lectures 3–5. https://arxiv.org/abs/1612.01009
J. Gutt, Generalized Conley–Zehnder index, Ann. Fac. Sci. Toulouse Math. (6) 23 (2014), 907–932, arXiv:1307.7239. https://arxiv.org/abs/1307.7239
D. Salamon, Lectures on Floer homology, in Symplectic Geometry and Topology, IAS/Park City Math. Ser. 7, AMS (1999), 143–229.
M. Schwarz, Cohomology operations from S¹-cobordisms in Floer homology, PhD thesis, ETH Zürich (1995).

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Economics Operations Research Optimization·Captain: mikedeng1

The Bright Side of the GDPR: Welfare-Improving Privacy Management I: The Equilibrium Price Is Higher After the GDPR for Every Opt-in Benefit and Data RevenueResearch Paper

Motivation

The European Union's General Data Protection Regulation (GDPR, in force since 2018) requires a firm to obtain a consumer's explicit opt-in consent before it collects her personal data. Most of the economic literature reads such consent rules as a cost to data-driven firms. Choe, Matsushima and Shekhar (ISER Discussion Paper No. 1246, Osaka University, June 2024; published in Management Science, 2025, doi:10.1287/mnsc.2024.06653) ask what happens to the price of a digital service when consumers gain the right to buy it without sharing data. Their model is a monopoly with two-dimensional consumer heterogeneity, simple enough to be solved in closed form, and it lets the answer be stated for every value of the firm's opt-in benefit and every mix of usage-based and data-based revenue.

This mission formalizes the paper's general price comparison, Proposition A1: the equilibrium price is higher after the GDPR.

Setting

A monopolist sells an online service at zero marginal cost and price $p \in \mathbb{R}$ . A consumer has a type $(v, c) \in [0,1]^2$ , where $v$ is her valuation for the service and $c$ her privacy cost of sharing data; types are independent and uniform on $[0,1]^2$ , so the mass of a set of consumers is its area. A consumer who shares data receives an opt-in benefit $x \in (0, 1)$ , and the firm earns an exogenous data revenue $\alpha \ge 0$ per unit of shared data.

The utilities of opt-in purchase (buy and share) and opt-out purchase (buy without sharing) are

U_S(v,c) = v + (x - c) - p, \qquad U_N(v,c) = v - p.

Before the GDPR opt-out purchase is unavailable. A consumer buys, and shares data, iff $U_S \ge 0$ . The demand $\widehat D(p)$ is the area of $\{(v,c) \in [0,1]^2 : c \le v + x - p\}$ , and the profit is $\widehat\Pi(p) = (p + \alpha)\widehat D(p)$ .
After the GDPR a consumer opts in iff $U_S \ge \max\{U_N, 0\}$ and opts out iff $U_N \ge \max\{U_S, 0\}$ . The opt-in set is $\{c \le v + x - p,\ c \le x\}$ , the opt-out set is $\{v \ge p,\ c > x\}$ , their areas are $\widetilde D_S(p)$ and $\widetilde D_N(p)$ , and the profit is $\widetilde\Pi(p) = p\,(\widetilde D_S(p) + \widetilde D_N(p)) + \alpha\,\widetilde D_S(p)$ .

The firm chooses the price first and consumers respond, so the equilibrium price in each regime is a maximizer over $\mathbb{R}$ of the corresponding profit: $\hat p^\star$ for $\widehat\Pi$ and $\tilde p^\star$ for $\widetilde\Pi$ . The paper's appendix expresses them through the thresholds

\alpha_1(x) = \max\Big\{\tfrac{1-2x}{2}, 0\Big\}, \qquad \alpha_2(x) = \max\Big\{\tfrac{2-4x+x^2}{2x}, 0\Big\}

and the radicals $\beta = \sqrt{6 + (x+\alpha-1)^2}$ , $\Gamma = \sqrt{6 + (2-2x+\alpha)^2}$ .

Formalization targets

Goal: Proposition A1 (p. 34)

For every $0 < x < 1$ and $\alpha \ge 0$ , both profits attain a global maximum on $\mathbb{R}$ , and every pair of maximizers satisfies

\tilde p^\star > \hat p^\star.

Milestones

Display (4) (p. 11): $\widehat D(p) = (1+x-p)^2/2$ for $x < p < 1+x$ , and $1 - (1-x+p)^2/2$ for $x-1 \le p \le x$ .
Display (5) (p. 14): $\widetilde D_S(p)$ equals $(1+x-p)^2/2$ , $x^2/2 + x(1-p)$ or $x - p^2/2$ on $(1, 1+x)$ , $(x, 1]$ , $[0, x]$ .
Display (6) (p. 14): $\widetilde D_N(p) = (1-x)(1-p)$ for $0 \le p \le 1$ , and $0$ for $1 < p < 1 + x$ .
Online Appendix A.1 (OA p. 2): $\hat p$ maximizes $\widehat\Pi$ iff $\hat p = (1+x-2\alpha)/3$ when $\alpha < \alpha_1(x)$ and $\hat p = (2(x-1) - \alpha + \beta)/3$ otherwise.
Online Appendix A.2 (OA p. 4): $\tilde p$ maximizes $\widetilde\Pi$ iff $\tilde p = (2(1-\alpha x) + x^2)/4$ when $\alpha < \alpha_2(x)$ and $\tilde p = (\Gamma - (2 - 2x + \alpha))/3$ otherwise.
p. 14: $\alpha_1(x) \le \alpha_2(x)$ for all $x \in (0,1)$ .
Lemma A3 (pp. 33–34): the three possible pairs of equilibrium regimes (case 1a/2a, 1b/2a, 1b/2b), located by price intervals.
Online Appendix D, first display (OA p. 7): the closed form of $\tilde p^\star - \hat p^\star$ in each of the three regions.

Significance

The result. Proposition A1 says the price rise is robust: it holds for every opt-in benefit and every revenue model, from a firm that earns nothing from data ( $\alpha = 0$ ) to one whose data revenue dominates. The paper's welfare results (on demand, profit and consumer surplus) are comparisons of the same two equilibria, and the price comparison is the one among them that holds uniformly; the others change sign with $(x, \alpha)$ and are established in the paper only numerically. The main text's Proposition 1 is the instance $x = 1/4$ .

Formalizing it. The paper's proof is a chain of closed-form computations spread over the appendix and online appendix: areas of polygons, the maximization of piecewise cubic profits over all of $\mathbb{R}$ , and a sign argument with nested square roots. Several steps are stated as "one can verify", and the online appendix contains printed slips (a derivative written with respect to the wrong variable in §D, a pair of cases listed incorrectly on p. 14). A machine-checked proof turns the chain into a verified one. The milestones isolate the parts that are reusable on their own: the demand formulas (4)–(6) and the closed-form equilibria, on which the paper's companion profit comparison (Proposition A3) also rests. None of these results has been formalized before.

Difficulty

The obvious route, solving the first-order condition on each demand piece, gives candidates, not maximizers. The profits are only piecewise smooth, the pieces meet at $p = x$ and $p = 1$ , and the price ranges over all of $\mathbb{R}$ , including negative prices before the GDPR. Showing that the candidate is the unique global maximizer requires comparing values across pieces and controlling the profit outside the paper's displayed price ranges. The comparison step is the second difficulty: in the middle region $\alpha_1(x) \le \alpha < \alpha_2(x)$ the gap $(14 + 4\alpha + 3x^2 - 8x - 6x\alpha - 4\beta)/12$ mixes a polynomial with a square root, and its positivity is not a one-line estimate. The gap tends to $0$ as $x \to 1$ , so no uniform margin is available.

Formalization scope

Types are pairs q : ℝ × ℝ with q.1 = v, q.2 = c; the type square is Set.Icc 0 1 ×ˢ Set.Icc 0 1, and every demand is (volume S).toReal for the Lebesgue measure on ℝ × ℝ.
The after-GDPR sets are copied from p. 13; they send the tie $U_S = U_N \ge 0$ to opt-in, which changes only a null set.
Profits are functions ℝ → ℝ of the price, and an equilibrium price is IsMaxOn profit Set.univ p. The price is never restricted to the paper's case ranges, and $\tilde p^\star \ge 0$ is a consequence, not an assumption.
Standing assumptions on every theorem: $0 < x$ , $x < 1$ , $\alpha \ge 0$ (pp. 7–8). The data revenue is written α in Lean.
"Case 1a/1b/2a/2b equilibrium" means the maximizer lies in $(x, 1+x)$ , $[x-1, x]$ , $(x, 1+x)$ , $[0, x]$ respectively.
The thresholds $\alpha_1, \alpha_2$ and radicals $\beta, \Gamma$ are plain definitions with the paper's formulas.

A formalization that compares the two closed-form prices instead of maximizers of the modelled profits, that asserts a property of "every maximizer" without proving one exists, or that restricts prices to a bounded interval, would not state Proposition A1; the goal and the milestones rule all three out. Results of the paper established only numerically (Propositions 2–4, A2, the second and third bullets of A3, A4) are not posed, and neither are the comparative statics of Lemmas A1–A2.

A complete development needs areas of polygons under the Lebesgue measure on $\mathbb{R}^2$ (via MeasureTheory.volume_regionBetween or Fubini), elementary calculus on piecewise polynomials, and inequalities with square roots. The demand computations and the global-maximization arguments are reusable for the companion mission on the paper's profit comparison. Contributions of any milestone, or of further intermediate lemmas such as the continuity of the demands in the price, are welcome.

Selected references

C. Choe, N. Matsushima, S. Shekhar, The Bright Side of the GDPR: Welfare-Improving Privacy Management, ISER Discussion Paper No. 1246, Osaka University, June 2024 (the version formalized; theorem and page numbers refer to it).
C. Choe, N. Matsushima, S. Shekhar, The Bright Side of the GDPR: Welfare-Improving Privacy Management, Management Science, 2025. https://doi.org/10.1287/mnsc.2024.06653
Regulation (EU) 2016/679 (General Data Protection Regulation), Official Journal of the European Union L 119, 2016. https://eur-lex.europa.eu/eli/reg/2016/679/oj

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Algorithmic Game Theory Economics Operations Research+1·Captain: mikedeng1

Imperfect Targeting and Advertising Strategies: With Costless Advertising and A < 1/4, Equilibrium Industry Profit Falls as Targeting Improves, Jumps Up at α̂₂(A), Then Stays ConstantResearch Paper

Motivation

Online advertising lets a firm aim its messages at the consumers most likely to want its product, but the data behind that aim is noisy: a platform's prediction of who belongs to a demographic or interest segment is right only with some probability. A natural expectation is that sharper targeting raises firms' profits, since fewer messages are wasted. Karle and Reisinger (SSRN 4626076, March 2024; published in Management Science, doi:10.1287/mnsc.2023.03632) show that this can fail: when both firms chase the same valuable consumers, better targeting also increases the number of consumers who see both firms' advertisements, which intensifies price competition.

The model combines a targeting stage with the clearinghouse price-competition model of Varian (1980) and Narasimhan (1988), in which some consumers are informed about one firm only (loyal consumers) and others about both (shoppers), and equilibrium prices are dispersed. This mission formalizes the paper's result for costless advertising: industry profit first falls as targeting improves, then jumps up when firms switch to segmenting the market, then stays flat.

Setting

There are two consumer groups $a$ and $b$ , each of mass one. Group- $b$ consumers value the product at $1$ , group- $a$ consumers at $A$ , with $0 < A < 1$ . Two firms produce at zero cost. For every consumer each firm receives a signal of the consumer's group that is correct with probability $\alpha \in (1/2, 1]$ , independently across firms. Only consumers of the group a firm targets are interested in its product.

Stage 1. Each firm chooses a target group $k \in \{a, b\}$ and a message scope: expected (message only the consumers its signal places in group $k$ , which reaches a mass $\alpha$ of group $k$ ) or all (message every consumer, which reaches all of group $k$ ). Messages cost $c \ge 0$ each; in this mission $c = 0$ .

Stage 2. Having observed stage 1, firms set prices. For a stage-1 profile $s$ , if both firms target the same group, the shoppers $S(s) = r_1 r_2$ are the targeted consumers reached by both, where $r_i \in \{\alpha, 1\}$ is firm $i$ 's reach; firm $i$ 's loyal mass is $L_i(s) = r_i - S(s)$ . If they target different groups, $S(s) = 0$ and $L_i(s) = r_i$ . A consumer buys if the price does not exceed her valuation; shoppers buy from the cheaper firm and split equally on ties. Firm $i$ 's profit at prices $(p, q)$ is $p\,(L_i + S)$ , $p\,(L_i + S/2)$ or $p\,L_i$ according as $p < q$ , $p = q$ or $p > q$ , and $0$ if $p$ exceeds the valuation.

Firms may randomize prices: a price strategy is a probability measure on $[0,1]$ , and $U_i(s; \mu, \nu)$ is firm $i$ 's expected profit. A subgame perfect Nash equilibrium (SPNE) is a pure stage-1 profile $s^\star$ and a continuation that assigns to each of the sixteen stage-1 profiles a Nash equilibrium of its pricing subgame, such that no firm gains at stage 1 by changing its action. The sum of expected profits is $V_1(s^\star) + V_2(s^\star)$ .

Two cumulative distribution functions appear, on $[1-\alpha, 1)$ :

F(p) = \frac1\alpha - \frac{1-\alpha}{\alpha p}, \qquad H(p) = \frac{p - (1-\alpha)}{p},

both $0$ below $1-\alpha$ and $1$ from $1$ on, so $H$ has an atom $1 - \alpha$ at $p = 1$ . The thresholds are $\hat\alpha_1(A) = 1 - A$ and $\hat\alpha_2(A) = \bigl(1 + \sqrt{1 - 4A}\bigr)/2$ .

Formalization targets

Goal: Proposition 2 (p. 15)

For $0 < A < 1/4$ : $1/2 < \hat\alpha_2(A) < \hat\alpha_1(A)$ ; for every $\alpha \in (1/2, \hat\alpha_2(A))$ there is an SPNE with

\textstyle\sum_i E[\pi_i^\star(\alpha)] = 1 - \alpha^2,

which is strictly decreasing in $\alpha$ ; for every $\alpha \in (\hat\alpha_2(A), 1]$ there is an SPNE with sum $1 + A$ ; and $1 - \hat\alpha_2(A)^2 < 1 + A$ (the upward jump).

Milestones

Display (1), p. 11. If both firms target $b$ and message expected consumers, both pricing by $F$ is an equilibrium with profit $\alpha(1-\alpha)$ each.
Lemma 2 part 1, display (2), pp. 14–15. If firm 1 messages expected and firm 2 all group- $b$ consumers, $(F, H)$ is an equilibrium with profits $\alpha(1-\alpha)$ and $1-\alpha$ .
Bertrand, p. 13. If both message all group- $b$ consumers, pricing at $0$ is an equilibrium with zero profits.
Segmentation, pp. 9, 13. If firm 1 targets $a$ and firm 2 targets $b$ , prices $A$ and $1$ are an equilibrium.
Threshold, p. 14. For $\alpha \in (1/2, 1]$ and $A \le 1/4$ : $\alpha(1-\alpha) \ge A \iff \alpha \le \hat\alpha_2(A)$ .
Footnote 36, p. 15. $\hat\alpha_2(A) < \hat\alpha_1(A)$ for $0 < A < 1/4$ .
Lemma 2, pp. 14–15. The SPNE with $c = 0$ in both regimes.

Significance

The proposition shows that the non-monotonicity of industry profit in targeting precision, which the paper first derives for high advertising costs, does not rely on advertising being costly. With free messages a firm could inform every consumer, which is equivalent to perfect targeting; the equilibrium instead has one firm refrain from doing so, because the resulting asymmetry keeps prices high. For regulation of consumer data and for platforms selling targeting precision, the result says that better targeting technology can lower sellers' profits over a range of precisions and raise them discontinuously elsewhere.

The result is proved in the paper (main text and Online Appendix A). It has not been machine-checked. Formalizing it means verifying mixed-strategy equilibria of a pricing game with a continuum of prices, including an atom at the valuation, in all sixteen subgames of a two-stage game, a pattern that recurs throughout the clearinghouse and price-dispersion literature.

Difficulty

The stage-1 comparison is arithmetic once the stage-2 payoffs are known; the work is in the pricing subgames. Each mixed equilibrium must be verified against every real price deviation, which needs the expected profit at price $p$ against a distribution with an atom, where the tie-splitting rule matters exactly at the atom. Subgame perfection requires a pricing equilibrium in every off-path subgame too, including those where both firms target group $a$ (the group- $b$ equilibria rescaled by $A$ ) and the Bertrand subgames. The obvious shortcut, assigning each subgame its textbook profit as a number, is not available: the stage-1 values are defined as integrals against the equilibrium strategies.

Formalization scope

Lean 4 with Mathlib. The pricing game is a separate definition (a general clearinghouse duopoly with valuations $v_i$ , loyal masses $L_i$ and shopper mass $S$ ); the targeting game builds on it. Committed conventions:

price strategies are probability measures on $\mathbb R$ with no mass outside $[0,1]$ ; expected profits are Bochner integrals; deviations range over all real prices;
a consumer buys at a price equal to her valuation (the page says "lower than"; the atom of $H$ at $1$ needs the sale);
targeting is in pure strategies (the paper's footnotes 26 and 34); the message scope is binary, the paper's options (1) and (2); the no-message option and the continuous message masses of the appendix proof are not modelled;
cdfs are Mathlib's ProbabilityTheory.cdf; statements about all measures with cdf $F$ also assert that such a measure exists;
the paper's "firm $i$ / firm $-i$ " are firm 1 (the expected-only firm, and the group- $a$ firm) and firm 2;
for $A > 1/4$ Lean's square root gives $\hat\alpha_2(A) = 1/2$ , the reading Lemma 2's range $(\max\{1/2, \hat\alpha_2(A)\}, 1]$ needs;
pricing-stage profits exclude the sunk message cost; the goal and Lemma 2 are at $c = 0$ .

The goal is Proposition 2 in existence form: it asserts an SPNE with the stated total. The paper exhibits that equilibrium; uniqueness of equilibrium payoffs is not claimed. Propositions 1 and 3 (high and general costs) are not posed. Trivializing encodings are excluded by construction: stage-1 values are integrals of the continuation strategies, not a table; the equilibrium condition quantifies over all sixteen subgames, not just the path; strategies are probability measures on $[0,1]$ , so no integral takes a junk value; and statements quantifying over measures with a given cdf include the existence of one.

Contributions welcome: integration of the tie-split profit against cdfs with atoms, a reusable lemma that a clearinghouse with given loyal and shopper masses has an equilibrium, and the rescaling of equilibria by the valuation.

Selected references

H. Karle and M. Reisinger, Imperfect Targeting and Advertising Strategies, SSRN working paper 4626076, March 25, 2024. https://ssrn.com/abstract=4626076 ; published in Management Science, 2024. https://doi.org/10.1287/mnsc.2023.03632
H. R. Varian, A Model of Sales, American Economic Review 70(4), 651–659, 1980. https://www.jstor.org/stable/1805229
C. Narasimhan, Competitive Promotional Strategies, Journal of Business 61(4), 427–449, 1988. https://doi.org/10.1086/296442

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Completed

Economics Operations Research Optimization·Captain: mikedeng1

Asset Pricing with Costly Short Sales 1: The Lending Agent Charges Half the Pessimist's Frictionless Sharpe Ratio, and Lending Is Active Exactly When Her Net Sharpe Ratio Is NegativeResearch Paper

Why the price of borrowing a stock matters

To sell a stock short, an investor must first borrow it, and the owner who lends it charges a lending fee. Securities lending is a large market, and the fee varies across stocks: D'Avolio (2002) documents that shorting costs are positively related to the dispersion of investors' beliefs. Short-sale constraints have a long history in asset pricing (Miller 1977; Harrison and Kreps 1978; Jones and Lamont 2002), but most models either forbid short sales outright or take the fee as an exogenous parameter. Neither explains where the fee comes from, nor when a lending market is active at all.

Evgeniou, Hugonnier and Prieto build a dynamic general equilibrium model in which the fee is endogenous: two investors with logarithmic utility disagree about the expected return of a risky asset, and a lending agent, acting for the asset owners, sets the fee charged to short sellers to maximize the owners' fee revenue. Their §3 derives the equilibrium shorting cost in closed form at every state of the economy. This mission formalizes that derivation.

Setting

The economy has a riskless asset and two risky assets with strictly positive volatilities $\sigma_1,\sigma_2$ . Asset 1 can be shorted at a cost; asset 2 cannot be shorted at all. There are two investors, an optimist $o$ and a pessimist $p$ , with wealths $W^{(o)},W^{(p)}>0$ . Each investor $a$ perceives market prices of risk (Sharpe ratios) $\theta^{(a)}_1,\theta^{(a)}_2$ , and the optimist is more optimistic about asset 1: $\theta^{(p)}_1\le\theta^{(o)}_1$ , the disagreement $\Delta\ge0$ .

A short position in asset 1 pays a shorting cost $\Phi$ per dollar shorted, and a long position earns a lending yield $\Gamma$ . Both are measured per unit of volatility of asset 1: $\phi=\Phi/\sigma_1$ and $\gamma=\Gamma/\sigma_1$ . A position $m$ in asset 1 earns fees at the rate

\Lambda(m;\Phi,\Gamma)=m^+\Gamma-m^-\Phi ,

with $m^+=\max\{m,0\}$ and $m^-=\max\{-m,0\}$ . Investor $a$ chooses wealth fractions $p=(p_1,p_2)$ with $p_2\ge0$ to solve the mean-variance problem

\max_{p\in\mathbb R\times\mathbb R_+}\Big\{\Lambda(p_1;\Phi,\Gamma)+p_1\sigma_1\theta^{(a)}_1+p_2\sigma_2\theta^{(a)}_2-\tfrac12(p_1\sigma_1+p_2\sigma_2)^2\Big\},

which is what logarithmic utility reduces to at each instant. The lending agent's revenue per unit of volatility at fee $y$ is

R(y)=\sum_a y\,\big(\theta^{(a)}_1+y\big)^-\,W^{(a)} .

An equilibrium at a state consists of these quantities such that: the no-dominance condition $\theta^{(a)}_2=\theta^{(a)}_1+\gamma$ holds for both investors (equation (7)); each investor's portfolio is optimal; asset 1, in positive supply $S_1>0$ , clears; the lending market clears (fees paid by short sellers equal fees received by long holders); and $\phi$ is the least maximizer of $R$ over $y\ge\gamma$ . The inactive set is $\mathcal L=\{\gamma=0\}$ and the active set $\mathcal S$ is its complement.

Formalization targets

Goal: equations (15)–(17)

At every equilibrium state,

\gamma=0\iff\theta^{(p)}_2\ge0,\qquad \phi=-\tfrac12\,\theta^{(p)}_1\,\mathbb 1_{\{\theta^{(p)}_2<0\}} .

Lending is active exactly where the pessimist perceives a strictly negative net Sharpe ratio on long positions, and there the lending agent charges half the Sharpe ratio $-\theta^{(p)}_1$ the pessimist could earn by shorting absent frictions.

Milestones, in attack order

Proposition 1 (p. 17): the optimal portfolios of the mean-variance problem under (7) and $\phi\ge\gamma$ .
(11) (p. 18): the optimist is long, $\theta^{(o)}_2=\theta^{(o)}_1+\gamma>0$ .
(14) (p. 18): $\gamma\sigma_1S_1=(\phi-\gamma)(\theta^{(p)}_1+\phi)^-W^{(p)}$ .
(12)–(13) (p. 18): the maximizers of the revenue over $y\ge\gamma$ , and their least element $\max\{\gamma,-\tfrac12\theta^{(p)}_1\}$ .

Two companion statements are also included: (18) (p. 20), that the pessimist is strictly short on $\mathcal S$ , and the fee flow $(\pi^{(p)}_1)^-\Phi=\tfrac14(\theta^{(p)}_1)^2W^{(p)}\mathbb 1_{\mathcal S}$ (p. 19).

Significance

The goal is the paper's description of the short market. It identifies the active set by a condition on one observable-in-principle quantity, the pessimist's net Sharpe ratio, and it gives the fee as a fixed fraction of the pessimist's frictionless Sharpe ratio. The rest of the paper rests on it: the one-asset closed forms of §4.1, the trading regions of §3.3, and the pricing results of Proposition 2 all start from (15)–(16). The fee-flow identity shows that the optimist captures a quarter of the pessimist's frictionless excess return, which is the upper bound Remark 1 attributes to any alternative fee-setting mechanism.

The result is proved in the paper, partly in the text and partly in Appendix A. Some steps are argued informally: (11) rests on the sentence "agents cannot be short simultaneously in equilibrium", and (14) on "substituting the optimal portfolios of Proposition 1 into the market clearing conditions". As far as is known, none of it has been machine-checked. A formalization makes each step explicit, including the degenerate states the text passes over. In checking the statements against each other, three printed slips were found and are corrected here (see Formalization scope).

Difficulty

The goal combines optimality, clearing and revenue maximization, and the obvious argument fails at its degenerate edge. On the inactive set the shorting cost equals the lending yield ( $\phi=\gamma=0$ ). There Proposition 1's characterization of optimal portfolios no longer holds: portfolios that short asset 1 to buy asset 2 are also optimal. So the step "substitute the optimal portfolios" is not available, and (14) must be obtained from the clearing conditions directly in that case. Similarly the revenue maximizer is not unique on the inactive set; the shorting cost is pinned down only by the paper's selection of the least maximizer, and (16) is false for any other selection. Finally, (11) requires an argument that the two investors cannot both be short while asset 1 is in positive supply. This uses optimality of both portfolios, not just of one.

Formalization scope

Everything is stated at one state $(\omega,t)$ of the paper's continuous-time economy, about real numbers. The continuous-time economy of §2 (Brownian filtrations, consumption streams, the dynamics of wealth) is the paper's motivation and is not formalized; the formalized statements are its instantaneous equilibrium conditions. The time index is dropped throughout, and the paper's "arbitrary process $x_t$ " in Proposition 1 becomes an arbitrary real number.

Conventions:

Rates are per unit of volatility; the dollar rates are $\Phi=\phi\sigma_1$ and $\Gamma=\gamma\sigma_1$ , never obtained by division.
$m^+$ and $m^-$ are written as $\max\{m,0\}$ and $\max\{-m,0\}$ ; indicators are if … then … else 0.
Dollar positions are $\pi^{(a)}_1=W^{(a)}p^{(a)}_1$ .
The standing assumptions are hypotheses of every equilibrium statement: $\sigma_1,\sigma_2>0$ (the standing assumption of §3), $S_1>0$ , $W^{(o)},W^{(p)}>0$ , and $\theta^{(p)}_1\le\theta^{(o)}_1$ .
The shorting-cost condition (4) is stated in its per-volatility form (10), over $y\ge\gamma$ , with the least maximizer selected as in (13). A single lending agent is built in.
Consumption clearing and asset-2 clearing are not part of the equilibrium predicate; no statement uses them, so leaving them out makes every statement more general.

Corrected slips of the page, each disclosed in the item concerned:

Proposition 1's "the optimal portfolio satisfies (9)" fails when $\phi=\gamma$ ; the converse direction is stated under $\phi>\gamma$ , and the constraint $0\le x\le\theta_2$ only where $\theta_2\ge0$ .
The indicator in (12) should be $\mathbb 1_{\{\theta^{(p)}_1+\gamma\ge0\}}$ , not $\mathbb 1_{\{\theta^{(p)}_1\ge0\}}$ ; the least maximizer (13) is unaffected.
(18) prints " $\neq\emptyset$ "; its proof shows the set is empty, which is what is stated.

The results (11), (13) and (14) are not built into the equilibrium predicate. With them assumed, the goal would reduce to a four-line computation; they are separate items to be proved. The definitions are short and self-contained, needing only real arithmetic from Mathlib. Proofs of the milestones, and proofs of the goal that use the milestones, are welcome.

Selected references

T. Evgeniou, J. Hugonnier, R. Prieto, Asset Pricing with Costly Short Sales, Swiss Finance Institute Research Paper N°22-21, version of September 22, 2022 — the source of this mission. Published as J. Hugonnier, R. Prieto, Asset Pricing with Costly Short Sales, Management Science (2024), https://doi.org/10.1287/mnsc.2023.01887, whose numbering is not used here.
G. D'Avolio, The market for borrowing stock, Journal of Financial Economics 66 (2002), 271–306.
C. Jones, O. Lamont, Short-sale constraints and stock returns, Journal of Financial Economics 66 (2002).
R. Radner, Existence of equilibrium of plans, prices and price expectations in a sequence of markets, Econometrica 40 (2) (1972), 289–303.

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Functional Analysis·Captain: savarin

Sharp diagonal Hlawka constants: lower the cutoff to 50Open Problem

The 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. The question is how large a comparison constant is needed to make this inequality hold.

This mission asks whether the best possible constant for complex diagonal matrices, already proved in Lean for every real $p\ge62$ , also holds for every real $p\ge50$ . This is an open problem: no proof is known, and the technique that reached 62 cannot go further. The constant is the one from the foundation mission: the largest comparison constant required by the cyclic family of three $3\times3$ diagonal matrices. Because the cutoff-62 mission already covers every $p\ge62$ , the new work is the range from 50 to 62.

The cutoff came down from 256 to 62 across nine missions. The step from 90 to 80 used a weighted pair-curvature and confinement estimate introduced by moona3k, and further refinement pushed it from 80 through 70 to 62. The confinement method is now exhausted: at $p=61$ , every confinement radius whose tangent model stays positive already makes the budget-7 box factor larger than 1. Reaching 50 requires a fundamentally new idea. The goal theorem below gives the exact statement.

This is an entry in the sharp diagonal Hlawka campaign, which asks for the smallest cutoff at which the same formula holds. Any proof for a cutoff of 50 or lower also settles this mission.

The broader question of optimal constants for Schatten norms appears in Audenaert and Kittaneh's Problem 7. Extending the sharp diagonal constant to general matrices is a separate challenge.

References

K. M. R. Audenaert and F. Kittaneh, Problems and Conjectures in Matrix and Operator Inequalities, arXiv preprint, 2012, §8.2, Problem 7. arXiv:1201.5232
Ezzeri Esa, Hlawka–Schatten inequalities: sharp diagonal construction, Lean source repository, 2026, revision 79aa498bfcf7b22bd91d771fb32ec278e2d4704b. Source library
Ezzeri Esa and project contributors, The cyclic bound for every real p ≥ 90, research note with appendices and exact certificates, 2026. Research note

Established results on Prove2Me

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Functional Analysis Mathematical Physics·Captain: Lucas

Heathcote 1990: A Normalized State Outside the Domains of Position and MomentumResearch Paper

Motivation

Quantum mechanics represents the states of a system by unit vectors (normed rays) of a complex Hilbert space $\mathcal H$ and its observables by self-adjoint operators on $\mathcal H$ . The most important observables — position, momentum, energy — are unbounded. A. Heathcote's paper Unbounded Operators and the Incompleteness of Quantum Mechanics (Philosophy of Science 57, 1990, 523–534) draws attention to a mathematical fact that is routinely glossed over in physical texts: an unbounded self-adjoint operator cannot be defined on all of $\mathcal H$ , and there are concrete normalized wave functions on which the position and momentum operators do not act at all. The paper uses this to argue that the standard postulates of quantum mechanics are, in a precise sense, incomplete.

This mission isolates the mathematical content of that argument, which is a clean and checkable piece of functional analysis on $L^2(\mathbb R)$ .

Setting

Let $\mathcal H$ be a complex Hilbert space. A (partially defined) linear operator $A$ has a domain $D(A)\subseteq\mathcal H$ ; it is bounded if there is $k\in\mathbb R$ with $\|A\psi\|\le k\|\psi\|$ for all $\psi\in D(A)$ , and unbounded otherwise. It is self-adjoint if it equals its Hilbert adjoint, $A=A^*$ , which includes $D(A)=D(A^*)$ .

On $\mathcal H=L^2(\mathbb R)$ :

The position operator acts by $(Q\psi)(x)=x\,\psi(x)$ on

D(Q)=\Big\{\psi\in L^2(\mathbb R): \int_{\mathbb R}x^2|\psi(x)|^2\,dx<\infty\Big\}\qquad\text{(eq. (3))}.

The momentum operator $P\psi=\hbar i^{-1}\,d\psi/dx$ has, in the paper's description, domain $D(P)$ consisting of the absolutely continuous functions in $L^2(\mathbb R)$ .
Heathcote's wave function (eq. (4)) is

\psi(x)=\begin{cases}x^{-1}, & x\ge 1,\\ 0, & x<1.\end{cases}

Formalization targets

Goal

There is a unit vector of $L^2(\mathbb R)$ , namely the class of $\psi$ , lying in neither domain:

\exists\,\varphi\in L^2(\mathbb R):\quad \varphi=\psi\ \text{a.e.},\qquad \|\varphi\|=1,\qquad \varphi\notin D(Q),\qquad \varphi\notin D(P).

Milestones

Hellinger–Toeplitz (p. 2, footnote 1): a symmetric operator defined on all of a Hilbert space is bounded (an existing platform theorem).
Unbounded self-adjoint operators (p. 2, footnote 2): if $A=A^*$ is unbounded then $D(A)$ is dense and $D(A)\neq\mathcal H$ .
Normalization (p. 5): $\psi\in L^2(\mathbb R)$ and $\int|\psi|^2=\int_1^\infty x^{-2}dx=1$ .
Not in $D(Q)$ (p. 5): $\int x^2|\psi(x)|^2dx=\infty$ .
Not in $D(P)$ (pp. 5–6): no a.e.-representative of $\psi$ is absolutely continuous on every compact interval.
Infinite position expectation (p. 9): $\int|x|\,|\psi(x)|^2dx=\int_1^\infty x^{-1}dx=\infty$ .

Significance

The result. The goal shows that the two postulates "every normed ray is a state" and "observables are self-adjoint operators" force the existence of states on which the position operator (and, independently, the momentum operator) is undefined. Milestone 5 is what the paper uses to separate this phenomenon from the canonical commutation relations: the failure of $\psi\in D(Q)$ is not explained by $\psi$ being "sharp in momentum", because $\psi$ is not even in $D(P)$ . Milestone 6 records the physical reading: the mean position of this normalized state is infinite.

Formalizing it. The individual facts are standard and elementary; none of them is new. What the mission produces is a machine-checked version of the paper's argument stated against the actual $L^2$ space and Mathlib's notion of unbounded operators and adjoints, with the domain conventions made explicit.

Difficulty

The calculus is easy; the subtleties are in the bookkeeping. Elements of $L^2(\mathbb R)$ are almost-everywhere classes, so " $\psi$ is discontinuous at $1$ " does not by itself show $\psi\notin D(P)$ : one must rule out every a.e.-modification of $\psi$ being absolutely continuous. Similarly, the Hellinger–Toeplitz argument gives $D(A)\neq\mathcal H$ but not density, which comes from the adjoint being a genuine operator.

Formalization scope

Operators are Mathlib's partially defined linear maps H →ₗ.[ℂ] H; self-adjointness is IsSelfAdjoint with respect to the adjoint A†, over a complex complete inner product space.
$L^2(\mathbb R)$ is Lp ℂ 2 volume on ℝ; $D(Q)$ and $D(P)$ are subsets of it. Integrals of non-negative quantities are Lebesgue integrals valued in $[0,\infty]$ , so " $=\infty$ " is literal and not a junk value.
$D(P)$ follows the paper (absolutely continuous elements, i.e. having a representative absolutely continuous on every $[a,b]$ ) and omits the standard requirement $\psi'\in L^2$ . This set contains the standard domain, so non-membership is a stronger statement.
The paper's further claim that $\psi\notin D(H)$ for a Hamiltonian $H=P^2/2m+V$ is stated there without proof and for an unspecified potential $V$ , and is not included.
Trivialization is excluded: the goal pins $\varphi$ to the explicit function $\psi$ and requires $\|\varphi\|=1$ .

Selected references

A. Heathcote, Unbounded Operators and the Incompleteness of Quantum Mechanics, Philosophy of Science 57 (1990), 523–534.
M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, revised ed., Academic Press, 1980.

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Machine Learning Quantum Information·Captain: Goku

Quantum Algorithms and Learning Theory — Part II 7: Identifying a codeword state needs Ω(k) copiesTextbook

Motivation

This mission develops Chapter 7 of Quantum Algorithms and Learning Theory — Part II by Srinivasan Arunachalam. 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

For a matrix $M\in\mathbb F_2^{n\times k}$ of rank $k$ (the generator matrix of an $[n,k,d]_2$ linear code) and $x\in\{0,1\}^k$ define the codeword state

|\psi_x\rangle=\frac{1}{\sqrt n}\sum_{i\in[n]}|i,(Mx)_i\rangle .

For every rate $c_1>0$ there is a constant $c_0>0$ such that the following holds for all $n,k$ and all rank- $k$ matrices $M\in\mathbb F_2^{n\times k}$ with $k\ge c_1 n$ : if a measurement (POVM $\{N_y\}_{y\in\{0,1\}^k}$ ) on $T$ copies identifies $x$ from $|\psi_x\rangle^{\otimes T}$ , for $x$ drawn uniformly from $\{0,1\}^k$ , with average success probability

\frac{1}{2^k}\sum_{x\in\{0,1\}^k}\operatorname{Tr}\!\big(N_x\,|\psi_x\rangle\langle\psi_x|^{\otimes T}\big)\ \ge\ \frac45,

then $T\ge c_0\,k$ .

This improves the Holevo-type bound $\Omega(k/\log n)$ to the optimal $\Omega(k)$ for constant-rate codes, showing that codeword fingerprint states cannot be identified from few copies.

Formalization Note " $k=\Omega(n)$ " is read as a fixed rate $k\ge c_1 n$ , and the constant of $\Omega(k)$ may depend on $c_1$ only. Identifying the state means outputting $x$ , which is equivalent since $x\mapsto Mx$ is injective for rank $k$ . The code distance $d$ plays no role and is not a hypothesis.

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 7.3.1: Barnum–Knill: the pretty good measurement is at most quadratically worse than optimal.
Claim 7.3.5: Fourier coefficients of f composed with a linear map over F2.
Claim 7.3.6: The maximum of (c/√t)^t on [1, c²] is e^{c²/(2e)} (for c ≥ √e).
Fact 7.3.7: Binary entropy: H(ε) = O(ε log 1/ε) and 1 − H(1/2+ε) ≤ 2ε²/ln 2 + O(ε⁴).
Fact 7.3.8: Binomial coefficients and their partial sums are at most 2^{nH(·/n)}.
Fact 7.3.9: Helstrom bound for guessing a uniform bit from |ψ_b⟩, and its corollary.
Fact 7.3.10: Subadditivity of von Neumann entropy.
Lemma 7.4.1: Quantum PAC learners of a non-trivial class need Ω(log(1/δ)/ε) examples.
Lemma 7.4.2: Quantum agnostic learners of a non-trivial class need Ω(log(1/δ)/ε²) examples.
Theorem 7.4.3: Classical PAC sample complexity is Ω(d/ε + log(1/δ)/ε).
Theorem 7.4.4: Classical agnostic sample complexity is Ω(d/ε² + log(1/δ)/ε²).
Theorem 7.4.5: ε-average agnostic learners need (d/ε²)(1/62 − log(2d+2)/(4d)) samples.
Theorem 7.4.6: Quantum PAC sample complexity lower bound (information-theoretic).
Theorem 7.4.7: Quantum agnostic sample complexity lower bound (information-theoretic).
Claim 7.5.2: Eigenvalues of an XOR-convolution matrix are scaled Fourier coefficients.
Claim 7.5.3: Diagonal of √A in terms of Fourier coefficients of f.
Lemma 7.5.4: Fourier coefficients of (1 − β|z|/m)^T are nonnegative and small.
Claim 7.5.5: Counting index sequences whose basis vectors sum to S.
Theorem 7.5.6: Optimal quantum PAC sample complexity lower bound.
Lemma 7.5.7: PGM success bound for the PAC hard ensemble.
Theorem 7.5.8: Optimal quantum agnostic sample complexity lower bound.
Lemma 7.5.9: PGM success bound for the agnostic hard ensemble.
Theorem 7.6.1: Quantum PAC lower bound under random classification noise.
Theorem 7.6.2: Identifying a codeword state needs Ω(k) copies.

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

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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Optimization Quantum Information·Captain: Goku

Quantum algorithms for optimizers 1: Schmidt decompositionTextbook

Motivation

This mission develops Chapter 1 of Quantum algorithms for optimizers by Giacomo Nannicini. The chapter's selected results describe the mathematical guarantees below.

Setting and conventions

Chapter goal

Let $AB$ be a register made of a register $A$ of $m_a$ qubits followed by a register $B$ of $m_b$ qubits. Let $|\psi\rangle$ be a pure state of $AB$ , that is, a unit vector. Then there exist a finite index set $\{1,\dots,r\}$ , an orthonormal family $|\phi^A_1\rangle,\dots,|\phi^A_r\rangle$ of states of $A$ , an orthonormal family $|\phi^B_1\rangle,\dots,|\phi^B_r\rangle$ of states of $B$ , and nonnegative reals $\lambda_1,\dots,\lambda_r$ with $\sum_j\lambda_j^2=1$ , such that

|\psi\rangle=\sum_{j=1}^r\lambda_j\,|\phi^A_j\rangle|\phi^B_j\rangle .

This is the singular value decomposition rephrased for bipartite pure states. Classical simulation algorithms for quantum circuits use it.

Formalization Note The source leaves the index set unspecified; here it is Fin r for an existentially chosen r.

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 1.4: Algebraic properties of the tensor (Kronecker) product.
Proposition 1.19: Basis states are exactly tensor products of single-qubit basis states.
Proposition 1.27: Measuring all qubits one at a time, in any order, yields j with probability |α_j|².
Theorem 1.32: No-cloning principle.
Proposition 1.37: Pauli matrices: a Hermitian basis of C^{2×2} with XYZ = iI.
Proposition 1.38: Hadamard on every qubit of |0⟩ gives the uniform superposition.
Proposition 1.41: Three CX gates implement SWAP.
Proposition 1.46: Errors of a product of unitaries add up.
Proposition 1.49: Close states give close measurement distributions.
Proposition 1.50: Principle of implicit measurement.
Theorem 1.60: Density matrices are exactly the unit-trace positive semidefinite matrices.
Proposition 1.67: The reduced density matrix gives the correct outcome probabilities on register A.
Theorem 1.70: Schmidt decomposition.

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 4: Quantum minimum finding (Dürr–Høyer), with explicit bit width, interruption and gate modelTextbook

Motivation

This mission develops Chapter 4 of Quantum algorithms for optimizers by Giacomo Nannicini. The chapter's selected results describe the mathematical guarantees below.

Setting and conventions

Chapter goal

Let $f : \{0,1\}^n \to \mathbb{Z}$ take values of bit width $b$ , i.e. $|f(j)| < 2^b$ for all $j$ , and let $U_f : |j\rangle|k\rangle \mapsto |j\rangle|k \oplus f(j)\rangle$ evaluate $f$ in binary. Algorithm 2 with budget $T$ starts from a uniformly random incumbent $\ell$ and repeatedly runs Algorithm 1 (search with an unknown number of marked items) with the marking unitary $U_m : |j\rangle|y\rangle \mapsto |j\rangle|y \oplus \mathbb{I}(f(j) < f(\ell))\rangle$ , replacing $\ell$ by the returned index $k$ whenever $f(k) < f(\ell)$ ; a call of Algorithm 1 that would exceed the budget is interrupted at the $T$ -th application of $U_f$ , and the algorithm then returns $\ell$ . Each application of $U_m$ counts as one application of $U_f$ and is charged $3n + 2b + 1$ additional gates; each round of Algorithm 1 with $m$ qubits of precision is charged $m + m(m+1)/2$ further gates (Hadamards and the inverse QFT).

There are absolute constants $c, C > 0$ such that for every $n$ , $b$ , every such $f$ and every $\delta \in (0,1)$ the following holds. Run Algorithm 2 independently $k = \lceil \log_3(1/\delta) \rceil$ times with budget $T = \lceil c\sqrt{2^n} \rceil$ and output an $f$ -minimal one of the $k$ returned indices. Then:

with probability at least $1 - \delta$ the output is a global minimizer of $f$ ;
on every outcome, the total number of applications of $U_f$ (including the $k$ evaluations used to select the output) is at most

C\sqrt{2^n}\,\bigl(1 + \log(1/\delta)\bigr),

and the number of additional gates is at most $C\sqrt{2^n}\,(n + b + 1)\,(1 + \log(1/\delta))$ .

This is the quadratic quantum speedup for unstructured minimization, the basis of the optimization applications of Grover search.

Formalization Note The source's $\tilde O(\sqrt{2^n})$ gate bound is stated with the suppressed polylogarithmic factors $n + b + 1$ and $1 + \log(1/\delta)$ explicit, the interruption rule for the budget is made explicit, and $T$ , $k$ are the choices of the source's proof. Algorithm 1 is the shared formalization of the earlier corollary; the outcome law is a weight on (returned index, applications of $U_f$ , gates).

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 4.6: Amplitude amplification: overlap 2/3 with the good state using O(1/√p) applications of S and R.
Corollary 4.9: Search with a known number of marked elements in O(√(2^n/|M|)) oracle calls.
Corollary 4.10: Finding all marked elements with O(√(|M| 2^n)) oracle calls.
Lemma 4.11: Oblivious amplitude amplification: the partner state Ψ⊥ is orthogonal to |0⟩|ψ⟩ and has no flag-0 support.
Theorem 4.12: Oblivious amplitude amplification: each iteration rotates by 2θ.
Proposition 4.15: Eigenstates of the Grover operator in amplitude estimation.
Proposition 4.19: Angle error to probability error for sin².
Proposition 4.21: Error bound of amplitude estimation with m qubits.
Corollary 4.22: Amplitude estimation to additive error ε with O(log 1/ε) qubits and O(1/ε) queries.
Lemma 4.28: Near-orthogonality of the two amplitude-estimation output states.
Theorem 4.29: Quantum search without knowing the number of marked items: O(1/θ) expected queries.
Corollary 4.30: Search without knowing |M| with O(√(2^n/|M|)) expected oracle calls.
Theorem 4.31: Quantum minimum finding (Dürr–Høyer), with explicit bit width, interruption and gate 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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Number Theory·Captain: Lucas

Class Number Formula: Continuation of the Dedekind Zeta Function and Dirichlet's FormulaTextbook

Motivation

The class number formula connects the analytic behaviour of the Dedekind zeta function $\zeta_K(s)$ of a number field $K$ at $s = 1$ with the basic arithmetic invariants of $K$ : its class number, regulator, number of roots of unity and discriminant. It is the main tool for computing and bounding class numbers analytically, and its quadratic case, Dirichlet's class number formula, came out of Dirichlet's proof of his theorem on primes in arithmetic progressions.

Timeline.

1801: Gauss appears to have known the quadratic case, stated for binary quadratic forms (MathOverflow 109330).
1839: Dirichlet publishes a proof of the formula for quadratic discriminants, in the language of binary quadratic forms.
Later in the 19th century: Dedekind introduces $\zeta_K$ for an arbitrary number field, and the residue formula is established in general.

Setting

Let $K$ be a number field of degree $n = r_1 + 2r_2$ , where $r_1$ is the number of real embeddings and $2r_2$ the number of complex embeddings of $K$ . Write $h_K$ for the class number (the order of the ideal class group), $\mathrm{Reg}_K$ for the regulator, $w_K$ for the number of roots of unity in $K$ , and $D_K$ for the discriminant of $K/\mathbb Q$ . The Dedekind zeta function is the Dirichlet series

\zeta_K(s) = \sum_{n \ge 1} \frac{a_n}{n^s}, \qquad a_n = \#\{\mathfrak a \subseteq \mathcal O_K : N(\mathfrak a) = n\}.

For the quadratic case, let $d$ be a fundamental discriminant, let $h(d)$ be the number of proper ( $\mathrm{SL}_2(\mathbb Z)$ ) equivalence classes of binary quadratic forms $ax^2 + bxy + cy^2$ of discriminant $b^2 - 4ac = d$ (positive definite ones if $d < 0$ ), and let $\chi(m) = \left(\frac{d}{m}\right)$ be the Kronecker symbol, a Dirichlet character modulo $|d|$ with $L$ -function $L(s,\chi)$ . For $d < 0$ let $w = 2, 4, 6$ according as $d < -4$ , $d = -4$ , $d = -3$ . For $d > 0$ let $t, u > 0$ solve $t^2 - du^2 = 4$ with $u$ minimal, and put $\varepsilon = \frac12(t + u\sqrt d)$ .

Formalization targets

Goal: the class number formula

\zeta_K(s) \text{ converges absolutely for } \operatorname{Re}(s) > 1, \text{ and extends meromorphically to } \mathbb C \text{ with a single simple pole at } s=1,

\lim_{s\to 1}(s-1)\,\zeta_K(s) = \frac{2^{r_1}(2\pi)^{r_2}\,\mathrm{Reg}_K\,h_K}{w_K\sqrt{|D_K|}} .

Milestones

Dirichlet's unit theorem, and the reduction of the residue to the average of the coefficients of $\zeta_K$ .
The case $K = \mathbb Q(i)$ : coefficients as sums of two squares, the lattice-point count in a quarter circle, and the residue $\pi/4$ .
The general ideal count $\#\{\mathfrak a : N(\mathfrak a) \le N\} \sim \rho_K N$ , and the residue at $s = 1$ approached from the right.
Dirichlet's formula:

h(d) = \frac{w\sqrt{|d|}}{2\pi}\,L(1,\chi)\ \ (d<0), \qquad h(d) = \frac{\sqrt d}{\ln\varepsilon}\,L(1,\chi)\ \ (d>0),

with the supporting facts that $\chi$ is a Dirichlet character, that $\zeta_K(s) = \zeta(s)L(s,\chi)$ for a quadratic field, that the Pell equation $t^2 - du^2 = 4$ is solvable, and that $\varepsilon$ is a fundamental unit or its square. 5. Finite forms of $L(1,\chi)$ for prime conductor $q$ :

L(1,\chi) = -\frac{\pi}{q^{3/2}}\sum_{m=1}^{q-1} m\left(\frac mq\right)\ (q\equiv3\bmod 4), \qquad L(1,\chi) = -\frac{1}{q^{1/2}}\sum_{m=1}^{q-1}\left(\frac mq\right)\ln\sin\frac{m\pi}{q}\ (q\equiv1\bmod 4).

Significance

The result. The formula makes $h_K \mathrm{Reg}_K$ computable from $L$ -values. In the quadratic case it gives closed formulas for class numbers, and it implies that $L(1,\chi) \neq 0$ for real characters, which is the hardest step in Dirichlet's theorem on primes in arithmetic progressions. Its refinements for cyclotomic fields relate class numbers to cyclotomic units and lead into Iwasawa theory.

Formalizing it. The result is classical. Mathlib already proves the residue as a one-sided real limit, $(s-1)\zeta_K(s) \to \rho_K$ as $s \to 1^+$ , and that statement is available on the platform as a proved theorem. What is not formalized is the meromorphic continuation of $\zeta_K$ to all of $\mathbb C$ with a single simple pole, which is what the goal requires. Dirichlet's formula in terms of binary quadratic forms, its supporting facts, and the finite forms of $L(1,\chi)$ are not formalized either.

Difficulty

The residue formula follows from counting ideals of bounded norm, but that only controls $\zeta_K$ for real $s > 1$ . Continuing $\zeta_K$ to all of $\mathbb C$ needs a separate analytic input, such as theta functions or Hecke's integral representation with a functional equation, or a decomposition into Hecke $L$ -functions. Mathlib has such continuations for the Riemann zeta function and Dirichlet $L$ -functions, but not for $\zeta_K$ in general. For Dirichlet's formula, the hard part is connecting classes of binary quadratic forms (proper equivalence, positive definiteness for $d<0$ ) with ideal classes of the quadratic order, keeping track of the unit and automorph conventions that decide the constants $w$ and $\varepsilon$ .

Formalization scope

The goal uses Mathlib's NumberField.dedekindZeta (the Dirichlet series, meaningful for $\operatorname{Re}(s) > 1$ ) together with nrRealPlaces, nrComplexPlaces, regulator, classNumber, torsionOrder and discr. The continuation is stated as an entire function $G$ with $\zeta_K(s) = G(s)/(s-1)$ on $\operatorname{Re}(s)>1$ and $G(1)$ equal to the residue. Since the residue is positive, this rules out the trivial reading where $\zeta_K$ is continued with no pole.
$L(s,\chi)$ is DirichletCharacter.LFunction, the analytic continuation, so $L(1,\chi)$ is its value at $s=1$ and not the (conditionally convergent) series. The character is a hypothesis agreeing with the Kronecker symbol, and a separate milestone shows that such a character exists.
$h(d)$ counts $\mathrm{SL}_2(\mathbb Z)$ -classes of integral forms of discriminant $d$ , positive definite when $d < 0$ (Davenport's convention, which the source follows).
Corrections to the source. Two displayed constants in the source are off by a factor $2$ and are corrected here: the $d>0$ case of Dirichlet's formula uses $\ln\varepsilon$ in place of $2\ln\varepsilon$ (as in Davenport, Ch. 6), and the $q \equiv 1 \pmod 4$ finite form uses $-q^{-1/2}$ in place of $-\frac12 q^{-1/2}$ . Each affected milestone checks the case $d = q = 5$ explicitly.
Not included: the sections on Galois and abelian extensions (Artin $L$ -functions, Kummer's cyclotomic formulas). Mathlib has no Artin $L$ -functions, and the source states those results only informally.
Reusable outputs: the Kronecker symbol, binary quadratic forms with proper equivalence, the Pell unit $\varepsilon$ , and the continuation of $\zeta_K$ .

Selected references

Wikipedia, "Class number formula", revision 1366411311. https://en.wikipedia.org/w/index.php?title=Class_number_formula&oldid=1366411311
H. Davenport, Multiplicative Number Theory, 3rd ed., rev. H. L. Montgomery, Graduate Texts in Mathematics 74, Springer, 2000, Ch. 6, pp. 43–53. https://books.google.com/books?id=U91lsCaJJmsC
W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, 2nd ed., Springer/PWN, 1990, pp. 324–355. https://archive.org/details/elementaryanalyt0000nark
"Did Gauss know Dirichlet's class number formula in 1801?", MathOverflow, 2012. https://mathoverflow.net/q/109330

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Number Theory·Captain: Lucas

Ramanujan–Petersson Conjecture (Wikipedia): Coefficient Bounds for Holomorphic Cusp FormsTextbook

Motivation

The Fourier coefficients of modular forms are among the most studied arithmetic functions in number theory. In 1916 Srinivasa Ramanujan studied the coefficients $\tau(n)$ of the discriminant modular form $\Delta$ and conjectured three properties: multiplicativity, a three-term recursion on prime powers, and the bound $|\tau(p)|\le 2p^{11/2}$ for primes $p$ (Ramanujan 1916). Hans Petersson extended the size bound to the coefficients of holomorphic cusp forms for congruence subgroups (Petersson 1930). The resulting Ramanujan–Petersson conjecture for holomorphic cusp forms is now a theorem of Deligne (weight $k\ge 2$ ) and Deligne–Serre (weight $k=1$ ). Its analogues for Maass forms and for general automorphic representations remain open.

Timeline (as recorded in the source article, Wikipedia: Ramanujan–Petersson conjecture):

1916: Ramanujan conjectures that $\tau$ is multiplicative, that it satisfies the prime-power recursion, and that $|\tau(p)|\le 2p^{11/2}$ .
1917: Mordell proves the first two statements, using what are now called Hecke operators.
1930: Petersson formulates the coefficient bound for holomorphic cusp forms for congruence subgroups.
1937: Hecke extends Mordell's method to holomorphic cusp forms in general.
1971: Deligne, building on Kuga, Sato, Shimura and Ihara, reduces the conjecture for weight $k\ge 2$ to the Weil conjectures via the Eichler–Shimura isomorphism (Deligne 1971).
1974: Deligne proves the Riemann hypothesis part of the Weil conjectures, which settles the case $k\ge 2$ and in particular $|\tau(p)|\le 2p^{11/2}$ (Deligne 1974). Deligne and Serre treat weight $k=1$ (Deligne–Serre 1974).

Setting

Let $\mathbb H=\{z\in\mathbb C:\operatorname{Im}z>0\}$ and $q=e^{2\pi i z}$ . The Dedekind eta function is $\eta(z)=q^{1/24}\prod_{n\ge1}(1-q^n)$ . The discriminant modular form is

\Delta(z)=q\prod_{n>0}(1-q^n)^{24}=\eta(z)^{24}.

It is a holomorphic cusp form of weight $12$ and level $1$ . Ramanujan's tau function $\tau:\mathbb N\to\mathbb C$ is defined by the Fourier expansion

\Delta(z)=\sum_{n\ge1}\tau(n)q^n=q-24q^2+252q^3-1472q^4+4830q^5-\cdots.

Its Ramanujan $L$ -function is the Dirichlet series $L(s,\tau)=\sum_{n\ge1}\tau(n)n^{-s}$ .

More generally, let $\Gamma\subseteq SL(2,\mathbb Z)$ be a congruence subgroup, meaning that $\Gamma$ contains the principal congruence subgroup $\Gamma(N)$ for some $N\ge1$ . A holomorphic cusp form of weight $k$ for $\Gamma$ is a holomorphic function $f:\mathbb H\to\mathbb C$ with $f(\gamma z)=(cz+d)^k f(z)$ for every $\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\Gamma$ , which vanishes at every cusp. Such an $f$ is periodic with some least period $h>0$ , and it has a Fourier expansion $f(z)=\sum_{n\ge1}a_n q_h^n$ with $q_h=e^{2\pi i z/h}$ . When $\Gamma$ contains $z\mapsto z+1$ (for example $\Gamma_0(N)$ , $\Gamma_1(N)$ , or $SL(2,\mathbb Z)$ itself), $h=1$ and $q_h=q$ .

Formalization targets

Goal: the Ramanujan–Petersson conjecture for holomorphic cusp forms

For every congruence subgroup $\Gamma$ , every weight $k\ge1$ , every cusp form $f$ of weight $k$ for $\Gamma$ with coefficients $a_n$ , and every $\varepsilon>0$ ,

a_n=O_\varepsilon\!\left(n^{(k-1)/2+\varepsilon}\right).

The goal splits by weight into Deligne's theorem ( $k\ge2$ ) and the Deligne–Serre theorem ( $k=1$ ), both listed as milestones.

The tau function

Values: $\tau(1)=1$ , $\tau(2)=-24$ , $\tau(3)=252$ , $\tau(4)=-1472$ , $\tau(5)=4830$ .
Multiplicativity (Mordell): $\tau(mn)=\tau(m)\tau(n)$ whenever $\gcd(m,n)=1$ .
Recursion (Mordell): $\tau$ is not completely multiplicative, and $\tau(p^{j+1})=\tau(p)\tau(p^j)-p^{11}\tau(p^{j-1})$ for primes $p$ and $j\ge2$ .
Ramanujan's bound (Deligne): $|\tau(p)|\le 2p^{11/2}$ for all primes $p$ .
Consequences: $|\tau(n)|\le d(n)n^{11/2}$ for all $n\ge1$ , where $d(n)$ is the number of divisors, and hence $\tau(n)=O(n^{11/2+\varepsilon})$ .

The Ramanujan $L$ -function

$\sum_{k\le n}\tau(k)=O(n^{13/2+\varepsilon})$ , and $L(s,\tau)$ converges absolutely for $\operatorname{Re}s>13/2$ .
$L(s,\tau)$ continues to an entire function satisfying $\dfrac{\Gamma(s)L(s,\tau)}{(2\pi)^s}=\dfrac{\Gamma(12-s)L(12-s,\tau)}{(2\pi)^{12-s}}$ .
Euler product: $L(s,\tau)=\prod_p\sum_{n\ge0}\tau(p^n)p^{-ns}=\prod_p\big(1-\tau(p)p^{-s}+p^{11-2s}\big)^{-1}$ for $\operatorname{Re}s>13/2$ .

Significance

The bound $|\tau(p)|\le 2p^{11/2}$ says that the roots of $1-\tau(p)u+p^{11}u^2$ have equal absolute value $p^{-11/2}$ , an analogue of the Riemann hypothesis for the local factors of $L(s,\tau)$ . The bounds for general cusp forms are used throughout analytic number theory to control sums of Fourier coefficients. They are also used in the explicit construction of Ramanujan graphs by Lubotzky, Phillips and Sarnak, which are optimal expander graphs. The generalized conjecture for $GL(n)$ implies Selberg's $1/4$ conjecture on Laplace eigenvalues.

All targets in this mission are proved theorems in the literature. At the pinned Mathlib revision, none of them is formalized except the identity $\tau(1)=1$ . Mathlib does contain the definitions they need: $\eta$ , $\Delta$ as a cusp form of weight 12 and level 1, $q$ -expansions, congruence subgroups, and the $L$ -function of a cusp form. Mathlib also proves the analytic continuation of that $L$ -function. The remaining work is to formalize Hecke theory for $\Delta$ , the elementary consequences of Deligne's bound, the $L$ -function statements, and ultimately Deligne's and Deligne–Serre's theorems.

Difficulty

The algebraic statements about $\tau$ (multiplicativity, the recursion, the Euler product) and the functional equation are classical, but formalizing them needs infrastructure that Mathlib does not yet have, such as Hecke operators on cusp forms. The size bounds are of a different kind. Estimates from the Petersson inner product or from Hecke's method give only the exponent $k/2$ (Hecke's bound). That exponent is attainable by complex analysis alone, but the improvement to $(k-1)/2$ is not. In the literature it depends on arithmetic geometry: Deligne's proof of the Weil conjectures for $k\ge2$ , and Galois representations for $k=1$ . The analytically similar statement for Maass forms is still open.

Formalization scope

$\tau(n)$ is defined as the $n$ -th coefficient of Mathlib's $q$ -expansion of ModularForm.discriminant with period $1$ , as a complex number. The source's description of $\tau$ is taken as the definition, and integrality is not assumed.
Absolute values are complex norms. $n^{11/2}$ and similar powers are real powers of nonnegative reals.
$O(\cdot)$ is Mathlib's big-O along atTop on $\mathbb N$ , with the constant allowed to depend on $\varepsilon$ .
In the general goal, the coefficients are taken with respect to $q_h=e^{2\pi i z/h}$ , where $h$ is the strict width of the cusp $\infty$ for $\Gamma$ (Subgroup.strictWidthInfty). This equals $1$ whenever $z\mapsto z+1$ lies in $\Gamma$ . A congruence subgroup is CongruenceSubgroup.IsCongruenceSubgroup.
The functional equation is stated only at points $s$ where neither $s$ nor $12-s$ is a pole of $\Gamma$ . This avoids Mathlib's convention that $\Gamma$ is $0$ at its poles.
The recursion is stated for $j\ge2$ , exactly as in the source.
The Maass-form and automorphic-representation versions, the function-field results and the bounds towards Ramanujan over number fields are outside this mission's scope. Mathlib lacks the definitions they need.

Selected references

S. Ramanujan, On certain arithmetical functions, Trans. Cambridge Philos. Soc. 22 (1916), 159–184. http://ramanujan.sirinudi.org/Volumes/published/ram18.pdf
H. Petersson, Theorie der automorphen Formen beliebiger reeller Dimension …, Math. Ann. 103 (1930), 369–436. https://doi.org/10.1007/BF01455702
P. Deligne, Formes modulaires et représentations l-adiques, Sém. Bourbaki 1968/69, LNM 179 (1971). https://doi.org/10.1007/BFb0058801
P. Deligne, La conjecture de Weil. I, Publ. Math. IHÉS 43 (1974), 273–307. https://doi.org/10.1007/BF02684373
P. Deligne, J.-P. Serre, Formes modulaires de poids 1, Ann. Sci. ÉNS 7 (1974), 507–530. https://doi.org/10.24033/asens.1277
P. Sarnak, Notes on the generalized Ramanujan conjectures, Clay Math. Proc. 4 (2005). http://www.claymath.org/publications/Harmonic_Analysis/chapter9.pdf

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Actuarial Science Probability·Captain: WillR

Actuarial Mathematics XC: Structural Corporate Default and Rating-Migration Credit ValuationTextbook

Motivation

Credit risk matters to insurer bond holdings, pension scheme fixed-income investment and counterparty guarantees. Merton's corporate debt model derives terminal shareholder and bondholder payoffs from limited liability and asset value. Jarrow–Lando–Turnbull introduces observable ratings with Markov state migrations and default, motivating a second, independently defined view of risky fixed-income valuation.

Setting

A firm has nonnegative terminal assets and a single promise of face-value debt. Creditors are paid no more than the promise and no more than available assets; equity gets the residual. Finite genuine probability masses attach to terminal asset scenarios and yield discounted debt and equity claim valuations. Separately, a three-state rating matrix contains two non-default ratings and a terminal default state, with nonnegative row probabilities and full row-sum constraints.

Formalization targets

The formalisation develops a connected set of actuarial identities, beginning with precise model definitions and ending with the prospective valuation result.

Merton limited liability and terminal default payoffs

Strict asset deficit is equivalent to zero equity except for solvent zero-cashflow special cases.

E_T=(V_T-D)^+,\quad B_T=\min(V_T,D)

Finite default risk premiums and risk-neutral debt pricing

A bond priced equally to risk-free debt has zero model spread.

B_0=e^{-rT}\mathbb E^{\mathbb Q}[\min(V_T,D)]

Three-state rating migration and recoverable default claims

Structural terminal solvency and an independent absorbing-rating transition system reconcile riskier defaultable debt prices, nonnegative credit value deductions and genuinely two-period probability conservation.

\sum_j P_{ij}=1,\quad P_{\text{default,default}}=1

Capstone theorem

\sum_j P_{ij}=1,\quad P_{\text{default,default}}=1

Significance

As discussed in Merton's corporate-credit model, The mission connects terminal firm claims to an actual finite risk-neutral weighted value: equity plus debt exhaust firm assets, expected discounted defaultable debt is bounded above by its risk-free counterpart, and credit value deductions are nonnegative. The genuine two-step rating migration law preserves total probability mass. These independently defined results jointly give an actuarial foundation for pricing defaultable corporate securities and recognising credit exposures in asset-liability valuation.

Difficulty

It must be proved that the limited-liability payoffs partition total assets, with exact boundary conditions where assets equal or fall below face value. The pricing theorem must distinguish actual discounted expected creditor payments from assuming a closed-form Black–Scholes price; no lognormal risk-neutral firm distribution is postulated. Credit spreads require positive debt prices and maturities. A rating transition matrix must be proved stochastic under composition, and an absorbing default state is a model condition rather than automatically true for arbitrary matrices.

Formalization scope

Limited liability and rating-migration probabilities are distinct mathematical constructs in this model. The complete diffusion-based Merton option value and continuous-time rating default models remain future mathematical research. The current target is nontrivial because it derives the bond valuation from limited-liability terminal payouts and an actual probability space, and the migration law from an independent finite transition kernel. Existing bond-present-value and state-reward missions are useful, but they do not identify shareholder and debt recovery claims. Its expected payoff and finite rating matrix infrastructure is reusable in asset-liability modelling and insurer counterparty credit valuation.

A firm's terminal assets must be partitioned consistently between equity and creditor payoffs. Above the promised debt face amount bondholders receive the promise and owners receive remaining assets. At the strict default threshold the owners' limited liability makes equity worthless and bondholders recover the remaining assets. The source values these payoffs using a proper finite risk-neutral scenario law, proving that discounted risky debt is no more valuable than the risk-free debt promise. The other branch builds a genuinely stochastic three-state rating transition matrix and derives two-step migration probabilities from matrix multiplication. Those observed or risk-neutral migration probabilities are not interchangeable without an additional measure-change argument. This is precise finite corporate credit mathematics; it is not a fabricated full closed-form structural diffusion model or a continuous-time default-intensity theory.

The new credit mission establishes exact one-period finite-state structural debt/equity payoff pricing and two-step rating probability conservation. It does not claim Merton's Black–Scholes closed-form equity value, stochastic diffusion replication, or the full continuous-time JLT risk-neutral generator model. The default-state absorbing property needs additional row restrictions and is not asserted for unrestricted transition matrices. Those are substantial extensions, and the current theorem bodies remain unproved.

These are original derived finite/analytic Actuarial formalisation targets rather than reproductions of a numbered textbook theorem. Exact Lean binder declarations are authoritative. The complete source uses targeted Mathlib imports and no local Lean compilation. All proof bodies remain genuinely open and do not constitute accepted results. Assumptions about negative rates, survival, policyholder independence, discounting and benefit timing must be preserved exactly.

Selected references

Robert C. Merton (1974), On the Pricing of Corporate Debt: The Risk Structure of Interest Rates, Journal of Finance 29(2), 449–470. https://doi.org/10.1111/j.1540-6261.1974.tb03058.x
Robert A. Jarrow, David Lando and Stuart M. Turnbull (1997), A Markov Model for the Term Structure of Credit Risk Spreads, Review of Financial Studies 10(2), 481–523. https://doi.org/10.1093/rfs/10.2.481

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Actuarial Science Probability·Captain: WillR

Actuarial Mathematics LXXXIX: Expected Utility, Certainty Equivalents and Stochastic Risk OrdersTextbook

Motivation

Insurance economics prices risk using the utility of wealth rather than merely its variance. Pratt's seminal work derives risk aversion and the certainty-equivalent insurance premium, while Rothschild and Stiglitz connect comparative risk to probability order. Earlier actuarial entropic premium missions contain special finite retention objectives but do not derive an independently constructed expected wealth utility with premium indifference and finite stochastic dominance.

Setting

Losses are monetary outcomes on a genuinely finite scenario probability model, whose masses are nonnegative and sum to one. The exponential CARA wealth utility is strictly increasing and concave for positive aversion. The certainty-equivalent premium is constructed from the actual exponential moment of loss. Full insurance replaces the uncertain loss with a fixed premium and its utility is compared with the truly weighted uninsured outcome.

Formalization targets

The formalisation develops a connected set of actuarial identities, beginning with precise model definitions and ending with the prospective valuation result.

Expected exponential utility and absolute risk aversion

Utility of deterministic wealth equals its scenario expectation.

u_a(w)=-e^{-aw},\quad A(w)=-u''_a(w)/u'_a(w)=a

Actuarial insurance certainty equivalents and premium indifference

Risk-averse exponential expected utility demands at least the expected claim cost.

\pi_a(X)=\frac{1}{a}\log\mathbb E[e^{aX}]

Genuine pathwise coupling and finite first-order stochastic dominance

The actuarial insurer and policyholder indifference premium follows from genuinely averaged utility, with positive risk loading and a finite-scenario stochastic risk ordering.

X\le Y\ \text{a.s.}\Longrightarrow \mathbb E[\phi(X)]\le\mathbb E[\phi(Y)]

Capstone theorem

The actuarial insurer and policyholder indifference premium follows from genuinely averaged utility, with positive risk loading and a finite-scenario stochastic risk ordering.

X\le Y\ \text{a.s.}\Longrightarrow \mathbb E[\phi(X)]\le\mathbb E[\phi(Y)]

Significance

As discussed in Pratt's utility-risk treatment, The capstone proves a complete indifference premium identity, not merely its algebraic definition, and derives risk loading above expected loss from convexity of the exponential. A coupled stochastic order implies both the ordering of actual finite distribution functions and higher expected utility for increasing utility functions. The same ordering is related to excess-of-loss premiums through statewise monotonicity. These results supply core insurer and policyholder pricing foundations.

Difficulty

The logarithmic certainty equivalent requires strictly positive exponential moments and a nonzero risk-aversion parameter. The actual expected utility factorization must account for the sign of negative exponential utility. Risk loading requires Jensen's inequality or an equivalent discrete exponential-convexity argument, not the assertion that every risk is profitable to insure. The comparison of finite CDFs must reverse the inequality for higher wealth, while a higher loss severity raises stop-loss costs. Pathwise coupling is stronger than the entire stochastic-dominance order and is identified as such.

Formalization scope

Certainty-equivalent premium and stochastic risk ordering are distinct mathematical constructs in this model. A future generalisation will derive second-order stochastic dominance via integrals of real CDFs and concave expected utility, not merely a pathwise coupling. The current project is nontrivial because the CARA indifference identity follows by differentiating and averaging the actual utility, and the insurer's risk loading is derived from Jensen's inequality. Existing specialised exponential risk-price examples provide a useful benchmark but do not replace these general sources. The premium and coupling functions are reusable for nonproportional reinsurance, risk classification and comparative risk analysis.

Under constant absolute risk aversion the negative exponential utility function is increasing and concave in wealth when risk aversion is positive, even though its numerical utility level is negative. A fair premium is not obtained by simply assuming a desired level of profit. The expected uninsured utility is calculated from true scenario probabilities, the utility with full insurance is calculated using a fixed premium and the resulting indifference equation is solved from the loss exponential moment. The log moment premium must then be compared against expected claim cost using convexity, and the sign of this loading depends on risk aversion. Finite distribution functions and expected excess claims are independently defined for comparison. The pathwise coupling order in the source is stronger than first-order stochastic dominance, so no false claim of characterising all stochastic orders is made. Second-order dominance requires a separate distribution-integral analysis.

This is a genuine finite-probability CARA model with exactly derived insured/uninsured premium indifference. The stochastic order is proven for explicit statewise couplings and thus yields a strong sufficient condition for first-order stochastic dominance, not a complete characterisation of arbitrary second-order stochastic dominance. General concave utility risk orders, mean-preserving spreads, Arrow–Pratt comparative aversion for all utility families, and infinite random variables are future developments. All proof obligations are open.

Selected references

John W. Pratt (1964), Risk Aversion in the Small and in the Large, Econometrica 32, 122–136. https://doi.org/10.2307/1913738
Michael Rothschild and Joseph E. Stiglitz (1970), Increasing Risk: I. A Definition, Journal of Economic Theory 2, 225–243. https://doi.org/10.1016/0022-0531(70)90038-4

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Actuarial Science Probability·Captain: WillR

Actuarial Mathematics XCI: Joint-Life Mortality Profit and Contingent Survivor BenefitsTextbook

Motivation

Mortality experience in joint-life assurance and pensions depends not only on whether someone died, but on which insured life survives and which survivor or joint reserve must then be held. Standard multiple-life actuarial valuations motivate last-survivor, reversionary and guaranteed pension benefits. Earlier actuarial missions XLV and XLVI treat basic reversionary annuities and single-life mortality profit separately; this connects them through a genuine four-outcome death-strain model.

Setting

Two members start alive and experience independent yearly survival outcomes. Exactly four mutually exclusive joint states follow: both survive, only the first survives, only the second survives, or neither survives. Their probabilities use product and complement survival factors. The year-end reserve and any benefits can differ for every one of these states. Opening reserves plus premiums earn the annual return before payment and the next state contingent reserve is recognized.

Formalization targets

The formalisation develops a connected set of actuarial identities, beginning with precise model definitions and ending with the prospective valuation result.

Independent joint-life outcomes and probabilities

Each valid joint-life outcome has nonnegative mass.

p_{xy}=p_xp_y,\quad p_{x\dagger y}=(1-p_x)p_y

Outcome-dependent reserves, benefits and death strain

Additional state-specific benefit claims add to expected death strain.

D_{s}=B_s+V_s-(V_0+P)(1+i)

Joint mortality profit and limited-contingent survivor benefits

An exhaustive, correctly normalized two-life outcome model makes the mortality experience profit mean zero, retaining separate survivor reserves, death benefits and guaranteed payment conventions.

\Pi_{\text{mort}}=\sum_s p_s D_s-D_{\text{observed}}

Capstone theorem

An exhaustive, correctly normalized two-life outcome model makes the mortality experience profit mean zero, retaining separate survivor reserves, death benefits and guaranteed payment conventions.

\Pi_{\text{mort}}=\sum_s p_s D_s-D_{\text{observed}}

Significance

As discussed in Dickson, Hardy and Waters' multiple-life analysis, The capstone derives actual four-state normalization and reconciles joint mortality profit as expected death strain less observed state-specific death strain, whose expected result is zero under its own properly normalized law. The state-specific reserve functions distinguish the two possible remaining single-life liabilities rather than collapsing them into a single average. Limited-reversionary and certain guarantee payments are represented as separate conditional benefits.

Difficulty

Four disjoint probabilities must be normalized with exact endpoint treatment. Reserve releases after each death outcome may have opposite signs to death-benefit payments, so the state strain must include both payout and next-state reserve less the accumulated starting fund. Expected mortality profit cannot be asserted by defining it to have zero mean: it must follow from the actual weighted expectation of the observed strain. Independence is an explicit current assumption; arbitrary dependent lives need a separate four-state probability law.

Formalization scope

Joint-life mortality profit and state-dependent death strain are distinct mathematical constructs in this model. A future mission could introduce dependent lifetimes and within-year order of deaths, then compute first-to-die and last-survivor benefit present values directly from the joint survival law. The present work is nontrivial because both survivor reserves and actual death claims differ across the four states, while the mean mortality profit emerges from their normalised probability distribution. Existing joint-life annuity and single-life profit missions do not establish this combination. The outcome-specific reserve and mortality gain functions are reusable for spouse's pensions, death-in-service benefits and multi-life experience analysis.

Four disjoint annual outcomes must be considered for a policy whose two insured lives begin the year alive. They are both alive, only the first alive, only the second alive and both dead. Each outcome has a separate year-end reserve and may have a different death or survivor benefit. Accordingly, its actual death strain includes the benefit paid plus the new reserve less accumulated beginning-year funds. Expected strain is a genuine probability-weighted average of those four observed strains. The mortality experience profit for any one realised outcome equals the expected strain less that outcome's actual strain. Its expectation must be shown to vanish under the normalised four-outcome probability distribution, rather than by assuming zero mean. This framework supplies an actuarial bridge from joint-life/reversionary benefits to experience profit analysis and guarantee-period pensions. Within-year timing and dependent life outcomes require additional formal work.

This is a complete one-year joint mortality-experience model with explicit separate survivor reserves, not a full continuous-time ordering of first and second deaths inside the same policy year. Guarantee-period timing is represented by a separate deterministic indicator input. Exact age differences, policy options, death-payment timing and dependence-copula extensions can build on these definitions later. All theorem statements remain unproved.

Selected references

David C. M. Dickson, Mary R. Hardy and Howard R. Waters (2020), Actuarial Mathematics for Life Contingent Risks, Cambridge University Press. https://www.cambridge.org/core/books/actuarial-mathematics-for-life-contingent-risks/
S. David Promislow (2015), Fundamentals of Actuarial Mathematics, Wiley. https://www.wiley.com/en-us/Fundamentals+of+Actuarial+Mathematics%2C+3rd+Edition-p-9781118782521

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Actuarial Science Probability·Captain: WillR

Actuarial Mathematics LXXVII: Compound Aggregate Masses and the Full Finite Panjer RecursionTextbook

Motivation

Panjer's original 1981 result derives the aggregate claims recursion from a special frequency probability law and arbitrary nonnegative-integer severity mass distribution, including claims worth zero. The original article and the Sundt–Jewell extension underpin the mission. Earlier actuarial mission LIV handles direct recurrence algebra, while LXXIII derives the Bernoulli first positive aggregate mass identity. This mission formalises a full positive-aggregate-severity recursion for a genuinely constructed compound law.

Setting

A claim count N has probability weights p_n supported on {0,...,m}, with no artificial truncation of an otherwise infinite count distribution. Independent severity masses f_j on nonnegative integer amounts admit a genuine n-fold convolution, with count-zero convolution equal to a unit atom at zero. Thus g_k=Σ_{n=0}^m p_n(f^{*n})k is defined independently of Panjer's formula. The count recursion p_n=(a+b/n)p{n−1} is assumed only for 1≤n≤m+1, including its boundary p_{m+1}=0, and p_n=0 above m. No conclusion about the aggregate g is assumed.

Formalization targets

Aggregate claims from an actual frequency distribution

Write $p_n$ for the probability of exactly $n$ claims and $f_j$ for the probability that one claim has size $j$ . The frequency law is genuinely supported on $0,\ldots,m$ ; an infinite Poisson distribution is not silently truncated. The empty portfolio has aggregate size zero, while $f^{*n}$ denotes the distribution obtained by summing $n$ independent severities.

The first group constructs the aggregate probability $g_k$ directly from the frequency and severity laws:

g_k=\sum_{n=0}^{m}p_n(f^{*n})_k,\qquad (f^{*0})_0=1.

It also proves the zero-severity convolution identities, nonnegative mass preservation, and finite frequency and severity generating-function relationships.

The Panjer law and its upper support boundary

The count probabilities obey a Panjer recurrence with coefficients $a$ and $b$ . Crucially, the finite-support model includes the transition from $m$ to the compulsory zero probability at $m+1$ :

p_n=\left(a+\frac{b}{n}\right)p_{n-1}, \qquad 1\le n\le m+1,\qquad p_{m+1}=0.

This makes the assumed count law compatible with the finite aggregate distribution. For a claim size that may equal zero, the aggregate zero atom involves every possible claim count:

g_0=G_N(f_0)=\sum_{n=0}^{m}p_n f_0^{\,n}.

Corrected aggregate recursion and valuation

The central target is the entire positive-aggregate-size Panjer recurrence, not merely its first positive atom. For every integer $k\ge1$ , the result is

(1-af_0)g_k= \sum_{j=1}^{k}\left(a+\frac{bj}{k}\right) f_jg_{k-j}.

The factor $1-af_0$ is essential because a positive number of claims may still produce aggregate size zero. Further targets establish properties of cumulative aggregate probabilities, capital exceedance tails and finite stop-loss expectations. Every aggregate probability is defined independently of the recurrence it is subsequently required to satisfy.

Significance

The original Panjer paper gives the classical aggregate-loss setting. The capstone derives (1-a f_0)g_k=Σ_{j=1}^k(a+bj/k) f_j g_{k-j} for every positive aggregate amount from the actual random sum, together with g_0=G_N(f_0). The generator gives a count-supported finite Panjer family; it does not claim an infinite Poisson/negative-binomial count law. The exact zero claim-size correction is essential even when count N is positive, because multiple claims may all have zero monetary size. The authoring graph also includes frequency PGFs, severity convolution, CDF/tail and stop-loss identities.

Difficulty

Existing actuarial probability and Panjer steps are established subject areas. Every formal result introduced in this mission is an open Lean theorem target; none has an accepted proof. A vacuous or trivial definition of the aggregate as the recurrence solution is explicitly ruled out. The proof must rearrange nested severity convolutions, differentiate coefficient identities or establish the equivalent frequency recursion, and manage the zero-severity term that multiplies g_k itself. The count-support cutoff must satisfy the Panjer multiplier at m+1 to avoid invalid boundary terms. The recurrence at k=0 is not defined due to division by k; zero aggregate mass follows separately from the frequency PGF. Because f_0 may be positive, truncating the count sum at arbitrary m with an infinite count law would be mathematically invalid; we avoid that by using a genuine finite count distribution.

Formalization scope

This mission supplies a full k≥1 Panjer relation under an exact finite-support frequency model, not general infinite Poisson or geometric counts and not continuous claim severities. Positivity and mass normalisation require additional explicit probability conditions beyond the purely algebraic count recurrence, so the capstone is a valid algebraic identity even before imposing proper probability weights. All formal results remain Lean theorem targets without accepted proofs. The finite model is a reusable foundation for formal-power-series and analytic infinite-frequency cases.

Selected references

Harry H. Panjer (1981), Recursive Evaluation of a Family of Compound Distributions, ASTIN Bulletin 12(1), 22–26. https://doi.org/10.1017/S0515036100006796
Bjørn Sundt and William S. Jewell (1981), Further Results on Recursive Evaluation of Compound Distributions, ASTIN Bulletin 12(1), 27–39. https://doi.org/10.1017/S0515036100006802
S. David Promislow (2015), Fundamentals of Actuarial Mathematics, third edition, Chapter 21, section 21.11. https://www.oreilly.com/library/view/fundamentals-of-actuarial/9781118782521/ftoc.xhtml

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Differential Geometry Mathematical Physics·Captain: Lucas

Uhlenbeck, Equations of Gauge Theory I: Self-Dual Connections Minimize Yang-MillsTextbook

Motivation

The Yang–Mills functional measures the total curvature of a connection on a principal bundle. Its critical points, the Yang–Mills connections, are the non-abelian generalization of solutions of Maxwell's equations, and in four dimensions the most important among them are the self-dual and anti-self-dual connections (instantons), whose moduli spaces underlie Donaldson theory of smooth four-manifolds. A basic and frequently used fact, recorded as Proposition 3.1.1 in K. Uhlenbeck's lectures Equations of Gauge Theory (notes by L. Fredrickson, Temple University, 2012), is that instantons are not merely critical points: within a fixed topological class and with fixed boundary data they are absolute minimizers of the Yang–Mills energy. The same "energy identity" argument is the template the notes later apply to the Kapustin–Witten equations (Proposition 3.1.5).

Timeline of the ideas involved: Yang and Mills introduce the equations (1954); Belavin, Polyakov, Schwartz and Tyupkin observe the topological lower bound on the energy and its saturation by self-dual fields (1975); Atiyah, Hitchin and Singer study the deformation theory of instantons (1977–78); Donaldson uses the instanton moduli space in four-manifold topology (1983).

Setting

Work on the trivial principal $SU(n)$ -bundle over a closed box $\Omega=[a,b]=\prod_{\mu=0}^{3}[a_\mu,b_\mu]\subset\mathbb R^4$ with the Euclidean metric. A connection is $D_A=d+A$ with $A=\sum_\mu A_\mu\,dx^\mu$ , where each $A_\mu$ is a smooth map into the Lie algebra $\mathfrak{su}(n)$ of traceless skew-Hermitian $n\times n$ complex matrices. Its curvature is the $\mathfrak{su}(n)$ -valued two-form $F_A=dA+A\wedge A$ with coefficients

(F_A)_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu+[A_\mu,A_\nu].

The Hodge star on two-forms is $(\star F)_{\mu\nu}=\tfrac12\sum_{\rho,\sigma}\varepsilon_{\mu\nu\rho\sigma}F_{\rho\sigma}$ , with $\varepsilon$ the Levi-Civita symbol. A two-form is self-dual if $\star F=F$ and anti-self-dual if $\star F=-F$ . The pointwise norm is $|F|^2=\sum_{\mu<\nu}\operatorname{Re}\operatorname{tr}(F_{\mu\nu}F_{\mu\nu}^{*})$ , and the Yang–Mills functional on $\Omega$ is

\mathrm{Ym}_\Omega(A)=\int_\Omega |F_A|^2\,d\mathrm{vol}.

The Chern–Weil density $\operatorname{tr}(F\wedge F)$ is the coefficient $\tfrac14\sum\varepsilon_{\mu\nu\rho\sigma}\operatorname{tr}(F_{\mu\nu}F_{\rho\sigma})$ of $dx^0\wedge dx^1\wedge dx^2\wedge dx^3$ , and the Chern–Simons three-form is $CS(A)=\operatorname{tr}(A\wedge dA+\tfrac23A\wedge A\wedge A)$ , with flux components $K^\mu=\sum\varepsilon_{\mu\nu\rho\sigma}\operatorname{tr}(A_\nu\partial_\rho A_\sigma+\tfrac23A_\nu A_\rho A_\sigma)$ .

Formalization targets

Goal (Proposition 3.1.1, box version)

If $A$ and $B$ are smooth $SU(n)$ -connections on $\Omega$ that agree on $\partial\Omega$ , and $F_A$ is self-dual on all of $\Omega$ or anti-self-dual on all of $\Omega$ , then

\mathrm{Ym}_\Omega(A)\le \mathrm{Ym}_\Omega(B).

Milestones

Lemma 3.1.2. For an $\mathfrak{su}(n)$ -valued two-form $F$ : $|F\pm\star F|^2=2|F|^2\mp2\operatorname{tr}(F\wedge F)$ .
Fact 3.1.3, local form. $\operatorname{tr}(F_A\wedge F_A)=\sum_\mu\partial_\mu K^\mu$ , i.e. $\operatorname{tr}(F_A\wedge F_A)=d\,CS(A)$ .
Fact 3.1.3, box form. $\int_\Omega\operatorname{tr}(F_A\wedge F_A)=\int_{\partial\Omega}CS(A)$ (the topological term vanishes for the trivial bundle).
Energy bound (proof of Proposition 3.1.1). $\mathrm{Ym}_\Omega(A)\ge \mp\int_{\partial\Omega}CS(A)$ .
Equality case (proof of Proposition 3.1.1). On a non-degenerate box, $\mathrm{Ym}_\Omega(A)=-\int_{\partial\Omega}CS(A)$ iff $F_A$ is self-dual on $\Omega$ , and $\mathrm{Ym}_\Omega(A)=\int_{\partial\Omega}CS(A)$ iff $F_A$ is anti-self-dual on $\Omega$ .

Significance

The energy identity turns a second-order variational problem into a first-order one: any connection satisfying the first-order self-duality equation is automatically a Yang–Mills connection, and in fact a global minimizer in its class. This is the reason instanton moduli spaces are the natural objects of four-dimensional gauge theory, and it is the model for the corresponding minimization statements for the Kapustin–Witten equations treated later in the same notes. The result is classical and its proof is known; the remaining work is formal: a machine-checked development of the curvature, Hodge star, Chern–Simons transgression and the boundary integration on boxes, none of which are currently packaged in this form in Mathlib.

Difficulty

The algebraic identity of Lemma 3.1.2 is a finite computation. The substantive step is the transgression formula $\operatorname{tr}(F_A\wedge F_A)=d\,CS(A)$ , a cancellation between derivative terms and cubic terms that depends on cyclicity of the trace and on the antisymmetry of $\varepsilon$ , followed by Stokes' theorem on the box with correct face orientations. The naive approach of comparing $|F_A|^2$ and $|F_B|^2$ pointwise fails: the two connections agree only on the boundary, and only the integrated topological term is boundary-determined.

Formalization scope

The general setting of the notes (a $G$ -bundle over a compact oriented Riemannian four-manifold with boundary, topological class $c_2(P)=k$ ) is replaced by the trivial $SU(n)$ -bundle over a closed axis-parallel box in Euclidean $\mathbb R^4$ , so the topological term $8\pi^2c_2(P)$ is zero. Connections are smooth on all of $\mathbb R^4$ (equivalently, smooth up to the boundary of the box) and every coefficient takes values in $\mathfrak{su}(n)$ . Two-forms are stored as antisymmetric $4\times4$ families of matrices; partial derivatives are taken entrywise; integrals are Lebesgue integrals over the box and over its faces. "Fixed boundary data" means $A=B$ at every point of $\partial\Omega$ . Self-duality is $\star F=F$ , matching $F=\star F$ in Section 2.2 of the notes.

Two sign/normalization corrections relative to the printed notes are made deliberately: Lemma 3.1.2 is stated with the sign obtained in its own proof on p. 32 ( $\mp$ rather than the printed $\pm$ , because $B^*=-B$ on $\mathfrak{su}(n)$ ), and the Chern–Simons form uses the standard coefficient $\tfrac23$ in $\operatorname{tr}(A\wedge dA+\tfrac23A^3)$ , i.e. $\operatorname{tr}(A\wedge F_A)-\tfrac13\operatorname{tr}(A^3)$ , for which $d\,CS=\operatorname{tr}(F\wedge F)$ holds, instead of the printed $\tfrac16$ .

A trivializing formalization would make the boundary condition or the self-duality hypothesis unsatisfiable; here the zero connection is self-dual and satisfies every hypothesis, and the statements are not vacuous.

Reusable infrastructure: Levi-Civita symbol and Hodge star on $\Lambda^2\mathbb R^4$ with matrix coefficients, curvature of matrix-valued connections, the Chern–Simons transgression, and Stokes' theorem for flux forms on boxes (Mathlib's divergence theorem on boxes is the natural tool). Contributions to any milestone are welcome.

Selected references

K. Uhlenbeck, Equations of Gauge Theory, notes by L. Fredrickson, Emil Grosswald Lectures, Temple University, 2012 (the source of all statements; Section 3.1, pp. 29–32).
C. N. Yang, R. L. Mills, Conservation of isotopic spin and isotopic gauge invariance, Phys. Rev. 96 (1954), 191–195. https://doi.org/10.1103/PhysRev.96.191
A. A. Belavin, A. M. Polyakov, A. S. Schwartz, Yu. S. Tyupkin, Pseudoparticle solutions of the Yang–Mills equations, Phys. Lett. B 59 (1975), 85–87. https://doi.org/10.1016/0370-2693(75)90163-X
S. K. Donaldson, An application of gauge theory to four-dimensional topology, J. Differential Geom. 18 (1983), 279–315. https://doi.org/10.4310/jdg/1214437665

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Combinatorics Group Theory Number Theory·Captain: Lucas

Landau's Function: Growth of the Maximal Order of a PermutationResearch Paper

Motivation

How large can the order of a single permutation of $n$ objects be? The answer, Landau's function $g(n)$ , is a basic invariant of the symmetric group $S_n$ : it is the maximum number of times a permutation can be iterated before it returns to the identity. Because the order of a permutation is the least common multiple of its cycle lengths, $g(n)$ is also the largest least common multiple of the parts of a partition of $n$ , which ties the question to the distribution of prime numbers.

Timeline.

1902–1903. E. Landau proved that $\ln g(n)\sim\sqrt{n\ln n}$ .
1968. J.-L. Nicolas showed that $g$ is constant on arbitrarily long runs of consecutive integers.
1984. J.-P. Massias proved the explicit bound $\ln g(n)\le 1.05313836\sqrt{n\ln n}$ for all $n\ge 1$ .
1988. Massias, Nicolas and Robin obtained a three-term asymptotic expansion of $\ln g(n)$ , its expression through the inverse logarithmic integral, and the equivalence of a one-sided inequality with the Riemann hypothesis.
2002. K. Ford's bounds for the Riemann zeta function give the error term $x\exp(-c(\ln x)^{3/5}(\ln\ln x)^{-1/5})$ in the prime number theorem, which can be fed into the 1988 result.

Setting

For a natural number $n$ , let $S_n$ be the group of all permutations of an $n$ -element set. The order of $\sigma\in S_n$ is the least $k\ge 1$ with $\sigma^k=\mathrm{id}$ . Landau's function is

g(n)=\max_{\sigma\in S_n}\operatorname{ord}(\sigma).

A partition of $n$ is a multiset of positive integers summing to $n$ . The prime counting function $\pi(x)$ is the number of primes $p\le x$ , the (offset) logarithmic integral is $\mathrm{Li}(x)=\int_2^x\frac{dt}{\ln t}$ , and $\mathrm{Li}^{-1}$ is its inverse on $[2,\infty)$ . Throughout, $\ln$ is the natural logarithm.

Formalization targets

Goal: Landau's theorem (1902)

\lim_{n\to\infty}\frac{\ln g(n)}{\sqrt{n\ln n}}=1.

Supporting milestones

Partition characterization: $g(n)=\max\{\operatorname{lcm}(\lambda):\lambda\vdash n\}$ .
Initial values $g(0),\dots,g(8)=1,1,2,3,4,6,6,12,15$ .
Nicolas (1968): for every $m$ there is $n$ with $g(n)=g(n+1)=\dots=g(n+m)$ .
Massias–Nicolas–Robin (1988): $\ln g(n)=\sqrt{n\ln n}\Bigl(1+\frac{\ln\ln n-1}{2\ln n}-\frac{(\ln\ln n)^2-6\ln\ln n+9}{8(\ln n)^2}+O\bigl((\tfrac{\ln\ln n}{\ln n})^3\bigr)\Bigr)$ .
Ford (2002): $\pi(x)-\mathrm{Li}(x)=O\bigl(x\exp(-c(\ln x)^{3/5}(\ln\ln x)^{-1/5})\bigr)$ for some $c>0$ .
Massias–Nicolas–Robin (1988): if $\pi(x)-\mathrm{Li}(x)=O(R(x))$ with $R(x)=x\exp(-c(\ln x)^{3/5}(\ln\ln x)^{-1/5})$ , then $\ln g(n)=\sqrt{\mathrm{Li}^{-1}(n)}+O\bigl(R(\sqrt{n\ln n})\ln n\bigr)$ .
Riemann hypothesis criterion: $\ln g(n)<\sqrt{\mathrm{Li}^{-1}(n)}$ for all large $n$ $\iff$ RH.
$g(n)\le e^{n/e}$ for all $n$ , with equality only at $n=0$ .
Massias (1984): $g(n)\le\exp(1.05314\sqrt{n\ln n})$ .

Significance

Landau's theorem pins down the growth rate of the largest element order in $S_n$ , which bounds, for example, how long any fixed shuffle of $n$ cards can be repeated before the deck returns to its starting order. The refined statements quantify the second-order behaviour and connect it to the error term in the prime number theorem; the criterion in milestone 7 is one of the known elementary reformulations of the Riemann hypothesis, and is included as a faithful statement of the source, not as an expected deliverable. The other results are proved in the literature; formalizing them requires building the partition/lcm description of element orders in $S_n$ and quantitative prime-counting estimates on top of Mathlib.

Difficulty

The order of a permutation is easy to compute from its cycle type, but maximizing an lcm over all partitions is a non-linear optimization: the optimum uses prime powers whose sizes are dictated by the distribution of primes, so even the leading-order asymptotics need Chebyshev-type or prime number theorem estimates, and the refined milestones need the prime number theorem with an explicit error term.

Formalization scope

$S_n$ is Equiv.Perm (Fin n); $g(n)$ is the supremum of orderOf σ over the finite group (a natural number; $g(0)=1$ ).
Partitions are Mathlib's Nat.Partition n; the lcm of a partition is the Multiset.lcm of its parts.
Real logarithms use Real.log (which returns $0$ at $0$ and is defined for negative input); all asymptotic statements are about $n\to\infty$ , so the values at small $n$ are irrelevant there. The bounds in milestones 8–9 are stated for every $n$ and are true at $n=0,1$ .
$\pi(x)$ for real $x$ is Nat.primeCounting ⌊x⌋₊; $\mathrm{Li}$ is the offset logarithmic integral $\int_2^x dt/\ln t$ ; $\mathrm{Li}^{-1}(y)$ is the least $x\ge2$ with $\mathrm{Li}(x)\ge y$ , which for $y\ge0$ is the inverse function value. Replacing $\mathrm{Li}$ by the full integral $\mathrm{li}$ changes the inverse by $O(\ln n)$ and does not affect any of the asymptotic statements.
The Riemann hypothesis is Mathlib's RiemannHypothesis.

Contributions of reusable infrastructure (cycle-type lcm, prime-counting error terms, the logarithmic integral) are welcome.

Selected references

E. Landau, Über die Maximalordnung der Permutationen gegebenen Grades, Arch. Math. Phys. Ser. 3, vol. 5, 1903.
J.-L. Nicolas, Sur l'ordre maximum d'un élément dans le groupe $S_n$ des permutations, Acta Arithmetica 14 (1968), 315–332.
J.-P. Massias, Majoration explicite de l'ordre maximum d'un élément du groupe symétrique, Ann. Fac. Sci. Toulouse Math. (5) 6 (1984), 269–281.
J.-P. Massias, J.-L. Nicolas, G. Robin, Évaluation asymptotique de l'ordre maximum d'un élément du groupe symétrique, Acta Arithmetica 50 (1988), 221–242.
K. Ford, Vinogradov's integral and bounds for the Riemann zeta function, Proc. London Math. Soc. 85 (2002), 565–633. https://doi.org/10.1112/S0024611502013655
W. Miller, The maximum order of an element of a finite symmetric group, Amer. Math. Monthly 94 (1987), 497–506.
OEIS A000793. https://oeis.org/A000793

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Economics·Captain: Lucas

Marginal Utility I: The Law of Diminishing Marginal UtilityTextbook

Motivation

Marginal utility is the change in utility (pleasure or satisfaction) that results from consuming one more unit of a good or service. It is the basic quantity of the marginalist theory of value, which explains prices and consumer choice by the additional satisfaction provided by each extra unit rather than by total usefulness. The law of diminishing marginal utility states that, other things being equal, the utility a consumer derives from additional units of a commodity goes on decreasing as the amount consumed increases; economists regard it as reflected in the concave shape of most utility functions.

Timeline. Gabriel Cramer (private letter, 1728) and Daniel Bernoulli (Specimen theoriae novae de mensura sortis, 1738) proposed that the desirability of money decreases as it accumulates, measuring it by the square root (Cramer) and the natural logarithm (Bernoulli), in order to resolve the St. Petersburg paradox. W. F. Lloyd (1833), N. W. Senior (1836), J. Dupuit (1844) and H. H. Gossen (1854) stated marginal-utility ideas with little contemporary influence. The theory entered the mainstream in the Marginal Revolution of Jevons (1862/1871), Menger (1871) and Walras (1874), and was consolidated by Marshall's Principles of Economics (1890), which treats utility as quantified. Ordinalist reformulations (Edgeworth 1881, Slutsky 1915, Hicks–Allen 1934) moved to indifference-curve analysis; cardinal utility returned with expected-utility theory (Ramsey 1926, von Neumann–Morgenstern 1944, Savage 1954), where diminishing marginal utility is linked to risk aversion.

Setting

A utility function of one quantified variable is a map $U:\mathbb R\to\mathbb R$ , where $g$ is the quantity of a single good and every other variable is held fixed (ceteris paribus, "c.p."). The change in utility from state $s_1$ to state $s_2$ is

\Delta U(s_1\to s_2)=U(s_2)-U(s_1).

The marginal utility at $g$ is the limit of the ratio of the change in utility to the size of the change in $g$ ,

\frac{\partial U}{\partial g}=\lim_{\Delta g\to 0}\left.\frac{\Delta U}{\Delta g}\right|_{c.p.},

which mainstream neoclassical economics assumes to exist. The function $U$ has diminishing marginal utility on a set $D$ of quantities when

\frac{\partial^2 U}{\partial g^2}<0\quad\text{on }D.

In the Lean development these are utilityChange, marginalUtility and HasDiminishingMarginalUtility in the namespace MarginalUtility.

Formalization targets

Goal: the law of diminishing marginal utility

For $D$ an open interval and $U$ with diminishing marginal utility on $D$ :

g_1<g_2 \text{ in } D\;\Longrightarrow\; \frac{\partial U}{\partial g}(g_2)<\frac{\partial U}{\partial g}(g_1),\qquad U(x+2h)-U(x+h)<U(x+h)-U(x)\ \ (h>0,\ x,x+2h\in D).

The first clause is the law as stated ("marginal utility goes on decreasing"); the second is its reading in terms of successive equal gains.

Milestones

Marginal utility is the limit of $\Delta U/\Delta g$ wherever $U$ is differentiable.
Positive marginal utility on an interval makes utility strictly increasing there; negative marginal utility makes it strictly decreasing.
Diminishing marginal utility makes marginal utility strictly decreasing.
Diminishing marginal utility makes $U$ strictly concave.
For a strictly concave $U$ , each equal gain adds strictly less utility than the previous one.
If marginal utility reaches zero at $x_0$ , total utility is maximal at $x_0$ and marginal utility is negative beyond $x_0$ .
Bernoulli's $\log x$ and Cramer's $\sqrt x$ have diminishing marginal utility on $(0,\infty)$ .

Significance

The law underlies the marginalist explanation of demand, the resolution of the water–diamond paradox, and, through the concavity of utility, the modelling of risk aversion in expected-utility theory. The concave-shape and satiation statements are the forms in which the law is used in consumer theory: under diminishing marginal utility, total utility is maximal exactly where marginal utility vanishes.

The mathematical content is classical single-variable calculus and is fully known. What this mission adds is a precise, machine-checked dictionary between the economic vocabulary ("marginal utility", "diminishing", "concave shape", "satiation") and standard analytic statements, on which later missions about consumer choice and risk aversion can build.

Difficulty

None of the statements is deep. The care required is in the hypotheses: the law is a statement about an open interval of quantities, marginal utility only makes sense where $U$ is differentiable, and mere (non-strict) concavity gives only non-strict decrease of successive gains. The source's verbal statements must be matched exactly in strength (strict versus non-strict inequalities) and in domain.

Formalization scope

Utility is a function $U:\mathbb R\to\mathbb R$ of one real quantity; the domain of interest is an open convex set $D\subseteq\mathbb R$ (an open interval, possibly unbounded), passed as a hypothesis. Marginal utility is deriv U, which Lean sets to $0$ where $U$ is not differentiable; every statement that uses it assumes the needed differentiability. Diminishing marginal utility means $U$ and $U'$ are differentiable on $D$ and $U''<0$ on $D$ . A statement that only assumed $U''<0$ without differentiability would be weaker than intended, because of the value- $0$ convention; the definition rules this out. Multivariate utility, ordinal utility and the qualitative assumptions listed in the source (identical units, constant tastes, and so on) are outside the scope of this mission. Proofs may use Mathlib's one-variable calculus and convexity libraries freely.

Selected references

Wikipedia, Marginal utility. https://en.wikipedia.org/wiki/Marginal_utility
D. Bernoulli, Specimen theoriae novae de mensura sortis (1738); English translation: Exposition of a New Theory on the Measurement of Risk, Econometrica 22 (1954) 23–36. https://doi.org/10.2307/1909829
A. Marshall, Principles of Economics, Macmillan, 1890.

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Mathematical Physics·Captain: Lucas

Second Law of Thermodynamics (Wikipedia) I: Carnot's Theorem and the Thermodynamic Temperature ScaleTextbook

Motivation

The second law of thermodynamics is a physical law based on universal empirical observation concerning heat and energy interconversions. Two of its classical statements are impossibility statements about machines. Rudolf Clausius (1854): "Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time." Lord Kelvin: "It is impossible for any system to operate in a thermodynamic cycle and deliver a net amount of energy by work to its surroundings while receiving energy by heat transfer from a single thermal reservoir." The Wikipedia article Second law of thermodynamics records that these two have been shown to be equivalent.

The law is not only prohibitive. Sadi Carnot's analysis of heat engines (1824) yields an upper bound on the efficiency of any engine running between two heat reservoirs, attained by reversible engines and strictly missed by irreversible ones, and — crucially — the same for all reversible engines whatever the working substance. That substance-independence is what makes a thermodynamic temperature possible: as the article puts it, "The second law allows a distinguished temperature scale, which defines an absolute, thermodynamic temperature, independent of the properties of any particular reference thermometric body."

This mission formalizes that chain: from the two impossibility statements, through Carnot's theorem, to the existence and uniqueness of the absolute temperature scale normalized by the Kelvin convention at the triple point of water.

Setting

A cyclically operating device is recorded by the heat it exchanges with thermal reservoirs over one complete cycle. Reservoirs are labelled by empirical temperature, an element of a linearly ordered type $\Theta$ ( $s<t$ : the reservoir at $s$ is colder). A cycle is a finitely supported function $c:\Theta\to\mathbb R$ , where $c(t)$ is the net heat absorbed by the device from the reservoir at $t$ (negative if rejected). Because the device returns to its initial state, the first law makes the net work delivered equal to the total heat absorbed:

W(c)=\sum_{t}c(t).

The cycles that can actually be realized form a set $\mathcal C$ , assumed to be a convex cone: the null cycle is realizable; two realizable devices can be run side by side, their heat exchanges adding; and a realizable device can be scaled by any factor $r\ge0$ . A cycle is reversible when $c\in\mathcal C$ and $-c\in\mathcal C$ , and operates between $h$ and $k$ when it exchanges heat with no other reservoir. A Carnot engine between a hot $h$ and a cold $k$ is a reversible cycle operating between them with $c(h)>0$ , $c(k)<0$ and $W(c)>0$ . The efficiency relative to the hot reservoir is $\eta=W(c)/c(h)$ .

In this language:

Clausius statement. For $k<h$ : every realizable cycle operating between $h$ and $k$ with $W(c)=0$ satisfies $c(k)\le0$ .
Kelvin statement. Every realizable cycle exchanging heat with a single reservoir satisfies $W(c)\le0$ .
$\mathcal C$ admits Carnot engines when a Carnot engine exists between every pair of reservoirs at different temperatures.

Formalization targets

Goal: the thermodynamic temperature scale

Assume $\mathcal C$ is a convex cone satisfying the Clausius statement and admitting Carnot engines, and fix a reference reservoir $T_0$ . Then there is exactly one $T:\Theta\to\mathbb R$ with

T>0,\qquad T(T_0)=273.16,\qquad \frac{|e(k)|}{|e(h)|}=\frac{T(k)}{T(h)}\quad\text{for every Carnot engine } e \text{ between } k<h.

The content is that the left-hand ratio, a property of an engine, depends on the engine only through its two reservoirs, and does so as a ratio of numbers attached to the reservoirs. The normalization $273.16$ at $T_0$ is the Kelvin convention; as the article notes, "any reference temperature and any positive numerical value could be used".

The two impossibility statements

\text{Clausius}(\mathcal C)\iff\text{Kelvin}(\mathcal C)

for a convex cone admitting Carnot engines over at least two reservoir temperatures.

Carnot's theorem

For $k<h$ and a Carnot engine $e$ between them, every realizable engine $c$ between the same two reservoirs satisfies $\eta(c)\le\eta(e)$ , with strict inequality when $c$ is irreversible ( $-c\notin\mathcal C$ ); and any two Carnot engines between $h$ and $k$ have equal efficiency.

Significance

The goal makes explicit which features of the realizable processes are used when a textbook defines absolute temperature. The three hypotheses — convex cone, Clausius statement, existence of Carnot engines — are exactly what the derivation needs; the working substance, the equation of state, and the mechanism of a device never appear. Uniqueness is the part that is usually left implicit: the Kelvin normalization at one reservoir pins the scale completely.

The milestones follow the article's own order: equation (1) for the efficiency of an arbitrary engine; the equivalence of the Clausius and Kelvin statements; the three clauses of Carnot's theorem; the composition law $f(T_1,T_3)=f(T_1,T_2)f(T_2,T_3)$ for the reversible heat ratio across an intermediate reservoir; the conversion of that law into a scale; and the resulting Carnot efficiency $\eta=1-T_{\mathrm C}/T_{\mathrm H}$ .

Difficulty

The mathematics is elementary — convex cones in a space of finitely supported real functions, and one functional equation — but the bookkeeping is where faithfulness is won or lost. Carnot's theorem is proved by scaling a candidate engine against a reversed Carnot engine so that the composite does no net work, which makes the Clausius statement applicable; the equality case has to identify the candidate as a positive multiple of the Carnot engine before concluding that it is reversible. The composition law requires the intermediate reservoir to be exactly balanced, so that the composite is again a two-reservoir engine — the article's remark that "the heat $q_2$ flown from the engine 2 to the intermediate reservoir must be equal to the heat $q_2^*$ flown out from the reservoir to the engine 3". The functional equation must be solved without assuming the reference reservoir is the hottest or the coldest, so the scale is defined by $f(T_0,\cdot)$ below $T_0$ and by $1/f(\cdot,T_0)$ above it, and the composition law is applied in four different orderings.

Formalization scope

Reservoir temperatures form an arbitrary linearly ordered type; the order is the empirical hotter/colder relation and nothing else is assumed of it.
Heat is positive into the device. Work is the sum of the heats, which is the first law for a cycle; the first law is therefore built into the model rather than stated.
A device is recorded only by its heat exchanges. Working substance, the states traversed, quasi-staticity and the internal construction do not appear; "reversible" means exactly that the reversed heat exchange pattern is also realizable.
The Clausius statement is formalized for a cycle with zero net work exchanging heat with exactly two reservoirs: that is the article's "without some other change, connected therewith, occurring at the same time".
Out of scope for this mission: the Clausius inequality $\oint \delta Q/T_{\mathrm{surr}}\le0$ , the definition of entropy as a state function, Carathéodory's and Planck's formulations, the statistical-mechanical derivations, and the Clausius–Duhem inequality for continua.

Selected references

Wikipedia, Second law of thermodynamics, revision oldid=1378607758. https://en.wikipedia.org/w/index.php?title=Second_law_of_thermodynamics&oldid=1378607758
S. Carnot, Réflexions sur la puissance motrice du feu, 1824.
R. Clausius, Über eine veränderte Form des zweiten Hauptsatzes der mechanischen Wärmetheorie, Annalen der Physik 93 (1854), 481–506.

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Differential Geometry Mathematical Physics·Captain: Lucas

Einstein Field Equations I: Local Energy–Momentum Conservation in D DimensionsTextbook

Motivation

The Einstein field equations (EFE) are the field equations of general relativity. They relate the curvature of spacetime, encoded in the metric $g_{\mu\nu}$ , to its matter–energy content, encoded in the stress–energy tensor $T_{\mu\nu}$ :

G_{\mu\nu}+\Lambda g_{\mu\nu}=\kappa T_{\mu\nu},\qquad \kappa=\frac{8\pi G}{c^4}.

They were published by Einstein in 1915. A structural requirement on any such equation is that it be compatible with the local conservation of energy and momentum, $\nabla_\beta T^{\alpha\beta}=0$ . For the EFE this compatibility is automatic: the left-hand side is identically divergence-free by the contracted Bianchi identity, so the field equations force conservation of the right-hand side. This mission formalizes that implication together with the standard algebraic reformulations of the equations, following the Wikipedia article Einstein field equations (revision 1374672993).

Setting

Everything is written in one coordinate chart. A point is $x\in\mathbb{R}^D$ , where $D$ is the spacetime dimension (not fixed to $4$ ). A 2-tensor field is a matrix-valued function $x\mapsto T_{ab}(x)$ . A metric on an open set $U$ is a 2-tensor field $g$ whose components are $C^\infty$ on $U$ and which is symmetric and nondegenerate at each point of $U$ ; $g^{ab}$ denotes the inverse matrix. Partial derivatives $\partial_a$ are coordinate (Fréchet) derivatives. Using the Misner–Thorne–Wheeler $(+\,+\,+)$ convention adopted by the source:

Christoffel symbols $\Gamma^c_{ab}=\tfrac12g^{cs}(\partial_ag_{sb}+\partial_bg_{sa}-\partial_sg_{ab})$ ;
Riemann tensor $R^\mu{}_{\alpha\beta\gamma}=\partial_\beta\Gamma^\mu_{\alpha\gamma}-\partial_\gamma\Gamma^\mu_{\alpha\beta}+\Gamma^\mu_{\sigma\beta}\Gamma^\sigma_{\gamma\alpha}-\Gamma^\mu_{\sigma\gamma}\Gamma^\sigma_{\beta\alpha}$ ;
Ricci tensor $R_{\mu\nu}=R^\alpha{}_{\mu\alpha\nu}$ , scalar curvature $R=g^{\mu\nu}R_{\mu\nu}$ , Einstein tensor $G_{\mu\nu}=R_{\mu\nu}-\tfrac12Rg_{\mu\nu}$ ;
covariant derivative $\nabla_\lambda T_{\mu\nu}=\partial_\lambda T_{\mu\nu}-\Gamma^\sigma_{\lambda\mu}T_{\sigma\nu}-\Gamma^\sigma_{\lambda\nu}T_{\mu\sigma}$ and divergence $(\operatorname{div}T)_\mu=g^{\nu\lambda}\nabla_\lambda T_{\mu\nu}$ , the index-lowered form of $\nabla_\nu T^{\mu\nu}$ ;
the field equations on $U$ : $G_{\mu\nu}(x)+\Lambda g_{\mu\nu}(x)=\kappa T_{\mu\nu}(x)$ for all $x\in U$ .

Formalization targets

Goal: local energy–momentum conservation

For a smooth nondegenerate symmetric metric $g$ on an open $U\subseteq\mathbb{R}^D$ , constants $G,c>0$ , $\kappa=8\pi G/c^4$ , any $\Lambda$ , and any $T$ such that the field equations hold on $U$ :

g^{\beta\lambda}\nabla_\lambda T_{\alpha\beta}=0\quad\text{on }U.

Milestones

Symmetry of the Einstein tensor, $G_{\mu\nu}=G_{\nu\mu}$ .
Metric compatibility, $\nabla_\lambda g_{\mu\nu}=0$ .
The differential Bianchi identity $R^\mu{}_{\alpha[\beta\gamma;\varepsilon]}=0$ .
The contracted Bianchi identity, $G^{\alpha\beta}{}_{;\beta}=0$ .
The trace of the EFE, $R-\tfrac D2R+D\Lambda=\kappa T$ .
Equivalence with the trace-reversed form $R_{\mu\nu}-\tfrac2{D-2}\Lambda g_{\mu\nu}=\kappa(T_{\mu\nu}-\tfrac1{D-2}Tg_{\mu\nu})$ for $D\neq2$ .
The vacuum equations $R_{\mu\nu}=\Lambda g_{\mu\nu}/(\tfrac D2-1)$ when $T=0$ , $D\neq2$ .
The cosmological term as vacuum stress–energy $T^{(\mathrm{vac})}_{\mu\nu}=-\tfrac\Lambda\kappa g_{\mu\nu}$ , with $\rho_{\mathrm{vac}}=-p_{\mathrm{vac}}=\Lambda/\kappa$ .
Minkowski space $\operatorname{diag}(-1,1,1,1)$ is a vacuum solution.

Significance

Conservation of $T_{\mu\nu}$ is a consistency condition that any relativistic field equation for gravity has to satisfy; it is also the starting point for deriving the motion of matter (for dust, the geodesic equation) from the field equations alone. The trace-reversed and vacuum forms are the ones used in practice to study vacuum solutions such as Schwarzschild and Kerr, and Einstein manifolds in geometry.

All of these results are classical and fully proved in the literature. What is not yet available is a machine-checked development of the Levi-Civita curvature calculus in coordinates suitable for these statements: Mathlib currently has no Riemann or Ricci tensor of a pseudo-Riemannian metric. The mission produces that layer and the proofs on top of it.

Difficulty

The algebraic milestones (trace, trace reversal, vacuum form, vacuum energy) only need matrix algebra and the symmetry of $g$ ; without symmetry the identity $g^{\mu\nu}g_{\mu\nu}=D$ fails, so symmetry is essential. The analytic core is the contracted Bianchi identity: it needs the symmetry of mixed second and third partial derivatives (Schwarz's theorem on an open set), derivatives of the matrix inverse $g^{-1}$ , and lengthy index bookkeeping with sums over $\operatorname{Fin} D$ . Since all objects are total functions, every derivative step must be justified by differentiability on the open set $U$ , not taken for granted.

Formalization scope

Coordinates: $\mathbb{R}^D$ is represented as Fin D → ℝ; tensors are Matrix (Fin D) (Fin D) ℝ-valued functions; no manifold structure. $D$ is arbitrary, and no Lorentzian signature is imposed (the results hold for any nondegenerate symmetric metric, which includes the Lorentzian case).
Regularity: the metric components are $C^\infty$ on the open set $U$ ; $T$ carries no assumptions, as the field equations determine it on $U$ .
Total-function conventions: the inverse of a singular matrix is $0$ and the derivative of a non-differentiable function is $0$ ; nondegeneracy on $U$ and openness of $U$ make these junk values irrelevant at points of $U$ .
Divergence is taken on the second index; for symmetric tensors such as $G$ this is the same as the first.
Not formalized: the Newtonian limit (it rests on approximations), the Einstein–Maxwell system, and the polynomial form.
Reusable output: coordinate Christoffel/Riemann/Ricci calculus, metric compatibility, Bianchi identities.

Selected references

Wikipedia, Einstein field equations, revision 1374672993. https://en.wikipedia.org/w/index.php?title=Einstein_field_equations&oldid=1374672993
C. W. Misner, K. S. Thorne, J. A. Wheeler, Gravitation, W. H. Freeman, 1973.
S. Carroll, Spacetime and Geometry: An Introduction to General Relativity, Addison Wesley, 2004.

12 thms3 active usersReviewed

Mathematical Physics Quantum Error Correction Quantum Gravity+1·Captain: Lucas

Harlow TASI 2017: Entanglement Wedge Reconstruction, Quantum RT and JLMS are EquivalentResearch Paper

Motivation

In the AdS/CFT correspondence a quantum theory of gravity in an asymptotically anti-de Sitter spacetime is equivalent to a conformal field theory on its boundary. A central question is how local bulk physics is encoded in the boundary theory, and in particular which bulk region is described by a given boundary subregion $R$ . Harlow's TASI 2017 lectures (arXiv:1802.01040) review the answer that emerged from quantum error correction: the boundary Hilbert space is a code, bulk effective field theory lives in a code subspace, and the boundary subregion $R$ can reconstruct every bulk operator in its entanglement wedge. The lectures state as their main technical target the theorem that this entanglement wedge reconstruction, the quantum Ryu–Takayanagi (RT) formula with a central area operator, and the JLMS formula (equality of boundary and bulk relative entropies) are mathematically equivalent. All of these statements are finite-dimensional linear algebra, which makes them formalizable now.

Timeline. Almheiri, Dong and Harlow proposed the quantum-error-correction interpretation of AdS/CFT in 2014 (arXiv:1411.7041). Jafferis, Lewkowycz, Maldacena and Suh derived the JLMS relation in 2015 (arXiv:1512.06431), and Dong, Harlow and Wall established the implications between JLMS and reconstruction for factors in 2016 (arXiv:1601.05416). Harlow proved the general three-way equivalence for arbitrary finite-dimensional von Neumann algebras in 2016 (arXiv:1607.03901, Comm. Math. Phys. 2017). The TASI lectures (2017, published 2018) present this theorem as Theorem 5.2.

Setting

All Hilbert spaces are finite-dimensional. The boundary space factorizes as $\mathcal H=\mathcal H_R\otimes\mathcal H_{\overline R}$ . A code subspace $\mathcal H_{code}\subseteq\mathcal H$ is described by an isometry $V$ whose columns $|\tilde i\rangle$ are an orthonormal basis of $\mathcal H_{code}$ . An operator $O_R$ on $\mathcal H_R$ represents a logical operator $\tilde O$ if $O_R|\tilde\psi\rangle=\tilde O|\tilde\psi\rangle$ and $O_R^\dagger|\tilde\psi\rangle=\tilde O^\dagger|\tilde\psi\rangle$ for every code state.

A von Neumann algebra $M$ on $\mathcal H_{code}$ is a set of operators closed under sums, products and adjoints and containing all multiples of the identity; $M'$ is its commutant and $Z_M=M\cap M'$ its center. Every such $M$ is block diagonal, $\mathcal H_{code}=\bigoplus_\alpha\mathcal H_{r_\alpha}\otimes\mathcal H_{\overline r_\alpha}$ with $M=\bigoplus_\alpha\mathcal L(\mathcal H_{r_\alpha})\otimes I_{\overline r_\alpha}$ . For a state $\tilde\rho$ with diagonal blocks $p_\alpha\tilde\rho_{r_\alpha\overline r_\alpha}$ , the algebraic entropy is $S(\tilde\rho,M)=-\sum_\alpha p_\alpha\log p_\alpha+\sum_\alpha p_\alpha S(\tilde\rho_{r_\alpha})$ , and the algebraic relative entropy $S(\tilde\rho|\tilde\sigma,M)$ is defined analogously (eqs. (5.16), (5.18)).

Formalization targets

Goal: Theorem 5.2 (entanglement wedge reconstruction theorem)

For $\mathcal H_{code}\subseteq\mathcal H_R\otimes\mathcal H_{\overline R}$ and a von Neumann algebra $M$ on $\mathcal H_{code}$ , the following are equivalent:

\text{(1) every } \tilde O\in M \text{ is represented on } R \text{ and every } \tilde O'\in M' \text{ on } \overline R;

\text{(2) } \exists\,\mathcal L_R\in Z_M:\ S(\tilde\rho_R)=\mathrm{Tr}(\tilde\rho\mathcal L_R)+S(\tilde\rho,M),\ \ S(\tilde\rho_{\overline R})=\mathrm{Tr}(\tilde\rho\mathcal L_R)+S(\tilde\rho,M')\ \ \forall\tilde\rho;

\text{(3) } S(\tilde\rho_R|\tilde\sigma_R)=S(\tilde\rho|\tilde\sigma,M),\ \ S(\tilde\rho_{\overline R}|\tilde\sigma_{\overline R})=S(\tilde\rho|\tilde\sigma,M')\ \ \forall\tilde\rho,\tilde\sigma.

Milestones

Theorem 4.1: four equivalent characterizations of a correctable erasure of $\overline R$ (operator reconstruction on $R$ ; $PX_{\overline R}P=\lambda P$ ; decoupling of a reference system from $\overline R$ ; the explicit encoding $|\tilde i\rangle=U_R(|i\rangle_{R_1}|\chi\rangle_{R_2\overline R})$ ).
Eq. (4.25): the quantum Singleton bound $m\ge\frac{n+k}{2}$ for codes whose accessibility depends only on the number of qudits.
Theorem 5.1: every von Neumann algebra on a finite-dimensional Hilbert space is a block-diagonal direct sum of factors, with the matching descriptions of $M'$ and $Z_M$ .

Significance

Theorem 5.2 turns three separately motivated features of holography into a single mathematical statement. Since the RT formula is independently derived from the gravitational path integral, the theorem implies subregion duality in the entanglement wedge; it also shows that a non-trivial area operator requires an algebra with non-trivial center. Theorem 4.1 is the subsystem-code special case, and Theorem 5.1 is the structure result that gives the algebraic entropies their meaning.

The results are proved in the literature (Harlow 2017 for Theorem 5.2; Theorems 4.1 and 5.1 are standard), but the lectures state Theorem 5.2 without proof, and none of these statements has a machine-checked proof that this mission is aware of. The mission produces formal statements and, eventually, formal proofs of the finite-dimensional structure theory of operator-algebra quantum error correction.

Difficulty

The three conditions live at different levels: (1) is a statement about operators, (2) about von Neumann entropies of single states, and (3) about relative entropies of pairs of states. Passing between them requires control of the full block structure of $M$ relative to the bipartition $R\overline R$ , not only of a single factor, and the classical term $\sum_\alpha p_\alpha\log(p_\alpha/q_\alpha)$ of (5.18) couples different blocks, so the general case does not reduce to the factor case treated earlier in the literature. Relative entropies can be infinite (when supports are not nested), so the equalities in (3) are equalities of extended reals and every argument must remain valid at $+\infty$ .

Formalization scope

Hilbert spaces are coordinate spaces $\mathbb C^{\iota}$ for finite index types, operators are complex matrices, tensor products are Kronecker products, and $\mathcal H_{code}$ is the image of an isometry $V$ (an operator $O_R$ acts on code states as $O_R\otimes I_{\overline R}$ applied to $V$ ). States are density matrices (positive semidefinite, trace one). Von Neumann entropy and relative entropy are defined from eigen-decompositions; relative entropy takes values in the extended reals. A von Neumann algebra is a $*$ -subalgebra of matrices; the algebraic entropies are computed in a block decomposition $U$ of $M$ , which every statement takes as an arbitrary decomposition realizing $M$ (the value does not depend on the choice). The RT identity is stated as an equality of complex numbers. Theorem 4.1 assumes a nonzero code subspace; the Singleton bound assumes $d\ge2$ and $k\ge1$ ; both conditions are implicit in the lectures and are needed for the statements to be true.

A trivializing reading is excluded: the hypotheses of every statement are satisfiable, and the goal is a full equivalence of three non-trivial conditions rather than any single implication.

Useful reusable infrastructure: partial traces and their interaction with Kronecker products, entropies of block-diagonal states, relative entropy (Klein's inequality and its equality case), and the structure theory of finite-dimensional $*$ -algebras of matrices. Contributions of proofs of the individual implications (1)⇒(2), (2)⇒(3), (3)⇒(1) as sketches are welcome.

Selected references

D. Harlow, TASI Lectures on the Emergence of Bulk Physics in AdS/CFT, PoS(TASI2017)002, 2018. https://arxiv.org/abs/1802.01040
D. Harlow, The Ryu–Takayanagi Formula from Quantum Error Correction, Comm. Math. Phys. 354 (2017) 865–912. https://arxiv.org/abs/1607.03901
A. Almheiri, X. Dong, D. Harlow, Bulk Locality and Quantum Error Correction in AdS/CFT, JHEP 04 (2015) 163. https://arxiv.org/abs/1411.7041
D. Jafferis, A. Lewkowycz, J. Maldacena, S. J. Suh, Relative entropy equals bulk relative entropy, JHEP 06 (2016) 004. https://arxiv.org/abs/1512.06431
X. Dong, D. Harlow, A. C. Wall, Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality, Phys. Rev. Lett. 117 (2016) 021601. https://arxiv.org/abs/1601.05416

13 thms3 active usersReviewed