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Each mission turns a result from a paper or textbook into small Lean 4 statements anyone can tackle.
Campaigns group missions around a shared mathematical goal. Each one tracks a quantity, such as an upper or lower bound. Have a good candidate in mind? Ping us on Slack, Zulip, or WeChat.
Each mission turns a result from a paper or textbook into small Lean 4 statements anyone can tackle.
Campaigns group missions around a shared mathematical goal. Each one tracks a quantity, such as an upper or lower bound. Have a good candidate in mind? Ping us on Slack, Zulip, or WeChat.
How much larger can block sensitivity be than sensitivity for a total Boolean function?
For a total Boolean function , a block of coordinates is sensitive at if flipping every bit in that block changes . Let count sensitive singletons and be the maximum number of pairwise disjoint sensitive blocks. Define and .
This campaign seeks larger exponents for which there exists an infinite family of total Boolean functions , indexed by , satisfying
Here denotes a quantity tending to zero as . Higher is better. The classical exponent is , and Huang's bound gives a ceiling of .
OpenAI's construction achieves an exponent strictly greater than .
One sufficient way to establish an exponent is to find a single total Boolean function with and . Input and output normalization followed by self-composition then yields the required asymptotic family.
Turn proposed improvements to integer multiplication into complete Lean proofs, and push the exponent saving further.
Harvey and van der Hoeven established an algorithm in 2021. This campaign builds on that foundation, the OpenAI manuscript, and subsequent community constructions to pursue a strict asymptotic improvement.
For two -bit integers, the target is
A positive beats asymptotically; larger is better. Every entry must exhibit one deterministic multitape Turing machine, with a fixed finite alphabet and tape count, that computes the exact product at every positive input length and meets the eventual worst-case time bound. The tracked number measures an asymptotic exponent saving.
Classical algorithms solve 3SUM in time. In a 2026 breakthrough, Alman and Vassilevska Williams gave a deterministic algorithm, refuting the integer 3SUM hypothesis. How low can the exponent go?
Building on existing Lean formalizations, this campaign tracks upper bounds for 3SUM on polynomially bounded integers, using a word RAM with -bit words, and pursues smaller exponents.
Classical algorithms solve all-pairs shortest paths in time. In a 2026 breakthrough, Alman and Vassilevska Williams refuted the APSP conjecture with a deterministic algorithm. How low can the exponent go?
Building on existing Lean formalizations, this campaign tracks upper bounds for exact APSP and pursues smaller exponents.
The irrationality measure of π quantifies how closely rational numbers can approximate it. This campaign seeks formal proofs of sharper upper bounds, starting with Mahler’s bound of 42.
The sharp Hlawka inequality for Schatten -norms is a cousin of the triangle inequality: it relates the norms of three matrices to the norms of their pairwise sums and their total sum. For complex diagonal matrices, an exact formula for the best possible comparison constant has been proved in Lean for every real . We conjecture that the same formula holds for all .
What is the smallest cutoff for which this formula holds for every real ?
References:
Is every odd number a sum of primes? This campaign tracks formalized proofs of the smallest that suffices.
Schnirelmann (1930) showed some finite works. Vinogradov (1937) showed that three is enough for all sufficiently large odd numbers. Tao (2012) proved unconditionally. Helfgott (2013) proved that every odd number greater than is a sum of three primes, though the proof is still unrefereed. Ideally, we can formalize this statement here. Note that three is optimal: is neither prime nor + prime.
Schoolbook matrix multiplication takes operations. The exponent is the infimum of all such that two matrices can be multiplied in arithmetic operations; trivially , and is conjectured but open.
Strassen gave the first nontrivial bound, , in 1969, and introduced the laser method in 1986 to reach . Coppersmith and Winograd's 1990 bound of stood for two decades. Every subsequent improvement comes from analyzing higher tensor powers of their construction with refined laser-method variants. That line reached in 2025, and the current record is , from August 2026. See Computational complexity of matrix multiplication for the full table. Can we formalize these results and even improve on them?
Blood platelets, fresh produce, bakery items and many pharmaceuticals have a fixed lifetime: a unit that is not used within periods of arrival must be discarded. Replenishing such a product is a classical problem of inventory theory (Nahmias 1982), and its optimal policy depends on the whole age profile of the stock, so it cannot be computed for realistic lifetimes. Practice therefore relies on base-stock policies, which raise total stock to a fixed level every period.
Bu, Gong and Chao give the first theoretical guarantees for such a policy. Under first-in-first-out (FIFO) issuance they choose the level by minimizing an explicit surrogate of the long-run cost and show that its optimality gap vanishes in four regimes. This mission is the fourth: a large unit outdating cost . The regime matters in practice because outdating is expensive. Slonim et al. (2014) put the component cost of a unit of blood sold to U.S. hospitals at about $570, and disposing of expired products often requires costly handling.
Demands are i.i.d. copies of a non-negative random variable with and distribution function . The product lives periods. At the start of period the state is , where is the stock with remaining lifetime at most . The firm orders , raising the total stock to (lead time zero). Demand is served oldest first and unmet demand is lost. The quantity outdates, and the state evolves by
The system starts empty. The period cost is
with holding cost , lost-sales penalty and outdating cost .
A policy is admissible if each period's order is a non-negative measurable function of the current state. Its long-run average cost is , and is the infimum of over admissible policies. The base-stock policy orders , and is its long-run average cost. The heuristic level is any minimizer over of
The lower support of demand is .
for every selection of minimizers, with , , and the demand law fixed. The statement holds for every demand law, whether is zero or positive.
The goal's proof also uses inequality (21), . That inequality, together with Lemmas 1–2 and Proposition 2 behind it, is posed in mission 1 of this series and is not restated here.
Theorem 4 says that when outdating is expensive, a policy computed from a one-dimensional convex function is as good as the optimal age-dependent policy. A base-stock policy does not need to track the age distribution, and its level can be found by simulation and golden-section search between explicit bounds. Together with Theorems 1–3 it shows that is asymptotically optimal in all four regimes of the paper. Remark 2 of the paper notes that the plain newsvendor level fails in this regime.
The result is proved in the paper (Management Science, 2022). It has not been formalized. This mission adds a machine-checked model of the FIFO perishable system with lost sales: the age-structured dynamics, admissible policies, long-run average costs and the optimal cost over all policies. The model is reusable for the other missions of the series and for the wider perishable-inventory literature. The paper's proof leaves several steps to the reader: the monotonicity of , the identity , and continuity of the newsvendor cost. Each of these becomes an explicit statement.
The obvious argument compares the heuristic with an optimal policy directly. That fails because the optimal policy has no tractable form. The proof instead goes through (21), a lower bound on obtained by a sample-path argument, and then has to show that the outdating term of the heuristic level vanishes faster than grows. That needs uniform control of as . The level must eventually fall into , where no stock survives periods of demand, and the newsvendor cost there must match the restricted minimum. A naive reading of " decreases in " holds only for the least minimizer. For an arbitrary selection, the argument has to work through the bound (EC.16) alone.
PerishBaseStock.Outdating and builds on the published demand layer CappedBaseStock_Model (i.i.d. demand path, demand law with ) and the newsvendor functions of LostSalesOrderUpTo.Asymptotic.Backorder.Fin (m - 1) → ℝ≥0 of cumulative levels, with . When the state is empty and everything left over outdates. Levels range over .atTop there. The goal asserts that is finite and that the difference of real values tends to , so no convention enters.When customers request a customized product, their requests can reveal information about what they value. A seller can use that information when setting a price, while a customer can choose a request with the resulting price in mind. The question is whether customers with different preferences can make different requests in equilibrium. Liang Guo's The Voice of Customers in Customization, §3.3, studies this question in a two-type model of buyer-led product design, called O-Learning. It compares O-Learning with a separate game in which the seller chooses quality after a customer's cheap-talk message. This mission concerns only the O-Learning game.
Proposition 3 of the SSRN author copy identifies when a particular cheap-signaling equilibrium exists: each type asks for the quality it would prefer if its type were known. Section 3.3 then asks whether a different pair of requests could separate the types when that equilibrium fails. The Appendix, under “No Other Separating Equilibrium under the O-Learning Customization,” establishes nonexistence below the same threshold. It is this latter claim, for every request pair and every admissible off-path belief, that is the mission's target.
A buyer has one-unit demand and a privately known quality preference , where . A fraction of buyers have the high preference. Quality is a real number . Producing or delivering a product of quality costs , while preparing to sell it costs , with . The latter is sunk before the price is determined. These are the assumptions of §2 of the paper, pp. 5–9.
Under O-Learning, the buyer requests a quality. The seller observes the request, forms a perceived type , and accepts or rejects. If rejected, both parties receive zero. If accepted, the price-setting stage follows the paper's reduced-form bargaining rule: the seller has the opportunity to make the price offer with probability and the buyer has it with probability . The buyer's expected payoff, conditional on acceptance, is equation (9):
The seller accepts precisely when , including equality. A separating pure-strategy equilibrium consists of two distinct, nonnegative requests and , plus a belief about the buyer's type after each possible request. The seller identifies the correct type at and ; away from those two requests its belief can vary. Given that belief and the seller's acceptance rule, neither buyer type can improve its payoff by making any other nonnegative request. The pricing stage is represented by the payoff formula rather than by a separate bargaining game, as in §3.3 of the source.
For a buyer whose type is known, equation (10) gives the complete-information request
Proposition 3 defines three ratios and their combined threshold:
The prior is part of the model, although the separation threshold does not depend on it.
The goal says that, whenever the standing assumptions hold,
Equivalently, the existence of any such equilibrium requires . The goal quantifies over all pairs of nonnegative requests and all belief functions consistent at those requests. It does not prescribe the requests or Proposition 3's supporting off-path beliefs.
The milestone list follows the two failure regimes stated in the Appendix, pp. 28–29. When , it first identifies the low type's equilibrium request and then rules out separation. When , it records the form of and the bounds on acceptable equilibrium requests before ruling out separation in either order of and . These are the author's stated intermediate claims, with their source text attached to each milestone.
Proposition 3 shows when the two complete-information requests can be sustained as a separating equilibrium. The Appendix result strengthens the interpretation of its threshold: below , changing the requests or the off-path beliefs cannot restore separation. This matters for the paper's comparison of who controls quality design and when customization can reveal preferences. Without the nonexistence result, the failure of the cheap-signaling request profile would leave open another separating profile with distorted qualities.
The result is proved in the SSRN author copy, and this mission asks for a machine-checked version of that proof. The proposed definitions and statements compile as Lean declarations, but their proofs are open. A completed development would also supply reusable calculations about the seller's acceptance range and buyer payoffs with quadratic costs and endogenous beliefs. The milestones identify these specific pieces of work; they do not claim that a general signaling-game library is already formalized.
Ruling out just the request profile would be insufficient. An equilibrium candidate can use another pair of qualities and can assign either type to requests outside its path. Acceptance changes the payoff discontinuously: a request that would be attractive if accepted instead pays zero when rejected. The buyer's conditional payoff also contains two terms, so a polynomial formula obtained on an accepted range need not hold at every quality. The main challenge is to control all deviations and beliefs under the actual acceptance rule, while maintaining the strict threshold inequalities in both failure regimes.
Lean uses a two-constructor buyer type, a model with real parameters , and a predicate for the paper's standing assumptions: , , , , and . Requests and deviations are restricted to . Beliefs are total functions from real requests to the two named types, constrained only at the equilibrium requests. Seller acceptance is the printed inequality and accepts at equality; a rejected request pays zero. The conditional payoff retains both maxima from equation (9), and is the printed expression, not a value defined through equilibrium existence. No fixed cost for learning or customization is included, as in the paper's main analysis.
The equilibrium predicate does not fix off-path beliefs, require the requests to equal , or replace seller acceptance with a derived threshold. The proof therefore cannot become a restatement of Proposition 3. The local O-Learning definitions duplicate the draft model of the companion mission because unpublished drafts cannot be imported. The paper's last sentence on p. 19 calls the cheap-signaling equilibrium “the only separating equilibrium.” The Appendix proves nonexistence when the ratio is below ; it does not establish global uniqueness when the ratio is above it, and global uniqueness is false. This mission states the Appendix's precise claim.
News publishers depend on direct visits for advertising revenue, while social platforms can show publishers' content inside their own feeds. That arrangement can move reading time from the publisher's site to the platform without changing how much news a consumer reads. It can therefore change the publisher's return from improving content quality. De Cornière and Sarvary study this question in a sequential model of a monopoly newspaper, a social platform, and consumers who divide a fixed amount of attention between the two sites. Their Proposition 1 says that unilateral content bundling lowers the publisher's profit, yet can either raise or lower its equilibrium quality depending on the cost of quality. De Cornière and Sarvary, working paper, pp. 7–12.
This mission concerns the paper's unilateral content bundling regime: the platform can incorporate news into its feed without a payment or a negotiated agreement with the newspaper. The source for theorem numbering is the Toulouse School of Economics Working Paper n° 1152, August 2021. The article later appeared in Management Science in 2023. The working paper's Proposition 1, Lemmas 1–4, and equation (2) are the targets here. The paper separately studies subscription pricing, voluntary participation, and negotiated bundling; those are separate settings. De Cornière and Sarvary, working paper, §§4–6.
A newspaper first chooses news quality . A platform next chooses a news share in its feed. Consumers finally allocate one unit of time between the newspaper's site and the platform. Write for newspaper time and for platform time. The newspaper shows only news. A fraction of platform time exposes the consumer to news; the remaining fraction exposes the consumer to user-generated content (UGC). News and UGC consumption are
The representative consumer obtains utility . When unconstrained by the available mix, the consumer wants news consumption . The platform earns , where one unit of UGC time yields one unit of revenue and one unit of news time yields . The newspaper earns : direct time pays per unit and indirect news time pays . Producing quality costs , giving newspaper profit . De Cornière and Sarvary, working paper, pp. 8–9, equation (1).
A subgame-perfect equilibrium (SPE) specifies consumer attention after every feasible pair , a platform news share after every feasible , and an initial quality. At each stage the chosen action maximizes that stage's payoff given the continuation strategies. The no-bundling benchmark is the same game with , which forces . Consumer surplus is utility at the equilibrium attention allocation. De Cornière and Sarvary, working paper, pp. 9–10, §4.1.
The goal is Proposition 1, comparing every bundling SPE with every benchmark SPE. Both sorts of equilibrium must exist. Publisher profit is strictly lower under bundling. Away from the crossover cost, quality and consumer surplus increase exactly in the low-cost regime:
where . A further clause records that at there is an equilibrium with quality below the benchmark. De Cornière and Sarvary, working paper, p. 12, Proposition 1; p. 31, proof.
The milestones follow the paper's named results and displays: Lemma 1 gives benchmark quality and profit ; Lemma 2 gives consumer attention ; Lemma 3 identifies the platform share ; equation (2) gives the publisher's two-branch profit; the displayed definitions on p. 12 identify the branch maximizers and ; and Lemma 4 characterizes equilibrium quality. The proof of Proposition 1 also states the ordering and relates surplus to the optimal content mix. De Cornière and Sarvary, working paper, pp. 10–12, 31.
The profit and quality effects answer different questions. Even if total news consumption stays at the consumer's preferred level, moving some of that consumption onto the platform reduces the publisher's direct traffic and hence its profit. Quality responds to a comparison of the publisher's gains from direct and indirect readers. The cost threshold distinguishes when the equilibrium moves toward a higher-quality strategy that attracts direct visits and when it moves toward a lower-quality strategy consumed through the platform. Because surplus in this model rises with equilibrium quality, the same threshold governs consumer surplus. De Cornière and Sarvary, working paper, pp. 12–13, 31.
The paper proves these claims mathematically; the Lean theorems in this proposal are open statements awaiting machine-checked proofs. Formalizing the complete sequential game would produce reusable specifications for attention allocation, contingent strategies, and equilibrium comparisons in platform markets. The first-stage profit identity and the best-response theorems are also usable independently of the final comparative-static result. This mission does not assert that the article's other models have already been formalized.
Checking only the realized quality and platform share does not establish subgame perfection. A different quality chosen by the newspaper changes the platform's subsequent action and then the consumer's allocation; those continuation choices must be valid after every feasible deviation. The publisher's resulting objective also changes form at . At the cost the two candidate qualities both maximize profit, so an assertion about “the” equilibrium quality requires care. The article prints a low-quality choice at equality, while the game itself admits both. De Cornière and Sarvary, working paper, pp. 11–12, equation (2), Lemma 4; p. 31, proof.
The Lean model uses real parameters . The standing assumptions are exactly Table 1: , , , and . Quality, attention, and platform share range over , , and respectively. The paper's earlier convention is superseded here by its Table 1 interior range; appears only as the benchmark instance. Real division in the candidate formulas is used under assumptions that make every denominator nonzero. De Cornière and Sarvary, working paper, pp. 8, 10, Table 1; p. 31, proof of Lemma 4.
The pin applies only to Proposition 1's two “if and only if” comparisons. At equality, Lemma 4 is stated with both possible equilibrium qualities, and the goal retains the strict publisher-profit comparison. The platform's share theorem excludes , where platform time is zero and its choice is indifferent. The consumer-attention theorem assumes before dividing by . The benchmark is an SPE with cap zero, and surplus is evaluated from utility; neither is inserted as a closed formula. Equilibrium strategies are defined by best responses after all feasible histories, and explicit existence clauses prevent claims about all SPEs from holding vacuously. These choices rule out defining equilibrium quality, benchmark quality, or surplus by the expressions that the milestones are meant to establish.
The development needs real arithmetic and interval sets from Mathlib, a model-specific strategy profile, utility and revenue definitions, and the three stagewise best-response conditions. Contributions establishing the consumer, platform, or publisher stages, the crossover case, and the final profit and surplus comparison are within scope. Subscription pricing and negotiated bundling require different games and are outside this mission.
Blood platelets, fresh food and many pharmaceuticals have a fixed lifetime: a unit that has not been used within periods of its arrival must be discarded. A retailer of such a product faces the usual trade-off between leftover stock and lost sales, plus the cost of discarding outdated units. The issuance policy, the order in which units of different ages are handed to customers, changes the problem. Under first-in-first-out (FIFO) the oldest units go first, as in a blood bank. Under last-in-first-out (LIFO) the newest units go first, as when shoppers pick the freshest milk at the back of the shelf, so that old stock tends to accumulate and outdate.
The optimal ordering policy of a perishable system depends on the full age profile of the stock (Nahmias 1975) and is intractable to compute except for small lifetimes. Practice uses base-stock policies, which raise the total inventory to a fixed level in every period. Bu, Gong and Chao (SSRN 3724966, accepted in Management Science, 2022) give the first theoretical guarantees for such policies in perishable systems. This mission covers their LIFO results (§4.1).
Demands are i.i.d. copies of a nonnegative random variable with and distribution function . Its lower support is , and is unbounded when for every . Units live for periods. Leftover stock costs per unit, a lost sale costs , and an outdated unit costs an extra .
At the start of period the state is , where is the stock whose remaining lifetime is at most periods, and . The firm orders , which raises the total level to . Demand is served newest-first and unmet demand is lost. The amount outdated and the next state are
The system starts empty. A policy is admissible if each order is a nonnegative measurable function of the current state. Its long-run average cost is
and is the infimum of over all admissible policies. The base-stock policy orders , and its cost is .
The paper bounds between two explicit functions (Lemma 3):
The heuristic level is any minimizer of on . The newsvendor level is . The proof also uses and its minimizer .
All three parts are stated for every minimizer , and in (b) and (c) for every selection of minimizers as the parameter varies.
Part (a) gives a gap of order that holds for every demand law with finite mean. Parts (b) and (c) identify regimes in which a single explicit base-stock level is asymptotically optimal. Part (c) is sharp: the companion Proposition 1 (mission 6 of this series) shows that when and , even the best base-stock policy stays a positive distance from optimal as . Under FIFO the corresponding gap decays exponentially in (Theorem 1, mission 1), so the two issuance rules behave differently. Whether the LIFO gap also decays exponentially is left open in the paper.
The results are proved in the paper but are not machine-checked anywhere. Prove2Me has no perishable lost-sales model with LIFO issuance. This mission provides one, built on the published demand layer and newsvendor functions, together with the sample-path recursion (EC.18), which is specific to LIFO.
is a long-run average over a Markov chain on an -dimensional state, and no closed form is known. The upper bound needs an accounting argument over windows of periods. The lower bound needs the exact recursion (EC.18) for the outdating of a base-stock policy under LIFO, which differs from the FIFO recursion. Treating the system as a newsvendor problem with a correction term fails: the outdating cost of lies strictly between the two bounds, and the minimizer of one bound is not a minimizer of the other. In (b) the absolute gap does not vanish, so the argument needs . In (c) the levels and move with , and the limits must be controlled uniformly over all minimizers.
Fin (m - 1) → ℝ of cumulative levels, with . The case (empty state) is included.leadTimeLaw.No trivializing reading is admitted. OPT ranges over all admissible policies, not only base-stock ones. is not a formula. No gap can hold through or division by zero. The parameters , , and the demand law are not specialized.
Proposition 2 holds under any issuance policy. Its FIFO instance and the FIFO lemmas are posed in mission 1 of this series, and only the LIFO instance is posed here. Lemma 3 and (EC.22) are also used in mission 6. Proofs of any milestone are welcome. Reusable infrastructure includes the law of the maximum of i.i.d. demands, convexity and continuity of the newsvendor cost, and existence of minimizers of and .
Social services such as public housing, food banks and free clinics are congested: the users who arrive wait in a queue they cannot see, and some of them could do almost as well elsewhere. A provider who sees the queue can decide what to tell arriving users about it. Telling low-need users that the queue is long sends them to their outside option and shortens the wait for everyone else; telling them too much may send away users the provider would rather serve. Information design (Bayesian persuasion, Kamenica and Gentzkow 2011) studies what a sender who commits to a signaling rule in advance can achieve.
Anunrojwong, Iyer and Manshadi (2023) build on the queueing persuasion model of Lingenbrink and Iyer (2019) and ask how much welfare information design can add in a congested service. Their answer depends on whether users are heterogeneous. This mission formalizes the homogeneous case, Proposition 1: when every user is low-need, the simplest policy, full disclosure of the queue length, is already within an explicit factor of the best signaling mechanism.
A single server with exponential service times of rate one serves users first come, first served. Low-need users (type L) arrive as a Poisson process of rate and may leave for an outside option worth ; high-need users (type H, rate ) always join. The total rate is . A type- user who joins with users present receives utility , where , is strictly decreasing, for large , and the increments are non-increasing (Assumption 1).
By Lemma 1 of the paper, an obedient signaling mechanism is described by the stationary distribution of the queue length on it induces. The admission policies are the probability distributions with
and the signaling mechanisms are those that also satisfy the obedience constraints
The welfare of low-need users is .
A threshold policy with , , , admits L-users below queue length , admits them with probability at , and turns them away above . denotes . The full-information threshold is the least with , and the full-information mechanism is . The homogeneous setting is .
In the homogeneous setting, with
the goal asserts three facts:
The paper phrases parts 1 and 3 for "the optimal signaling mechanism sm". They are stated here for every element of , which implies the paper's form for any maximizer and requires no existence result.
Proposition 1 bounds what persuasion can do when users are alike. The ratio is at most , so the gain from optimal signaling is small whenever full information already admits a few users. The paper contrasts this with the heterogeneous setting (§4.2), where information design can Pareto-improve on full information substantially. The homogeneous bound is the baseline for that comparison: it shows that the gains there come from the presence of two types, not from pooling queue states.
The result is proved in the paper. Nothing in this model has been machine-checked so far. A formal proof would check the reduction from an arbitrary signaling mechanism to a threshold mechanism (Theorem 1), the unimodality of threshold welfare (Lemma 5(2)), and the obedience calculation on p. 40. It would also confirm that these hold at , where the stationary distribution of the always-join policy no longer exists. The model layer defined here is shared with three sibling missions on Theorems 1 and 6 and Proposition 3 of the same paper.
The two finite inequalities, the bound on and the comparison of with , are elementary. The difficulty lies in part 1 for an arbitrary , which is a distribution on an infinite state space. Restricting attention to threshold policies is not free: Theorem 1 only says that Pareto-efficient mechanisms are thresholds. An arbitrary obedient must either be shown to be dominated by a threshold mechanism, or bounded directly. The sets are not compact in the obvious topology at , so the existence of an optimal mechanism cannot simply be assumed.
Part 3 needs a strict comparison. When , the proof must exhibit an obedient threshold policy strictly better than full information, which requires the obedience constraint to hold for thresholds slightly below .
Lean represents queue lengths as ℕ and distributions as functions ℕ → ℝ, inside the namespace NeedPersuasion.Homog. Every statement carries the standing assumptions M.Standing: , , , , for some , strictly decreasing, and non-increasing. The homogeneous statements add . The type-H utility stays a field of the model so that the definitions coincide with those of the sibling missions; with it does not enter .
Conventions the formalization commits to:
The following trivializations are excluded: defining "sm" as , defining it through sSup of an unbounded set or Classical.choose, computing from Lemma 5's closed form, or stating part 3 with both sides about the same threshold. A sorry-free sanity check exhibits a model satisfying all hypotheses (, , , ) with .
A complete development needs:
ℕ.The perturbation arguments behind Theorem 1 and the lemmas about threshold policies are reusable in the sibling missions. Proofs of any milestone are welcome. So are intermediate lemmas, such as closed forms for and of a threshold policy in the homogeneous case.
Social services such as public health clinics, legal aid offices and food banks are congested: users wait in a queue whose length they cannot see. Some users have no alternative (high need) and others have an outside option (low need). A provider cannot price access, but can decide what to tell arriving users about the queue. Anunrojwong, Iyer and Manshadi (Management Science, 2023; read as arXiv:2005.07253v3) study how far such information design can raise the welfare of both groups. Their model builds on the obedient signaling framework of Lingenbrink and Iyer (Operations Research 67(5), 2019). This mission formalizes their Section 5. It compares the best signaling mechanism with the first-best: the admission policy a provider would choose if it could see user types and order users to join or leave.
A single server with exponential service times of rate one serves an unobservable first-come-first-served queue. High-need users (type H) arrive at Poisson rate and always join. Low-need users (type L) arrive at rate and either join or take an outside option worth . Write . A type- user who joins behind users gets utility . The standing assumptions are , , each strictly decreasing, and the increments non-increasing (Assumption 1).
An admission policy is a distribution on with
their set is . It is the steady state of a queue in which a low-need user arriving at length is admitted with probability . A signaling mechanism is an admission policy that low-need users obey when it is announced as a join/leave recommendation. The obedience constraints are
and the signaling mechanisms form . The welfares are and . For a weight , . Then is the set of maximizers of over , and the set of maximizers over . Under no information (ni) every low-need user joins with the same equilibrium probability . Finally,
is a low-need user's expected utility in an M/M/1 queue with load . It has a unique root .
The goal does not fix or give it in closed form. It asserts only that exists, that is positive, and that the optimal signaling mechanism splits into these two regimes.
In the order the proof in Appendix D uses them:
Theorem 6 says that over a large range of weights the obedience constraints cost nothing. The optimal signaling mechanism coincides with the first-best admission policy, even though that benchmark presumes the provider sees user types and can enforce decisions. In that range the signal acts only as a coordination device, and no user is indifferent between the recommended action and the alternative. This differs from typical Bayesian persuasion, where optimal schemes make some receiver indifferent. The two regimes of the theorem coincide with those of Theorem 5. Where no information is already Pareto-efficient, efficient signaling and efficient admission agree for all weights. Elsewhere, signaling falls short of the first-best only for weights that favour high-need users.
The paper proves these results on paper. To our knowledge none of them has a machine-checked proof. The mission produces a formal model of obedient binary signaling in an unobservable queue, with admission policies, obedience constraints, welfare and Pareto efficiency within a class. It also formalizes the structural theorems and the first-best comparison. Formal proofs would also check the steps the paper calls "straightforward" or leaves out, such as Theorem 2's omitted proof and the coupling arguments.
Each step of the proof uses a different tool. The threshold monotonicity in is a revealed-preference argument, and it needs of threshold policies to be strictly decreasing. Obedience propagates upward only along threshold policies, through the shape of in Lemma 6. The positivity of needs the optimal admission policy for small to violate (LEAVE). Below , the binding (LEAVE) constraint must pin a single threshold. The tempting shortcut is to argue only with the scalarized problem , and it fails for two reasons. At maximizers need not be Pareto-efficient (footnote 12). Also, and are sets, so set equality needs both inclusions and non-emptiness. Existence (Lemma 2) rests on a weak-compactness argument over distributions on , which is delicate at .
Hypotheses added relative to the page, all stated in the items:
Encodings that would trivialize the goal are excluded:
The work needs:
The model layer and the threshold structural theorems can be reused by the companion missions of this series. Proofs of individual milestones are welcome independently.
Perishable products (blood platelets, fresh food, pharmaceuticals) have a fixed lifetime: units not sold within periods expire and are discarded. Managing such inventory well is a classical problem in operations research, going back to Nahmias (1975). The difficulty is that the state of the system is the whole age profile of the stock, so the optimal policy depends on an -dimensional state and is intractable for realistic lifetimes. In practice firms use base-stock policies, which raise total stock to a fixed level every period, but until recently there were no theoretical guarantees for them in the perishable setting.
Bu, Gong and Chao (SSRN 3724966, accepted by Management Science, 2022) give the first such guarantees. Under FIFO issuance with zero lead time they define a simple heuristic level and bound its optimality gap explicitly (Theorem 1). This mission formalizes their Theorem 2: when demand is generated by a large population of customers, the optimality gap of the heuristic base-stock policy decays exponentially in the population size . This regime corresponds to high-volume products, where demand is the sum of many small individual requests.
The inventory system. Time is divided into periods . The product lasts periods. At the start of period the state is , where is the stock whose remaining lifetime is at most periods. The firm orders , raising total stock to (zero lead time). Demand is served oldest-first (FIFO); unmet demand is lost. The quantity expires, and the remaining stock ages by one period:
The system starts empty. Each period costs with , , . A policy chooses by a non-negative measurable function of the current state. Its cost is , and . The base-stock policy orders ; its cost is .
The heuristic level. With i.i.d. copies of the one-period demand , let
and let be any minimizer of over . is the newsvendor level of .
Demand population form (7). For a positive integer , the one-period demand is
where the customer demands are i.i.d., non-negative, with mean and variance , and the are i.i.d., non-negative, independent of the , with , variance , and for some . Thus is the number of renewals in ; when , and is compound Poisson with mean when the are exponential. , , , denote the objects above for demand , with fixed.
There exist constants , such that, for all sufficiently large and every minimizer of ,
The constants may depend on and the laws of and . The goal asserts only the exponential rate, not the explicit constants of the proof.
The proof also uses the FIFO results (Lemmas 1–2, Proposition 2 and inequality (21)) that are posed in mission 1 of this series; they are not restated here, and (23) is their consequence at the population demand.
The result. Theorem 2 shows that the heuristic base-stock policy, which is computed from a one-dimensional convex function, is nearly optimal for high-volume perishable products: the loss from ignoring the age profile vanishes exponentially as demand scales. Together with Theorem 1 (exponential decay in the lifetime) it explains why simple base-stock rules perform well in practice, and it gives a quantitative criterion for when a firm can stop worrying about the optimal policy.
Formalizing it. The result is proved in the paper; no part of it is machine-checked. A formalization adds a checked perishable inventory model (FIFO dynamics, admissible policies, long-run average cost), a random-sum demand model built on a renewal counting process, and checked versions of three classical probabilistic tools that the proof cites: Scarf's minimax bound for the newsvendor, Wald's identity with the law of total variance for random sums, and a lower-deviation bound for renewal counts from Bernstein's inequality for sub-exponential variables. The paper's justification of cites convergence in distribution, which does not imply convergence of variances; the statement is still true, and a formal proof has to supply the renewal variance argument.
The cost comparison itself is inherited from Theorem 1; what is new is controlling both factors of (23) as . The level grows linearly in , so the factor grows exponentially, and must be beaten by the decay of . Neither factor is explicit: is a quantile of a compound distribution, and has a random number of summands. Simply replacing by its mean is not possible, because the lower tail of must be shown to be exponentially small with a rate that does not degrade with ; this needs a concentration inequality for the renewal count, not only a law of large numbers. When the right side of (23) is zero, so the substantive case is , where the constants must be balanced to make the exponent negative. None of the needed tools (renewal theory, sub-exponential Bernstein, Scarf's bound) is in Mathlib.
PerishBaseStock.Population. Demand laws are CappedBaseStock.DemandLaw (non-negative, finite positive mean); the newsvendor functions come from LostSalesOrderUpTo.Asymptotic.Backorder; the renewal count is the counting variable N of SmithRenewal.Elementary at time .Fin (m - 1) → ℝ≥0 holding cumulative levels, with ; is allowed.Perishable products, among them fresh food, blood platelets and pharmaceuticals, lose their value after a fixed number of periods. Their inventory control has been studied since the 1970s (Nahmias, Perishable inventory theory: a review, Operations Research 1982). The optimal policy of the standard lost-sales model depends on the whole age profile of the stock and is computationally intractable when the lifetime is long. Practitioners therefore use base-stock policies, which raise the total inventory to a fixed level every period. Before Bu, Gong & Chao (SSRN 3724966, accepted at Management Science, 2022), no guarantee was known on how far the best such policy, or any explicitly computable one, is from optimal.
This mission concerns the regime of a large unit penalty cost. Firms in retail and manufacturing often run at very high service levels (Huh et al. 2009; Wei et al. 2021), which in a cost-based model means that the unit lost-sales penalty is much larger than the unit holding cost . The question is whether a simple base-stock policy remains near-optimal as grows.
A product has a fixed lifetime of periods. Demands are i.i.d. copies of a non-negative random variable with and distribution function . At the start of period the state is , where is the stock whose remaining lifetime is at most periods, and . The firm orders and the total level becomes (zero lead time). Demand is served oldest first (FIFO issuance) and unmet demand is lost. The outdated quantity is , and the dynamics are
starting from the empty state . The period cost is , with , and an additional outdating cost . An admissible policy orders with each a non-negative measurable function of the state. Its long-run average cost is , and over all admissible policies.
The base-stock policy orders , and is its average cost. is its long-run average outdated quantity. The heuristic level is any minimizer over of
Two further objects appear: the upper support (demand is bounded when ), and the floor level . A subscript marks dependence on the penalty: , , .
with , , and the demand law fixed, for every selection of minimizers. No boundedness assumption on demand is made.
Theorem 3 shows that the explicitly computable policy is asymptotically optimal in the high-service regime. The optimal policy is a function of the whole age vector, whereas is the minimizer of a one-dimensional convex function. Together with the companion results of the paper, in the lifetime , the demand population and the outdating cost , this gives a uniform picture of when a single stock level suffices for a perishable product. Lemma 6 and Proposition 3 are of independent interest: a uniform floor on the optimal order-up-to level and a lower bound on the optimal average cost that does not require solving the dynamic program.
The result is proved in the paper; it has not been formalized. A machine-checked development requires a perishable inventory model, long-run average costs over an infinite i.i.d. demand path, and, for Lemma 6, the existence of an average-optimal policy. The last step requires a vanishing-discount argument.
The unbounded-demand case rests on (25). The lower bounds it combines (Lemma 2 and Proposition 2 of the paper) are posed in mission 1 of this series and only cited here. The bounded-demand case is harder. One would like to compare with an optimal policy directly, but a priori there is no optimal policy at all: is an infimum over a large class of state-feedback policies on a continuous state space. Lemma 6 asserts that the infimum is attained, and attained by a policy whose order-up-to level never drops below . The paper obtains it from the discounted problem (Nandakumar & Morton 1993) and a vanishing-discount limit, verifying the conditions of Schäl (1993, Theorem 3.8) and in particular a uniform bound on relative discounted values (EC.34). Schäl's theorem is posed separately on Prove2Me as a statement (SchalAvg.AvgOpt.theorem_3_8, not proved), which is context and not part of this mission. Bounding the lost-sales term by a quantity independent of , as (EC.13) does, needs the first-order condition at a minimizer of , which is a non-smooth convex function.
Fin (m - 1) → ℝ of cumulative levels, with , so (empty state, everything left over outdates) is included, not excluded.CappedBaseStock.demandPathLaw), valued in . Long-run averages are limsups of Cesàro averages.Trivializing formalizations are ruled out. OPT is not restricted to base-stock policies, is not defined by its formula (3), is not replaced by a closed form, and , , and the demand law are kept general.
The FIFO lemmas the proof uses, namely Lemma 1, Lemma 2, Proposition 2 and the recursion (19), are posed in mission 1 of this series and are not restated. The development reuses the published demand layer CappedBaseStock_Model and the newsvendor functions LostSalesOrderUpTo.Asymptotic.Backorder. A general vanishing-discount theorem for Borel-state average-cost control would be reusable well beyond this mission; contributions toward it, and toward the convexity and first-order conditions of newsvendor-type costs, are welcome.
Public and nonprofit services such as food banks, shelters, free clinics and legal aid often run with too little capacity and serve two kinds of users: people with no alternative, and people who have a reasonable outside option. When the queue is long, a user who has an alternative but joins anyway imposes waiting costs on everyone behind them, including the users who cannot go elsewhere. Pricing is usually ruled out in such settings, which leaves the provider one lever: what it tells arriving users about congestion.
Anunrojwong, Iyer and Manshadi (arXiv:2005.07253v3; Management Science, 2023) study this lever as a Bayesian persuasion problem in a queue, building on the revenue-maximization model of Lingenbrink and Iyer (Operations Research 67(5):1397–1416, 2019). The natural benchmark is full information: announce the queue length and let users decide. This mission formalizes the paper's answer to when full information can be improved for both user types at once, and by how much.
A single server serves users first come, first served, with exponential service times of rate . High-need users (type H) arrive as a Poisson stream of rate and always join. Low-need users (type L) arrive at rate and either join or take an outside option worth . Write . A type- user who joins with users already present receives utility . The standing assumptions are , for some , each strictly decreasing, and the increments non-increasing (Assumption 1).
By a revelation-principle argument (Lemma 1 of the paper, from Lingenbrink and Iyer), a recommendation mechanism that users obey is described by the queue's stationary distribution . An admission policy is a distribution with for all ; their set is . A signaling mechanism is an admission policy that also satisfies the obedience constraints
called (JOIN) and (LEAVE); their set is . The welfare of the two types is and . A policy Pareto-dominates if it is at least as good for both types and strictly better for one; is Pareto-efficient within a class if no member of the class dominates it.
A threshold policy , , admits L users when fewer than users are present, admits them with probability at , and turns them away above . Write , and for its welfare and (LEAVE) value. The full-information threshold is the smallest with , and the full-information mechanism is .
For , with fixed, there is a threshold such that some signaling mechanism Pareto-dominates if and only if , and
If moreover , , , and , then a signaling mechanism exists with
where and .
The proposition gives a clean regime split. Full information is already Pareto-efficient when low-need demand is light. Once crosses an explicit level, revealing everything leaves welfare on the table for both types, so a provider who cares about either group has a reason to hide some queue information. The lower bound on is explicit in the primitives, and the factors give a guaranteed improvement, not only an existence statement. Theorem 4, the milestone behind part 1, reduces the comparison to two adjacent thresholds.
The paper proves these results on paper. To our knowledge none of them has a machine-checked proof. A formal development checks the "little algebra" steps that the paper compresses (the equivalence with , the welfare identities of p. 41–42). It also produces a reusable layer for persuasion in birth–death queues: obedient mechanisms as distributions, threshold policies, and Pareto efficiency within a class. The companion missions of this series (threshold structure, the homogeneous case, first-best) are stated over the same layer.
The threshold comes from analysing the sign of on . Translating "fi is not dominated" into that sign condition is where the work lies. The obvious attempt, comparing only with nearby threshold policies, settles one direction: if , the threshold dominates. The converse needs every competitor in , not just threshold policies. That requires the structure theorem for efficient admission policies, the existence of maximizers, and unimodality of , followed by a transfer from to that only works because (LEAVE) is slack at . The welfare series are infinite, and the threshold family is parametrised by a real whose fractional part enters nonlinearly, so the comparisons are not finite-dimensional linear algebra.
Everything is stated over a structure Model with real fields and utilities , with a predicate Standing holding the standing assumptions ( is encoded as ). Queue lengths are natural numbers, and the two types are separate fields. Membership in includes summability of and : the paper treats all welfare as finite, and an unrestricted tsum would assign to divergent welfare. is the right-hand form of (2), with no division by . A finite threshold requires , and a joining fraction in , as in Definition 2. Threshold policies are defined from their unnormalised weights, is as a distribution, and Pareto efficiency is Definition 1 within an explicit class. "Exactly one" is exclusive or.
Added hypotheses, each disclosed in the item that carries it:
The page's range is intersected with . In part 2, "" is read existentially.
Ruled out as trivializing: choosing after (the goal quantifies it once, with fixed); defining through its welfare; replacing Pareto efficiency by maximization of a weighted welfare; replacing exclusive or by inclusive or; and any constant that differs from the page's , or lower bound symbol for symbol.
A complete development needs geometric and arithmetico-geometric series, rearrangement of infinite sums over , and convexity arguments in the space of summable sequences. Proofs of any milestone are welcome, as are lemmas about threshold policies that the other missions of the series can reuse.
Many social services are rationed by waiting rather than by price: public health clinics, food banks, shelters, legal aid. Users who need the service most often have no outside option, while users with less need could be served elsewhere but join the queue anyway and lengthen everyone's wait. The provider usually cannot force anyone to leave, and charging money defeats the purpose of the service. What the provider does control is information: it observes the queue and decides what to tell arriving users about it.
Anunrojwong, Iyer and Manshadi (2023) model this as a Bayesian persuasion problem in a queue, building on the revenue-maximization model of Lingenbrink and Iyer (2019, Operations Research 67(5):1397–1416). Their first structural result, Theorem 1, says that whatever trade-off the provider makes between the two user groups, the efficient way to signal is a simple threshold rule, and that the threshold never exceeds the one under full information. This mission formalizes Theorem 1 and the steps of its proof.
A single server serves users first-come-first-served; service times are exponential with rate one. Users of two types arrive by independent Poisson processes: low-need users (type ) at rate and high-need users (type ) at rate , with total rate . A type- user who joins with users already present receives utility , where , for large , is strictly decreasing, and the increments are non-increasing (Assumption 1). High-need users always join; low-need users take an outside option worth unless joining is at least as good in expectation.
By the revelation principle and Lemma 1 of the paper, an obedient binary signaling mechanism is identified with its steady-state distribution of the queue length. The admission policies are the probability distributions on satisfying the detailed-balance constraints
The signaling mechanisms additionally satisfy the obedience constraints
The welfare of the two groups is and . A mechanism Pareto-dominates if for both , with at least one strict inequality; is Pareto-efficient within if no dominates it (Definition 1).
A distribution has a threshold structure (Definition 2) if for some , for and for : low-need users are told to join below , to leave above , and to join with probability at . The threshold is . The full-information threshold is the smallest with : when the queue is revealed, low-need users join exactly below it.
For every that is Pareto-efficient within , there is a finite such that
No weight is fixed: the statement covers the whole Pareto frontier.
Theorem 1 reduces an infinite-dimensional design problem (a distribution on subject to obedience) to a one-parameter family of threshold policies . Every later result of the paper is computed in that family: the bound of the optimal signaling mechanism against full information when all users are low-need (Proposition 1), the condition under which information design Pareto-dominates full information (Theorem 4 and Proposition 3), and the conditions under which signaling achieves the first-best admission policy (Theorems 5 and 6). The bound also has a direct reading: efficient persuasion only ever discourages low-need users relative to full information, never encourages them.
The theorem is proved in the paper (App. A). This mission produces a machine-checked version of the proof, on top of a reusable Lean model of obedient signaling in an -type queue with two user types. To our knowledge no part of this model or of its results has been formalized before.
The statement quantifies over the full Pareto frontier, so it cannot be obtained from first-order conditions of a single weighted objective. Both steps are perturbation arguments on infinite sequences: each must build an explicit competitor, keep it inside the detailed-balance constraints and both obedience constraints at once, and show weak improvement for one group with strict improvement for the other. The obedience constraint (LEAVE) cuts against (JOIN): moving mass toward shorter queues raises but can raise , and the bound on is where is needed. Since is allowed, the geometric tails are not automatically summable and the infinite sums have to be handled with care. The threshold must be excluded explicitly; it is not ruled out by the structure argument alone.
All definitions live in NeedPersuasion.Threshold (file Def_NeedPersuasion_Threshold_Model). Queue lengths are ℕ, everything else is ℝ; the two types are two fields of a Model structure. The standing assumptions are the predicate Model.Standing: , , , , for some (equivalent to for a strictly decreasing sequence, and allowing ), both utilities strictly decreasing, and the -increments non-increasing. Every theorem assumes it; no other hypothesis is added, and in particular is not assumed.
Committed conventions:
HasSum, and finite welfare: and summable. The paper treats the welfare sums as real numbers; without this, a policy with welfare (possible at ) would receive welfare from Lean's tsum.HasThresholdStructure; HasThresholdAtMost π y asks for a finite with .ParetoEfficientIn), never maximization of .Trivializing encodings are ruled out: the threshold-structure predicate is not satisfied by every admission policy, efficiency is taken over all of (including the competitors the proof builds), and no sum whose value is compared can be a divergent tsum. A sorry-free sanity file exhibits a model satisfying Standing whose full-information mechanism lies in , so the hypotheses are not vacuous.
A complete development needs manipulation of tsum over shifted and truncated sequences, geometric tails, and first-order stochastic dominance for sequences on ; these pieces are reusable for the other missions of this series (missions 2–4 restate the same model). Contributions of proofs of individual milestones, or of general lemmas on detailed-balance sequences, are welcome.
Many models in statistics, machine learning and engineering are nonconvex, nonsmooth and coupled by constraints: a sparsity penalty such as SCAD, MCP or capped- on one block of variables, a smooth loss coupling the blocks, and a nonlinear equality constraint tying them together. Augmented-Lagrangian methods and ADMM-type splittings are the workhorses for such problems, but most convergence theory for them assumes linear constraints or convexity. Zhu, Zhao and Zhang (arXiv:2007.12219v1, published in Mathematics of Operations Research, 2024) propose a first-order primal-dual method, NAPP-AL (nonconvex auxiliary problem principle with augmented Lagrangian), for a nonconvex problem with a nonlinear coupling constraint, and prove that it reaches approximate stationarity at a sublinear rate. This mission targets that rate result and the chain of lemmas behind it.
Fix dimensions . Problem (P) is
where and are differentiable with Lipschitz gradients, is lower semicontinuous and possibly nonconvex and nonsmooth, is closed and convex, is Lipschitz with constant , each component of has a Lipschitz gradient (constants , ), and is an matrix with . These are Assumption 1 (H1)–(H7); (H7) asks to be bounded below and coercive on the feasible set .
The augmented Lagrangian of (P) at is
Its stationarity set (p. 6) collects the triples with, for ,
where is the limiting subdifferential and the normal cone of . A stationary point is a with , and , with .
NAPP-AL takes a -strongly convex with Lipschitz gradient (Bregman distance ), a penalty (Assumption 2) and . With it sets to a global minimizer over of the linearized model
to the minimizer of a strongly convex quadratic, and . The step lies in , with given by the explicit formula (8), which depends on . The analysis runs through the potential
with explicit constants .
Under Assumptions 1 and 2, with set by (8),
The result gives NAPP-AL an iteration complexity of to reach an -stationary point (Remark 4.1), for problems with a nonlinear, nonconvex coupling constraint and a nonsmooth nonconvex regularizer, using only gradient evaluations and one proximal-type subproblem per iteration. It is also the base layer for the paper's later results: linear convergence under an error bound (Theorem 5.1) and the sufficient conditions for that error bound (Propositions 6.1–6.2) all start from Theorem 4.1 and Lemma 4.3.
The result is proved in the paper; nothing here is formalized elsewhere. A formal development adds a machine-checked Lyapunov analysis of an augmented-Lagrangian method with nonlinear constraints, with every constant explicit. It also records a correction: Proposition 3.1 as printed is false by a factor of at most , and the corrected bound is what the paper's proof establishes.
The difficulty is the dual variable. The multiplier update has no step size of its own, so the augmented Lagrangian itself need not decrease along the iterates, and the obvious descent argument for fails; the step of (8) depends on the current multiplier estimate , so even the step sizes are not uniformly bounded a priori. Boundedness of the iterates is not a hypothesis, and the iterates are infeasible, so coercivity of on the feasible set does not apply to them directly. The rate needs the residual bound of Proposition 3.1 with a constant uniform in , while depends on and . Passing to cluster points involves the limiting subdifferential of a nonconvex, nonsmooth , which is not closed under arbitrary convergence of its arguments.
EuclideanSpace ℝ (Fin k). The norm of a triple is the Euclidean norm of the stacked vector, defined explicitly; Mathlib's sup norm on products is never used. is a continuous linear map with its operator (spectral) norm and adjoint; is the Rayleigh minimum over the unit sphere.Metric.infDist (zero on ) is not used; with it, every residual bound would hold whenever is empty. The goal is stated as in , not as a big-O bound.NonconvexSplitting.Shared.LimitingSubdiff) and the normal cone of , which is empty off , so that stationarity forces .LocallyLipschitzOn); (full column rank, invoked in the proof of Lemma 4.2); (H7) is coercivity on the feasible set only.Contributions welcome: proofs of the milestones in any order, general facts about limiting subdifferentials and normal cones of convex sets (sum rule with a locally Lipschitz function, closedness under attentive convergence), and the "non-increasing summable sequence is " lemma. These are reusable beyond this mission. The source is the preprint arXiv:2007.12219v1, not the published version; all labels and constants are those of v1.
Retailers and restaurants increasingly deliver orders from their own stores within an hour or less. Orders arrive throughout the day, a fixed pool of drivers is based at the store, and in each dispatch wave the operator must decide how many drivers to send out now and along which routes. Sending many drivers now gives short delivery times for the current orders but leaves few drivers for orders that arrive while those drivers are still on the road. Liu and Luo (arXiv:2107.13058, Manufacturing & Service Operations Management, 2023) model this trade-off as a finite-horizon stochastic dynamic program and propose a rollout policy built on a tractable approximation of its cost-to-go. Their Theorem 1 justifies the approximation: with suitable parameters it lies below the true expected cost-to-go, with other parameters it lies between that cost and the cost of any feasible simple policy, and in the last period it is exact. This mission formalizes that theorem.
A store (the depot ) serves a finite set of customer locations over periods with strictly increasing decision epochs . A fleet of drivers carries at most items per trip; travel times , an on-site service time and a delivery target are given. At epoch the operator observes the demand of period : the set of locations with an order and the quantities , drawn from a finitely supported law , independently across periods.
The driver status records how many drivers are still en route in each period because of earlier dispatches, so drivers are available at . A decision is a set of depot-to-depot routes , one per dispatched driver. It is feasible if it uses at most drivers, serves every current order exactly once, loads at most items per route, and delivers every order within . The delivery times are and ; the cost of is . A driver whose route lasts longer than is counted as en route in period ; this is the transition (9) from to .
The cost-to-go is the minimum over feasible of the period's cost plus at the resulting status, with the period cost alone ((7)–(8)). The single-period cost is the minimum cost of serving the demand with at most drivers.
The approximation (10) replaces the future by single-period problems coupled only through driver counts:
where and the nonnegative parameters , , estimate how many of the drivers dispatched in period are still en route in period . A static myopic policy dispatches a state-independent number of drivers in each period and routes them to minimize the single-period cost; it is feasible from if it never asks for more drivers than are available and its single-period problems are always feasible.
For and a status with :
and for every .
The rollout policy AJRP of the paper solves, at each epoch, the current routing problem together with as a stand-in for the future. Theorem 1 is what connects that stand-in to the dynamic program: it shows that the en route parameters move the approximation between a valid lower bound and the cost of an implementable policy, and its exactness part yields the paper's Corollary 1 (AJRP is optimal for ). Lemma 1 shows the approximation keeps the monotonicity of the true value in the driver status.
The paper's proofs are in its Online Appendix (Appendix D). They are short but informal, and the formal statement here differs from the printed one in three pinned-down respects: the single-period cost is read with at most drivers, as the proofs use it; the upper bound is capped by the policy's cost, since otherwise it is vacuous; and the parameter choice printed for the upper bound has to be corrected in the last period. No part of the result has a machine-checked proof. A formal proof makes these readings definitive, and it produces a reusable Lean model of multi-period dispatching with routes, en route drivers and a Bellman recursion over a random demand.
Every part is a backward induction on the remaining periods, but the inductions run over a state that grows: the status depends on the durations of the routes chosen in period , and the infimum in (7) is over finite lists of routes, not over a finite action set. The lower bound needs that a larger status can only raise the cost; the upper bound needs that a policy which is feasible on its reachable states stays feasible after every realized demand. Exchanging the expectation with the minimum over the driver count in the last period is not a general fact and holds only because the single-period cost is nonincreasing in the number of available drivers. The obvious choice of parameters for the upper bound, the one printed in the proof, does not bound the last period's count and fails.
Locations are a type with the depot as none; periods are 1-based natural numbers and every statement carries . Costs and values lie in ; every minimum is an infimum, equal to on an empty feasible set; expectations are sums against the probability mass function of . A decision is a list of nonempty routes covering the current orders exactly once. The parameters are nonnegative reals by their type, with the diagonal written into (10). The available drivers in the dynamic program are in ; in APT the right-hand side is the real .
Standing assumptions, shared by every statement: ; strictly increasing epochs; finitely supported, period-independent demand laws; off . Added hypothesis: for (the en route drivers belong to the fleet). Disclosed readings: travel times do not depend on the period; delivery times follow the recursion of Appendix C with equality, starting from the depot leg; a policy's expected cost is defined recursively along its decisions.
Ruled out: defining or through the APT objective (exactness would become definitional); decision sets that may leave orders unserved (every cost would vanish); an upper bound without the cap by the policy's cost; negative parameters .
The development needs the model and its recursion (provided as one definition file), monotonicity of in , and induction principles for the well-founded recursions. Proofs of the milestones in any order are welcome; Lemma 1 is not used by the paper's proof of Theorem 1 but is a natural first step.
Repair access changes more than the cost of fixing a broken product. It also changes how many used products reach a resale market, what those products sell for, and whether a manufacturer prefers to sell another new unit or earn a repair margin. Jin, Yang, and Zhu study these connected pricing decisions in a two-period durable-goods model in Right to Repair: Pricing, Welfare, and Environmental Implications. Their Proposition 1 identifies the manufacturer's prices and the resulting market outcome when independent and manufacturer repair restore the same product quality. This mission formalizes that proposition, including all seven rows of its Table 1.
The question is relevant to work on durable-goods markets because a repaired product can compete with a new one in the next period. A lower independent-repair cost can affect the manufacturer's behavior even if consumers do not choose independent repair in the final equilibrium. The paper's setting makes that effect precise through the used-product price, new-product demand, and the manufacturer's repair decision.
A new product provides one period of use and then fails with probability . If it remains functional, or if a failed unit is repaired, it can be used for one more period as a used product. After two periods of use it has no value. Consumers have types uniformly distributed on . A new product gives per-period gross utility ; a functional used product gives , with . This mission takes the paper's identical-repair-quality case and zero resale transaction cost .
The manufacturer chooses a new-product price and a manufacturer repair price . Producing a unit costs and repairing it costs . A consumer can instead pay for independent repair. A manufacturer repair price of means that service is unavailable. The used-product price emerges from the secondary market. Consumers who face a failure repair with probability : they repair whenever the resale gain exceeds , do not repair when it falls short, and may mix at equality.
Consumers can be inactive, buy a used unit every period, or buy a new unit every period. Their corresponding per-period utilities are , , and
The thresholds partition these choices. In a market with a positive used price, used demand equals the supply generated by new buyers: . At , used supply may exceed demand. The manufacturer's per-period profit combines new-unit margin with its repair margin on units it actually repairs. These primitives and Assumption 1 appear in §4.1 of the paper.
Lemma 1 states that consumer utility has the three-segment form above. The appendix then compares reduced pricing problems for no repair, full repair, and partial repair. One milestone records the full-repair optimum:
at a feasible full-repair maximizer. Other milestones record the profit-gap identities, the ordering of two production-cost cutoffs, and the appendix results for the two higher ranges of independent-repair cost. Their statements follow Online Appendix G.
The goal is the complete equilibrium characterization of Proposition 1 and Table 1. When , the optimal outcome changes at and from no repair to partial repair to full repair. When , three cutoffs give four rows: two no-repair regimes, partial repair, and full repair. Each row specifies , , , and ; the partial-repair rows require . The target fixes the paper's explicit formulas rather than only the number or order of the regimes.
The theorem gives a simultaneous answer for manufacturer pricing and the secondary-market outcome. It identifies when manufacturer repair is unused, offered to some consumers, or offered to all consumers facing failure. In the low- case it also distinguishes two no-repair regimes with different used prices. Those distinctions matter for subsequent welfare and environmental comparisons in the paper: a price change can alter new production and resale quantities even when the repair count does not change.
The paper proves Proposition 1 analytically; this mission asks for a machine-checked proof of its exact statement. A complete development would also establish reusable interfaces for a market-clearing equilibrium with a zero-price excess-supply corner, price optimization across several feasible regimes, and comparisons between algebraic cutoff formulas. The draft theorem statements are open Lean goals; the local sanity check proves that one specified market equilibrium is inhabited, not the proposition itself.
The manufacturer's choice affects both the direct repair margin and the resale price that determines demand for new and used goods. Optimizing the new-product price in isolation therefore does not give the global result. The appendix's candidate solutions come from distinct repair and market-clearing regimes, and their feasible sets meet at cutoffs where different prices can tie. A global claim has to compare each candidate against all admissible price pairs and their equilibria. The radical and rational cutoff formulas add boundary cases: a formula may be used only in the regime where its denominator and square-root argument have the intended meaning.
Lean uses real prices, costs, probabilities, and consumer thresholds. The standing assumptions are , , , and the paper's Assumption 1, and . It also pins the three unit costs to , consistent with the paper's nonnegative-cost appendix domain. An unavailable manufacturer repair service is represented by none, not by an arbitrary large real number. A real repair price must be nonnegative.
The equilibrium predicate requires actual utility-maximizing consumer segments, the repair rule, and market clearing. It allows excess used-product supply only at . It also admits the appendix's zero-demand outcome, in which neither product yields positive utility, no units sell, and profit is zero. An optimal outcome contains an equilibrium and beats every equilibrium at every admissible competing price pair. Thus an empty equilibrium relation cannot make the goal true. At cutoff ties the theorem asserts that the displayed prices are an optimum, without claiming uniqueness. The two partial-repair rows assert strict bounds on , excluding a vacuous value.
The appendix's Problems 1, 4, and 5 are separate reduced objectives and feasible sets. The rational partial-repair objective is accompanied by its polynomial continuation for the profit-gap identity; that continuation is used only where the printed rational form could produce . The source's second identity in (G.41) omits a factor in its denominator. The Lean statement uses the corrected algebraic identity, while the milestone preserves the printed line. Contributions that prove the consumer, reduced-problem, cutoff, and global-optimality milestones are all within scope. The remaining propositions on welfare, quality differences, and environmental effects are outside this mission.
b869b33a21376a298a2355fb4de697d59b1362d5895f166f3255348812bc1763.On-demand delivery platforms for meals and groceries promise delivery within minutes, and the main operational lever they control is how a courier is dispatched. A courier can carry each order as soon as it is ready (dedicated delivery), or wait until two orders are available and deliver them in one trip (batch delivery). Batching saves travel per order but makes the first customer wait for a second order to arrive. Which mode gives the shorter wait, and hence the higher revenue, is a question practitioners face daily; it is the subject of M. Chen and M. Hu, Courier Dispatch in On-Demand Delivery (SSRN 3675063; published in Management Science, DOI 10.1287/mnsc.2023.4858).
The paper answers the question in closed form for a stylized disk model. This mission formalizes its first headline result, for the case in which the demand rate is fixed exogenously (for example by a growth target or by kitchen capacity), as stated in Propositions 1 and 2 of the authors' version dated 2022-09-21.
A vendor sits at the centre of a disk of radius and employs one courier with unit travel speed, so travel time equals travel distance. Customer orders arrive as a Poisson process of rate , the demand rate, and each customer's location is independent and uniformly distributed on the disk.
Under dedicated delivery each trip is a round trip to one customer, of length with uniform on the disk. Its moments are and , so the courier is an M/G/1 server with
where is the service rate and the summed coefficients of variation. The paper defines the expected wait under dedicated delivery as the M/G/1 in-line delay
Under batch delivery the courier leaves only with two orders and visits both, so a trip has length for two independent uniform points, with . This gives
and a variability coefficient that the paper obtains by numerical integration. The paper defines the batch wait as the time for a second order to arrive, plus Kingman's approximation of the E/G/1 in-line delay, plus the expected delay of being second in a batch:
Customers have valuations with c.d.f. on and a wait cost per unit time; at price the potential arrival rate thins to . The vendor's revenue rate at demand rate and expected wait is
Dedicated delivery is stable exactly when ; batch delivery when .
For every there is a constant such that for all , with ,
and the same two equivalences hold for the revenues and , for every , every with , and every . The paper reports ; the goal asserts only the existence and the location in .
The result says that, with demand fixed, the choice between dedicated and batch delivery depends on the market only through the product : dedicated delivery is better when demand is low or the area small, batching when demand is high or the area large. The threshold is a single universal number, independent of the valuation distribution and of the wait-cost rate, because at a fixed demand rate the strategy with the shorter wait always earns more. This is the efficiency-driven benchmark against which the paper's second main result (endogenous demand in a crowded market, where the order reverses) is contrasted.
To our knowledge none of this has been machine-checked. The mission produces a checked version of the paper's closed-form comparison and, in milestones 1–2, checked computations of classical geometric-probability moments of the disk (the mean distance between two uniform points). The paper's own argument for the single crossing is numerical ("solved using standard methods"); a formal proof must replace it by an exact argument with rigorous bounds on .
The reduction to one variable is algebra. The difficulty is the sign analysis of on : it is a difference of two rational functions with poles at and , involving , and clearing denominators gives a cubic whose roots the paper only reports numerically. The natural first idea, showing is monotone on the whole interval, does not apply directly, since starts at , first decreases, and only later rises to ; a proof must establish that it crosses zero exactly once. A further point is that is not a closed-form number, so the crossing must be shown uniformly over an interval of values.
The moment milestones need integration in polar coordinates on the plane; the mean distance between two uniform points of a disk is a classical but nontrivial four-dimensional integral.
All objects live in the namespace CourierDispatch.Exogenous. Real numbers are ℝ, is lam, is Lam, and is an arbitrary function Q : ℝ → ℝ (no property of it is needed, since at equal the price terms cancel). The plane is EuclideanSpace ℝ (Fin 2) and the uniform law on the disk is Lebesgue measure conditioned on the closed ball.
Committed conventions:
Q stands for ; its properties cancel when the two strategies are compared at the same demand rate.The threshold (and , ) is existential in every statement; it is never a definition and never an infimum of a sign set, and , are never defined through , so the goal cannot be made true by construction.
Needed infrastructure: polar-coordinate integration on the plane (Mathlib's integral_comp_polarCoord_symm), and rigorous rational bounds on (Real.pi_gt_d6, Real.pi_lt_d6). The goal does not depend on the moment milestones 1–2, which ground the constants , , in the geometry of the disk and are welcome as separate contributions.
Blood platelets, fresh produce, dairy and many pharmaceuticals have a fixed shelf life: a unit that has not been used within periods of its arrival must be discarded. Managing such stock is a classical problem of inventory theory, going back to Nahmias (1975), and it remains of practical interest for hospital blood banks and grocery retail. The difficulty is structural. The state of the system is the whole age profile of the stock, a vector of dimension , and the optimal policy depends on all of it. Dynamic programming over this state is intractable even for moderate lifetimes, so practice relies on simple heuristics, above all base-stock policies, which raise the total stock to a fixed level every period.
For decades the performance of base-stock policies in this system was understood only numerically. Chazan and Gal (1977) and Cooper (2001) bounded the long-run outdating of a base-stock policy; Nandakumar and Morton (1993) built a base-stock policy from bounds on the optimal order-up-to level and showed numerically that it performs close to optimal. Bu, Gong and Chao (SSRN 3724966, accepted by Management Science, 2022) give the first theoretical guarantees, comparing a simple base-stock policy with the optimal policy in several asymptotic regimes. This mission formalizes the first of them: the optimality gap decays exponentially in the lifetime .
Time is discrete, . Demands are i.i.d. copies of a nonnegative random variable with and distribution function ; .
At the start of period the state is , where is the stock whose remaining lifetime is at most periods. The firm orders , which arrives immediately and raises the total level to (with ). Demand is served oldest-first (FIFO issuance) and unmet demand is lost. Stock reaching the end of its life is outdated:
The system starts empty. Each period costs
with holding cost , lost-sales penalty and additional outdating cost . A policy is admissible if each order is a nonnegative measurable function of the current state; is the set of admissible policies. The long-run average cost is , and .
The base-stock policy orders ; its cost is . The heuristic level is a minimizer over of
and is the newsvendor level.
For every and ,
The bound holds for every minimizer and every demand law with positive finite mean. Since , the gap decays exponentially in .
Theorem 1 says that a single explicit base-stock level, computable from the demand law by a one-dimensional convex search, is nearly optimal once the lifetime is moderately long, with an explicit, distribution-dependent rate. It explains why the non-perishable newsvendor intuition carries over to long-lived products, and it is the first of seven guarantees in the paper. Its ingredients are reused throughout: Lemmas 1 and 2, Proposition 2 and the gap bound (21) are posed in this mission and used, without restatement, by missions 2 to 4 of this series (demand population size, penalty cost, outdating cost).
The result is proved on paper; nothing about perishable inventory has been machine-checked. A formal development would provide a reusable model of FIFO perishable dynamics with long-run average costs, sample-path inequalities for outdating, and the first verified optimality-gap bound for an inventory heuristic in a high-dimensional state space.
Most of the argument is sample-path bookkeeping plus Jensen's inequality, but two steps are not routine. Lemma 1 asserts that the long-run average outdating of exists and does not depend on the initial state. The natural route through a stationary distribution fails: when the chain can be periodic and has no steady state, so the identity must hold for Cesàro averages directly; in the aperiodic case the paper uses a minorization condition (Meyn and Tweedie, Theorem 16.0.2). Proposition 2 compares every admissible policy with the heuristic cost, which needs the independence of the order-up-to level from current and future demands, and a conditional Jensen step for nonlinear expectations of a policy-dependent quantity.
The Lean development fixes the following conventions.
Fin (m - 1) → ℝ holding the cumulative levels, with , so (empty state, all leftover stock outdates) is included; , , , are fields of the parameter record.ℝ≥0).CappedBaseStock_Model and LostSalesOrderUpTo.Asymptotic.Backorder.Trivializing readings are excluded: is the simulated cost of , not a formula; ranges over all admissible policies, not only base-stock ones; is the argmin predicate, not a closed form; , and the demand law are general; and no gap is stated with truncated subtraction. Contributions of any of the milestones, of general facts about Cesàro limits of Markov chains, or of the convexity of , are welcome.
Many problems in machine learning, signal processing and engineering design minimize a nonconvex, possibly nonsmooth objective subject to nonlinear equality constraints. Augmented Lagrangian methods are the standard tool for such constraints, but each classical iteration solves a hard nonconvex subproblem. Zhu, Zhao and Zhang (arXiv:2007.12219v1, published in Mathematics of Operations Research, 2024) propose NAPP-AL, a first-order primal-dual method in which each iteration takes one linearized proximal step in each primal block and one multiplier step. Their §4 shows that the best stationarity residual decays like . That rate is sublinear, and the question addressed by §5 is when the same method converges linearly. The answer is a local error bound, the value proximity error bound (VP-EB), under which the iterates have finite length, converge to a stationary point, and do so at a geometric rate. Error-bound conditions of this kind are the usual route from sublinear to linear rates for first-order methods (Luo–Tseng; Attouch–Bolte; Bolte–Nguyen–Peypouquet–Suter), and VP-EB is the paper's version adapted to its potential function.
The problem (P) is
with , and smooth with Lipschitz gradients (, ), lower semicontinuous and possibly nonsmooth and nonconvex, closed and convex, Lipschitz with constant , each component with -Lipschitz gradient, , and an matrix with and (Assumption 1). A point is , with the Euclidean norm of the stacked vector.
The augmented Lagrangian is . The stationarity set (p. 6) collects the triples
with the limiting subdifferential and the normal cone; is stationary when .
NAPP-AL (p. 8) fixes a -strongly convex with -Lipschitz gradient and its Bregman distance , and a penalty above (Assumption 2). With , it sets to a global minimizer over of , to the minimizer of a strongly convex quadratic, and . The step lies in , with given explicitly by rule (8).
The analysis uses the potential (18)
with explicit . VP-EB at with constants (Definition 5.1) asks that whenever and .
Under Assumptions 1–2 and rule (8), let be an accumulation point of an NAPP-AL run with , and let VP-EB hold at with . Then (i) eventually and ; (ii) ; (iii) and is stationary; (iv) for some and ,
and the iterates converge R-linearly: for some , .
No constant is hard-coded in the conclusions: , , and the thresholds are existential.
The theorem upgrades the method's guarantee from sublinear stationarity to a geometric rate for both the potential and the iterates, and it gives whole-sequence convergence rather than subsequential convergence. VP-EB is checkable through standard conditions: §6 of the paper shows that the Kurdyka–Łojasiewicz property of with exponent , and metric subregularity of , each imply it; those implications are the companion missions of this series. The result is proved on paper; no machine-checked proof of it or of NAPP-AL's convergence theory is known. Formalizing it produces a checked localization-and-induction argument of the kind that recurs throughout the error-bound literature, written against an explicit potential with explicit constants.
Accumulation alone does not make the iterates stay near : VP-EB only applies while is inside the ball and below the level , so its use is circular unless one first shows the iterates never leave the region. The obvious argument — apply the error bound at every step — therefore fails; localization (part (i)) is the core of the proof, and it must be carried out with the paper's explicit constants and the sufficient decrease of (Lemma 4.3 and Theorem 4.1, pp. 15–16, milestones of the first mission of this series). A second subtlety is that the page's proof treats as strictly decreasing, while the sufficient-decrease lemma gives only non-increase; the case where reaches in finite time must be handled separately.
Spaces are EuclideanSpace ℝ (Fin k); triples use the explicit Euclidean norm tnorm of the stacked vector, never the sup norm of the product type. is a continuous linear map with its operator (spectral) norm, never a Matrix with the entrywise norm; is the Rayleigh minimum over the unit sphere. is an infimum in , equal to on , never Metric.infDist. The normal cone is empty off , so stationary points lie in . NAPP-AL is a relation on given sequences (any global minimizers of (5); existence is not asserted), with arbitrary . is junk in Lean, so statements about concern .
Disclosed readings of the page: are real valued with gradients Lipschitz on the whole space; (H3)'s local Lipschitz clause means each has a neighbourhood in on which is Lipschitz (not Mathlib's relative LocallyLipschitzOn); ; for ; Definition 5.1's phrase "which converges to " is dropped, since Theorem 5.1 proves convergence, and keeping it would assume conclusion (iii); "" is the hypothesis . Using Metric.infDist, the product sup norm, an entrywise matrix norm, a relative local Lipschitz condition, or a normal cone nonempty off would each change or trivialize the statements and is ruled out.
The source is the preprint arXiv:2007.12219v1, not the published version; all labels, pages and constants are v1's. The definitions are restated in this mission's namespace NAPPAL.Linear and coincide with those of the other missions of the series. A complete development needs the sufficient-decrease lemma and the convergence theorem of the first mission (Lemma 4.3, Theorem 4.1), plus elementary facts on summable sequences and geometric bounds. Contributions to the localization induction, which is reusable for the other error-bound arguments of §6, are welcome.
Decentralized exchanges on public blockchains do not match buyers with sellers through an order book. Most of them trade against a liquidity pool: a smart contract that holds reserves of two tokens and quotes a price by a fixed formula. The most widely used formula is the constant product rule of Uniswap (Adams, Zinsmeister and Robinson, 2020), under which a trade must leave the product of the two reserves unchanged. Pools of this kind settle a large share of on-chain token trading, and their pricing rule is fixed in code, visible to every participant, and executed in the order in which transactions are included in a block.
That last feature creates a market-structure question. Pending transactions are public before they execute, so another trader can see an order and place her own trade ahead of it. The empirical literature documents this behaviour at scale (Daian et al., 2020; Zhou et al., 2021). Park's paper asks which properties of the pricing rule itself make such strategies profitable, independently of any information the front-runner has. The paper compares the constant product rule with the cost function of a traditional market maker whose marginal price is linear in quantity, in the tradition of inventory models such as Biais (1993), and then derives four properties of constant product pricing in its Proposition 3.
Timeline:
A pool holds units of a token and units of a token . A trader who takes units of out of the pool pays units of , where is fixed by the constant product invariant
A positive is a buy of ; a negative is a sale of units of into the pool, and then is negative: the trader receives . Solving the invariant gives the paper's cost function, equation (3):
After a buy of the pool holds units of and units of . Selling units of into a pool therefore earns units of . The paper writes for on the current pool.
For the companion comparison the mission also defines the linear market maker's cost , where is the fundamental value, the number of market makers, their excess inventory and a liquidity factor.
For , and :
Part (1) says that splitting an order in time does not change its cost; part (3) that splitting liquidity into identical pools, with the order split pro rata, does not change it either. Part (2) is the front-runner's profit: she buys , the victim buys , and she sells into the pool left by both buys. Part (4) is the profit of selling into a pool just moved by a buy of and buying on an identical untouched pool, for every .
A companion item poses Proposition 1: with , , and , constant product pricing has a larger signed cost than linear pricing for , and a smaller one for , .
Parts (1) and (3) are the consistency properties one asks of any market maker: an order's cost does not depend on how it is sliced in time, or on how liquidity is fragmented across identical venues. Parts (2) and (4) show that the constant product rule, despite these properties, rewards anyone who can trade ahead of an observed order or across two identical pools. These are properties of the pricing curve alone, with no informational advantage, which is what makes them relevant to the design of on-chain exchanges, transaction ordering rules and fee schedules.
All statements are proved in the paper by direct algebra, and they are not open. What this mission adds is a machine-checked record of the exact statements, with the paper's printed sign slips resolved explicitly, and a reusable definition of the constant product cost function in Lean on which statements about fees, multi-hop routing or other invariant curves can build. No machine-checked version of Proposition 3 is known.
The mathematics is rational-function algebra; the difficulty is in stating it correctly. Three points need care. First, the sale convention: the paper's cost function returns a negative number for a sale, while the printed statements of parts (2) and (4) add that number as if it were the positive proceeds. Read literally, both printed left-hand sides are negative. Second, the pool after a trade must be tracked exactly: part (4) prints the first pool's reserve as , whereas after a buy of it is . Third, every quantity is a ratio with a denominator such as , or , and the statements hold only on the domain where those are positive. A solver who simplifies before checking signs will prove the wrong inequality.
Everything lives in the namespace CPAMMFlaws.Properties over the real numbers. The definition pcmm x X Y := x * Y / (X - x) is equation (3) verbatim, argument order ; ptmm x V ℓ N I is the printed closed form of . Conventions committed to:
Trivializing formalizations are ruled out: Lemma 2 is the invariant characterization over all , not the defining equation of pcmm; parts (2), (4) and equation (4) are strict inequalities on the signed proceeds, not and not absolute values; and no statement drops .
Out of scope: the paper's CARA-normal market-maker equilibrium (Lemma 1), its results on the linear market maker (Proposition 2), the fee analysis of §IV and the calibration of §V. The linear cost enters only through the companion Proposition 1. Contributions welcome beyond the milestones: proofs of the goal's parts directly, and further invariant-curve properties built on pcmm.
Best subset selection fits a linear predictor using at most of available columns. Its explicit limit on the number of selected variables makes the model easy to interpret, but the search can still fit noise: among many subsets of the same size, one can align unusually well with a noisy response. Mazumder, Radchenko and Dedieu study this effect in Subset Selection with Shrinkage. They add a penalty on the size of the selected coefficients and show how this changes prediction error when the signal-to-noise ratio is low.
An penalty is familiar from the Lasso. The estimator here also keeps the subset-size constraint, so its comparison class and its tuning parameter have a different role from those in an unconstrained Lasso. The paper's Theorem 3 gives three deterministic error bounds for this estimator, each valid on a specified event of the Gaussian noise. These bounds are the target of this mission. The separate probability estimates for the events are Theorem 1 of the paper and belong to another mission in this series.
Let be a fixed design matrix whose columns have Euclidean norm one. The observation is , where is a fixed mean and the coordinates of are independent random variables, with . The result below fixes one realized noise vector and assumes it lies in an event stated below. No assumption that is itself a linear combination of columns of is required.
For , write , where counts nonzero coordinates. An -regularized best subset estimator is any minimizer
The objective has no factor . The assertion applies to every minimizer, so tied solutions are covered. The sparse minimum singular value is . When and its domain is nonempty. Write for the decreasing rearrangement of .
The paper defines three noise events used here. bounds the largest coordinate of by . On , every satisfies
On , every satisfies . Here and all logarithms are natural. These definitions, including the constant in , are the printed deterministic events of p. 13.
At , Theorem 3(A) states
For , a positive universal constant gives
For every , a positive universal constant gives
The multiplying distinguishes this rate from the -penalized estimator's bound. The three claims together, with their separate tuning conditions, are Theorem 3 on p. 16. The milestones follow the displays of its proof in Supplement B.6, pp. 42–43.
Each inequality compares the prediction of any selected minimizer with that of every sparse competitor. The first bound has an exact leading constant and no additive noise term on . The second keeps a linear penalty term when meets a fixed noise-dependent threshold. The third replaces the linear term with a quadratic term controlled by the sparse minimum singular value. This distinguishes regimes in which coefficient shrinkage changes the error rate. The paper also combines deterministic bounds with event probabilities in subsequent corollaries; those probability statements are separate from this goal.
The paper proves Theorem 3. The formalization task is to verify its model definitions, event interpretation and all three inequalities in Lean, with the same constants and quantifiers. The milestone statements isolate the basic inequality, the bound, the rearranged-coordinate bound, the case split and the sparse step. A complete proof would also provide reusable finite-dimensional norm and matrix estimates. This mission does not include the paper's numerical experiments, data-driven model-size selection, the -penalized estimator, or the event-probability theorem.
The constraint makes the estimator a minimizer over a nonconvex set. A standard unconstrained Lasso argument does not directly supply the paper's bound for every feasible sparse comparator. The slow rates rely on two different noise controls: a coordinatewise bound on and a maximum of a sorted-coordinate term and a prediction-norm term on . The fast rate also needs the sparse minimum singular value to relate coefficient error to prediction error. The constants in these steps matter: replacing the exact first tuning equality or dropping the factor in the fast rate would state a different theorem.
Lean uses Fin n → ℝ and Fin p → ℝ for vectors and Matrix (Fin n) (Fin p) ℝ for . Norms are explicit finite sums, rather than the function type's default supremum norm. The paper's one-based sorted index is j.val + 1; the sum using is encoded as a maximum over coordinate permutations, which agrees with the decreasing rearrangement. The events are predicates on realized noise vectors. The Gaussian law explains the model but is not needed in these deterministic statements. The estimator is an argmin predicate, never a chosen minimizer.
The universal constants precede all dimensions and data in the Lean goal. The infima in the three displayed bounds are expressed as inequalities for every , avoiding the default value of an empty or unbounded real infimum. The standing conditions are , unit-norm columns, and . The first slow-rate branch retains the paper's equality for . The fast branch additionally requires or : this makes the term meaningful under Lean's total division, which returns zero at a zero denominator. The displayed events use the paper's original constant; this mission makes only a deterministic claim on that event.
The sparse domain for is nonempty under these conditions. All square-root radicands and logarithm arguments in the target are in their intended ranges. The formalization must rule out a universal constant that depends on , , , or , and must apply to every minimizer. Contributions to the shared norm inequalities, rearrangement identity and matrix calculations are useful beyond this theorem.
A retailer can pay a supplier when it places an order or after the supplier delivers it. When production yield is uncertain, the choice changes who bears the cost of missing units. It can also change the supplier's production decision and, through the amount available for sale, the surplus received by consumers. Geng, Guo, Xiao and Yang study these effects in a decentralized supply chain and show that payment postponement can help all three parties, hurt all three, or benefit some while hurting others, depending on production cost and the retailer's pricing rule. Their Proposition 4 gives the precise cost intervals for these outcomes in the SSRN author version, pp. 19–20.
The result matters to contract design in supply chains where realized output can be below planned production. Paying for units ordered and paying for units delivered allocate that shortage differently. A comparison based only on the retailer's expected profit would miss the effects on the upstream supplier and final consumers. Proposition 4 keeps all three outcomes visible while holding the retailer's pricing policy fixed.
There is one supplier and one retailer. The supplier chooses a wholesale price and plans production at cost . The retailer orders and sells to a market with linear demand , where and is the retail price. The yield factor equals with probability and with probability , with . Thus . The retailer receives units; the production cost applies to planned , including units lost to yield risk §3.1–3.2.
Under pay-by-order, the retailer transfers to the supplier and the supplier produces . Under pay-by-delivery, the transfer is and the supplier can choose a larger production quantity. Under ex ante pricing, the retailer fixes before yield is known. Under responsive pricing, the retailer sets after observing delivery. Combining payment and pricing gives OA (order, ex ante), OR (order, responsive), DA (delivery, ex ante), and DR (delivery, responsive). The supplier first sets , the retailer next orders and possibly fixes a price, the supplier chooses production, yield is realized, and responsive pricing occurs last Table 1, p. 9.
Both firms maximize expected profit at every stage. A subgame-perfect equilibrium is a complete plan of choices after every history in which each choice is optimal at that history. For scenario , write and for equilibrium retailer and supplier profit. The consumer surplus is expected surplus from sales: for delivered stock and market price , the model integrates inverse demand less from zero to . This is the paper's priority allocation convention for consumers with the highest willingness to pay p. 10.
The main target is Proposition 4, which compares OA with DA and OR with DR. Let mean the delivery-payment outcome minus its order-payment counterpart. A plus sign below means a strict increase, a minus sign a strict decrease, and zero equality. The admissible cost range is .
For ex ante pricing, with no restriction , the paper gives a threshold and the following pattern:
For responsive pricing, assuming , the pattern is on , on , on , and on . The thresholds are those specified in the paper: is equation (6), is equation (8), and the scenario cost thresholds come from Appendix B. The first ex ante interval can be empty for some parameters; its printed endpoints are retained. The milestones establish the supplier's production and responsive pricing rules, the four scenario outcomes, and the threshold ordering used in the comparison Appendices B–C.
The theorem separates the effect of changing the payment contract from the effect of changing the pricing rule. For advance pricing it identifies an interval where payment postponement helps both firms and consumers, then an interval where consumers lose despite both firms gaining. For responsive pricing the direction of every effect can reverse as cost passes successive thresholds. At high cost the change in payment timing leaves all three equilibrium outcomes unchanged Proposition 4, pp. 19–20.
The paper derives the equilibrium formulas in Appendix B and proves the comparisons in Appendix C. The result is therefore a known mathematical claim, while this mission asks for a machine-checked development from the sequential game through the cost-dependent equilibrium outcomes and final comparisons. The definitions of a two-point yield game and its complete contingent strategies can support other formal studies of payment or pricing postponement. There is no claim that Proposition 4 already has a machine-checked proof.
The supplier's production payoff has kinks at and . The retailer's best order changes when those production choices change, and responsive pricing adds another kink when delivered stock crosses . As wholesale price varies, the supplier must compare outcomes on both sides of these changes. Consequently a single first-order condition for a smooth objective does not characterize the equilibrium. The cost thresholds also depend on square roots and comparisons among several formula pieces, including the larger root chosen for Appendix B, pp. 33–37.
Lean represents expectation by the two weighted yield outcomes. Production cost is a separate argument, so thresholds depend only on . Parameters satisfy , , and . Wholesale and retail prices obey the strict admissibility bounds of equation (3). Orders are nonnegative; pay-by-delivery production is nonnegative; pay-by-order production equals the order. Consumer surplus is a real interval integral. Every comparison quantifies over actual subgame-perfect equilibria, and the goal separately asserts that the compared equilibria exist at every admissible cost. Equilibrium profits are computed from strategies and transfers, rather than defined to equal the Appendix B formulas.
The paper's statements require care at ties. The model selects when that is a profit-maximizing pay-by-delivery production choice, including the indifferent DR regime. The printed middle DR consumer-surplus expression combines outcomes from different production choices; the equilibrium-consistent expression is used here, and Proposition 4(R)'s comparison remains valid under either expression. Equation (2) swaps the scenario labels of committed and responsive prices, so consumer surplus uses the actual ex ante price in OA and DA and the price after delivery in OR and DR. The printed in Appendix B.1 is read as , as in Lemma 1. Outcome claims use open cost intervals because uniqueness at exact thresholds fails; equilibrium existence remains part of the targets there. The threshold is the larger root of the relevant quadratic factor, and and use the paper's explicit formulas. These conventions are recorded in the local moderation notes.
The development calls for real inequality reasoning, optimization of piecewise objectives, and careful handling of interval integrals. Contributions that prove the production and price best responses, establish the four equilibrium lemmas, or verify the threshold order each advance the final comparison.
A retailer that buys from a supplier with unreliable production has two contractual levers against supply risk, and both postpone a decision until after the yield is known. Payment postponement means paying only for the units actually delivered (pay-by-delivery) instead of the units ordered (pay-by-order). This moves the yield risk upstream. Pricing postponement means setting the retail price after the delivery is observed (responsive pricing) instead of committing to it in advance (ex ante pricing). This moves the risk downstream to consumers. Both practices are common in procurement, and the natural intuition is that two ways of shedding the same risk should be substitutes: once one lever is in use, the other should be worth less.
Geng, Guo, Xiao and Yang (MSOM 25(3), 2023; author version SSRN 3546872) compute the subgame-perfect equilibria of all four combinations of the two levers in a stylized supplier–retailer game. Their Proposition 3 shows that the intuition fails at intermediate production cost. When full yield is likely and yield loss is mild, the two postponements are complements there: postponing payment is worth more when pricing is also postponed. This mission formalizes that result together with the four equilibrium characterizations it rests on.
A supplier sells one product to a retailer, who sells it in a market with linear demand , . The supplier plans production at unit cost and realizes the fraction of it, where
and is the mean yield. The retailer orders and receives .
The four scenarios are OA, OR, DA and DR. The first letter is the payment rule: by order (O, the transfer is ) or by delivery (D, the transfer is ). The second letter is the pricing rule: ex ante (A, the price is fixed before the yield) or responsive (R, the price is set after is observed). Each scenario is a sequential game. The supplier posts a wholesale price . The retailer orders , and also commits to a price in the ex ante scenarios. The supplier then chooses ; under pay-by-order it produces exactly . The yield realizes, and under responsive pricing the retailer then sets . The retailer earns minus the transfer, and the supplier earns the transfer minus . Both are risk neutral. Admissible actions follow the paper's standing ranges: , , and between (or under pay-by-delivery) and .
Write for the retailer's expected profit in a subgame-perfect equilibrium of scenario . The paper's Definition 1 measures the interaction of the two postponements by
the value of payment postponement under responsive pricing, minus its value under ex ante pricing. The postponements are substitutes if and complements if .
The parameter region of the result is together with , where is the largest real root of the cubic . The cost thresholds (scenario DR) and (scenario DA) are explicit algebraic functions of given in the paper's Appendix B.
For and , every scenario has an equilibrium at every cost , and there is a cost such that, at every choice of the four equilibria,
is existential, as in the paper, and depends only on the parameters.
The result separates the first-order question (which lever the retailer prefers, the subject of the paper's Proposition 2) from the second-order one (whether one lever raises the value of the other). It shows that two risk-shifting instruments can reinforce each other. On , adopting one postponement raises the value of the other. So whenever the retailer gains from adopting either lever alone, it gains even more from adopting both. Below and above the reverse holds. Above , payment postponement is irrelevant, because the supplier never inflates production there.
The paper proves Lemmas 1–4 by backward induction in Appendix B and the proposition by comparing closed forms in Appendix C. None of it is machine-checked. A formal proof requires the four backward inductions with every kink and tie handled, and the polynomial inequalities between the thresholds over the parameter region. The formalization also corrects the record: the paper's explicit formula (7) for does not locate the sign change of (at it exceeds ), while the existential statement of Proposition 3 holds at every sampled point.
Once Lemmas 1–4 are available, is a difference of explicit piecewise functions of . The difficulty lies in those lemmas and in the thresholds. Each lemma is a three-stage optimization. The supplier's profit is piecewise linear in , with kinks at and . The retailer's profit is piecewise in , with kinks where it becomes worth inflating the order. The supplier's value function in is piecewise quadratic, and its maximum can sit at a kink. In scenario DR it sits exactly at the price where the supplier is indifferent to inflating production. A naive reading of the paper's tie rule removes the equilibrium there altogether. On the middle DR interval the sign of changes inside the interval, at a root of a polynomial with nested square roots. On the other pieces, the signs of and the ordering are polynomial inequalities in , with defined implicitly as a cubic root.
The equilibrium profits are outcomes of the equilibrium predicate. They are never defined by their closed forms, and the goal carries the existence of an equilibrium in each scenario. Without that, the comparison would be vacuous whenever the predicate is unsatisfiable.
Every object is elementary: real arithmetic, square roots, min/max, one interval integral and one sSup. The equilibrium predicate and the four lemmas are shared with the three companion missions of this paper (Propositions 2, 4 and 5). Proofs of the lemmas, of the threshold inequalities and of the goal are all welcome.
Machine-learning systems are increasingly placed beside human experts — physicians reading scans, loan officers screening applications, judges assessing risk — rather than replacing them. The machine processes the structured data it can see; the human also has access to information the algorithm cannot observe, but her attention is limited. Whether such a partnership improves the decision is a question about how the human reallocates scarce cognitive effort once part of the information is handed to her for free.
Boyacı, Canyakmaz and de Véricourt (ESMT Working Paper 20-02, 2020; published in Management Science, 2024) model the human as a rationally inattentive decision-maker in the sense of Sims (2003): she chooses how much information to process, and pays a cost proportional to the mutual information between the state and her decision. In the binary-choice setting, Matějka and McKay (2015) showed that such problems reduce to choosing conditional choice probabilities, with logit-form solutions. The paper uses this reduction to compare a decision-maker who decides alone with one who first sees a machine's assessment. This mission formalizes its first comparison, Proposition 1: the machine never lowers the decision-maker's accuracy or her expected value.
A decision-maker (DM) must announce whether the state of the world is good () or bad (). A correct choice pays , an incorrect one . Her prior that the state is good is , and is the information cost.
A strategy is a pair of conditional probabilities of choosing in state and in state . It induces the choice probability , the accuracy , the cognitive cost
(the mutual information between state and action, in nats) and the expected value . The DM maximizes . The optimal accuracy and optimal value are the accuracy and value at a maximizer. Two thresholds organize the solution:
The machine. Information is aggregated into two binary variables: , which the machine and the DM can both evaluate, and , which only the DM can assess. Realizations are and the state is good iff . The DM's prior on is , with for all four realizations, so . The machine reveals at no cost; the DM updates by Bayes' rule to and then solves her problem with prior . The machine-assisted accuracy and value are the ex-ante expectations
For every and every prior with positive entries,
The result. Proposition 1 is the baseline against which the paper's other findings are measured: although accuracy and value always improve, the machine can lower the probability of choosing the good state, raise false positives, and increase the DM's cognitive effort (Theorems 2–4 of the paper). Without Proposition 1 those effects would not be separable from a simple loss of decision quality. The closed forms (13) and (15) also show that, where the DM processes information, her accuracy depends on alone and not on her prior.
Formalizing it. The proposition and its milestones are proved in the paper, in Appendix A, and the closed forms rely on the logit characterization of the optimum. None of this is machine-checked. A complete development produces a verified closed-form solution of the binary rational-inattention problem with entropy cost, with accuracy and value as explicit functions of the prior, and a verified value-of-information inequality for that problem. Both are reusable by the other four missions of this series and by any binary rational-inattention model.
The obvious argument — more information cannot hurt — is not available off the shelf: the DM is not a Bayesian with free information but optimizes an entropy-penalized program, and the comparison is between optimal values of two such programs at different priors. and are values at an optimum of a non-linear program over . Their closed forms require identifying exactly when the corner solutions and are optimal and handling the degenerate priors , where the maximizer is not unique. Both functions are piecewise in , so their shape on does not follow from the shape of each piece: the behaviour at the kinks and decides it. Accuracy is a component of the objective, not the objective itself, so statements about do not follow from optimality alone.
Real.binEntropy (natural logarithm); is lam.Classical.epsilon); every maximizer yields the same values. They are defined for every real and used on .Bool with true ; the prior is π : Bool → Bool → ℝ with positive entries summing to . Posteriors are defined by Bayes' rule; , are expectations over .Real.binEntropy (derivative, concavity) is in Mathlib. Needed: existence of a maximizer on the square, the first-order characterization of the interior optimum, and convexity of piecewise functions glued with matching slopes. Contributions of general lemmas on these points are welcome.When a supplier's production is subject to random yield, only part of a planned production run turns into usable output. A retailer sourcing from such a supplier bears the resulting shortage risk unless the contract or the selling strategy moves it elsewhere. Random yield is a classical topic in production planning (Yano and Lee, 1995), and postponing a decision until uncertainty resolves is a classical risk-mitigation tool (Van Mieghem and Dada, 1999).
Geng, Guo, Xiao and Yang study two postponements available to a retailer facing an unreliable supplier. Payment postponement replaces payment for the ordered quantity by payment for the delivered quantity, which shifts the yield risk to the supplier. Pricing postponement (responsive pricing) sets the retail price after the delivery is known, which shifts the risk to consumers. Their Proposition 2 answers which of the four combinations the retailer prefers, as a function of the production cost, when the yield loss is not too severe. This mission formalizes that proposition and the equilibrium results it rests on.
A supplier sells to a retailer, who sells to a market with linear demand , . The supplier's yield factor takes the value with probability and with probability , with , and has mean . The supplier plans production , pays per planned unit, and delivers against an order . Throughout, .
There are four scenarios, written :
Each scenario is a Stackelberg game. The supplier sets the wholesale price , the retailer orders (and, ex ante, prices), the supplier produces, the yield realizes, and under responsive pricing the retailer prices. Both firms are risk neutral. The solution concept is subgame-perfect equilibrium, and denotes the retailer's expected equilibrium profit in scenario . Admissible prices are those of the paper's standing assumption (3): , and under pay-by-order or under pay-by-delivery.
The cost thresholds of the comparison are
For and every , equilibria of OR, DA and DR exist, and every choice of equilibria satisfies
The proposition describes a preference cycle. Payment postponement (DA) is best when production is cheap, because the supplier then absorbs yield risk by inflating production. Pricing postponement (OR) is best at intermediate cost. Using both (DR) is strictly best on , where the two postponements are worth more together than either alone. Above the supplier stops inflating, DR and OR coincide, and pricing postponement suffices. These are the paper's managerial conclusions on contract and pricing choice under supply risk, and the other three missions of the series (strategic complementarity, and the effect of each postponement on the whole chain) rest on the same equilibria.
The paper derives the four equilibria in its Appendix B and proves the propositions in Appendix C as direct comparisons of closed forms. None of this has been machine-checked. The mission produces a formal model of the four games and a verified link from the game to the closed forms. Lemmas 2–4 are reusable by every statement about this model.
Once the closed forms are known, the comparison is polynomial algebra in with square roots. The work lies upstream. Each equilibrium requires a backward induction over piecewise-linear objectives in and and piecewise-quadratic objectives in and . Optima occur at kinks such as the inflation threshold , and the supplier's value function is discontinuous there, so the maximum can sit at a boundary rather than at the center of a piece. In scenario DR the equilibrium wholesale price on sits exactly at , where the supplier is indifferent among all productions in , so existence depends on how that tie is resolved. Uniqueness of outcomes, which the comparison needs, holds only on open cost intervals, because at each threshold two wholesale prices tie with different outcomes. The threshold inequalities for are not visible from the formulas and need careful estimates of the radicals.
Parameters are a structure with validity predicate , ; the cost is a separate real argument. The expectation over is the two-term sum , with no measure theory. A strategy profile consists of a wholesale price and rules for production , the order , the ex ante price and the ex post price . Equilibrium requires optimality at every admissible history over the admissible sets of (3). Consumer surplus is the integral , evaluated at the ex post price under responsive pricing and at the committed price under ex ante pricing.
Conventions and disclosed departures from the page:
is the retailer's profit along an equilibrium of the game, never its closed form; a formalization that defines the profits by the Appendix B formulas, or an equilibrium predicate that no profile satisfies, would make the goal trivial. The goal therefore also asserts existence of equilibria of the three scenarios.
Contributions welcome: proofs of the milestones; general lemmas on maximizing piecewise-linear and piecewise-quadratic functions with kinks; the threshold inequalities. Proposition 1 (OA versus the others) is not part of this mission.
A life annuity pays instalments conditional on an annuitant surviving to specified instants. As discussed in Dickson, Hardy and Waters (2020), actuarial value depends on the exact timing of the payments. For a pension payable in arrears, the payment event is at the end of the period. A contractual guarantee can preserve payments for the estate after death, but only once the entitlement has commenced. Those conditions must be explicitly modelled before calculating actuarial factors.
Take a finite distribution over last survived anniversaries with nonnegative probabilities summing to one. The final state is right-censored: it does not assert death at the terminal age. Define . A schedule has its first index , up to instalments, an amount for each instalment, and an annual discount factor with . Advance instalments fall at and arrears instalments at .
The first scheduled instalments are guaranteed only when survival to commencement activates the annuity. They continue to the estate after death during that period. Later instalments depend on survival at their actual payment instant. No spouse benefit is assumed.
Define the realised random variable by summing actual payment indicators across the finite schedule. Derive its expectation independently from the lifetime-state mass function. Define a second valuation using survival to each payment instant, and prove the equivalence by interchanging finite sums.
The instalment sum is truncated at the finite horizon. The timing offset equals zero for advance and one for arrears.
Partition a non-guaranteed whole-horizon annuity into temporary and deferred periods. For a guaranteed annuity, split payments into the activated period-certain amount and the strictly later survival-contingent amount. If , its entire value is survival to commencement multiplied by the corresponding annuity-certain present value. The benefit kernel supports level, arithmetic increasing or decreasing, geometric and piecewise step-up instalments.
The exact finite level-annuities relationship includes the terminal payment:
Expected squares of realised present values yield second moments and variance. The uniform distribution of deaths (UDD) provides a separate linear interpolation of observed annual survival probabilities at fractional instants.
At a fixed payment frequency, split each annual death-state mass equally across its subperiods under UDD. The same resulting micro-lifetime distribution can be used for fractional instalments and continuous benefit-rate integrals, so their actual cashflow representations share exact event boundaries. The construction agrees with annual survival at integer anniversaries. Placing the unsplit annual masses directly at integer death times would give a different, stepwise survival model within a year.
The zero-interest level annuity has an exact finite correction to the integral of linearly interpolated UDD survival. If is that integral, annual absolute survival and , the due value equals , and the arrears value equals . Both have an explicit error and converge to . Under the fixed micro-atom lifetime law, continuously paid GBP 1 per year instead equals the same-frequency arrears value at zero interest, because the real-time survival indicator is constant between atom times. The limit of these finite-atom laws recovers the UDD linear-survival valuation.
A separate real-time lifetime-state model represents a continuous benefit rate in money per year rather than discrete money per instalment. Its actual realised annuity present value is integrated across the entitlement interval, with a guarantee activated by survival at commencement and payments then continuing to the estate before expiry. A finite probability expectation of the realised integral is reconciled with independently integrated survival weights. The continuous certain value is a comparison under the same deterministic discount force.
The discrete and real-time laws are connected at annual and fractional instalment boundaries through exact survivor-event identities. A bounded continuous payment-rate function admits an appropriate Riemann-sum limit under the specific regularity assumptions in the formal declaration. A twice differentiable integrand with uniformly bounded second derivative admits the two-term Woolhouse-type endpoint correction with a finite-horizon quadrature error bound. These limiting and approximation statements remain open formal theorem obligations, with no universal error claim for arbitrary discontinuous mortality laws.
The unified capstone reconciles actual annual advance and arrears payments and UDD subperiod payments with independent survival-weighted sums. It also reconciles the same finite lifetime-state law with real-time survival at anniversaries and continuous integrated benefit valuations, including an activated guarantee. Separate milestones handle second moments, terminal-censored assurance-annuity identity, Riemann convergence and the explicit quadrature error bound. The earlier finite due-minus-arrears relationship remains a supporting target. These obligations follow from the separately defined probability masses and cashflow events, rather than assuming any valuation equality.
The classical distinction between annuity-certain and life-annuity present values is explained by Promislow (2015) through an explicit interest-and-survivorship discount mechanism. The present model makes this accounting precise at the level of individual payment events. It is consequently possible to compare a certain payment stream with a contingent one using exactly the same instalment schedule. It is also possible to alter the instalment amount without silently altering the survival law or guarantee entitlement.
The resulting mathematical foundation is suitable for retirement income and defined benefit pension cashflows where timing and benefit continuation clauses materially affect value. It separates annual unit amount from each instalment, and survival at commencement from survival at later payment events.
A death before commencement cannot activate a deferred guarantee. A death during the guarantee permits payments to the estate. An arrears instalment requires survival at the subsequent instant and the final one cannot silently disappear. Proving these relationships from true payment indicators is harder than defining annuity factors so the desired equations hold automatically. Finite lifetime censoring and the last surviving anniversary must remain visible in all derivations.
The proposed Lean representation is an indexed real-valued finite probability mass function, a survival indicator and Finset-based cashflow sums. The canonical scope is a single namespace and targeted imports of Mathlib's real-number and finite-big-operator files. The environment revision is pinned in the manifest, but remains subject to live remote verification. Periods are integer anniversaries, amounts are per actual instalment, and the valuation origin is zero. Only finite sums are asserted without convergence hypotheses. The existing classical annuity formulae appear in the cited books. Their underlying mathematics is known, while the original finite cashflow encoding and its Lean theorem obligations are open and unproved.
These primitives are intended to supply later work on DB pension guarantee periods, future life-insurance premium annuities and pension escalation. A future separate mission could reuse the actual realised-cashflow event semantics to value survival-contingent spouse benefits with a two-life entitlement law, while keeping spouse entitlement fixed at the policyholder's death under the scheme rules. The current work limits itself to the single-life estate-continuation benefit and therefore does not risk silently importing joint-life assumptions.
The extended finite discrete formalisation constructs an actual UDD subperiod death-state law. Its fractional annuity valuations follow from realised subperiod payment events, with each instalment equal to the nominal annual pension divided by frequency. Separate theorems reconcile second moments with pairwise survival-event intersections, derive the incremental value of guarantees and reconcile finite annual assurance with the life-annuity due while retaining the terminal censored-state residual. The separate continuous law and a quantitative two-term Woolhouse-type error inequality are now included as remotely elaborated open theorem declarations, still requiring genuine proofs. General commutation-column identities, noninteger-year deferrals under arbitrary interpolation, unbounded whole-life convergence and classical higher-order Woolhouse estimates need further formal models. An unconfirmed Foundation A owner Draft is not an importable dependency. Every theorem declaration is an open proof obligation and no accepted proofs are asserted.
A quantum LDPC code is a CSS code whose two check matrices and have rows and columns of bounded weight, so that every check measures only a few qubits. Such codes are the main candidates for fault-tolerant quantum memories, because their checks are local, but a code is only useful together with a decoder: an efficient procedure that recovers the error, up to stabilizers, from its syndrome. For classical LDPC codes built from expander graphs, Sipser and Spielman showed in 1996 that a simple bit-flipping algorithm corrects a constant fraction of the minimum distance in linear time (Sipser–Spielman 1996). In the quantum setting the obvious analogue fails, and before this work no family of quantum LDPC codes was known to admit an efficient decoder correcting a number of adversarial errors proportional to its minimum distance.
Timeline.
Let be a finite bipartite graph, , , with , and write for adjacency. is -biregular if every vertex of has neighbours and every vertex of has . For let be its neighbourhood. is -left-expanding if for every with ; right expansion is the same on the side, and left-right expanding means both. denotes the unique neighbours of , adjacent to exactly one vertex of .
Let be the binary biadjacency matrix of . The quantum code has qubits, indexed by (written and ), and check matrices
with rows indexed by and . The row of has support the Z-generator . An error is , its syndrome is , and is the space of sums of Z-generators. The weight of a quantum error counts the qubits where or is nonzero.
The decoder. On a syndrome , one iteration picks a vector whose support lies in some generator , such that has and the ratio is maximal among all such vectors. It iterates on the current syndrome, fails if it gets stuck at a nonzero syndrome, and otherwise outputs the sum of the flipped vectors. The decoder is the same with and exchanged.
Assume is -biregular and -left-right expanding with
and let be a quantum error with
Then on every run, the decoder on input never outputs a failure and its output satisfies . The decoder on behaves symmetrically, with .
In the paper's order of proof:
The result. Theorem 2 gives the first family of quantum LDPC codes with an efficient decoder that corrects a number of adversarial errors proportional to the minimum distance. Here that number is of order . The same analysis yields a robustness property: low-weight errors have syndromes of proportional weight (Corollary 9 of the paper). Later work on quantum expander codes builds on this analysis, for example on their decoding under random noise (arXiv:1711.08351) and on fault tolerance with constant overhead (arXiv:1808.03821).
Formalizing it. The result is proved on paper. As far as is known, it has no machine-checked proof. The mission produces a formal statement of the decoder and its correctness, and it also produces reusable pieces:
The obvious quantum analogue of Sipser–Spielman flips single qubits that lower the syndrome weight. It fails because of degeneracy: an error and its sum with a generator have the same syndrome. A small error can therefore look locally like a large one, and no single qubit need be worth flipping. The paper's decoder instead flips arbitrary patterns inside one generator. The difficulty is to show that some pattern always lowers the syndrome weight by a constant fraction of its size. The proof needs a combinatorial structure, the critical generator, whose existence uses unique-neighbour expansion on both sides of the graph, and a case analysis on its six parts. A second difficulty is that the decoder minimizes the syndrome weight, not the error weight, so the intermediate error can grow. Bounding that growth is what limits the guarantee to rather than the distance.
adj : A → B → Prop on finite types, and = bipMatrix adj : Matrix B A (ZMod 2).TillichZemor.HGP.Setting is reused. The paper's is TillichZemor.HGP.hz H H and its is hx H H; the X/Z names are swapped relative to the 2009 paper, and this is checked against eq. (5) in a sanity file.(A × A) ⊕ (B × B), weights are hammingNorm, and is CodingTheory.dualCode of ker H_Z.Candidate, Step, Reach, DecodesCorrectly), generic in the syndrome and generator matrices. Its ratio is computed in .Reach contains every state reachable under any tie-breaking. Correctness requires, at every reachable state, a further step while the syndrome is nonzero, and in the row space of the generator matrix once it is zero. Termination is automatic, because each step lowers a natural number.Additions and omissions.
Trivializing formalizations ruled out. The goal names the paper's own decoder, not "there exists a decoder", which a brute-force coset decoder would satisfy. It quantifies over every run, not a single favourable one. It asserts both never-failing and correctness, not partial correctness alone. It uses = hz H H, not the swapped matrix.
Welcome contributions. Proofs of the counting milestones (Lemma 3, Proposition 4, the column-weight bound, (19), (22)–(24)) are independent of one another. The distance bound of Tillich–Zémor, open on the platform, is the deepest dependency. Unique-neighbour expansion and the hypergraph-product distance bound are useful beyond this mission.
Best subset selection fits a linear model using at most of available predictors, choosing the that minimize the residual sum of squares. Mixed-integer optimization made it computable at moderate scale (Bertsimas, King & Mazumder, 2016), which reopened the statistical question of when it should be used. Empirical comparisons (Hastie, Tibshirani & Tibshirani, 2020) found that best subsets predicts poorly when the signal-to-noise ratio (SNR) is low: by fitting the noise exactly within its chosen support it overfits, and shrinkage methods such as the Lasso do better there.
Mazumder, Radchenko & Dedieu propose to add a shrinkage penalty to the cardinality-constrained problem. Their Theorem 2 quantifies the effect of an (Euclidean-norm) penalty: it proves prediction-error bounds that are linear in the penalty level and therefore remain small in the low-SNR regime where best subsets fails. This mission formalizes that theorem.
The data follow the model
where , is an unknown deterministic mean and is a noise vector (in the paper, with independent entries, ). The design is deterministic and its columns have unit Euclidean norm. Write for the Euclidean norm, for the number of nonzero coordinates of , and .
For a model size and a penalty level , the -regularized best-subsets estimator is any solution
The penalty is the norm , not its square, and the loss carries no factor . The quantity of interest is the prediction error .
The minimal -sparse eigenvalue of is . For , is the non-increasing rearrangement of . Given , three noise events are sets of noise vectors:
The paper's Theorem 1 shows these events have high probability; Theorem 2 is a deterministic statement on each of them. The symbol means "up to a positive universal multiplicative constant", one that does not depend on or .
Let .
(A) Slow rate. If , then on
and on
(B) Fast rate. For every , on
The bounds hold for every minimizer and for misspecified models: need not lie in the column span of .
The steps of the printed proof (Supplement B.2, pp. 39–40), in the order the proof uses them: the basic inequality (29); the bound (30) on and its consequence (31); the dichotomy on , where (30) implies (31) and (32) implies (33); the elementary inequality (34); the basic inequality on (p. 40); and the bound of through (p. 40).
The result. The slow-rate bound of part (A) is linear in . When the signal is small compared with , the term is smaller than the rate that best subsets attains, and the paper's Theorem 4 shows that in this regime the prediction error of best subsets is at least of order . Together the two results explain the empirical finding that shrinkage helps best subsets at low SNR. Part (B) shows nothing is lost at high SNR: the fast rate is the minimax rate over -sparse vectors plus the shrinkage term . Combined with Theorem 1, the bounds give the high-probability statements of the paper's Corollaries 1 and 3.
Formalizing it. The theorem is proved in the paper; no machine-checked version is known to exist. The mission produces a checked deterministic core: once the events are granted, every constant (, the factor , the threshold ) is verified exactly. Together with the companion mission on Theorem 1 it would yield fully checked high-probability oracle inequalities for a cardinality-constrained estimator, which no current formal library contains.
Every step is elementary, but the bookkeeping is exact. The obvious reading "insert the event's inequality into (29)" does not by itself work on : the event's right-hand side is a maximum of a sorted weighted sum and a prediction-norm term, and the first branch has to be compared with for the -sparse difference , including when exceeds . That comparison is a combinatorial estimate on sorted sums, not a substitution. On , the penalty difference is not controlled by the event at all and has to be converted into prediction errors through . In both parts the final constant must not depend on , or .
All objects are explicit Lean definitions in the namespace SubsetShrink.L2:
Fin n and Fin p with 0-based indices, and is a real Matrix (Fin n) (Fin p);Fin m → ℝ is the sup norm;The model consists of with unit columns, , , and . The estimator is a predicate that every minimizer satisfies; no minimizer is chosen and existence is not assumed. "On the event" is a hypothesis that the noise vector satisfies the event's printed inequality, so no probability measure appears. is the event as printed on p. 13; the correction of its probability bound belongs to the mission on Theorem 1 and does not affect these deterministic statements.
Committed conventions:
Part (B) assumes or . When the paper's bound is vacuous (), but Lean's would turn it into a false claim.
A formalization that lets the constant depend on , or , fixes a particular minimizer, weakens an event (for example drops "for all "), or computes the infimum as a real ⨅ over an unbounded set would trivialize the theorem and is ruled out.
Not formalized here: the probability bounds of Theorem 1, the -penalized Theorem 3, the lower bound of Theorem 4, the data-driven choice of (Theorem 5), and the corollaries and numerical experiments. These are separate missions or follow-ups. Contributions of reusable lemmas (the rearrangement bound for the sorted sum, properties of , the elementary absorption inequality (34)) are welcome.