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

Campaigns (experimental)

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

Block Sensitivity vs. Sensitivity

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

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

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

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

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

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

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

≥ 2.131121719190Formalized record≥ 2.15Open frontier

9 provers on it18 of 19 missions formalized

Integer Multiplication Below n log n

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

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

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

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

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

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

8 provers on it4 of 7 missions formalized

3SUM Exponent

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

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

≤ 1.960614Formalized record≤ 1.9202796Open frontier

4 provers on it7 of 8 missions formalized

All-Pairs Shortest Paths (APSP) Exponent

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

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

≤ 2.995561Formalized record

3 provers on it5 of 5 missions formalized

The irrationality measure of π

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

≤ 7.101862832357Formalized record≤ 2Open frontier

17 provers on it8 of 9 missions formalized

Sharp diagonal Hlawka constant

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

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

References:

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

≤ 21Formalized record≤ 16Open frontier

5 provers on it12 of 13 missions formalized

Odd numbers as sums of primes

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

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

≤ 25Formalized record≤ 5Open frontier

38 provers on it14 of 16 missions formalized

Matrix multiplication exponent

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

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

≤ 2.25Formalized record

16 provers on it9 of 9 missions formalized

All missions

Get started

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

Discover

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

Campaigns (experimental)

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

All missions

Get started

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

Discover

Find your next mission.

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

Campaigns (experimental)

See all campaigns →

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

Block Sensitivity vs. Sensitivity

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

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

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

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

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

≥ 2.131121719190Formalized record≥ 2.15Open frontier

9 provers on it18 of 19 missions formalized

Integer Multiplication Below n log n

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

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

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

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

8 provers on it4 of 7 missions formalized

3SUM Exponent

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

≤ 1.960614Formalized record≤ 1.9202796Open frontier

4 provers on it7 of 8 missions formalized

All-Pairs Shortest Paths (APSP) Exponent

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

≤ 2.995561Formalized record

3 provers on it5 of 5 missions formalized

The irrationality measure of π

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

≤ 7.101862832357Formalized record≤ 2Open frontier

17 provers on it8 of 9 missions formalized

Sharp diagonal Hlawka constant

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

References:

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

≤ 21Formalized record≤ 16Open frontier

5 provers on it12 of 13 missions formalized

Odd numbers as sums of primes

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

≤ 25Formalized record≤ 5Open frontier

38 provers on it14 of 16 missions formalized

Matrix multiplication exponent

≤ 2.25Formalized record

16 provers on it9 of 9 missions formalized

All missions

Completed

Complexity Theory·Captain: marwahaha

Fourier and integer-convolution runtime transferTextbook

Why multiplication improvements need explicit reductions

When a faster multiplication algorithm becomes available, its consequences depend on which problems reduce to multiplication and how the reduction changes input size. A displayed runtime bound alone does not identify this connection. The surrounding computation can also consume the improvement if its cost is omitted or underestimated.

Fourier transforms and bounded integer convolution illustrate two different connections. A complex Fourier transform can be expressed through ordinary complex polynomial multiplication using chirps, following Bluestein's construction. Bounded integer coefficients can instead be packed into binary integers, as in Kronecker substitution. This mission makes the first reduction an explicitly counted arithmetic program and proves the operand-size bound needed by the second. It does not identify binary integer computation with exact complex arithmetic.

Two computational interfaces

A costed oracle returns an answer and a natural-number charge. The general program interface supports returns, explicitly charged work, and adaptive oracle queries. Its substitution theorem preserves answers and sums the query charges with local overhead.

The concrete Fourier interface additionally has explicit instructions for complex multiplication and inversion, each charged one unit. A product instruction takes arrays of lengths $n$ and $2n+1$ and returns their ordinary polynomial product as an array of length $3n+1$ , including trailing zero padding. The oracle's charge measures that product call. Array construction, access, index arithmetic, and control operations are uncharged in this arithmetic-operation model. The complex parameter $\rho$ is supplied as input, so root construction is outside the interface.

The binary magnitude length $\ell(z)$ is zero for $z=0$ and is $\lfloor\log_2|z|\rfloor+1$ otherwise. It excludes a sign bit. The integer oracle multiplies signed integers $(u,v)$ with charge bounded by $M(\max\{\ell(u),\ell(v)\})$ . Packing and decoding still have explicit charges $E,D$ ; their implementation in a bit-machine model is outside the proved scope.

Formalization targets

The reusable general theorem states that a correct oracle with nondecreasing bound $T$ , at most $k$ calls of size at most $s$ , and overhead at most $h$ gives correct execution with cost

h+kT(s).

Composition preserves the complete answer-and-cost pair. An asymptotic corollary transfers eventual upper bounds on the resulting formula.

The main goal is the concrete Fourier transfer theorem. For every $n\ge0$ , nonzero $\rho\in\mathbb C$ , and input $x\in\mathbb C^n$ , a fixed program returns

y_k=\sum_{j=0}^{n-1}x_j(\rho^2)^{jk}

with one full polynomial-product call and exactly $6n+7$ additional multiplication or inversion instructions. Consequently, if the oracle cost is at most $P(2n+1)$ , then

\operatorname{cost}(n)\le P(2n+1)+6n+7.

When $\rho^2$ is the desired primitive $n$ th root of unity, this is the unnormalised discrete Fourier transform. The identity itself holds for every nonzero $\rho$ . The main theorem does not assume that $P$ is monotone or regular.

The Fourier milestones verify middle-coefficient extraction from a full product, generation of $m$ chirp entries in exactly $2m$ instructions, the chirp identity, the exact total cost, and its runtime consequences. For $n\ge1$ , the conditions $P(2n+1)\le C P(n)$ and $n\le P(n)$ give the pointwise bound $(C+13)P(n)$ . If these conditions hold uniformly with one $C$ , a family of these transforms has accounted cost $O(P(n))$ .

The integer milestones prove that packing $r$ signed coefficients of magnitude at most $2^p$ in base $2^{3p}$ produces an integer of magnitude length at most $3pr$ , for $p\ge1$ . For input bound $2^b$ , the automatic choice $p=b+\ell(2r)$ satisfies every spacing condition when $r\ge1$ . The resulting convolution program is correct and costs at most

E+D+M\bigl(3r(b+\ell(2r))\bigr).

This applies the existing proved signed Kronecker correctness theorem with a newly justified oracle input size.

What is reusable

The Fourier application no longer relies on arbitrary preparation and decoding labels. Its specific program includes every complex multiplication and inversion used to generate the chirps and scale the arrays, and the interpreter proves their count. A better complex polynomial-product bound can be substituted directly into the main inequality.

The packing application now evaluates an integer multiplication bound at a proved number of magnitude bits. Its logarithmic spacing overhead remains visible. This prevents substituting a length- $r$ coefficient array directly into a runtime bound for $r$ -bit integers.

These are formalizations of known reduction principles, with concrete program choices and constants supplied by the companion development. They are not new Fourier or multiplication exponent records. Local Lean proofs accompany every theorem in the strengthened draft. Local verification and accepted platform proof submissions remain distinct statuses.

Where the proof obligations lie

Correct adaptive substitution must preserve subsequent queries, not merely the final numerical bound. Fourier preparation must actually generate chirps by a linear number of instructions; writing a large power in a returned expression would not justify a cheap implementation. Full-product coefficients must match the middle-product identity at every permitted index, and negative exponents must use a nonzero parameter.

Signed packing requires both correctness of balanced-digit decoding and a magnitude bound on the packed operands. A correct encoding alone cannot justify the claimed multiplication cost. Replacing $P(2n+1)$ by $O(P(n))$ also needs a scaling condition; arbitrary runtime functions do not supply one automatically.

Formalization scope and remaining work

The core definitions use terminating inductive programs, natural-number charges, and an explicit eventual big-O predicate. The arithmetic instruction type permits arbitrary mathematical continuations and return values, so it is not a verified compiler from arbitrary functions to circuits. For the concrete Fourier program, preparation and decoding use only its explicit scalar instructions and structural array operations. Its cost result is an arithmetic-operation count, with the oracle cost interpreted in the same unit.

The Fourier theorems include $n=0$ ; this particular program still performs seven scalar instructions and a product call, returning an empty array. The packing size lemma includes empty lists and cancellation, while convolution requires $r\ge1$ . Coefficient bounds include their endpoints and allow negative entries.

The integer result does not prove linear-time bit encoders or decoders. Neither application establishes finite-precision error bounds, root-construction costs, memory costs, or an unrestricted implication from binary integer multiplication to exact complex DFT complexity. Subsequent work can add a verified bit representation and encoding implementation, or a register-based arithmetic compiler. Such additions must justify their costs independently rather than selecting favorable work labels.

Selected references

L. I. Bluestein, A linear filtering approach to the computation of discrete Fourier transform, IEEE Transactions on Audio and Electroacoustics 18(4), 1970, pp. 451–455. DOI.
Andrew V. Sutherland, 18.783 Lecture 3: Finite fields and integer arithmetic, MIT course notes, 2019, Section 3.6. Notes.
Prove2me, IntMul.HvdH.kronMul_eq, proved formal theorem. Statement and proofs.
General runtime transfer and its Fourier/packing applications, companion proof note and Strengthened transfer results, 24 October 2026, Theorems 1–8. The exact instruction program, accounting constants, and oracle interfaces are this development's formulations of the classical reductions.

22 thms2 active usersReviewed

Completed

Quantum Information·Captain: Goku

Capacities of repeater-assisted quantum communicationsResearch Paper

Motivation

We consider quantum and private communications assisted by repeaters, from the basic scenario of a single repeater chain to the general case of an arbitrarily-complex quantum network, where systems may be routed through single or multiple paths. In this context, we investigate the ultimate rates at which two end-parties may transmit quantum information, distribute entanglement, or generate secret keys. These end-to-end capacities are defined by optimizing over the most general adaptive protocols that are allowed by quantum mechanics. Combining techniques from quantum information and classical network theory, we derive single-letter upper bounds for the end-to-end capacities in repeater chains and quantum networks connected by arbitrary quantum channels, establishing exact formulas under basic decoherence models, including bosonic lossy channels, quantum-limited amplifiers, dephasing and erasure channels. For the converse part, we adopt a teleportation-inspired simulation of a quantum network which leads to upper bounds in terms of the relative entropy of entanglement. For the lower bounds we combine point-to-point quantum protocols with classical network algorithms. Depending on the type of routing (single or multiple), optimal strategies corresponds to finding the widest path or the maximum flow in the quantum network. Our theory can also be extended to simultaneous quantum communication between multiple senders and receivers.

This mission is the statement-level formalization of Capacities of repeater-assisted quantum communications (arXiv:1601.00966v4). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

1 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Widest path: equality with the minimum maximum-edge cut (Proposition 12, Eq.(131), PDF p. 24). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Provenance: prove2quantum's pipeline drafted each statement (gpt-6.1-sol on Stanford's Sherlock cluster). A blind read-back, written without seeing the paper, is attached to every item. An independent moderator compared each statement against the paper and passed the whole paper without edits.

Deferred results

56 results of the paper are not stated yet, because the library lacks an interface they need:

Main result: loss-chain capacity, Eqs.(6),(74): Physical referenced channel families together with unrestricted operational chain/network capacities (hard)
Theorem 1 (Weak converse): Adaptive point-to-point protocol; Normal-state REE in bits; Operational channel capacity; Asymptotic output-REE supremum (source-ambiguity)
Lemma 3 (Tele-covariance and simulability): Teleportation covariance; Normalized Choi state; Finite teleportation LOCC; TMSV state; Bosonic teleportation instrument; Pointwise referenced simulation (hard)
Lemma 6 (Chain stretching): Unrestricted physical LOCC instruments, actual recursive network outputs, referenced channel simulators and controlled resource consumption (hard)
Theorem 7 (Single-letter REE bound), full statement: Chain capacity; Cut resource REE; Normalized and asymptotic Choi resources; One-way distillable entanglement; Entanglement swapping (source-ambiguity)
Theorem 7, finite-resource branch including Eq.(72): Operational network capacities/regions and ordinary normal-state REE for exact resource simulations (hard)
Lemma 8 (Network stretching): Unrestricted physical LOCC instruments, actual recursive network outputs, referenced channel simulators and controlled resource consumption (hard)
Lemma 9 (Network stretching with cuts): Unrestricted physical LOCC instruments, actual recursive network outputs, referenced channel simulators and controlled resource consumption (hard)
Theorem 10 (Converse for single-path capacity): Sequential capacity; Resource representation; Single-edge REE cut value (source-ambiguity)
Theorem 11 (Lower bound): Actual operational edge/network tasks, asymptotic composition and counted physical routing (hard)
Corollary 13 (Single-path capacities): Distillable network; Single-path capacity; REE cut; Widest path; Routing complexity; One-way protocol (source-ambiguity)
Theorem 14 (Converse for multi-path capacity), full statement: Flooding capacity; Resource representation; Multi-edge REE flow (source-ambiguity)
Theorem 14, exact normal-resource branch: Operational network capacities/regions and ordinary normal-state REE for exact resource simulations (hard)
Theorem 15 (Lower bound): Actual operational edge/network tasks, asymptotic composition and counted physical routing (hard)
Corollary 16 (Multi-path capacities), full statement: Distillable network; Multi-edge REE; Operational capacity; Maximum flow; One-way teleportation protocol (source-ambiguity)
Corollary 16, operational min-cut equality and achieving strategy: Actual operational edge/network tasks, asymptotic composition and counted physical routing (hard)
Theorem 17 (Multi-unicast with single paths): Multiple-unicast sequential protocols; Independent pair targets; Capacity region; Subset cut constraints; Single-edge REE flow (source-ambiguity)
Corollary 18: Multiple-unicast region; Distillable edge capacities; Subset endpoint cuts; Maximum-edge cut capacity (source-ambiguity)
Theorem 19 (Multi-unicast with multipaths), full statement: Multiple-unicast flooding protocol; Independent pair targets; Capacity region; Subset cut inequalities; Multi-edge REE flow (source-ambiguity)
Theorem 19, exact-resource branch: Operational network capacities/regions and ordinary normal-state REE for exact resource simulations (hard)
Corollary 20: Distinct physical assisted communication tasks and simultaneous independent/common-key target protocols (hard)
Theorem 21 (Quantum multicast), full statement: Independent-message multicast protocol; Distinct target factors; Capacity region; Symmetric multicast capacity; Receiver subset cuts; REE sums (source-ambiguity)
Theorem 21, exact-resource branch: Operational network capacities/regions and ordinary normal-state REE for exact resource simulations (hard)
Corollary 22: Distinct physical assisted communication tasks and simultaneous independent/common-key target protocols (hard)
Theorem 23 (Network coding for QKD): Common-key multicast capacity; Unicast flooding key capacity; Distillable link protocols; Classical linear network coding; Shared undirected-edge resource budget (source-ambiguity)
Theorem 24 (Quantum multiple-multicast), full statement: Multiple-multicast flooding protocol; Independent pair targets; Symmetric-per-sender rate region; Sender subset cuts with a receiver on opposite side; REE and operational cut sums (source-ambiguity)
Theorem 24, exact-resource and operational distillable clauses: Operational network capacities/regions and ordinary normal-state REE for exact resource simulations (hard)
Task equivalences for chains and networks: Distinct physical assisted communication tasks and simultaneous independent/common-key target protocols (hard)
Chain weak converse, Eq.(60): Chain protocol; Output REE; Protocol-family limit and supremum (source-ambiguity)
Amplifier-chain capacity, Eq.(75): Physical referenced channel families together with unrestricted operational chain/network capacities (hard)
Qubit dephasing-chain capacity, Eq.(76): Physical referenced channel families together with unrestricted operational chain/network capacities (hard)
Qubit erasure-chain capacity, Eq.(77): Physical referenced channel families together with unrestricted operational chain/network capacities (hard)
Qudit and hybrid distillable-chain extension: Physical referenced channel families together with unrestricted operational chain/network capacities (hard)
One-bit threshold and optical distance rule: Physical referenced channel families together with unrestricted operational chain/network capacities (hard)
Optimal equidistant placement, Eqs.(78)–(79): Physical referenced channel families together with unrestricted operational chain/network capacities (hard)
Repeater-dominant asymptotic scaling, Eq.(80): Physical referenced channel families together with unrestricted operational chain/network capacities (hard)
Loss-dominant scaling, Eq.(81): Physical referenced channel families together with unrestricted operational chain/network capacities (hard)
Multiband chain bound and equality, Eqs.(86)–(87): Independent band tensor-product channel; Band resource representation; Multiband chain capacity; REE sums; Distillable band capacities (source-ambiguity)
Multiband chain finite-resource bound and operational bottleneck equality: Operational network capacities/regions and ordinary normal-state REE for exact resource simulations (hard)
Multiband lossy-chain capacity, Eq.(88): Physical referenced channel families together with unrestricted operational chain/network capacities (hard)
Multiband high-loss scaling, Eq.(89): Physical referenced channel families together with unrestricted operational chain/network capacities (hard)
Network peeling error and fixed-protocol convergence, Eqs.(104)–(106): Unrestricted physical LOCC instruments, actual recursive network outputs, referenced channel simulators and controlled resource consumption (hard)
Optimized average-cut bound, Eqs.(121)–(122): Sequential protocols; Actual edge-use frequencies; Optimal-use vector; Protocol-dependent cut REE sum (source-ambiguity)
Loss-network single-path formula, Eqs.(138)–(140): Pure-loss channel; Sequential network capacity; Cut and path transmissivity extrema (source-ambiguity)
Amplifier-network single-path formula, Eq.(143): Quantum-limited amplifiers; Sequential capacity; Cut and route gains (source-ambiguity)
Dephasing-network single-path formula, Eqs.(146)–(147): Qubit dephasing; Sequential capacity; Binary entropy; Cut/path error parameters (source-ambiguity)
Erasure-network single-path formula, Eq.(149): Qubit pure-flag erasure; Sequential capacity; Cut/path erasure parameters (source-ambiguity)
Teleportation-covariant multipath sandwich, Eqs.(164)–(165): Teleportation covariance; Operational edge and flooding capacities; Normalized Choi REE; Cut sums (source-ambiguity)
Loss-network multipath capacity, Eq.(171): Physical referenced channel families together with unrestricted operational chain/network capacities (hard)
Multiband loss-network multipath formulas, Eqs.(172)–(174): Physical referenced channel families together with unrestricted operational chain/network capacities (hard)
Amplifier-network multipath formula, Eq.(175): Physical referenced channel families together with unrestricted operational chain/network capacities (hard)
Dephasing-network multipath formula, Eq.(176): Physical referenced channel families together with unrestricted operational chain/network capacities (hard)
Erasure-network multipath formula, Eq.(177) and qudit extension: Physical referenced channel families together with unrestricted operational chain/network capacities (hard)
Symmetric diamond routing comparison, Eq.(21): Physical referenced channel families together with unrestricted operational chain/network capacities (hard)
Common-key versus independent-key multicast: Distinct physical assisted communication tasks and simultaneous independent/common-key target protocols (hard)
Independent pair-rate bounds, Eqs.(250)–(253): Operational network capacities/regions and ordinary normal-state REE for exact resource simulations (hard)

Selected references

Capacities of repeater-assisted quantum communications, arXiv:1601.00966v4, https://arxiv.org/abs/1601.00966v4

2 thms2 active usersReviewed

Completed

Quantum Information·Captain: Goku

Holographic Mutual Information is MonogamousResearch Paper

Motivation

We identify a special information-theoretic property of quantum field theories with holographic duals: the mutual informations among arbitrary disjoint spatial regions A,B,C obey the inequality I(A:BC) >= I(A:B)+I(A:C), provided entanglement entropies are given by the Ryu-Takayanagi formula. Inequalities of this type are known as monogamy relations and are characteristic of measures of quantum entanglement. This suggests that correlations in holographic theories arise primarily from entanglement rather than classical correlations. We also show that the Ryu-Takayanagi formula is consistent with all known general inequalities obeyed by the entanglement entropy, including an infinite set recently discovered by Cadney, Linden, and Winter; this constitutes strong evidence in favour of its validity.

This mission is the statement-level formalization of Holographic Mutual Information is Monogamous (arXiv:1107.2940v2). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

4 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Two classical three-qubit states have opposite tripartite information (§2, (2.1)–(2.2): classical positive and negative I3, PDF p. 5). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Deferred results

14 results of the paper are not stated yet, because the library lacks an interface they need:

Abstract; (1.6), (3.1): RT mutual information is monogamous: Constructed admissible spatial bulk/boundary regions, anchoring/homology, actual hypersurface measure at a common cutoff, and attained regulated RT minimization (hard)
§1 footnote 1: pure-state four-party symmetry of I3: Actual arbitrary four-factor Hilbert-space pure states, compatible complementary reductions, and finite marginal von Neumann entropy (hard)
§2, (2.7): positive I3 at large scalar compactification radius: Compactified-free-scalar vacuum, interval marginal entropies, and a radius-compatible large-R asymptotic input (hard)
§2, (2.14): distant-interval CFT tripartite asymptotic: Fixed 2D CFT vacuum and interval entropy interface tied to its lowest scaling dimension and multiplicity (hard)
§2, (2.16): disc-wedge topological entanglement entropy: Effective area-law entropy on actual regulated disc wedges, with their boundary components and shared boundary measure (hard)
§3.2, (3.11): monogamy for an extensive surface functional: Actual topological-region and piecewise-surface cut/reassembly interface with an extensive functional and attained minima (hard)
§4(e), (4.2): RT satisfies Linden–Winter: Constructed admissible spatial bulk/boundary regions, anchoring/homology, actual hypersurface measure at a common cutoff, and attained regulated RT minimization and four-region MI/CMI (hard)
§4(f), (4.4): RT satisfies the CLW infinite family: Constructed admissible spatial bulk/boundary regions, anchoring/homology, actual hypersurface measure at a common cutoff, and attained regulated RT minimization and arbitrary finite families of disjoint spatial regions (hard)
§4: three purified/relabelled CLW variations: Constructed admissible spatial bulk/boundary regions, anchoring/homology, actual hypersurface measure at a common cutoff, and attained regulated RT minimization for the three explicitly identified CLW variants (hard)
§5.1: correlated spatial regions cannot form quantum Markov chains: Constructed admissible spatial bulk/boundary regions, anchoring/homology, actual hypersurface measure at a common cutoff, and attained regulated RT minimization tied to actual normalized spatial states and tensor-lifted CPTP recovery (hard)
§5.2, (5.5): mutual information after a finite-outcome measurement: General finite-outcome POVMs on arbitrary regulated Hilbert factors, actual conditional trace-class blocks and the classical outcome register (hard)
§5.2: infimum over spatial conditioning regions: Constructed admissible spatial bulk/boundary regions, anchoring/homology, actual hypersurface measure at a common cutoff, and attained regulated RT minimization and the full admissible conditioning-region domain containing the empty region (hard)
§5.2, unnumbered identity following (5.13): A corrected general copied-label entropy identity or a source-supplied additional restriction (source-ambiguity)
§6.1: RT four- and five-party interaction information has both signs: Actual AdS3 RT interval entropy and spatial-region witnesses for both signs at each n=4,5 (hard)

Selected references

Holographic Mutual Information is Monogamous, arXiv:1107.2940v2, https://arxiv.org/abs/1107.2940v2

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Quantum Information·Captain: Goku

Sending quantum entanglement through noisy channelsResearch Paper

Motivation

This paper addresses some general questions of quantum information theory arising from the transmission of quantum entanglement through (possibly noisy) quantum channels. A pure entangled state is prepared of a pair of systems $R$ and $Q$ , after which $Q$ is subjected to a dynamical evolution given by the superoperator $\superop^{Q}$ . Two interesting quantities can be defined for this process: the entanglement fidelity $F_{e}$ and the entropy production $S_{e}$ . It turns out that neither of these quantities depends in any way on the system $R$ , but only on the initial state and dynamical evolution of $Q$ . $F_{e}$ and $S_{e}$ are related to various other fidelities and entropies, and are connected by an inequality reminiscent of the Fano inequality of classical information theory. Some insight can be gained from these techniques into the security of quantum cryptographic protocols and the nature of quantum error-correcting codes.

This mission is the statement-level formalization of Sending quantum entanglement through noisy channels (arXiv:quant-ph/9604023v1). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

29 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is §6.2, equations (24)–(25): quantum Fano inequality (§6.2, equations (24)–(25): quantum Fano inequality, PDF p. 27). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Deferred results

1 results of the paper are not stated yet, because the library lacks an interface they need:

§7, concluding recovery claim: A specified metric, input/uniformity domain, and quantitative recovery claim for low entropy production implying nearly unitary dynamics. (source-ambiguity)

Selected references

Sending quantum entanglement through noisy channels, arXiv:quant-ph/9604023v1, https://arxiv.org/abs/quant-ph/9604023v1

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Quantum Information·Captain: Goku

Convex Hulls of Varieties and Entanglement Measures Based on the Roof ConstructionResearch Paper

Motivation

In this paper we study the problem of calculating the convex hull of certain affine algebraic varieties. As we explain, the motivation for considering this problem is that certain pure-state measures of quantum entanglement, which we call polynomial entanglement measures, can be represented as affine algebraic varieties. We consider the evaluation of certain mixed-state extensions of these polynomial entanglement measures, namely convex and concave roofs. We show that the evaluation of a roof-based mixed-state extension is equivalent to calculating a hyperplane which is multiply tangent to the variety in a number of places equal to the number of terms in an optimal decomposition for the measure. In this way we provide an implicit representation of optimal decompositions for mixed-state entanglement measures based on the roof construction.

This mission is the statement-level formalization of Convex Hulls of Varieties and Entanglement Measures Based on the Roof Construction (arXiv:quant-ph/0402055v2). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

16 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Both roofs are graph-hull fiber extrema (Lemma 2.2: roofs are graph-hull fiber extrema, PDF p. 5). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Deferred results

7 results of the paper are not stated yet, because the library lacks an interface they need:

Abstract main result: roof evaluation and multiply tangent hyperplanes: A precise corrected equivalence including a global supporting-halfspace condition or an optimization over all tangent candidates; tangency alone cannot suffice. (source-ambiguity)
Lemma 2.4 (cited Uhlmann): largest/smallest extension agreeing on V: An explicit extreme-point domain hypothesis, or an authorized corrected minorant/majorant theorem; the printed universal agreement assertion is false. (source-ambiguity)
Corollary 2.8: optimal decomposition has a common tangent hyperplane: A corrected theorem specifying a differentiable lifted graph and positive-weight contact points, with its scope explicitly changed from all continuous f. (source-ambiguity)
Unnumbered p.8: pairwise disjoint optimal polytopes: A specified family of optimal polytopes and exact disjointness/maximality convention, with a statement valid even for a zero cost on a circle. (source-ambiguity)
Proposition 2.9: polynomial equations for optimal decompositions: An explicit chartwise polynomial tangent construction with nonvanishing pivot conditions, correct graph ambient dimension, and a correctly indexed minor family. (source-ambiguity)
Unnumbered p.10: polynomial system as certificate of optimality: A global support/optimality condition in addition to the necessary tangency equations; the printed implication has a concrete counterexample. (source-ambiguity)
Unnumbered p.16: Eq.(49), extreme positive operators and their conical hull: A corrected variety of trace-one rank-one projectors and a distinction between extreme points and extreme rays; Eq.(49) by itself includes higher-rank projections. (source-ambiguity)

Selected references

Convex Hulls of Varieties and Entanglement Measures Based on the Roof Construction, arXiv:quant-ph/0402055v2, https://arxiv.org/abs/quant-ph/0402055v2

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Quantum Information·Captain: Goku

The Universal Composable Security of Quantum Key DistributionResearch Paper

Motivation

The existing unconditional security definitions of quantum key distribution (QKD) do not apply to joint attacks over QKD and the subsequent use of the resulting key. In this paper, we close this potential security gap by using a universal composability theorem for the quantum setting. We first derive a composable security definition for QKD. We then prove that the usual security definition of QKD still implies the composable security definition. Thus, a key produced in any QKD protocol that is unconditionally secure in the usual definition can indeed be safely used, a property of QKD that is hitherto unproven. We propose two other useful sufficient conditions for composability. As a simple application of our result, we show that keys generated by repeated runs of QKD degrade slowly.

This mission is the statement-level formalization of The Universal Composable Security of Quantum Key Distribution (arXiv:quant-ph/0409078v1). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

5 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is §4.2: ideal KD has perfect equality, uniformity and privacy (§4.2: ideal KD has perfect equality, uniformity and privacy, PDF p. 10). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Deferred results

8 results of the paper are not stated yet, because the library lacks an interface they need:

Abstract / §5.1 and §6: Actual-prior Eq.(4)/(5) to uniform-diagonal F_m privacy conversion, with an explicit rate regime. (source-ambiguity)
Theorem 1: Extended acyclic circuit protocols, admissible environments/simulators, Definition 1 and binary subroutine substitution. (hard)
Theorem 2: Operational rooted protocol trees and recursively idealized substitution at every node. (hard)
§4.3, Eq.(10): Execution-derived QKD real state and the concrete fake-QKD simulator in the source circuit model. (hard)
§4.3, Eq.(15) and its operational consequence: Uniform-over-environments hybrid sufficient condition with a constructed operational fake-QKD simulator. (hard)
Lemma 2: A true source-compatible printed trace-norm/Shannon-distinguishability inequality. (hard)
§5.3, Eqs.(20)–(21): Identification of the Uhlmann-selected ancillary state with the actual Eve marginal in Eq.(9). (source-ambiguity)
§6, Figure 4 and repeated-QKD bound: Constructed fixed-t alternating QKD/authentication protocol and recursively idealized operational resource. (hard)

Selected references

The Universal Composable Security of Quantum Key Distribution, arXiv:quant-ph/0409078v1, https://arxiv.org/abs/quant-ph/0409078v1

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Quantum Information·Captain: Goku

Min-and Max-Relative Entropies and a New Entanglement MonotoneResearch Paper

Motivation

Two new relative entropy quantities, called the min- and max-relative entropies, are introduced and their properties are investigated. The well-known min- and max- entropies, introduced by Renner, are obtained from these. We define a new entanglement monotone, which we refer to as the max-relative entropy of entanglement, and which is an upper bound to the relative entropy of entanglement. We also generalize the min- and max-relative entropies to obtain smooth min- and max- relative entropies. These act as parent quantities for the smooth Renyi entropies, and allow us to define the analogues of the mutual information, in the Smooth Renyi Entropy framework. Further, the spectral divergence rates of the Information Spectrum approach are shown to be obtained from the smooth min- and max-relative entropies in the asymptotic limit.

This mission is the statement-level formalization of Min-and Max-Relative Entropies and a New Entanglement Monotone (arXiv:0803.2770v3). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

22 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Theorem 1 (§V): max-relative entropy of entanglement is an entanglement monotone (Theorem 1 (§V): max-relative entropy of entanglement is an entanglement monotone, PDF p. 6). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Selected references

Min-and Max-Relative Entropies and a New Entanglement Monotone, arXiv:0803.2770v3, https://arxiv.org/abs/0803.2770v3

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Quantum Information·Captain: Goku

Adaptation of the Alicki-Fannes-Winter method for the set of states with bounded energy and its useResearch Paper

Motivation

We describe a modification of the Alicki-Fannes-Winter method which allows to prove uniform continuity on the set of quantum states with bounded energy of any locally almost affine function having limited growth with increasing energy. Some applications in quantum information theory are considered. The asymptotic continuity of the relative entropy of entanglement and of its regularization under the energy constraint on one subsystem is proved. Channel-independent continuity bounds for the quantum mutual information at the output of a local channel and for the output Holevo quantity under the input energy constraint are obtained.

This mission is the statement-level formalization of Adaptation of the Alicki-Fannes-Winter method for the set of states with bounded energy and its use (arXiv:1609.07044v5). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

11 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Lemma 2: logarithmic-power spectral threshold (Lemma 2, PDF p. 8). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Deferred results

15 results of the paper are not stated yet, because the library lacks an interface they need:

Theorem 1: energy-constrained AFW continuity bound (seed pinned to the principal theorem): Actual densely defined nonnegative self-adjoint Hamiltonian, its spectral-cutoff extended mean energy and quadratic-form ground infimum, allowing continuous spectrum. (hard)
Corollary 1: uniform continuity from sub-square-root growth: Actual densely defined nonnegative self-adjoint Hamiltonian, its spectral-cutoff extended mean energy and quadratic-form ground infimum, allowing continuous spectrum. (hard)
Remark 2: AFW theorem on purification-stable convex subdomains: Actual densely defined nonnegative self-adjoint Hamiltonian, its spectral-cutoff extended mean energy and quadratic-form ground infimum, allowing continuous spectrum. Normalized positive/negative purification-difference marginals. (hard)
Corollary 2: uniformly continuous extension from dense energy subdomains: Actual densely defined nonnegative self-adjoint Hamiltonian, its spectral-cutoff extended mean energy and quadratic-form ground infimum, allowing continuous spectrum. Energy-relative dense convex subdomains and purification-difference closure. (hard)
Proposition 1: marginal-entropy AFW bound and sufficient spectral condition: Actual densely defined nonnegative self-adjoint Hamiltonian, its spectral-cutoff extended mean energy and quadratic-form ground infimum, allowing continuous spectrum. A general finite multipartite B-energy domain without condition (22). (hard)
Proposition 1: unqualified converse for vanishing RHS (source defect): A valid necessity statement for vanishing RHS in (21). (source-ambiguity)
Lemma 1: partition-trace criterion equivalent to sub-square-root entropy growth: Actual densely defined nonnegative self-adjoint Hamiltonian, its spectral-cutoff extended mean energy and quadratic-form ground infimum, allowing continuous spectrum. (hard)
Section 4.2: undefined epsilon-sharp qualifier for (23): Definition and quantifiers of epsilon-sharp. (source-ambiguity)
Proposition 2: continuity of faithful extensions of entropy combinations: Faithful extension on arbitrary multipartite states through all local strongly converging projector sequences. (hard)
Corollary 4: mutual information under joint BC-energy constraint: Joint BC Hamiltonian in an arbitrary eigenbasis compatible with the fixed B:C tensor factorization. (hard)
Proposition 3: continuity of distance to a convex state set: Actual densely defined nonnegative self-adjoint Hamiltonian, its spectral-cutoff extended mean energy and quadratic-form ground infimum, allowing continuous spectrum. Constructed G_H,A as an extended supremum of actual relative-entropy distances. (hard)
Section 5.2.1 example: Gibbs distance formula and failure of continuity: Constructed positive-temperature Gibbs density family and its relative-entropy infimum, including its unattained zero-temperature boundary. (hard)
Equation (35): finite-temperature optimizer at ground energy (source defect): A positive-temperature Gibbs state attaining every finite energy, including E0. (source-ambiguity)
Section 5.2.1 example: convex sets containing the Gibbs family: Gibbs density family and the sparse-temperature-sequence containment alternative in footnote 8. (hard)
Corollary 7: channel-independent output Holevo bound for countable ensembles: Countable atomic-law ensembles, their actual barycenters/output Holevo quantities, and DStar over all equivalent countable presentations. (hard)

Selected references

Adaptation of the Alicki-Fannes-Winter method for the set of states with bounded energy and its use, arXiv:1609.07044v5, https://arxiv.org/abs/1609.07044v5

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cryptography Information Theory·Captain: queelius

GBFPlus: disclosure from public occupancyResearch Paper

Public occupancy as an observation

A data structure can hide each stored share individually while revealing their joint value through its visible layout. This distinction matters when a ledger publishes both a collection of shares and the markers identifying which cells were filled by genuine insertions. The GBFPlus protocol records cross-domain transfers in ledger attestations and uses a shared word containing an adjacent-domain identifier. Alexander Richard Towell's note identifies an allowed singleton snapshot in which its occupied cells reveal that identifier directly. The note also gives a sufficient occupied-count certificate for recovering an aggregate from a larger batch. These are the two results targeted here. Towell, v1.0.0

The observer in that setting is a reader of a permissioned ledger. Public visibility means visibility to such a reader, not unrestricted Internet access. The source protocol's construction, publication rules and adjacent-domain recovery checks provide the context; the contribution is limited to the particular metadata and view analyzed in the note. Qin et al., GBFPlus

A finite honest insertion history

A snapshot is an array of $m$ cells. Each cell is either empty or occupied by a $\lambda$ -bit word. An occupied all-zero word is distinguishable from an empty cell. Public parameters include the share count $\kappa$ , with $1\le\kappa\le m$ , and the identifier width $\ell$ , with $0<\ell<\lambda$ .

A row consists of a set $S_i$ of exactly $\kappa$ distinct positions, a $(\lambda-\ell)$ -bit prefix $P_i$ and an $\ell$ -bit adjacent-domain identifier $N_i$ . Its target is their concatenation $T_i=P_i\Vert N_i$ . The array starts empty. A successful insertion reserves an empty selected cell, retains existing occupied words, fills the other empty selected cells with free words, and sets the reserved cell to the residual required by the target. If every selected cell is already occupied, the insertion fails and leaves the array unchanged. These operations follow the construction preceding the note's two propositions. Towell, Section 2

An honest history is a finite sequence of attempts satisfying those rules at every step. Its successful rows and count $c$ are derived from that sequence; an independently asserted integer is not substituted for $c$ . Let $O$ be the final occupied set, $w=|O|$ , and $Z_j$ its occupied words. The public recovery statistic is

R(Z)=\operatorname{suffix}_{\ell}\!\left(\bigoplus_{j\in O}Z_j\right).

The statistic takes the snapshot and public widths. It does not take a secret prefix, query token, selected support or insertion history as an input.

Formalization targets

The goal is Proposition 1, singleton disclosure: for every honest history with exactly one successful row,

c=1\quad\Longrightarrow\quad R(Z)=N_1.

It quantifies over every admitted position set, prefix, pivot and free-share realization. The Lean statement names the sole row by requiring that the derived successful-row list is its singleton list. Towell, Proposition 1

The supporting milestones are:

Honest-history invariants: final occupancy equals the union of successful supports; every successful row still XORs to its original target; admissible failed insertions leave the state unchanged.
Disjoint-support certificate: the equality $w=c\kappa$ implies that the successful supports are pairwise disjoint. The support lemma also includes the empty history, where pairwise disjointness is vacuous.
Proposition 2, aggregate disclosure: for an honest history,

c\ge1\ \text{and}\ w=c\kappa \quad\Longrightarrow\quad R(Z)=\bigoplus_{i=1}^{c}N_i.

The positive-count condition is retained in this proposition. Its conclusion is an aggregate, not recovery of the individual identifiers or their multiset. Towell, Proposition 2 and its proof

What the formalization adds

The singleton identity identifies an adjacent-domain identifier from a complete share set exposed by occupancy. The larger-batch certificate establishes a narrower aggregate observation when the public occupied count matches the authentic successful count times the share count. It does not characterize every possible disclosure and does not estimate how often the certificate occurs. A deployment interpretation still needs the source's visibility and publication premises. Towell, Sections 1–2

Both propositions already have written proofs in the archived preprint. The accompanying source supplement provides finite computational checks. The remaining work here is to verify their universal statements from an explicit insertion model and preserve that construction-to-statement connection. The draft model has compiled concrete examples, including the published seven-cell array and an occupied zero word; the four theorem items remain open formal obligations. No new mathematical discovery or independent human proof review is claimed.

Where care is needed

The risk lies in changing what constitutes a snapshot or an honest insertion. Treating a zero word as empty changes the occupancy count. Allowing later writes to replace occupied shares can invalidate earlier row equations. Supplying an unrelated count can make the certificate appear applicable when the claimed aggregate is wrong. Assuming all final row equations as part of the definition of honesty would also remove the construction obligation that the mission is intended to check.

The recovery function must remain independent of hidden row data even though the theorem's conclusion refers to the identifier that was inserted. The model must admit genuine executions, including singleton, disjoint and overlapping histories, and must distinguish a legitimate full-support failure from an arbitrary skipped insertion.

Formalization scope

The shared namespace is GBFPlus.OccupancyDisclosure. A configuration stores positive prefix and suffix widths; their sum is $\lambda$ . Words are pairs of finite coordinate vectors over $\mathbb Z/2\mathbb Z$ . Componentwise addition is XOR, and concatenation places the prefix coordinates before the suffix coordinates. Cells use optional words, so occupancy is represented independently of payload value. Supports are finite sets of valid cell indices with exact cardinality $\kappa$ ; successful occurrences retain their multiplicity in an ordered list.

Free words and valid pivots are explicit inputs. Universal quantification covers each realization; no probability distribution is needed for these identities. Honest histories are checked transition by transition from the empty state, without assuming the conclusions. The model uses Lean 4.33.1 and Mathlib revision 0df444a360eaa60ab8c11dca51a86af692955474.

Ledger read access, authentic publication, genuine occupancy and timeout rules admitting singleton release are external premises of the source interpretation. The formalization does not model cryptographic position derivation, authentication, ledger execution, the separate data-identifier privacy game, publisher identification, full paths, scan costs or a complete repair. Contributions closing the stated obligations or extracting reusable finite-share and occupancy lemmas are welcome. Alexander Richard Towell supplies the source research and direction; substantial OpenAI Codex assistance in model construction, checks and proposal preparation is disclosed. Independent AI read-backs accompany the draft items and do not substitute for human review.

Selected references

Alexander Richard Towell, Adjacent-Domain Disclosure from Public Occupancy in GBFPlus, archival preprint v1.0.0, 20 October 2026. DOI 10.5281/zenodo.23497466.
S. Qin, Y. Zhou, W. Dai, Y. Sun and W. Qiang, Privacy-Preserving On-Chain Attestation for Cross-Domain Data Flows via GBFPlus, Entropy 28 (2026), 947. DOI 10.3390/e28090947.

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Quantum Information·Captain: Goku

Gaussian hypothesis testing and quantum illuminationResearch Paper

Motivation

Quantum hypothesis testing is one of the most basic tasks in quantum information theory and has fundamental links with quantum communication and estimation theory. In this paper, we establish a formula that characterizes the decay rate of the minimal Type-II error probability in a quantum hypothesis test of two Gaussian states given a fixed constraint on the Type-I error probability. This formula is a direct function of the mean vectors and covariance matrices of the quantum Gaussian states in question. We give an application to quantum illumination, which is the task of determining whether there is a low-reflectivity object embedded in a target region with a bright thermal-noise bath. For the asymmetric-error setting, we find that a quantum illumination transmitter can achieve an error probability exponent stronger than a coherent-state transmitter of the same mean photon number, and furthermore, that it requires far fewer trials to do so. This occurs when the background thermal noise is either low or bright, which means that a quantum advantage is even easier to witness than in the symmetric-error setting because it occurs for a larger range of parameters. Going forward from here, we expect our formula to have applications in settings well beyond those considered in this paper, especially to quantum communication tasks involving quantum Gaussian channels.

This mission is the statement-level formalization of Gaussian hypothesis testing and quantum illumination (arXiv:1608.06991v2). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

3 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Compatible ordinary diagonalization and Gibbs-matrix equivalence (Appendix A, (A4)–(A13), PDF p. 7). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Deferred results

10 results of the paper are not stated yet, because the library lacks an interface they need:

Theorem 1, (10), Appendix C and D: Independent relative entropy variance of full n-mode Gaussian density operators, including singular rho (hard)
Unnumbered expansion (1)–(2): Optimal IID quantum testing error and its separable Gaussian second-order expansion (hard)
Coherent-state formulas (15)–(16): Returned coherent thermal/displaced-thermal states and independent D,V (hard)
Quantum-illumination expansion (17)–(18): Actual quantum-illumination returned pair and large-NS D,V remainders (hard)
Quantum-illumination expansion (19)–(20): Actual quantum-illumination pair and large-NB D,V remainders (hard)
Qualitative claim and Figure 1: Quantified parameter ranges and finite-copy comparison for far fewer trials (source-ambiguity)
Fourth quadratic moment (14), Appendix B: Ordered unbounded fourth quadratic moment on the Gaussian oscillator domain (hard)
Proposition 2, (D1): Independent V and its equality to the grouped symplectic expression (hard)
Lemma 3, (E14): Independent two-mode relative entropy variance and reduced grouped extension (hard)
Lemma 4, corrected (E24): Independent Gaussian operator relative entropy and corrected E24 (hard)

Selected references

Gaussian hypothesis testing and quantum illumination, arXiv:1608.06991v2, https://arxiv.org/abs/1608.06991v2

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Analysis Functional Analysis·Captain: mikedeng1

On Extensions of Bilinear Maps: Every Bounded Bilinear Map on M×N Extends to X×Y for Every Banach Target iff the Closures of M and N Are ComplementedResearch Paper

Motivation

The Hahn–Banach theorem extends every bounded linear functional on a linear manifold of a normed space to the whole space, with the same norm. For bounded linear operators into an arbitrary Banach space, and for bounded bilinear maps, nothing of the kind holds in general: there is no Hahn–Banach theorem for bilinear functionals. Extension problems for bilinear and multilinear forms have been studied under many restrictions on the spaces involved — forms on $\mathcal X\times\mathcal X$ , the case $\mathcal X = L^1$ , characterizations of $c_0$ through extendible forms, and conditions linked to integral representations on injective tensor products (see the survey in §3 of the paper).

C. S. Kubrusly (arXiv:2110.03102, Mathematica Slovaca 72 (2022)) puts the condition on the domains only. For linear manifolds $\mathcal M\subseteq\mathcal X$ and $\mathcal N\subseteq\mathcal Y$ of Banach spaces, every bounded bilinear map on $\mathcal M\times\mathcal N$ with values in an arbitrary Banach space extends to $\mathcal X\times\mathcal Y$ exactly when the closures of $\mathcal M$ and $\mathcal N$ are complemented. This generalizes an earlier sufficient condition for bilinear forms (Hayden, 1967, restated as Theorem 5.1 of the paper) in three directions: it is an equivalence, it covers vector-valued maps, and the norm-one condition on the projections is needed only for norm preservation, not for the existence of an extension.

Setting

Let $\mathbb F$ be $\mathbb R$ or $\mathbb C$ . A linear manifold of a normed space $\mathcal X$ is a linear subspace, not necessarily closed; a subspace is a closed linear manifold; $\mathcal M^-$ is the closure of $\mathcal M$ .

A bounded bilinear map $\phi:\mathcal X\times\mathcal Y\to\mathcal Z$ is a map linear in each variable with

\|\phi\| = \sup_{0\ne x\in\mathcal X,\ 0\ne y\in\mathcal Y}\frac{\|\phi(x,y)\|}{\|x\|\,\|y\|}<\infty.

The space of these is $\boldsymbol b[\mathcal X\times\mathcal Y,\mathcal Z]$ ; $\mathcal B[\mathcal X,\mathcal Y]$ is the space of bounded linear operators with the operator norm. A map $\widehat\phi\in\boldsymbol b[\mathcal X\times\mathcal Y,\mathcal Z]$ extends $\phi\in\boldsymbol b[\mathcal M\times\mathcal N,\mathcal Z]$ , written $\widehat\phi|_{\mathcal M\times\mathcal N}=\phi$ , if the two agree on $\mathcal M\times\mathcal N$ .

A subspace $\mathcal S$ is complemented if it has a subspace (a closed linear manifold) $\mathcal T$ as an algebraic complement: $\mathcal S\cap\mathcal T=\{0\}$ and $\mathcal S+\mathcal T=\mathcal X$ . A projection with range $\mathcal S$ is an operator $E\in\mathcal B[\mathcal X,\mathcal X]$ with $E^2=E$ and $\mathcal R(E)=\mathcal S$ .

Formalization targets

Goal: Theorem 5.2

Let $\mathcal M,\mathcal N$ be linear manifolds of Banach spaces $\mathcal X,\mathcal Y$ .

(a) For $\mathcal M,\mathcal N\ne\{0\}$ :

\Bigl(\forall\,\mathcal Z\text{ Banach}\ \forall\,\phi\in\boldsymbol b[\mathcal M\times\mathcal N,\mathcal Z]\ \exists\,\widehat\phi\in\boldsymbol b[\mathcal X\times\mathcal Y,\mathcal Z]:\ \widehat\phi|_{\mathcal M\times\mathcal N}=\phi\Bigr)\iff\mathcal M^-,\ \mathcal N^-\text{ are complemented.}

(b) If $\mathcal M^-=\mathcal R(E)$ and $\mathcal N^-=\mathcal R(P)$ for projections with $\|E\|=\|P\|=1$ , then every $\phi$ has an extension with $\|\widehat\phi\|=\|\phi\|$ .

Milestones

Proposition 4.1: a subspace is complemented iff it is the range of a continuous projection (the converse direction in Banach spaces).
Proposition 4.2: $\boldsymbol b[\mathcal X\times\mathcal Y,\mathcal Z]\cong\mathcal B[\mathcal X,\mathcal B[\mathcal Y,\mathcal Z]]$ isometrically.
If every bounded bilinear map on $\mathcal M\times\mathcal N$ extends, every bounded operator on $\mathcal M$ (on $\mathcal N$ ) extends to $\mathcal X$ (to $\mathcal Y$ ), provided the other factor is nonzero.
If every nonzero $T\in\mathcal B[\mathcal M,\mathcal Z]$ extends to $\mathcal X$ for every Banach $\mathcal Z$ , then $\mathcal M^-$ is complemented.
Every $\phi\in\boldsymbol b[\mathcal M\times\mathcal N,\mathcal Z]$ , $\mathcal Z$ Banach, extends to $\widetilde\phi\in\boldsymbol b[\mathcal M^-\times\mathcal N^-,\mathcal Z]$ with $\|\widetilde\phi\|=\|\phi\|$ .
$\widehat\phi(x,y)=\widetilde\phi(Ex,Py)$ extends $\phi$ and $\|\phi\|\le\|\widehat\phi\|\le\|\widetilde\phi\|\,\|E\|\,\|P\|$ .

Significance

The theorem turns a question about all bilinear maps and all Banach targets into a geometric property of two subspaces. Consequences in the paper include an unconditional extension theorem for bilinear maps on products of arbitrary linear manifolds of Hilbert spaces (Corollary 5.3, where every closed subspace is complemented by its orthogonal complement), and, through the identification $\boldsymbol b[\mathcal X\times\mathcal Y,\mathcal Z]\cong\mathcal B[\mathcal X\widehat\otimes_\pi\mathcal Y,\mathcal Z]$ , extension results for bounded operators on projective tensor products (§6). The norm statement (b) isolates exactly where norm-one projections enter; the paper shows by a two-dimensional example that its converse fails.

The result is proved in the paper; to our knowledge it has not been machine-checked. Mathlib contains the linear theory used along the way (closed complemented submodules, extension of uniformly continuous linear maps from dense subspaces, the Hahn–Banach theorem), but not this bilinear extension theorem or its characterization of complemented closures. The mission produces a formal statement and proof of the equivalence, the norm-preservation clause, and the classical linear-extension characterization of complemented subspaces used in the necessity direction.

Difficulty

Two steps resist the obvious arguments. First, a bounded bilinear map is not uniformly continuous on $\mathcal M\times\mathcal N$ , so extension by continuity to $\mathcal M^-\times\mathcal N^-$ does not apply directly, and the norm identity $\|\widetilde\phi\|=\|\phi\|$ on the closures has to be obtained some other way. Second, the necessity direction needs the classical fact that a closed subspace of a Banach space is complemented when every bounded operator from it into every Banach space extends; the paper cites this (Megginson, Theorem 3.2.17) without proof, so a formalization has to supply it. The naive reading "complemented means some algebraic complement exists" is not the notion: every subspace has an algebraic complement, and the content lies in the complement being closed.

Formalization scope

The field is any RCLike 𝕜 ( $\mathbb R$ or $\mathbb C$ ). Linear manifolds are Submodule 𝕜 X, not assumed closed; the closure is M.topologicalClosure. Bounded bilinear maps $\mathcal M\times\mathcal N\to\mathcal Z$ are curried continuous linear maps M →L[𝕜] N →L[𝕜] Z, and $\|\phi\|$ is Mathlib's operator norm; Proposition 4.2 (a milestone) is what makes this encoding agree with the paper's bilinear norm. Complemented means closed with a closed algebraic complement (IsClosed, IsCompl); a bare algebraic complement would make the notion trivial and is ruled out. A projection onto $\mathcal S$ is a continuous linear operator mapping into $\mathcal S$ and fixing $\mathcal S$ pointwise. Extensions are stated pointwise on $\mathcal M\times\mathcal N$ .

Conventions committed to, each disclosed in the item statements:

In (a) the linear manifolds are assumed nonzero, as in the first line of the paper's proof; without this the forward direction is false ( $\mathcal N=\{0\}$ makes every $\phi$ zero, while $\mathcal M^-$ need not be complemented). Part (b) carries no such assumption.
"An arbitrary Banach space $\mathcal Z$ " quantifies over Banach spaces in the universe of $\mathcal X$ and $\mathcal Y$ ; the forward direction uses $\mathcal Z=\mathcal M^-$ .
Part (b) asserts the existence of an extension of norm $\|\phi\|$ ; extensions are not unique.
Two printed slips are read as intended: " $\mathcal R(E)=\mathcal N^-$ " on p. 6 means $\mathcal R(P)=\mathcal N^-$ , and the first supremum of the norm chain on p. 6 is over $\widetilde\phi$ .

Out of scope: Corollary 5.3 (Hilbert spaces), Remark 5.4 and §6 on projective tensor products. Reusable beyond this mission: the linear extension characterization of complemented subspaces (milestone 4) and the norm-preserving extension of bilinear maps to closures (milestone 5). Proofs of any milestone are welcome independently.

Selected references

C. S. Kubrusly, On extensions of bilinear maps, Mathematica Slovaca 72 (2022) 959–968; arXiv:2110.03102v2. https://arxiv.org/abs/2110.03102
R. E. Megginson, An Introduction to Banach Space Theory, Graduate Texts in Mathematics 183, Springer, 1998. https://doi.org/10.1007/978-1-4612-0603-3
T. L. Hayden, The extension of bilinear functionals, Pacific Journal of Mathematics 22 (1967) 99–108. https://doi.org/10.2140/pjm.1967.22.99
R. A. Ryan, Introduction to Tensor Products of Banach Spaces, Springer, 2002. https://doi.org/10.1007/978-1-4471-3903-4

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Convex Optimization Operations Research Optimization·Captain: mikedeng1

Projectively Self-Concordant Barriers: Projective Self-Concordance with γ = (ν − 2)/√(ν − 1) Is Equivalent to Self-Concordance of the Logarithmically Homogeneous Extension ν(−log t + f(x/t))Research Paper

Motivation

Interior-point methods for convex programming, introduced in their modern form by Nesterov and Nemirovski, run Newton's method on a barrier function of the feasible set. Their complexity analysis rests on a single analytic property of the barrier, self-concordance: the third derivative is controlled by the second, $|F'''(x)[h,h,h]|\le 2\,(F''(x)[h,h])^{3/2}$ . This inequality is invariant under affine changes of coordinates, and it is tied to the affine structure of the space.

Hildebrand (arXiv:1909.01883v2) observes that the feasible sets of conic programs carry more structure than that. A feasible set $X=\{x : Ax=b\}\cap K$ of a conic program is a slice of a cone, and the barrier used on it is the restriction of a logarithmically homogeneous barrier of the cone. Restricted barriers of this kind satisfy a sharper, projectively invariant inequality, which the paper calls projective self-concordance. Estimates used in the analysis of interior-point methods (Dikin ellipsoids, outer approximations of the feasible set, short-step path-following parameters) improve when this extra information is used; Sections 3–8 of the paper develop those consequences.

All of those developments rest on one characterization, Theorem 1.5: a function on a convex set is projectively self-concordant with parameter $\gamma$ exactly when its logarithmically homogeneous extension to the cone over the set is self-concordant, with the sharp relation $\gamma=(\nu-2)/\sqrt{\nu-1}$ . This mission formalizes that theorem and the steps of its proof.

Setting

Let $D\subset\mathbb R^n$ be an open convex set and $f:D\to\mathbb R$ a $C^3$ function. Write $f'(x)[h]$ , $f''(x)[h,h]$ and $f'''(x)[h,h,h]$ for the first three derivatives of $f$ at $x$ in direction $h$ .

The affine metric of $f$ is the quadratic form $G(x)[h,h]=f''(x)[h,h]-(f'(x)[h])^2$ , and the cubic form is $C(x)[h,h,h]=f'''(x)[h,h,h]-6f''(x)[h,h]f'(x)[h]+4(f'(x)[h])^3$ .
Projective conditions (Definition 1.4, conditions (1) and (3)), with parameter $\gamma$ : for all $x\in D$ , $G(x)[h,h]>0$ for every $h\neq0$ , and $|C(x)[h,h,h]|\le 2\gamma\,(G(x)[h,h])^{3/2}$ for every $h$ .
The conic extension of $D$ is the open cone $K=\{(t,x)\in\mathbb R^{n+1} : t>0,\ t^{-1}x\in D\}$ .
For a real $\nu\ge2$ , the logarithmically homogeneous extension of $f$ is $F(t,x)=\nu\bigl(-\log t+f(t^{-1}x)\bigr)$ on $K$ . It satisfies $F(\tau y)=F(y)-\nu\log\tau$ for $\tau>0$ .
Affine conditions (Definition 1.1, conditions (1) and (3)): for all $y\in K$ , $F''(y)[h,h]>0$ for every $h\ne0$ , and $|F'''(y)[h,h,h]|\le 2\,(F''(y)[h,h])^{3/2}$ for every $h$ .
The proof uses the sextic $p_\mu(t)=4(\nu-1)t^6+12(\nu-2)t^4+4\mu t^3+12(\nu-3)t^2+12\mu t+4\nu-\mu^2$ in a real parameter $\mu$ .

In the Lean development these are ProjCond1, ProjCond3, AffCond1, AffCond3, conicExt, homExt and pmu in the namespace ProjSelfConcordant.ConicExt, with derivatives dk k g x h.

Formalization targets

Goal: Theorem 1.5

With $\gamma=(\nu-2)/\sqrt{\nu-1}$ and $\nu\ge2$ ,

\Bigl(f \text{ satisfies (1), (3) of Def. 1.4 on } D \text{ with } \gamma\Bigr)\iff\Bigl(F \text{ satisfies (1), (3) of Def. 1.1 on } K\Bigr).

Both directions are required, and each side is the conjunction of the two conditions, as on the page.

Milestones (Section 2, p. 5)

At $x=(1,\tilde x)$ and $h=(0,\tilde v)+\alpha(1,\tilde x)$ :

F''(x)[h,h]=\nu(\alpha^2-2\alpha f'[\tilde v]+f''[\tilde v,\tilde v]),\quad F'''(x)[h,h,h]=\nu(-2\alpha^3+6\alpha^2f'[\tilde v]-6\alpha f''[\tilde v,\tilde v]+f'''[\tilde v,\tilde v,\tilde v]).

Condition (1) of Definition 1.4 for $f$ is equivalent to condition (1) of Definition 1.1 for $F$ .
With $c_2=G[\tilde v,\tilde v]>0$ and $\mu=c_2^{-3/2}C[\tilde v,\tilde v,\tilde v]$ , the cubic inequality in $\alpha$ is equivalent to $p_\mu(t)\ge0$ for all $t$ .
For $\nu=\kappa^2+1$ , $\kappa\ge1$ : closed forms of $p_\mu(\mp\kappa^{-1})$ as products of two factors linear in $\mu$ .
Both $p_\mu(\pm\kappa^{-1})\ge0$ if and only if $|\mu|\le 2(\kappa^2-1)/\kappa=2\gamma$ .
$F''(\tau y)[\tau h,\tau h]=F''(y)[h,h]$ and $F'''(\tau y)[\tau h,\tau h,\tau h]=F'''(y)[h,h,h]$ for $\tau>0$ .
Factorizations of $p_{\pm2\gamma}(t)$ , showing both are nonnegative.
$|\mu|\le2\gamma$ implies $p_\mu(t)\ge0$ for all $t$ .

Significance

Theorem 1.5 identifies projective self-concordance with a familiar object, so every later result of the paper can be read either on the set or on the cone. Corollary 1.6 derives from it the passage between projectively and affinely self-concordant barriers on the same set. The improved inner and outer approximations of Section 3 and the duality theory of Section 5 are stated for projectively self-concordant functions, and Theorem 1.5 is what makes them applicable to the restrictions of standard conic barriers (log-det, logarithmic barriers of the orthant, the second-order cone barrier). The constant $\gamma=(\nu-2)/\sqrt{\nu-1}$ is sharp: for $|\mu|$ slightly above $2\gamma$ the sextic $p_\mu$ takes negative values.

The result is proved in the paper. As far as is known, neither it nor the general theory of projective self-concordance has a machine-checked proof; the platform's self-concordance vocabulary covers the affine notion only. A formal proof contributes reusable calculus for logarithmically homogeneous functions on cones (the derivatives of $(t,x)\mapsto f(t^{-1}x)$ , and the homogeneity of $F''$ and $F'''$ ) and a certified treatment of the sextic inequality that fixes the sharp constant.

Difficulty

The argument has two different parts. The first is multivariable calculus: computing the second and third Fréchet derivatives of $F(t,x)=\nu(-\log t+f(t^{-1}x))$ at points of the slice $t=1$ , along directions with a component in $t$ , and transferring them along rays by homogeneity. In Lean this means iterated derivatives of compositions with the nonlinear map $(t,x)\mapsto t^{-1}x$ , restricted to an open set on which $f$ is only $C^3$ , and with derivatives taken in a product space.

The second is real algebra. The affine condition (3) along a two-dimensional slice is a degree-six polynomial inequality in $\alpha$ ; after normalization it becomes nonnegativity of $p_\mu$ for all $t$ , and the content of the theorem is that the set of $\mu$ for which this holds is exactly $[-2\gamma,2\gamma]$ . Checking finitely many values of $t$ gives only necessity; sufficiency needs a certificate of nonnegativity at the endpoints and an argument for the interval in between. Direct expansion produces sextics whose nonnegativity is not visible term by term.

The equivalence is between conjunctions of conditions (1) and (3). Splitting it into one equivalence for each condition is not what the page claims: the conditions (3) are only meaningful where the forms of condition (1) are nonnegative.

Formalization scope

$\mathbb R^n$ is EuclideanSpace ℝ (Fin n) and $\mathbb R^{n+1}$ is ℝ × EuclideanSpace ℝ (Fin n), first coordinate $t$ . The product carries the sup norm; Fréchet derivatives do not depend on the choice of norm in finite dimensions.
$f:\mathbb R^n\to\mathbb R$ and $F:\mathbb R^{n+1}\to\mathbb R$ are total functions. Since $D$ and $K$ are open, derivatives at their points depend only on values inside them. Regularity is ContDiffOn ℝ 3 f D.
Derivatives are iterated Fréchet derivatives evaluated on the diagonal, iteratedFDeriv ℝ k g x (fun _ => h). Positive definiteness is strict positivity of the quadratic form on nonzero vectors. Tangent vectors range over the whole ambient space.
$(\cdot)^{3/2}$ is Real.rpow. Condition (3) appears only in conjunction with condition (1), under which its bases are nonnegative.
Only conditions (1) and (3) of Definitions 1.1 and 1.4 enter. The barrier condition (2), the parameter condition (4), regularity of a closed set, and convexity of $F$ are not hypotheses. No nonemptiness of $D$ and no $n\ge1$ is assumed; the statement holds as written in those cases.
A trivializing formalization is ruled out: $\gamma$ is not a free parameter but the explicit $(\nu-2)/\sqrt{\nu-1}$ with $\nu\ge2$ , both directions of the equivalence are required, and $F$ is the explicit extension of $f$ , not a function constrained by hypotheses.
The algebraic milestones 3–5, 7, 8 are stated on abstract reals (the page's $c_2$ , $c_3$ , $t$ , $\mu$ are exactly those expressions) and need no calculus. Milestones 1, 2 and 6 need derivative computations on product spaces that are reusable for any logarithmically homogeneous extension.

Contributions are welcome on either layer, and alternative proofs of the sextic bounds (for example sum-of-squares certificates) are as useful as transcriptions of the paper's argument.

Selected references

R. Hildebrand, Projectively self-concordant barriers, Mathematics of Operations Research, 2022; preprint arXiv:1909.01883v2 (2021). https://arxiv.org/abs/1909.01883
Y. Nesterov and A. Nemirovskii, Interior-Point Polynomial Algorithms in Convex Programming, SIAM Studies in Applied Mathematics 13, 1994. https://doi.org/10.1137/1.9781611970791
J. Renegar, A Mathematical View of Interior-Point Methods in Convex Optimization, MPS-SIAM Series on Optimization, 2001. https://doi.org/10.1137/1.9780898718812

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Economics Operations Research Optimization·Captain: mikedeng1

If It Is Surely Better, Do It More? Implications for Preferences Under Ambiguity: Ambiguity Aversion Toward an Event Breaks Monotonicity in Optimal Mixtures for c-Linearly Biseparable PreferencesResearch Paper

Motivation

Many economic and managerial decisions are choices of a mixing weight: how much effort to put on one sale rather than another, how much wealth to put in a risky asset, how to split a portfolio between two contracts. A natural comparative-statics requirement is that if one of the two options is improved in every state of the world, the decision maker should put at least as much weight on it. Ghili and Klibanoff (Management Science 67(12), 2021) call this property monotonicity in optimal mixtures and ask which models of decision making under ambiguity satisfy it.

Subjective expected utility satisfies it. The models used to describe ambiguity-averse behavior, such as the maxmin expected utility model of Gilboa and Schmeidler (1989), Choquet expected utility of Schmeidler (1989) and α-MEU of Ghirardato, Maccheroni and Marinacci (2004), are widely applied in contracting, finance and operations. Whether they predict that "surely better" options receive more weight is a basic question for any such application. The paper's first main finding is that, for the c-linearly biseparable class of Ghirardato and Marinacci (2001), which contains all three, the answer is negative as soon as the decision maker is ambiguity averse toward a single event.

Setting

The setting is an Anscombe–Aumann framework. $S$ is a finite set of states; an event is a subset $E\subseteq S$ and $E^c$ is its complement. $X$ is the set of simple lotteries over prizes. An act is a map $f:S\to X$ , and acts are mixed statewise: $(\alpha f+(1-\alpha)g)(s)=\alpha f(s)+(1-\alpha)g(s)$ . For lotteries $x,y$ , the binary act $xEy$ gives $x$ on $E$ and $y$ off $E$ .

All preferences are in the MBA class of Cerreia-Vioglio et al. (2011): an act $f$ is evaluated by $V\big((u(f(s)))_{s\in S}\big)$ , where $u:X\to\mathbb R$ is a nonconstant affine utility and $V:u(X)^S\to\mathbb R$ is normalized, monotone and sup-norm continuous. Under this representation an act may be identified with its utility profile $u\circ f:S\to U$ , where $U=u(X)$ is a convex subset of $\mathbb R$ with at least two points.

A capacity is a set function $\rho:2^S\to\mathbb R$ with $\rho(\emptyset)=0$ , $\rho(S)=1$ and $\rho(E)\le\rho(F)$ whenever $E\subseteq F$ . A preference is c-linearly biseparable if there is a capacity $\rho$ such that

W(xEy)=u(x)\rho(E)+u(y)(1-\rho(E))

represents the preference over all binary acts $xEy$ with $x\succsim y$ , for all events $E$ . The number $\rho(E)+\rho(E^c)$ classifies the attitude toward bets on $E$ : below $1$ is ambiguity aversion, equal to $1$ is neutrality, above $1$ is ambiguity loving.

For acts $f,g$ , the set of optimal mixtures $\alpha^*_{f,g}$ is the set of all $\alpha\in[0,1]$ such that $\alpha f+(1-\alpha)g$ is optimal in $\{\beta f+(1-\beta)g:\beta\in[0,1]\}$ . For $A,B\subseteq\mathbb R$ , $A\le_s B$ in the set order if $a\in A$ , $b\in B$ imply $\min\{a,b\}\in A$ and $\max\{a,b\}\in B$ . Monotonicity in optimal mixtures restricted to a set of acts $\mathcal G$ requires $\alpha^*_{f,g}\le_s\alpha^*_{f',g}$ for all $f,f',g\in\mathcal G$ with $f'(s)\succsim f(s)$ in every state $s$ .

Formalization targets

Goal: Theorem 2

Fix any event $E$ and a c-linearly biseparable preference with capacity $\rho$ .

\rho(E)+\rho(E^c)<1\ \Longrightarrow\ \text{monotonicity in optimal mixtures fails on }\{E,E^c\}\text{-measurable acts},

\big(\max\{\rho(E),\rho(E^c)\}<1\ \text{or}\ \min\{\rho(E),\rho(E^c)\}=0\big)\ \text{and}\ \rho(E)+\rho(E^c)\ge1\ \Longrightarrow\ \text{it holds there}.

Milestones

§5, p. 7625: given state-by-state monotonicity, the independence axiom implies monotonicity in optimal mixtures.
Proof of Theorem 2, p. 7627: if $\rho(E)+\rho(E^c)<1$ , the preference over $\{E,E^c\}$ -measurable acts is represented by $\min\{\rho(E)a+(1-\rho(E))b,\ (1-\rho(E^c))a+\rho(E^c)b\}$ , with $a$ , $b$ the utilities on $E$ and $E^c$ .
Proof of Theorem 2, p. 7627: for $f=xEy$ , $g=wEz$ whose connecting segment has slope strictly between the two linear pieces, the unique optimal mixture is $\alpha^*=1/\big(1+(u(x)-u(y))/(u(z)-u(w))\big)$ .
Theorem 2, first direction.
Proof of Theorem 2, p. 7628: if $\rho(E)+\rho(E^c)\ge1$ , the same acts are represented by the maximum of the two linear functionals.
Theorem 2, converse direction.

Significance

The theorem gives a sharp dividing line inside a large and heavily used family of ambiguity models. Ambiguity aversion toward even one event is incompatible with monotonicity in optimal mixtures for MEU, Choquet EU and α-MEU. The immediate corollary in the paper (Proposition 1) is that an MEU preference satisfies the property on bets on $E$ exactly when all its priors agree on the probability of $E$ . In applications such as the paper's sales-agent model, this means that improving the compensation for one task in every state can lead an ambiguity-averse agent to reduce effort on it. The result also separates these models from the smooth ambiguity model, for which the paper proves a compatibility result (Theorems 3–4).

Theorem 2 is proved in the paper; it has not been machine-checked. The mission produces a Lean development of the Anscombe–Aumann setting in utility space, capacities, MBA and c-linearly biseparable preferences, optimal-mixture sets and the set order, together with a checked proof of both directions. This layer is reusable for Proposition 1 and for other results on comparative statics under ambiguity.

Difficulty

The obvious route through comparative statics does not apply. The Milgrom–Shannon monotonicity theorem needs a single-crossing or supermodularity property of the objective in $(\alpha,f)$ , and the c-linearly biseparable functional restricted to a segment is only piecewise linear, with a kink where utilities on $E$ and $E^c$ coincide. Exhibiting a violation requires solving the maximization of a minimum of two affine functions on $[0,1]$ exactly, and constructing acts whose slope falls strictly inside a window determined by $\rho(E)$ and $\rho(E^c)$ , inside an arbitrary convex utility range.

The converse is not the mirror image of the first direction: the functional is then convex along segments, optimal sets can be whole intervals, and indifferences among $f$ , $f'$ and $g$ are where the argument is delicate. Remark 2 of the paper gives a capacity with $\rho(E)=1$ , $\rho(E^c)=1/2$ for which the converse fails, so the condition cannot be dropped.

Formalization scope

Acts are represented by utility profiles v : S → ℝ with values in a set U : Set ℝ. U is assumed convex and to contain two distinct points, which is exactly the range of a nonconstant affine utility on simple lotteries. Mixtures are α • f + (1 - α) • g, and " $f'(s)\succsim f(s)$ for all $s$ " is ∀ s, f s ≤ f' s.
S is a Fintype. An empty S makes the capacity conditions unsatisfiable, as on the page.
MBA is encoded as normalization on constants ( $V(c,\dots,c)=c$ ), monotonicity and continuity on U^S. On a finite S the product topology is the sup-norm topology.
c-linear biseparability is used through its representation (7), as in the paper. It is encoded as an order equivalence between comparisons of V and of W over binary profiles of all events, together with a capacity. The Ghirardato–Marinacci axioms are not encoded.
Optimal mixtures range over the closed interval $[0,1]$ for both the candidate and the competitors. The set order is the published Supermodularity.Lattices.InducedSetOrder on ℝ.
In milestone 3 the slope window is stated cross-multiplied, because $\rho(E^c)=0$ is allowed.
The min-representation is stated as a representation; the paper's strict inequality between the two pieces fails only on the diagonal $u(h(E))=u(h(E^c))$ , where the pieces coincide.
Ruled out: monotonicity with $\subseteq$ , or with a comparison of single chosen optima, in place of the set order; optimal mixtures over an open interval; (7) stated only one way or only for the event $E$ ; the converse without its side condition; and a preference class that is empty. A sorry-free sanity file exhibits a c-linearly biseparable preference with $\rho(E)+\rho(E^c)<1$ , and the Remark 2 preference, which satisfies all hypotheses except the side condition and violates the conclusion.
Not in scope: Theorem 1 (sales agent), Proposition 1 (MEU), Theorems 3–5 (smooth ambiguity and general MBA bounds), Axiom 3 and the remarks.

Contributions welcome: proofs of the milestones, and reusable lemmas on maximizers of minima and maxima of affine functions on intervals.

Selected references

S. Ghili and P. Klibanoff, If It Is Surely Better, Do It More? Implications for Preferences Under Ambiguity, Management Science 67(12):7619–7636, 2021. https://doi.org/10.1287/mnsc.2020.3890
P. Ghirardato and M. Marinacci, Risk, Ambiguity, and the Separation of Utility and Beliefs, Mathematics of Operations Research 26(4):864–890, 2001. https://doi.org/10.1287/moor.26.4.864.10002
S. Cerreia-Vioglio, P. Ghirardato, F. Maccheroni, M. Marinacci and M. Siniscalchi, Rational Preferences Under Ambiguity, Economic Theory 48(2–3):341–375, 2011. https://doi.org/10.1007/s00199-011-0643-3
I. Gilboa and D. Schmeidler, Maxmin Expected Utility with Non-Unique Prior, Journal of Mathematical Economics 18(2):141–153, 1989. https://doi.org/10.1016/0304-4068(89)90018-9
D. Schmeidler, Subjective Probability and Expected Utility without Additivity, Econometrica 57(3):571–587, 1989. https://doi.org/10.2307/1911053
P. Ghirardato, F. Maccheroni and M. Marinacci, Differentiating Ambiguity and Ambiguity Attitude, Journal of Economic Theory 118(2):133–173, 2004. https://doi.org/10.1016/j.jet.2003.12.004
P. Milgrom and C. Shannon, Monotone Comparative Statics, Econometrica 62(1):157–180, 1994. https://doi.org/10.2307/2951479
F. J. Anscombe and R. J. Aumann, A Definition of Subjective Probability, Annals of Mathematical Statistics 34(1):199–205, 1963. https://doi.org/10.1214/aoms/1177704255

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Algorithmic Game Theory Mechanism Design Operations Research·Captain: mikedeng1

Learning-Augmented Mechanism Design: Leveraging Predictions for Facility Location: The Coordinatewise Median with Predictions Is √(2c²+2)/(1+c)-Consistent and √(2c²+2)/(1−c)-RobustResearch Paper

Motivation

Learning-augmented algorithms use a machine-learned prediction to perform better when the prediction is accurate, while keeping a worst-case guarantee when it is not. The two guarantees are called consistency (performance under a correct prediction) and robustness (performance under an arbitrary prediction). Agrawal, Balkanski, Gkatzelis, Ou and Tan (arXiv:2204.01120v1; Mathematics of Operations Research, 2024) brought this framework to mechanism design, where the input is reported by strategic agents and the designer must also keep reports truthful.

Their test case is facility location in the plane. Agents report preferred locations and a mechanism places one facility. Without predictions, the coordinatewise median achieves a $\sqrt2$ -approximation for the average distance (Meir, 2019), which the paper reports as optimal among deterministic strategyproof mechanisms; the strategyproof mechanisms in the plane were characterized by Peters, van der Stel and Storcken (1993). A prediction of the optimal location can be used to beat $\sqrt2$ , but only by giving up robustness. This mission targets the paper's exact description of that trade-off.

Timeline.

1993. Peters, van der Stel and Storcken characterize the strategyproof mechanisms in the plane; the deterministic, strategyproof, anonymous and unanimous ones are generalized coordinatewise medians.
2019. Meir shows the coordinatewise median is a $\sqrt2$ -approximation for the average cost in $\mathbb R^2$ .
2021. Goel and Hann-Caruthers analyze the worst-case approximation ratio of the coordinatewise median as a function of $n$ for odd $n$ (for even $n$ the tight bound $\sqrt2$ was known).
2022. Agrawal et al. introduce the Coordinatewise Median with Predictions (CMP) and prove Theorem 9 below.

Setting

There are $n\ge1$ agents. Agent $i$ has a preferred location $p_i\in\mathbb R^2$ ; the profile is $P=(p_1,\dots,p_n)$ . The plane carries the Euclidean distance $d$ . Placing the facility at $f$ costs

C^u(f,P)=\frac1n\sum_{i=1}^n d(f,p_i),

the utilitarian social cost. An optimal location $o(P)$ is any point minimizing $C^u(\cdot,P)$ (a geometric median of $P$ ).

The lower median of $m$ real numbers is the $\lceil m/2\rceil$ -th smallest of them. The Coordinatewise Median with Predictions takes a prediction $\hat o\in\mathbb R^2$ and a confidence $c\in[0,1)$ such that $cn$ is an integer. It adds $cn$ copies of $\hat o$ to the agents' points and returns $f(P,\hat o,c)$ , the point whose $x$ -coordinate is the lower median of the $n+cn$ $x$ -coordinates and whose $y$ -coordinate is the lower median of the $n+cn$ $y$ -coordinates. At $c=0$ this is the coordinatewise median.

A mechanism is $\alpha$ -consistent if $C^u(f(P,o(P)),P)\le\alpha\,C^u(o(P),P)$ for every profile, and $\beta$ -robust if $C^u(f(P,\hat o),P)\le\beta\,C^u(o(P),P)$ for every profile and every prediction $\hat o$ . The approximation ratio of CMP on an instance is $r(P,\hat o,c)=C^u(f(P,\hat o,c),P)/C^u(o(P),P)$ .

Formalization targets

Goal: Theorem 9

For every $c\in[0,1)$ , every $n\ge1$ with $cn\in\mathbb Z$ , every profile $P$ , every optimal location $o$ of $P$ and every prediction $\hat o$ ,

C^u\big(f(P,o,c),P\big)\le\frac{\sqrt{2c^2+2}}{1+c}\,C^u(o,P),\qquad C^u\big(f(P,\hat o,c),P\big)\le\frac{\sqrt{2c^2+2}}{1-c}\,C^u(o,P).

The constants are exact: Lemma 8 shows both are attained, so no weaker goal would be the paper's theorem.

Milestones

The paper's proof reduces every instance to a one-parameter family. The milestones follow it:

Lemmas 26, 27, 28 (pp. 23–24): moving a point along its ray from the optimum keeps the optimum; moving a point towards the optimum without changing CMP's output worsens a ratio above $1$ ; two points moving towards each other get closer in total to a third point, strictly when not collinear.
Lemmas 15, 16 (p. 20): two conditions under which a strictly worse instance exists.
Lemma 17 (p. 20): a bound valid on the Optimal-on-Axes (OA) and Clusters-on-Axes (CA) families holds everywhere.
Lemmas 18, 24, 25 (pp. 20–22): OA and CA instances can be moved into the Clusters-and-Opt-on-Axes (COA) classes without improving the ratio.
Lemma 7 (p. 13): a bound on the COA classes holds everywhere.
Lemma 8 (p. 13): explicit COA instances attaining both constants.

The COA classes contain the instances with $o(P)=(0,1)$ , CMP output $(0,0)$ , and all points in $\{(0,1),(x,0),(-x,0)\}$ for one $x\ge0$ .

Significance

The result. Theorem 9 gives a one-parameter menu of guarantees. At $c=0$ both bounds equal Meir's $\sqrt2$ . As $c\to1$ consistency tends to $1$ while robustness grows without bound. The paper's Theorem 10 shows that no deterministic, strategyproof, anonymous mechanism improves on this menu (Pareto optimality), and its Theorem 12 refines the bound as a function of the prediction error, building on Theorem 9. It is one of the first exact consistency–robustness trade-offs for a mechanism design problem.

Formalizing it. The theorem is proved in the paper; nothing about it is machine-checked. A formal proof would check a reduction argument that runs over several pages and many cases (Lemmas 15–25, Appendix A). It would also check two delicate conventions on which the bounds rest: the lower median of $n+cn$ values, and the Euclidean metric. The model layer (Euclidean utilitarian cost, the geometric median as a minimizer, coordinatewise medians with phantom points) is reusable for other planar facility-location mechanisms.

Difficulty

The cost $C^u(f(P,\hat o,c),P)$ is not a convex or even continuous function of the profile, because the median jumps. So the worst case cannot be found by local optimization over the profile, and the geometric median $o(P)$ has no closed form. The paper instead transforms an arbitrary instance into a structured one step by step. Each step must keep three things under control at once: CMP's output, the optimal location (which can move when points move), and a ratio that may only increase. Several steps change the number of agents or regroup clusters (the four-fold copying after Lemma 7, the instances invoked in the proofs of Lemmas 21 and 22), so the argument is not an induction over a fixed $n$ . The obvious approach, the triangle inequality $d(f,p_i)\le d(f,o)+d(o,p_i)$ agent by agent, needs a bound on $d(f,o)$ in terms of $C^u(o,P)$ ; but the median is controlled by counts of points, not by distances, and bridging the two is exactly what the reduction does.

Formalization scope

All declarations live in the namespace LearnAugFL.CMP, in one definitions module and one module per theorem.

The plane is EuclideanSpace ℝ (Fin 2), with the Euclidean dist. Profiles are P : Fin n → Pt; CMP depends only on the multiset of points.
CMP is cmp P ô k: each coordinate is the lower median of the agents' coordinates together with k copies of the prediction's. Admissible c n k states $n\ge1$ and $k=cn$ , so the constants are functions of the same $c$ that fixes $k$ .
An optimal location is a hypothesis IsOpt P o (a minimizer of $C^u(\cdot,P)$ ), not a chosen function. Every statement quantifies over every optimal location; geometric medians of collinear profiles need not be unique.
Bounds are cross-multiplied. Ratios ratio P ô o k appear only in the reduction lemmas. There, the families force a positive optimal cost, so no ratio is a junk $0/0$ .
"CMP is $\alpha$ -consistent for $\alpha=\max$ over a family" (Lemmas 7 and 17) is stated as: every bound valid on the family is valid everywhere. This does not presuppose that the maximum is attained.
"There exists an instance $Q$ " allows $Q$ to have any number of agents, as in the paper's proofs.
Lemma 8 assumes that some $n_0$ makes $cn_0$ an integer, the standing assumption without which no instance exists.
Definition 13 (3) is read over points $p_i\ne o(P)$ , and its clause (e) uses the $y$ -clusters $(0,-y_1),(0,y_2)$ . Both printed versions are slips.

Trivializing encodings are ruled out explicitly:

ℝ × ℝ carries the sup metric, under which the bounds are a different statement.
The upper median, or the median of the agents alone, is a different mechanism.
Letting $k$ vary independently of $c$ decouples the constants from the mechanism.

Contributions welcome: proofs of the milestones in any order; facts about the lower median of a multiset under insertion of phantom values; and first-order optimality of the geometric median (a subgradient characterization), which Lemma 26 and the instances of Lemma 8 need and Mathlib does not have.

Selected references

P. Agrawal, E. Balkanski, V. Gkatzelis, T. Ou, X. Tan, Learning-Augmented Mechanism Design: Leveraging Predictions for Facility Location, arXiv:2204.01120v1, 2022; Mathematics of Operations Research, 2024. https://arxiv.org/abs/2204.01120v1
S. Goel, W. Hann-Caruthers, Optimality of the coordinate-wise median mechanism for strategyproof facility location in two dimensions, 2021 (as cited in arXiv:2204.01120v1).
R. Meir, Strategyproof facility location for three agents on a circle, International Symposium on Algorithmic Game Theory (SAGT), Springer, 2019, pp. 18–33.
H. Peters, H. van der Stel, T. Storcken, Range convexity, continuity, and strategy-proofness of voting schemes, ZOR Methods and Models of Operations Research 38(2):213–229, 1993.

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Quantum Information·Captain: Goku

Quantum trade-off coding for bosonic communicationResearch Paper

Motivation

The trade-off capacity region of a quantum channel characterizes the optimal net rates at which a sender can communicate classical, quantum, and entangled bits to a receiver by exploiting many independent uses of the channel, along with the help of the same resources. Similarly, one can consider a trade-off capacity region when the noiseless resources are public, private, and secret key bits. In [Phys. Rev. Lett. 108, 140501 (2012)], we identified these trade-off rate regions for the pure-loss bosonic channel and proved that they are optimal provided that a longstanding minimum output entropy conjecture is true. Additionally, we showed that the performance gains of a trade-off coding strategy when compared to a time-sharing strategy can be quite significant. In the present paper, we provide detailed derivations of the results announced there, and we extend the application of these ideas to thermalizing and amplifying bosonic channels. We also derive a "rule of thumb" for trade-off coding, which determines how to allocate photons in a coding strategy if a large mean photon number is available at the channel input. Our results on the amplifying bosonic channel also apply to the "Unruh channel" considered in the context of relativistic quantum information theory.

This mission is the statement-level formalization of Quantum trade-off coding for bosonic communication (arXiv:1105.0119v4). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

11 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Thermal entropy: convergent power series (Lemma 4: thermal entropy power series, PDF p. 7). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Deferred results

17 results of the paper are not stated yet, because the library lacks an interface they need:

Theorem 3 / (1), (28)–(30): pure-loss quantum dynamic achievability: Full-Fock energy-constrained signed (C,Q,E) block protocols and their independently defined asymptotic operational region (hard)
Proposition 1: regularized quantum dynamic formula (background reformulation): Canonical interpretation and full domain of unrestricted infinite-dimensional entropy differences (source-ambiguity)
Proposition 2: degradable private dynamic formula (background reformulation): Canonical interpretation and full domain of unrestricted infinite-dimensional entropy differences (source-ambiguity)
Theorem 7 / (2), (48)–(50): pure-loss private dynamic achievability: Full-Fock forward public/private/key consuming and generating block protocols, joint recovery/security, and independent asymptotic signed (R,P,S) region (hard)
Theorem 8: unconditional entanglement-assisted quantum slice: Full-Fock energy-constrained signed (C,Q,E) block protocols and their independently defined asymptotic operational region (hard)
Theorem 9: thermal-noise quantum dynamic achievability: Full-Fock energy-constrained signed (C,Q,E) block protocols and their independently defined asymptotic operational region (hard)
Theorem 10: thermal-noise private dynamic achievability: Full-Fock forward public/private/key consuming and generating block protocols, joint recovery/security, and independent asymptotic signed (R,P,S) region (hard)
Theorem 11: amplifier quantum dynamic achievability: Full-Fock energy-constrained signed (C,Q,E) block protocols and their independently defined asymptotic operational region (hard)
Theorem 12: amplifier private dynamic achievability: Full-Fock forward public/private/key consuming and generating block protocols, joint recovery/security, and independent asymptotic signed (R,P,S) region (hard)
§III B, (40)–(41): pure-loss classical–quantum achievable slice: Full-Fock energy-constrained signed (C,Q,E) block protocols and their independently defined asymptotic operational region (hard)
§III B, (42)–(43): pure-loss assisted/unassisted classical achievable slice: Full-Fock energy-constrained signed (C,Q,E) block protocols and their independently defined asymptotic operational region (hard)
§III F, (58)–(59): public/private and public/key achievable slices: Full-Fock forward public/private/key consuming and generating block protocols, joint recovery/security, and independent asymptotic signed (R,P,S) region (hard)
§III G: conditional full quantum dynamic capacity: Full-Fock energy-constrained signed (C,Q,E) block protocols and their independently defined asymptotic operational region (hard)
§III G.1: conditional CE and CQ capacity slices: Full-Fock energy-constrained signed (C,Q,E) block protocols and their independently defined asymptotic operational region (hard)
§III H, (77)–(79): conditional full private dynamic capacity: Full-Fock forward public/private/key consuming and generating block protocols, joint recovery/security, and independent asymptotic signed (R,P,S) region (hard)
§IV A, (81)–(82) and CE display: thermal-noise achievable slices: Full-Fock energy-constrained signed (C,Q,E) block protocols and their independently defined asymptotic operational region (hard)
§VI / Figure 7: mean-photon encoding versus single-excitation Unruh encoding: Physical two-use full-Fock vacuum amplifier precomposed with the dual-rail isometry, and both of its operational CQ/CE slices (hard)

Selected references

Quantum trade-off coding for bosonic communication, arXiv:1105.0119v4, https://arxiv.org/abs/1105.0119v4

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Quantum Information·Captain: Goku

The Quantum Schur Transform: I. Efficient Qudit CircuitsResearch Paper

Motivation

We present an efficient family of quantum circuits for a fundamental primitive in quantum information theory, the Schur transform. The Schur transform on n d-dimensional quantum systems is a transform between a standard computational basis to a labelling related to the representation theory of the symmetric and unitary groups. If we desire to implement the Schur transform to an accuracy of epsilon, then our circuit construction uses a number of gates which is polynomial in n, d and log(1/epsilon). The important insights we use to perform this construction are the selection of the appropriate subgroup adapted basis and the Wigner-Eckart theorem. Our efficient circuit construction renders numerous protocols in quantum information theory computationally tractable and is an important new efficient quantum circuit family which goes significantly beyond the standard paradigm of the quantum Fourier transform.

This mission is the statement-level formalization of The Quantum Schur Transform: I. Efficient Qudit Circuits (arXiv:quant-ph/0601001v1). Every numbered result of the paper was scanned. The results stated here are open sorry goals, ready for solvers to prove.

What is formalized

3 statements of the paper are in Lean 4 (Mathlib 0df444a). The goal theorem is Finite Gelfand–Zetlin labels fit in binary registers (III B: finite GZ labels and binary storage, PDF p. 11). The milestones are the other numbered results, in the paper's order, each quoting the paper.

The paper's own objects are the definition item of this mission. Objects shared across papers come from the prove2quantum library and from earlier papers' missions, listed as references.

Deferred results

8 results of the paper are not stated yet, because the library lacks an interface they need:

Abstract/I/V C: efficient qudit Schur transform: A phase-compatible GZ/YY Schur analysis map and a clean, uniform qubit compiler with charged reversible classical work. (hard)
III B, Eq.(29): reversible YY path compression: A bit-costed clean reversible rank/unrank program for valid one-box YY paths, implementing Eq.(29). (hard)
IV B: coherent Schur cascade and composable cost: A constructed physical-space-to-GZ/YY cascade and row-label reversible schedule, tied to local CG maps. (hard)
V opening: direct synthesis for fixed d: A fixed-d all-label GZ CG target with a uniform circuit-construction program and controlled-label synthesis costs. (hard)
V B, Eqs.(46)–(54), Figs.4–5: dimension-reducing CG factorization: Compatible subgroup branching coordinates and total padded reduced-Wigner blocks controlled on the retained lower output. (hard)
V C: efficient coherent reduced Wigner synthesis: An independently defined controlled reduced-Wigner target and an executable clean synthesis of it with numerical-error accounting. (hard)
V C, Algorithm: Clebsch–Gordan transform: A source-compatible finite GZ one-box CG analysis map, clean recursive circuit interpreter and uniform d^3 polylog cost contract. (hard)
III B: unfulfilled optimal-width GZ compression promise: A specified optimal-width GZ rank/inverse algorithm and a pinned meaning of efficient. (source-ambiguity)

Selected references

The Quantum Schur Transform: I. Efficient Qudit Circuits, arXiv:quant-ph/0601001v1, https://arxiv.org/abs/quant-ph/0601001v1

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Algorithmic Game Theory Convex Optimization Probability·Captain: mikedeng1

From Proper Scoring Rules to Max-Min Optimal Forecast Aggregation: Quasi-Arithmetic Pooling Is the Unique Report Maximizing the Worst-Case Profit Over Sub-Contracted ExpertsResearch Paper

Motivation

Forecast aggregation asks how to combine several experts' probability forecasts for the same event into one forecast. The two classical answers are linear pooling (average the probability vectors) and logarithmic pooling (average the log-probabilities and renormalize). Neither is justified by the way forecasts are usually evaluated: with a proper scoring rule, a payment rule under which reporting one's true belief maximizes expected reward. Neyman and Roughgarden (arXiv:2102.07081, Operations Research 2023) tie the aggregation method to the scoring rule. For each proper scoring rule $s$ they define quasi-arithmetic (QA) pooling with respect to $s$ , which recovers linear pooling for the quadratic (Brier) score and logarithmic pooling for the log score, and they justify it in several ways. This mission formalizes the first justification, the paper's Theorem 4.1: QA pooling is the unique max-min optimal way for an agent paid by $s$ to aggregate the forecasts of experts it has sub-contracted.

The result places the paper in a line of work on scoring rules and convex analysis: the Savage representation of proper scoring rules as tangent planes of a convex function (Gneiting and Raftery, JASA 2007, Theorem 2), the link between entropy maximization and worst-case expected loss (Grünwald and Dawid, Annals of Statistics 2004), and the result of Chun and Shachter (UAI 2011) that, for $n=2$ outcomes, experts rewarded with the same scoring rule cannot be prevented from colluding profitably, by all reporting what Neyman and Roughgarden call the QA pool of their reports.

Setting

There are $n$ outcomes. A forecast is a point of the standard simplex $\Delta^n=\{p\in\mathbb R^n: p_j\ge 0,\ \sum_j p_j=1\}$ , and $\delta_j$ is the vertex with a $1$ in coordinate $j$ . A scoring rule is a function $s:\Delta^n\times[n]\to\mathbb R\cup\{-\infty\}$ : an expert who reports $p$ receives $s(p;j)$ if outcome $j$ happens. The rule is proper if for every belief $p\in\Delta^n$ and every report $x\in\Delta^n$ ,

\sum_j p_j\,s(p;j)\ \ge\ \sum_j p_j\,s(x;j),

with equality only when $x=p$ . It is regular if $s(p;j)$ is real except possibly $s(p;j)=-\infty$ when $p_j=0$ . The forecast domain $\mathcal D$ is the set of $p$ at which every $s(p;j)$ is real, and the expected reward function is $G(p)=\sum_j p_j s(p;j)$ . The paper assumes throughout that each $s(\cdot\,;j)$ and $G$ are continuous on $\Delta^n$ .

An exposure function $g$ of $s$ satisfies the Savage representation

s(p;j)=G(p)+\langle g(p),\delta_j-p\rangle\qquad(p\in\mathcal D),

so $g(p)$ is a subgradient of $G$ at $p$ . The Bregman divergence is $D_G(p\,\|\,q)=G(p)-G(q)-\langle g(q),p-q\rangle$ .

Given forecasts $p_1,\dots,p_m\in\mathcal D$ and non-negative weights $w_1,\dots,w_m$ adding to $1$ , the QA pool $p^*=\bigoplus_{\mathbf g}^{i=1..m}(p_i,w_i)$ is the unique $p^*\in\Delta^n$ at which $\sum_i w_i g(p_i)$ is a subgradient of $G$ . An agent paid by $s$ who pays expert $i$ the amount $w_i s(p_i;j)$ and reports $q$ earns the profit

u(q;j)=s(q;j)-\sum_{i=1}^m w_i\,s(p_i;j).

Formalization targets

Goal: Theorem 4.1

For a regular, proper, continuous scoring rule $s$ with exposure function $g$ , forecasts $p_i\in\mathcal D$ and weights $w_i\ge 0$ adding to $1$ :

\min_j u(q;j)\ <\ \min_j u(p^*;j)\quad\text{for every } q\in\Delta^n,\ q\neq p^*;

the minimum $\min_j u(p^*;j)$ is attained at every $j$ with $p^*_j>0$ ; it is non-negative; and it is positive unless all reports with positive weight are equal. All four clauses are part of the goal.

Milestones

In the order the paper's proof uses them:

Proposition 3.1 (only-if direction, on $\mathcal D$ ): $G$ is convex on $\Delta^n$ , $s(p;\cdot)$ is an exposure function, and every exposure function gives subgradients of $G$ on $\mathcal D$ .
Proposition 3.4, first bullet: $D_G(x\,\|\,q)\ge 0$ , with equality only when $x=q$ , for $q\in\mathcal D$ .
Well-definedness of the QA pool (§3.2): exactly one $p^*$ exists.
Claim 3.10: $p^*\in\mathcal D$ .
KKT display in the proof of Theorem 4.1: $g(p^*)=\sum_i w_i g(p_i)+\lambda\mathbf 1_n+\mu$ with $\mu\ge 0$ and $\mu_j=0$ where $p^*_j>0$ .
Profit identity: $u(p^*;j)=\sum_i w_i D_G(p^*\,\|\,p_i)+\mu_j$ .

Significance

Theorem 4.1 says that an agent who sub-contracts experts and is paid by the same proper scoring rule can guarantee itself the largest worst-case profit, uniformly over outcomes, by reporting the QA pool, and that no other report achieves this guarantee. It gives an operational reason to aggregate with the method the scoring rule induces rather than a fixed one, and the strict positivity of the guarantee unless the weighted experts agree recovers, as a special case, the two-outcome collusion result of Chun and Shachter. The same objects (QA pools, exposure functions, Bregman divergences of expected rewards) carry the paper's later results on learning expert weights online and on axiomatizing QA pooling.

The theorem is proved in the paper; none of it is machine-checked. The mission produces a Lean statement and proof of Theorem 4.1 in full generality, including boundary pools where some outcome has probability zero, together with reusable statements about proper scoring rules on the simplex that are not in Mathlib: the Savage representation, non-negativity of the expected-reward Bregman divergence, existence and uniqueness of QA pools, and first-order optimality conditions on the simplex.

Difficulty

When $p^*$ lies in the interior of $\Delta^n$ and $G$ is differentiable, first-order optimality gives $g(p^*)=\sum_i w_i g(p_i)+\lambda\mathbf 1_n$ directly, and the rest is algebra. The difficulty is the boundary. The QA pool can have $p^*_j=0$ for some $j$ ; $G$ need not be differentiable there; and for rules such as the logarithmic score $s(p;j)=-\infty$ at points with $p_j=0$ , so $G$ 's subgradients may blow up near the boundary of $\Delta^n$ . Showing that the pool avoids the points where $s$ is $-\infty$ (Claim 3.10) and that the multipliers $\lambda,\mu$ exist with the right sign there requires the continuity of $s$ into the extended reals, not just convexity of $G$ . Uniqueness of the maximizer also needs the strictness built into properness.

Formalization scope

Forecasts are functions Fin n → ℝ, and $\Delta^n$ is Mathlib's stdSimplex ℝ (Fin n). The scoring rule takes values in EReal; only its values on $\Delta^n$ matter, the expected score uses $0\cdot(-\infty)=0$ , and inner products are finite sums. Properness is strict, belief first and report second. $G$ is the real part of the expected score and is only evaluated on $\Delta^n$ . Subgradients are taken relative to $\Delta^n$ , the domain of $G$ . The exposure function is real-valued and constrained by the Savage representation on $\mathcal D$ only; on $\mathcal D$ it is determined up to a multiple of $\mathbf 1_n$ , and no statement depends on that choice. The Bregman divergence uses the exposure function in place of a gradient, since $G$ need not be differentiable at boundary forecasts. Continuity of $s(\cdot\,;j)$ is in the order topology of the extended reals. The QA pool is a predicate (a point of $\Delta^n$ with the subgradient property), and its existence and uniqueness is a separate milestone. The profit is $-\infty$ when the agent's report scores $-\infty$ , and $\min_j$ is an infimum over the outcomes.

A trivial formalization is ruled out: the QA-pool hypothesis of the goal is satisfiable (milestone 3), exposure functions exist (milestone 1), and the continuity assumptions of the paper are kept as hypotheses rather than dropped.

Not formalized: the paper's results on learning expert weights (Theorems 5.1, 5.3), its axiomatizations (Theorems 6.6, C.7), which rules have convex exposure (Appendix D), the "if" direction of Proposition 3.1, and the second bullet of Proposition 3.4. Contributions of general lemmas on convex functions on the simplex (one-sided directional derivatives, first-order conditions at boundary minimizers) are welcome alongside the milestones.

Selected references

E. Neyman and T. Roughgarden, From Proper Scoring Rules to Max-Min Optimal Forecast Aggregation, Operations Research, 2023; arXiv:2102.07081v2. https://arxiv.org/abs/2102.07081
T. Gneiting and A. E. Raftery, Strictly Proper Scoring Rules, Prediction, and Estimation, Journal of the American Statistical Association 102(477), 2007. https://doi.org/10.1198/016214506000001437
P. D. Grünwald and A. P. Dawid, Game Theory, Maximum Entropy, Minimum Discrepancy and Robust Bayesian Decision Theory, Annals of Statistics 32(4), 2004. https://doi.org/10.1214/009053604000000553
S. Chun and R. D. Shachter, Strictly Proper Mechanisms with Cooperating Players, UAI 2011, pp. 125–134. https://dslpitt.org/uai/displayArticleDetails.jsp?mmnu=1&smnu=2&article_id=2168&proceeding_id=27
S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004, §5.5.3. https://doi.org/10.1017/CBO9780511804441

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Algorithmic Game Theory Convex Optimization Operations Research·Captain: mikedeng1

Approximate Nash Equilibria in Large Nonconvex Aggregative Games: existence of a 2H((√q+1)MΔ/n)^γ-pure Nash equilibrium, built by Shapley–Folkman disaggregation of the convexified gameResearch Paper

Motivation

Many large-population systems are modelled as aggregative games: each agent's cost depends on its own decision and on an average of everybody's decisions. Examples are electric-vehicle charging and demand response, where a household's cost depends on its own consumption profile and on the aggregate load through the electricity price, and congestion and routing, where a traveller's cost depends on the total flow on the links used. In these applications individual decisions are often non-convex: a charging profile may be on/off, a route is a discrete choice, an appliance runs in fixed blocks. Without convexity, the classical existence theorems for pure Nash equilibria (Rosen 1965, Debreu–Glicksberg–Fan) do not apply, and a pure Nash equilibrium may fail to exist.

The intuition that "non-convexity averages out when many small agents are aggregated" goes back to Starr's work on markets with non-convex preferences (1969), based on the Shapley–Folkman lemma. Liu, Oudjane and Wan (arXiv:2011.12604, Mathematics of Operations Research) make it quantitative for sum-aggregative games: they show that a non-convex aggregative game with $n$ players has an approximate pure Nash equilibrium whose error decays like $n^{-\gamma}$ , and they construct it from an equilibrium of a convexified game.

Setting

There are $n$ players $N=\{1,\dots,n\}$ . Player $i$ chooses an action $x_i$ in a compact, nonempty, possibly non-convex set $\mathcal X_i\subset\mathbb R^d$ , of diameter at most $\Delta>0$ . Write $\tilde{\mathcal X}_i=\operatorname{conv}\mathcal X_i$ for its convex hull. There are $q\times d$ matrices $A_1,\dots,A_n$ with spectral norm $\|A_j\|_2\le M$ , $M>0$ . The cost of player $i$ at a profile $x=(x_1,\dots,x_n)$ is

f_i(x_i,x_{-i})=\theta_i\Big(x_i,\ \frac1n\sum_{j\in N}A_jx_j\Big),

where $\theta_i$ is defined on $\mathcal X_i\times\Omega$ and $\Omega\subset\mathbb R^q$ is a neighbourhood of the set of aggregates $\frac1n\sum_jA_jy_j$ with $y_j\in\tilde{\mathcal X}_j$ .

Assumption 1 requires that (1) $x_i\mapsto\theta_i(x_i,y)$ is lower semicontinuous on $\mathcal X_i$ for every $y\in\Omega$ , and (2) $y\mapsto\theta_i(x_i,y)$ is $(H,\gamma)$ -Hölder on $\Omega$ uniformly in $x_i\in\mathcal X_i$ : $|\theta_i(x_i,y')-\theta_i(x_i,y)|\le H\|y'-y\|^\gamma$ , with $H,\gamma>0$ .

For $\epsilon\ge0$ , an $\epsilon$ -pure Nash equilibrium ( $\epsilon$ -PNE) is a feasible profile $x$ such that no player can lower its cost by more than $\epsilon$ through a unilateral deviation: $f_i(x_i,x_{-i})\le f_i(y_i,x_{-i})+\epsilon$ for all $i$ and $y_i\in\mathcal X_i$ .

The convexified game $\tilde\Gamma$ has action sets $\tilde{\mathcal X}_i$ and costs

\tilde f_i(x_i,x_{-i})=\inf\Big\{\sum_{k=1}^{d+1}\alpha^kf_i(z^k,x_{-i})\ \Big|\ \alpha\in\mathcal S_d,\ z^k\in\mathcal X_i,\ x_i=\sum_k\alpha^kz^k\Big\},

with $\mathcal S_d$ the probability simplex in $\mathbb R^{d+1}$ : $\tilde f_i(\cdot,x_{-i})$ is the convex envelope of $f_i(\cdot,x_{-i})$ . A generator $Z(i,x)$ is the set of points $z^k$ carrying positive weight in a minimizer of this infimum. A point $\tilde x$ satisfies the $\eta$ -stability condition with respect to generators $(Z(i,\tilde x))_i$ if $\tilde f_i(x_i,\tilde x_{-i})\le\tilde f_i(\tilde x_i,\tilde x_{-i})+\eta$ for all $i$ and $x_i\in Z(i,\tilde x)$ .

Formalization targets

Goal: Proposition 2.4

Under Assumption 1, with

\epsilon=2H\Big(\frac{(\sqrt q+1)M\Delta}{n}\Big)^\gamma,

the non-convex game $\Gamma$ has an $\epsilon$ -PNE. More precisely, if $\tilde x$ is a pure Nash equilibrium of $\tilde\Gamma$ and $Z(i,\tilde x)$ are arbitrary generators, every profile

x^*\in\operatorname*{arg\,min}_{x_i\in Z(i,\tilde x),\,i\in N}\Big\|\sum_iA_i\tilde x_i-\sum_iA_ix_i\Big\|^2

is an $\epsilon$ -PNE of $\Gamma$ . Both sentences are part of the goal.

Stronger: Theorem 2.7

If $\tilde x$ is only an $\epsilon$ -PNE of $\tilde\Gamma$ and satisfies the $\eta$ -stability condition with respect to a generator profile, the same disaggregation is an $\tilde\epsilon$ -PNE of $\Gamma$ with $\tilde\epsilon=\epsilon+\eta+2H\big(\frac{(\sqrt q+1)M\Delta}{n}\big)^\gamma$ .

Milestones

In attack order: Lemma 5.1 (a fixed point of the argmin map of a function continuous in $x$ , jointly lower semicontinuous and convex in $y$ ); Theorem 5.3 (existence of a PNE in games with convex compact action sets, convex lower semicontinuous costs and uniform equicontinuity in the other players' actions); Lemma 5.5 (1)–(2), (3) and (4) (properties of the convex envelope $\tilde f_i$ and of generators); Proposition 2.3 (the convexified game has a PNE); the Shapley–Folkman lemma (Lemma 5.4, first bullet, already proved on the platform); Starr's bound (Lemma 5.4, second bullet); Lemma 5.6 (1) ( $f_i=\tilde f_i$ at generator points); Theorem 2.7.

Significance

The result. Proposition 2.4 gives existence of approximate pure equilibria in a class of games where exact pure equilibria need not exist, with an explicit error bound that vanishes as the population grows, for fixed aggregate dimension $q$ . It also says how to obtain one: solve the convexified game, which is a convex game with lower semicontinuous costs, and round its equilibrium to generator points. Theorem 2.7 makes the construction robust to approximate solutions of the convexified game, which is what an iterative algorithm delivers. The paper's later sections (a randomized disaggregation and an application to flexible electricity consumption) rest on these two statements.

Formalizing it. The results are proved in the paper; none of them is formalized. The Shapley–Folkman lemma itself is proved on the platform and is reused here. The mission adds a quantitative Starr bound in Euclidean norm, an existence theorem for pure equilibria under lower semicontinuity (an extension of Rosen's theorem that the authors prove themselves because they found no reference), the basic theory of the $(d+1)$ -point convex envelope, and the disaggregation argument. Formalization also checks the paper's statements: the printed definition of a generator, read literally, makes Theorem 2.7 false (see Formalization scope).

Difficulty

The disaggregation step needs more than "average over many players". The convexified equilibrium gives each player a point $\tilde x_i$ in $\operatorname{conv}\mathcal X_i$ , but the deviation analysis has to happen in the original game, at points of $\mathcal X_i$ . Two facts must be combined: the generator points of a best response of $\tilde\Gamma$ are best responses of $\Gamma$ against the convexified profile, and a profile of generator points can be found whose aggregate is within $O(\sqrt q\,M\Delta/n)$ of the convexified aggregate. The first fails if a generator is allowed to contain points of zero weight. The second fails with the naive bound: picking arbitrary generator points for the at most $q$ exceptional players of the Shapley–Folkman lemma gives an error of order $q$ , not $\sqrt q$ ; the bound $\sqrt q$ requires a choice that exploits Euclidean geometry.

On the existence side, Rosen's theorem does not apply to $\tilde\Gamma$ : $\tilde f_i$ is convex and lower semicontinuous in the own action, but it need not be continuous even when $f_i$ is (the paper gives a three-dimensional example on p. 5).

Formalization scope

$\mathbb R^d$ is EuclideanSpace ℝ (Fin d) and players are Fin n. A $q\times d$ matrix is a continuous linear map from $\mathbb R^d$ to $\mathbb R^q$ ; its operator norm is the spectral norm $\|A\|_2$ of the paper. Deviations $(y_i,x_{-i})$ are Function.update x i y, and $f_i(z^k,x_{-i})$ inside $\tilde f_i$ is evaluated at the profile in which player $i$ plays $z^k$ , so the aggregate contains $A_iz^k$ . One $\epsilon$ -PNE predicate, parameterized by action sets and costs, serves both $\Gamma$ and $\tilde\Gamma$ . Hölder powers are real powers, and $\gamma\le1$ is not assumed. $\Omega$ is a neighbourhood of the aggregate set of $\tilde{\mathcal X}$ , encoded with nhdsSet. The infimum defining $\tilde f_i$ is a real infimum; at every profile of $\tilde{\mathcal X}$ its set is nonempty and bounded below, so Lean's default value for empty or unbounded sets is never reached.

Three readings differ from the printed text, and each is the one the paper's proofs use. (i) A generator is the set of components with positive weight, as in the paper's Lemma 5.7; keeping zero-weight components makes Lemma 5.6, Theorem 2.7 and Proposition 2.4 false (a two-player example with $d=q=1$ exists). (ii) Lemma 5.5 (2), printed "for all $x_i\in\mathcal X_i$ ", is stated on $\tilde{\mathcal X}_i$ , as its proof and Definition 2.2 require. (iii) Action sets are assumed nonempty. Lemma 5.1 is stated in any finite-dimensional real normed space; the uniform equicontinuity of Theorem 5.3 measures closeness coordinatewise, an equivalent norm. Starr's bound is stated with any upper bound $D$ on the diameters, which is equivalent to the maximal diameter.

A trivializing formalization is ruled out: $\tilde f_i$ is an infimum over representations of the player's own action, not over all points, so equilibria of $\tilde\Gamma$ are not arbitrary profiles; a sorry-free sanity check confirms that the definitions are satisfiable and agree with $f_i$ on a singleton action set.

Out of scope: Theorem 3.3, Propositions 2.8, 3.2, 3.4, Lemmas 2.6, 3.1, 5.6 (2), 5.7, 5.8, the full stability condition, mixed strategies, and the numerical study of §4. Contributions welcome: proofs of the milestones, in particular a reusable Starr bound and a reusable existence theorem for lower semicontinuous convex games.

Selected references

K. Liu, N. Oudjane, C. Wan, Approximate Nash equilibria in large nonconvex aggregative games, Mathematics of Operations Research; arXiv:2011.12604v3, 2022. https://arxiv.org/abs/2011.12604
R. M. Starr, Quasi-equilibria in markets with non-convex preferences, Econometrica 37(1), 25–38, 1969. https://doi.org/10.2307/1909201
J. B. Rosen, Existence and uniqueness of equilibrium points for concave n-person games, Econometrica 33(3), 520–534, 1965. https://doi.org/10.2307/1911749
S. Kakutani, A generalization of Brouwer's fixed point theorem, Duke Mathematical Journal 8(3), 457–459, 1941. https://doi.org/10.1215/S0012-7094-41-00838-4
J.-B. Hiriart-Urruty, C. Lemaréchal, Convex Analysis and Minimization Algorithms I, Springer, 1993. https://doi.org/10.1007/978-3-662-02796-7

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Economics Operations Research Probability·Captain: mikedeng1

Assortment Rotation and the Value of Concealment: offering products sequentially in decreasing price order earns at least the optimal fixed assortment's revenueResearch Paper

Motivation

Fast-fashion chains and flash-sale sites rotate their assortments: a product is on display for a short window and is then replaced. Retailers that do this face a choice that a classical assortment model cannot express. They can show the whole catalog at once, or they can conceal part of it and reveal products one at a time. In the classical discrete-choice models, which include multinomial logit, nested logit and Markov chain choice, each consumer buys at most one product, so the order in which products are revealed is immaterial.

Ferreira and Goh (HBS Working Paper 17-041; published in Management Science, 2021, doi:10.1287/mnsc.2020.3610) let consumers buy several products. Each additional purchase is worth less to the consumer than the previous one. This diminishing marginal utility goes back to Marshall and appears in the satiation model of Baucells and Sarin (2007). With it, the order of revelation matters, and the paper asks whether concealment pays.

Setting

A retailer has $N$ products with nonnegative prices $p_1,\dots,p_N$ . A consumer has a random valuation shock $X_i$ for each product $i$ . The shocks are mutually independent, continuous and integrable random variables on a probability space $(\Omega,\mathbb P)$ . The quantity $X_i-p_i$ is the price-adjusted valuation of product $i$ . Buying a set $S$ of products gives the consumer the utility

u(S)=\sum_{j\in S}(X_j-p_j)+\sum_{j=0}^{|S|-1}v(j),

where the marginal utility $v:\mathbb N_0\to\mathbb R$ is nonnegative and decreasing.

Fixed assortment. The retailer offers a subset $S$ of $M\le N$ products for the whole season. The consumer sees all realized shocks and buys a utility-maximizing set. Lemma 1 shows that she buys in an order $\sigma$ of decreasing price-adjusted valuation (Definition 1), so the retailer earns

R^f=\sum_{i=1}^{M}p_{\sigma(i)}\,\mathbb 1\{v(i-1)+X_{\sigma(i)}-p_{\sigma(i)}\ge 0\}.

Sequential assortment. The retailer shows one product per period. A consumer who has already bought $k$ items buys the current product $i$ iff $v(k)+X_i-p_i\ge 0$ . With $W_{i-1}$ purchases made before period $i$ , the revenue is $R^s=\sum_i p_i\,\mathbb 1\{v(W_{i-1})+X_i-p_i\ge 0\}$ . The quantity $R_k(n)$ is the revenue from the last $n$ products when the consumer enters them with $k$ purchases.
Optimized strategies. $\mathbb E[R^f_*]$ is the largest expected fixed revenue over all subsets $S$ , and $\mathbb E[R^s_*]$ the largest expected sequential revenue over all orders of all $N$ products. The value of concealment is $\mathrm{VoC}=\mathbb E[R^s_*]-\mathbb E[R^f_*]$ .

In Lean a product is a pair (price, realized shock). seqRev v L k is the sequential revenue of a list L for a consumer starting with k purchases, fixedRev v L' is (4) along a list L', and ERsStar, ERfStar are the two optima.

Formalization targets

Goal: Theorem 1 (p. 15)

Let $S$ be an optimized fixed assortment, and let $R$ be the revenue of the sequential assortment showing the same products in decreasing order of prices. Then

\text{(a)}\quad R(\omega)\ \ge\ R^f_*(\omega)\ \text{ for every outcome }\omega,\qquad\text{(b)}\quad \mathbb E[R^s_*]\ \ge\ \mathbb E[R^f_*].

Part (b) fixes no constants and asserts only the sign of the value of concealment, which can be zero (Theorem 2 of the paper).

Milestones

Lemma 1 (p. 11): buying the products $\sigma(1),\dots,\sigma(D^f)$ maximizes $u$ .
Lemma 2 (p. 13): $R_{k+1}(n)\le R_k(n)\le \rho^*_n+R_{k+1}(n)$ , with $\rho^*_n$ the highest remaining price.
Lemma 4 (p. 38): if the first product has the highest price and $X_1-p_1\le X_2-p_2$ , interchanging products 1 and 2 does not raise the revenue.
Proof of Theorem 1 (p. 39): in a Definition-1 order the sequential revenue equals $R^f$ .
Theorem 1(a) for every fixed assortment: $R\ge R^f$ pathwise.
Remark 1 (p. 12): appending a product never lowers $R^s$ .

Significance

The result. Theorem 1 says that concealment never hurts a retailer facing multi-purchase consumers with diminishing marginal utility. Part (a) also gives a concrete policy, independent of the shock distributions: take the optimal fixed assortment and reveal its products from the most to the least expensive. The paper reads the gain as structural: the retailer learns nothing from offering products sequentially, yet it still gains from them. The rest of the paper builds on it. Theorem 2 characterizes when the value of concealment is zero, Theorem 3 gives its asymptotics, and §6 measures the decreasing-price heuristic numerically.

Formalizing it. The theorem is proved in the paper; no machine-checked version exists. The mission produces a reusable model of a multi-purchase consumer under fixed and sequential presentation, together with its basic monotonicity facts (Lemma 2, Remark 1). The goal is also stated pathwise, which is stronger than the paper's "with probability 1".

Difficulty

The obvious approach compares the two strategies product by product, and it fails. Under the fixed assortment the purchase decision for a product depends on its rank among the realized price-adjusted valuations. Under the sequential assortment it depends on the number of earlier purchases, which in turn depends on every earlier shock. Neither revenue is monotone in the order of presentation. Swapping two adjacent products can raise or lower the revenue, depending on the realization and on which of the two is more expensive. Lemma 4 identifies the one swap whose sign is controlled, and Lemma 2 bounds how far an extra purchase can shift the remaining revenue. The difficulty is to arrange swaps of this kind into a sorting of the decreasing-price sequence into a Definition-1 order, and to keep the bookkeeping exact when prices or valuations tie.

Formalization scope

Products are pairs (price, realized shock), sequences are lists, and positions are 0-based: the paper's position $i$ with $v(i-1)$ is list position $i-1$ .
The marginal utility $v$ is taken directly, nonnegative and antitone, rather than through the concave $V$ of (2).
Prices are nonnegative. The paper leaves this implicit, and Lemma 2's lower bound needs it.
The purchase rule is $v(k)+X-p\ge 0$ everywhere, including $R_k(n)$ , for which (7) prints $>0$ . The two differ only on a null event for continuous shocks.
All "with probability 1" statements are proved for every realization. The milestones are stated for arbitrary lists, with ties in prices and in valuations allowed.
$R^f(S)$ uses the canonical Definition-1 order, with ties broken by smaller index. Every other order gives the same revenue off the null event of a tie.
$\mathbb E[R^s_*]$ is a maximum over the orders of all $N$ products, and $\mathbb E[R^f_*]$ a maximum over all subsets, the empty set and strict subsets included. Both maxima range over finite sets, and the revenues are bounded and measurable, so the integrals are genuine.
Independence, continuity and integrability of the shocks are the paper's standing assumptions and stay in the goal, although the inequality does not use them.

Trivializing encodings are ruled out: $R^f$ is the positional formula (4), not the sequential revenue in $\sigma$ -order. Restricting $R^s_*$ to the $M$ products of the fixed assortment, or $R^f_*$ to the full set, would change the claim and is not done.

The development needs only finite sums, lists and permutations from Mathlib, plus Bochner integrals of bounded random variables. The list-level lemmas about seqRev (Lemmas 2 and 4, Remark 1) apply to any sequential-purchase model with diminishing marginal utility. Contributions are welcome on every milestone, and on general list-sorting lemmas that carry an interchange argument from adjacent swaps to a full permutation.

Not formalized here: Lemma 3 and Corollaries 1–2 (optimal sequences under hazard-rate and IFR conditions), Theorems 2–3, the forward-looking consumer model of §5, and the numerical study of §6.

Selected references

K. J. Ferreira and J. Goh, Assortment Rotation and the Value of Concealment, Harvard Business School Working Paper 17-041 (2020 revision); Management Science, 2021. https://doi.org/10.1287/mnsc.2020.3610
M. Baucells and R. K. Sarin, Satiation in discounted utility, Operations Research 55(1), 2007. https://doi.org/10.1287/opre.1060.0322
F. Caro and V. Martínez-de-Albéniz, Product and price competition with satiation effects, Management Science 58(7), 2012. https://doi.org/10.1287/mnsc.1110.1484
M. Shaked and J. G. Shanthikumar, Stochastic Orders, Springer, 2007. https://doi.org/10.1007/978-0-387-34675-5

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Economics Operations Research Optimization·Captain: mikedeng1

A Simple Rule for Pricing with Limited Knowledge of Demand: for every concave inverse demand the linear price (P_m + c)/2 earns at least half the optimal profitResearch Paper

Pricing without knowing demand

A firm launching a new product (a drug, a software package, a digital download) must choose a price before it has data on its demand curve. Classical monopoly pricing needs the whole demand curve: the optimal price balances marginal revenue against marginal cost, and both depend on the curve's shape. Cohen, Perakis and Pindyck (Management Science; accepted version SSRN 2673810) ask how well a firm can do when it knows only two numbers: its constant marginal cost $c$ and the maximum price $P_m$ , the price at which consumers stop buying. Their answer is a rule that needs nothing else: charge the price that would be optimal if demand were linear through $(0, P_m)$ . They give closed-form performance guarantees for this rule under several demand families. The guarantee that holds for an entire nonparametric class is the one for concave inverse demand, and that is the subject of this mission.

The setting matters in practice when the firm need not plan output in advance (zero or near-zero marginal cost, no capacity decision): then the only decision is the price, and the slope of the demand curve, which determines how much will be sold, is immaterial for choosing it.

Setting

A market consists of

a maximum price $P_m$ and a marginal cost $c$ with $0 \le c < P_m$ ;
a saturation quantity $Q_{\max} > 0$ : no more than $Q_{\max}$ units can be sold, however low the price;
the actual inverse demand $P_A(Q)$ , the price at which exactly $Q$ units are demanded, defined on $[0, Q_{\max}]$ , with $P_A(0) = P_m$ , non-increasing and continuous.

Selling $Q$ units at the market-clearing price earns the profit $\Pi(Q) = (P_A(Q) - c)\,Q$ . A firm that knew $P_A$ would choose an optimal quantity $Q^{**}$ maximizing $\Pi$ on $[0, Q_{\max}]$ , charge the optimal price $P^{**} = P_A(Q^{**})$ , and earn $\Pi^{**} = \Pi(Q^{**})$ .

The demand at price $p$ is $Q_A(p) = \sup\{Q \in [0, Q_{\max}] : P_A(Q) \ge p\}$ , the quantity sold when the firm charges $p$ .

If demand were linear, $P(Q) = P_m - bQ$ with $b > 0$ , the profit $(P_m - bQ - c)Q$ would be maximized at $Q^*_L = (P_m - c)/2b$ , at the price

P^* = \frac{P_m + c}{2},

which does not depend on $b$ . This is the linear price. The linear pricing rule charges $P^*$ whatever the true demand is, and earns $\Pi^* = (P^* - c)\,Q_A(P^*)$ .

In Lean these objects are Market, Market.Standing, profit, IsOptQty, linPrice, demandAt and linProfit in the namespace LinearPriceRule.Concave.

Formalization targets

Goal: Proposition 5 (p. 13)

If $P_A$ is concave on $[0, Q_{\max}]$ , then an optimal quantity exists, $\Pi^* > 0$ , and every optimal quantity satisfies

1 \le \frac{\Pi^{**}}{\Pi^*} \le 2, \qquad 1 \le \frac{P^{**}}{P^*} \le 2 .

Milestones

Eq. (1), §1.4, p. 4. Under linear inverse demand $P_m - bQ$ , the profit is uniquely maximized at $Q^*_L = (P_m - c)/2b$ , and the price there is $(P_m + c)/2$ .
Theorem 1, concave case, p. 9. If $P_A$ is concave, every optimal price satisfies $P^{**} \ge P^*$ .
Proof of Proposition 5, p. 26. $Q_A$ is non-increasing on $(-\infty, P_m]$ , and $Q_A(P^{**}) = Q^{**}$ , so $\Pi^{**} = (P^{**} - c)\,Q_A(P^{**})$ .
Theorem 1, convex case, p. 9. If $P_A$ is convex and the optimum is interior ( $Q^{**} < Q_{\max}$ ), then $P^{**} \le P^*$ .
Remark after Proposition 5, p. 13. For a rectangular demand ( $P_A \equiv P_m$ on $[0, Q_{\max}]$ ), $\Pi^{**}/\Pi^* = 2$ , and $P^{**}/P^* = 2$ when $c = 0$ .

Significance

Proposition 5 is a distribution-free guarantee: across the whole class of concave inverse demands, which contains linear, concave quadratic and many other shapes, a rule that uses only $P_m$ and $c$ loses at most half of the optimal profit. The rectangle shows that the factor $2$ cannot be improved within the class. Theorem 1 says in which direction the rule errs, under-pricing for concave and over-pricing for convex inverse demand, so a firm with qualitative information about curvature knows which way to adjust. The paper's parametric results (quadratic, monomial, semi-log, log-log demand) and its welfare comparisons are built on the same model.

The results are proved in the paper. As far as is known, none of them has been formalized. This mission provides a machine-checked model of a monopolist facing an unknown inverse demand on a bounded quantity interval, which other robust-pricing statements can reuse, together with the paper's argument for the concave case, carried out without the differentiability its Taylor-expansion proof of Theorem 1 assumes.

Difficulty

The paper's proof of Theorem 1 writes $P_A(Q) = P_m - bQ + f(Q)$ , equates marginal revenue with marginal cost, and applies a second-order Taylor expansion. None of this is available for the statement as formalized: $P_A$ is only continuous and concave, it may have kinks (the rectangle has one at $Q_{\max}$ ), and the optimum may sit at the corner $Q^{**} = Q_{\max}$ , where the first-order condition fails. A proof must compare the profit at $Q^{**}$ with profits at other quantities using only the defining inequality of concavity.

The second subtlety is the demand $Q_A$ , a supremum over a set. Identifying $\Pi^*$ with the profit of an actual quantity, and $Q_A(P^{**})$ with $Q^{**}$ , needs continuity and the optimality of $Q^{**}$ : a flat stretch of $P_A$ at the optimal price would otherwise let $Q_A(P^{**})$ exceed $Q^{**}$ .

The convex half of Theorem 1 is false at a corner optimum. For $P_A(Q) = 5 - Q/10$ on $[0, 1]$ and $c = 1$ , the optimum is $Q^{**} = 1$ with $P^{**} = 4.9 > P^* = 3$ . That milestone therefore assumes $Q^{**} < Q_{\max}$ .

Formalization scope

$P_A$ is a function $\mathbb R \to \mathbb R$ . Only its values on $[0, Q_{\max}]$ matter: monotonicity, continuity, concavity and convexity are assumed on $[0, Q_{\max}]$ , and optimality and $Q_A$ range over $[0, Q_{\max}]$ . Concavity on all of $[0, \infty)$ together with nonnegativity would force $P_A$ to be constant, so it is not used.
The standing assumptions are $0 \le c < P_m$ , $0 < Q_{\max}$ , $P_A(0) = P_m$ , and $P_A$ non-increasing and continuous on $[0, Q_{\max}]$ . Nonnegativity of $P_A$ is not assumed. Continuity guarantees that an optimum exists and that $Q_A(P^*)$ is actually sold at $P^*$ .
"Concave demand curve" is read as concavity of the inverse demand $P_A$ in $Q$ . Proposition 5's proof invokes Theorem 1, which is about $P_A$ .
The firm optimizes over quantities, and $P^{**}$ is read off as $P_A(Q^{**})$ . All bounds are stated for every optimal quantity, and existence is a separate clause.
$Q_A(p)$ is the supremum of $\{Q \in [0, Q_{\max}] : P_A(Q) \ge p\}$ , and $\Pi^* = (P^* - c)\,Q_A(P^*)$ is the profit of charging $P^*$ , not the profit at some quantity.
Ratios are real divisions, and the goal asserts $\Pi^* > 0$ . Without the clauses "an optimum exists" and " $\Pi^* > 0$ ", the goal could hold vacuously or through the convention $x/0 = 0$ . These clauses are part of the statement and rule out that trivialization.
No differentiability is assumed anywhere.

Only Mathlib's convexity and order-theoretic infrastructure (ConcaveOn, ConvexOn, AntitoneOn, sSup on ℝ, the extreme value theorem on compact intervals) is needed. Proofs of the milestones are welcome in any order. The parametric Propositions 1–4, the welfare results (Propositions 6–7) and the uncertain- $P_m$ analysis of §3 are outside this mission.

Selected references

M. C. Cohen, G. Perakis, R. S. Pindyck, A Simple Rule for Pricing with Limited Knowledge of Demand, Management Science. https://doi.org/10.1287/mnsc.2020.3602 — accepted version: SSRN 2673810, https://ssrn.com/abstract=2673810 (the version cited by page in this mission).

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