Algorithms and the time, memory, queries and other resources needed to solve computational problems.
For k≥2 and m≥1, a Strassen combinatorial degeneration of a finite group tensor onto the mth tensor power of the balanced tripartitioning tensor T_k requires |G|≥binom(3k,⌊3k/2⌋)^m>τ_k^m, where τ_k=8^k binom(3k,k)binom(2k,k)/27^k. For single-copy group restrictions, |G|≥2^{3k−1}; the partition products factor through commuting elements even when G is nonabelian.
A single deterministic Turing machine outputs the smallest prime factor Nat.minFac N of every natural number N in binary. For each integer k≥1 there is a constant C_k such that its running time on every n-bit input is at most C_k·2^{⌊n/k⌋}; all stack alphabets are finite. Thus the stated deterministic running time is N^{o(1)}.
For SK and finite mixed Ising p-spin models with all interaction orders at least three, at a fixed margin below the second-moment threshold and homogeneous fields |h|≤ρ₀N^{-1/4}, algorithms approximate log Z_N to additive error ε using a number of exact operations polynomial jointly in N, ε^{-1} and δ^{-1}. Symmetric unit-variance disorder is allowed; failure is at most δ plus an explicit disorder-tail term, which vanishes with a finite fourth moment.
One deterministic finite-alphabet Mathlib TM2 machine exactly counts the satisfying assignments of k-CNF formulas with k≥1, m≥2 distinct clauses and n≥4 occurring variables in time 2^{O(min{k²n log log n/√log n, km/√log m})}, with base-two logarithms and an absolute hidden constant. A separately formalized decision statement refutes ETH.
For every positive integer pair a≥b, gcd(a,b) has a remainder-only derivation using at most (6+o(1)) log₂(a)/log₂log₂(a) remainder calls, uniformly in b. Previously computed values may be reused and the choice of operands is free. This refutes Moschovakis’s stated uniform-process lower-bound conjectures, not the Main Conjecture for algorithms.
For each n≥2, among ordered positive integer inputs u≥v>0 taking exactly n iterations of the Shallit–Sorenson left-shift binary GCD algorithm, u²+v² has a unique minimum at (p[n−1],p[n−2]). Here p[0]=1, p[1]=3, and p[j+2]=p[j+1]+p[j] for even j, or 2p[j+1]+p[j] for odd j.
The manuscript gives converse bounds that completely characterize the storage/repair-bandwidth tradeoffs of exact-repair regenerating codes for (n,k,d)=(6,5,5) and (7,6,6), without assuming linear codes. Previously known layered codes with space sharing attain the bounds.
For every fixed k≥3, random k-SAT with independent uniformly signed proper clauses sampled with replacement has a positive finite limiting density αₖ: satisfiability probability tends to one below αₖ and zero above it. The claim makes no assertion at αₖ.
Given unary n and a size bound s, a deterministic machine constructs in polynomial bit time a polynomial-size list of rational matrix tuples of polynomial total output length and common polynomial dimension. Every admissible nonzero noncommutative rational formula over Q of size at most s has a defined, invertible evaluation on a listed tuple.
For every closed simple polygon with n vertices, including collinear vertices and equal heights, an explicit exact-real-RAM program returns a conforming triangulation with n−2 triangles in at most C(n+1) steps and natural words bounded by K(n+1). It is a variant of Chazelle’s algorithm that completes the visibility map directly below n=2³⁰+2.
For each fixed finite simple graph shape in the globally injective dense independent-sign model and fixed p>0, the pth norm moment has matching upper and lower orders n^((h+i₀−s)/2)log(2n)^(b*/2), with h the role count, i₀ isolated interior roles, s the minimum boundary-separator size and b* the largest attached floating-component count over minimum separators.
For n=2^k, a fixed real orthonormal n×d frame, 0<ε<1 and 0<δ<1/2, two independent sign rerandomizations of the normalized Walsh transform followed by uniform row sampling preserve all squared norms within ε with probability at least 1−δ. The stated row count is min(n,⌈8192(d+27⌈log₄(4d/δ)⌉³)/ε²⌉), which is O(d/ε²) for fixed δ.
For a well-formed Mazur edge, cycle, series and join scheme with n≥2 vertices and j nontrivial joins, the span satisfies T+1≤2^j(n−2j), with equality at (n,j)=(2,0) or 2j+3≤n. Its worst case is T=2^(n/2)−1 for even n and T=3·2^((n−3)/2)−1 for odd n. Companion theorems give interior dart mass ≤aₛbₛT, Dirichlet row norm ≤2(n−1)²T and |det M|≤(2(n−1)²T)^n, bound the candidate products T²|det M| and T³|det M|, and prove z=adj(M)u with Mz=Δu and boundary zₛ=Δhₛ(T−hₛ) when hₛ≤T, where Δ=det M.
For every nonnegative rational n×n matrix, Anari’s formal development constructs one rational-output algorithm and an absolute constant c<√2 with a two-sided cⁿ approximation guarantee for the permanent. It also proves deterministic polynomial time in the ordinary binary input size.
Protti’s planar acyclic four-terminal instance needs upper overload at least 335 when the largest demand is 294 if cost cannot increase. A rational family gives a lower bound (299−41√41)/32≈1.139747 for the unrestricted planar constant.
For the paper’s one-layer, attention-only transformer on Boolean inputs with a dedicated query token and linear threshold readout, the autonomous core results characterize head complexity by sums of admissible linear-fractional functions, bound it below by threshold degree, and determine it exactly for symmetric Boolean functions by the number of sign changes. In particular, n-bit parity requires exactly n heads.
For each fixed k≥2, one deterministic finite-tape machine exactly counts satisfying assignments of every canonically encoded k-CNF formula on n variables. With encoding length L, its time is (L+1)^{O_k(1)} 2^{O_k(n(log log n)²/√log n)} and its visited tape space is polynomial in L at every execution prefix. In particular, fixed-width 3-SAT has a subexponential-time decision algorithm.
Anthropic reports recovering a reproducible lrand48-and-timestamp recipe for all five NIST prime-curve seeds, five additional ANSI prime-curve seeds and four NIST binary-curve seeds. The reported timestamps and candidate indices regenerate each 160-bit public seed. The NIST B-163 seed remains unexplained.
The AI-origin improvement in Appendix B gives non-expansive line embeddings with O(log h) average distortion for the stated vertex-weighted metrics of connected graphs excluding the one-subdivision of K_h as an induced minor. The paper derives balanced separators of size O(h log h √m) for the corresponding m-edge graphs.
For every s≥3, allowed base q and k≥1, the explicit N=q^(sk) digital net has covering radius ≤√s N^(−1/s); when q≥5a*_s+4 its point separation is ≥(1−a*_s/q)N^(−1/s), with the stated bound on a*_s.
For every commutative ring R and every k≥0, the exact, unscaled Walsh–Hadamard transform on N=2^k inputs has a straight-line circuit with at most (3/4) N log₂ N + N/4 lines. Each line creates u+c·v or c·v−u from available values, with a fixed ring constant c, and costs one. Reuse, output selection and permutations are free.
Deterministic integer 3SUM in O(n^1.99896) and APSP in O(n^2.99791) in the pinned upstream word RAM model, for polynomially bounded input numbers.
Proudfoot and Perrault report that the exact tensor-rank matrix-multiplication exponent is strictly below 2.3710449 over every commutative ring. Their formal theorem uses the exact rational bound 23710449/10000000; the accompanying paper rounds this to 2.371045.
Navot’s AI-assisted Lean developments extend OpenAI’s matrix multiplication exponent bound ω ≤ 9/4 from complex numbers to every field, including finite fields and positive characteristic, and subsequently to every associative unital ring, including noncommutative rings. The ring development establishes exact integer coefficient tensor-rank identities and, for every ε > 0, correct division-free arithmetic programs with cost at most C n^(9/4 + ε) at every positive matrix size, for one C independent of n. Palomar registered the field development on 8 October and the ring development on 11 October 2026.
Every regular language over a finite alphabet has a generalized regular expression over that same alphabet whose Kleene-star nesting height is at most thirteen, with complement taken relative to the same free monoid.
For each k≥4, an explicit construction maps a finite compatible-choice system to two equal-size simple uncolored graphs that are k-dimensional Weisfeiler–Leman equivalent exactly when the system has no successful choice; a successful choice is detected after two joint-refinement rounds or one separate-refinement round.
Every exact affine semidefinite lift of the perfect-matching polytope of the complete graph on n vertices has matrix size at least 2^{c n} for some absolute c>0 and all sufficiently large even n.
For every binary string of known length n and known rational deletion retention probability p, one randomized algorithm reconstructs the string with probability at least 2/3 using an explicit sample budget and bit time polynomial in that budget and the binary input length; for each fixed p, both are quasipolynomial in n.
Given a prime p in binary and positive n,s in unary, one deterministic polynomial-bit-time machine outputs a tuple of matrices over F_p of dimension O(n³s⁶) that detects every nonzero division-free noncommutative formula of size at most s in n variables over every field of characteristic p.
For every fixed r≥3, every vertex-relabeling-invariant Boolean property of simple r-uniform hypergraphs on n vertices satisfies Var_p(f)≤C_r I_p(f)/(log n)^(r/(r−1)) for all 0<p<1, without requiring monotonicity.
For every fixed C>1, approximating Uniform Sparsest Cut within factor C is NP-hard on graphs with nonnegative rational capacities and unit demand between every pair of distinct vertices.
With exact rank-value oracles, the number of common integer bases of two equal-rank polymatroids can be approximated by an FPRAS whose oracle calls and non-oracle bit work are polynomial on every run in ground-set size, binary input length, inverse relative error, and log inverse failure probability.
The truth of existential–universal sentences over the reals is decidable in one fixed level of the counting hierarchy, even when their integer polynomials are specified by arithmetic circuits; in particular, the existential theory of the reals lies in the counting hierarchy.
For every fixed d≥3 and every sufficiently large even n, a deterministic algorithm outputs in polynomial bit time a simple connected nonbipartite d-regular n-vertex graph whose nonconstant adjacency eigenvalues lie strictly between ±2√(d−1).
There are fixed positive rational ε and δ and a deterministic reduction mapping an End-of-Line instance of binary length N to a generalized circuit of length N(log N)^O(1). A solution to the original instance can be recovered in polynomial time from any polynomial-length rational assignment ε-satisfying all but a δ fraction of gates.
For every fixed A>0 and sufficiently large dimension d, a one-pass learner retaining at most Ad² bits between independent exact Gaussian linear measurements needs at least c_A d log(1/ε) measurements to estimate a uniformly random unit vector to angular error ε≤1/10 with success probability at least 2/3.
Approximating the electronic Coulomb spectral infimum in three-dimensional continuum space is QMA-hard when positive integer nuclear charges use binary encoding, nuclei have distinct rational positions and electron count is unary; a deterministic polynomial-time reduction produces a promise gap of at least one.
An explicit isomorphism-invariant query on all finite structures in one fixed vocabulary of eight relations is decidable in deterministic polynomial time but is not definable in choiceless polynomial time with counting.
For every C>0, some nonconstant Boolean function on a finite sign cube has the sum of its linear Fourier coefficients greater than C√deg(f), where degree is that of its real multilinear representation.
A fixed deterministic machine with one writable tape can be simulated through a supplied time cap T≥2 in O(T^(2/5) polylog(T+2)) work-space bits while preserving its finite-control and halting outcome.
Over every characteristic-zero field, for all sufficiently large n every syntactically homogeneous depth-five ΣΠΣΠΣ circuit computing the (1,1) entry of n independent n×n matrix products has at least n^(√n/400) gates.
For every n≥4, an explicit n-state two-way nondeterministic finite automaton has a complement requiring at least ½·2^⌊(n−4)/127⌋−1 states in any two-way nondeterministic finite automaton.
Every degree-at-most-d real polynomial threshold function on the uniform n-dimensional Boolean cube has average sensitivity at most 8d√n for 1≤d≤n.
For any nonnegative row and column margins with equal totals, an exact uniform sampler for integer contingency tables terminates almost surely and has expected bit complexity polynomial in both dimensions and the binary input length.
There are total Boolean functions with unbounded block sensitivity divided by sensitivity squared; in particular, a fixed exponent α>2 holds along a sequence with unbounded block sensitivity.
For a binary channel on a uniform discrete cube under independent symmetric noise, a noisy coordinate maximizes retained information at fixed initial information; for Boolean functions this gives I(f(X);Y) ≤ 1 − h₂(ε) at crossover probability 0≤ε≤1/2.
One fixed polynomial-time Boolean language has exact depth-three OR–AND–OR circuit size greater than 2^(A√n) for every fixed A>0 at all sufficiently large input lengths n.
For every finite matroid, one independent sample per element supports an online rule with expected reward at least 2^-310 times the expected offline optimum, even when the arrival-order adversary sees all samples, online values, and the rule’s seed.
For every graphical degree sequence on n≥4 vertices, the lazy simple-undirected switch chain that proposes switches uniformly on four vertices has total-variation mixing time at distance 1/4 at most 2n⁸.
A deterministic polynomial-time algorithm finds a minimum-makespan schedule for unit-length jobs with arbitrary acyclic precedence constraints on three identical parallel machines.
A randomized algorithm approximates the number of perfect matchings in every finite simple undirected graph to relative error ε with failure probability at most δ in worst-case bit time polynomial in the input length, ε⁻¹, and log δ⁻¹; it returns zero with certainty when there is no perfect matching.
The complex border determinantal complexity of the m×m permanent is Ω(m³), allowing affine-linear determinant representations and coefficientwise limits.
For every fixed 0<δ<1/3, it is NP-hard to distinguish three-colorable finite simple graphs from graphs whose largest independent set has fewer than δ times the vertices.
For every fixed rational 0<δ<1, it is NP-hard to distinguish satisfiable 2-to-1 games from unweighted games of value at most δ, with an alphabet depending only on δ.
The manuscript reports one deterministic algorithm computing the exact complex discrete Fourier transform at every length n in O(n(log n)^(1−10^−13)) operations, including scalar preparation, scheduling and indexing, in its specified arithmetic model.
OpenAI reports a deterministic polynomial-time reduction from 3SAT to fixed-alphabet Unique Games with completeness at least 1−ε and soundness at most δ for each fixed ε,δ in (0,1/2).
OpenAI reports a uniform deterministic logarithmic-space simulation of polynomial-time randomized logarithmic-space decision procedures with bounded error, yielding L=RL=BPL.
OpenAI reports uniform deterministic complete factorization of a densely represented polynomial over a binary-specified prime field in bit complexity polynomial in (degree+1) log p, including irreducible factors and multiplicities.
The manuscript reports a deterministic quasipolynomial algorithm for the nonnegative-value vertices of finite turn-based stochastic mean-payoff games.
Claude’s original sparse-output thin-matrix multiplication algorithm, as presented in Section 2 and related to the certified 3SUM/APSP theorems. Human-optimized numerical parameters, new reductions, data structures and Hinted OMv extensions retain human credit.
The manuscript reports a randomized classical Subset Sum algorithm in O(2^(0.49n)) time for polynomial-bit inputs.
For every ε>0, the OpenAI construction multiplies n×n complex matrices with Oε(n^{9/4+ε}) division-free arithmetic operations; consequently ω(C)≤9/4.
The manuscript reports that the complex matrix-multiplication exponent is at most 9/4.
For joint-update k-dimensional Weisfeiler–Leman equivalence, with k≥2 binary encoded as part of the input and graphs given by explicit adjacency matrices, the test is EXPTIME-complete. The hardness also holds for connected equal-order simple uncolored graphs of maximum degree three.
Every regular language over a finite alphabet has a generalized regular expression over that same alphabet with at most three nested Kleene stars, allowing union, concatenation and complement.
For each h≥2, an h+3-state one-way nondeterministic automaton recognizes liveness of products of binary relations, while every equivalent two-way deterministic automaton with s states satisfies 2^⌊(h−2)/31⌋≤4(s+2)². The finite alphabet grows with h.
For an ordinary mean-payoff game of complete binary input length L with arbitrary signed binary edge weights, the September 25 deterministic manuscript reports a 2^O((log(L+2))²)-bit-operation algorithm for the exact winning set, values and optimal positional strategies.
The electronic Coulomb ground-energy promise problem is QMA-hard for distinct rational positions of unit-charge nuclei, unary electron count and rational thresholds separated by at least an inverse polynomial, with minimization over the full three-dimensional spinful fermionic continuum.
For every Boolean function on the uniform discrete cube, including biased outputs, and every product-noise correlation ρ∈[−1,1], the manuscript claims H(Ef)−E H(Tρf)≤1−H(ρ), where H(t)=sqrt(1−t²). Signed coordinates attain equality.
For every fixed deletion probability q∈(0,1), worst-case reconstruction of deterministic binary strings from independent deletion traces needs n^{Ω(log log n)} samples for any fixed positive success probability, even with unrestricted computation.
For fixed local dimension q, interaction bound J and global spectral gap Δ>0, the manuscript claims a PEPS with bond dimension at most CLᶜ and normalized vector error at most L⁻¹ for the unique ground state of any on-site plus nearest-neighbour Hamiltonian on an open L×L square grid.
For every fixed ε>0, a deterministic polynomial-time algorithm on finite rational metric k-median instances with specified candidate facilities opens at most k facilities and attains factor 1+2/e+ε. Together with known P≠NP hardness, this gives infimum approximation factor 1+2/e.
For rational-coefficient affine linear matrix pencils, a universal nonadaptive polynomial-size list of rational matrix evaluations detects nonsingularity over the free skew field. The paper also gives a rank-detecting list for rectangular rational pencils, with polynomial construction and output bounds in the dimensions and variable count.
The manuscript reports a randomized algorithm taking (n+m)^(1+o(1)) time and returning a maximum matching with probability at least 2/3.
The manuscript reports a randomized (1+ε)-approximation in N^(1+o(1)) worst-case expected time for each fixed rational ε in (0,1).
For a doubly symmetric binary source with crossover p∈[0,1], the convex hull of rate triples from arbitrary binary channels equals that from binary symmetric channels. At p=0 and parameters a,d∈(0,1), no binary channel pair with rates no larger than the stated Z/S pair has greater I(U;V), and no pair with rates at least those of the stated Z/Z pair has smaller I(U;V). These are value bounds, not uniqueness claims.
The manuscript reports deterministic multiplication in O(n (log n)^(1−κ)) time with κ = 2^−182 on a fixed finite-tape Turing machine.
For weighted k-matroid intersection with k≥2, the natural LP has integrality gap at most k−1+1/k. A deterministic LP-relative algorithm achieves that factor; the paper separately extends the bound to p-matchoids.
For m≥2 labeled supplies on [0,1], independent uniform demands and independent uniform replacement after each online match, the project proves a policy whose one-period root-mean-square cost, from every fixed initial inventory, has limsup at most (72000/log 2)·log(m+1)/m. After a scheduled m-period replacement phase, expected average cost including that phase obeys the same bound for every horizon N≥2m².
Dong and Jain report that, for a nonnegative n-by-n matrix whose bipartite support graph has girth at least even g>=4, Theorem 1.3 proves Bethe(A)<=per(A)<=2^(2n/g)Bethe(A); disjoint unions of g-cycles attain the upper bound when g divides 2n. The model-developed intermediate proof had a weaker exp(n Theta(log g/g)) factor; the exact model-origin step retained in the final sharp proof is not identified.
For every finite metric voting profile, Shah’s v2 random-size stable-lottery rule has distortion at most 107200742653/50000000000 = 2.14401485306. This supersedes the v1 mixed-rule bound of 2.3282.
For each δ∈(0,1), sufficiently large maximum degree Δ and finite simple n-vertex graphs of girth at least 5, Theorem 1 gives irreducible proper-q-coloring Glauber dynamics with spectral gap Ωδ(1/n) and mixing time Oδ(n² log q+n log(1/ε)) when q≥(1+δ)Δ. Theorem 3 extends the method to antiferromagnetic Potts models under its stated β and q conditions; Theorem 5 gives a local spectral-contraction criterion for general multi-spin systems. The result is rapid mixing, not the conjectured exact q≥Δ+2 threshold or uniformly optimal O(n log n) bound.
For iid Bernoulli(p) bits with 0<p≤1/2 and a full-row-rank k×n binary linear encoder H, its optimal bitwise-MAP distortion D(H) obeys k/n≥h(p)(1−D(H)/p). A nonasymptotic entropy bound for every affine subspace drives the converse.
For a fixed n×d real matrix U with UᵀU=I, d≥1, 0<ε≤1 and 0<δ≤1, the fully independent signed-hash SparseStack construction gives failure probability at most δ with column sparsity below 1356 max(1,⌈log₂(d/δ)⌉)/ε and row count below 306456(d+max(1,⌈log₂(d/δ)⌉))/ε².
Huang, Liu and Liu give a finite two-receiver discrete memoryless broadcast channel T for which Marton’s two-use sum-rate quantity M(T⊗2) exceeds twice its one-use value M(T), with a certified gap above 1.8821×10⁻⁶ nats. Its one-letter Marton region is therefore strictly smaller than capacity.
A deterministic exact algorithm counts linear extensions of any n-element poset in O*(1.89^n) time.
Dupont and collaborators report ω<2.371177 by improving the combination-loss optimization and refining it with AlphaEvolve.
For the Geode G defined by S−1=(t₂+⋯+t₆)G, a coefficient-collapse identity reduces its diagonal coefficient to a one-dimensional integral and coefficient extraction. It enables an exact 8,367-digit computation of G[1000^(5)] using modular arithmetic and bounded Chinese remainder reconstruction.
For fixed matrix dimension, whether the mean of i.i.d. binary random matrices is diagonalizable over R or C can be predicted with only finitely many errors almost surely. The argument uses eventual predictability of semialgebraic properties of bounded means.
Fredslund-Hansen, Mozes and Weimann report that, for a connected undirected unweighted planar graph with a designated face whose facial walk has k edges, the number of distinct facial distance-difference patterns is O(k^2), matching the known Omega(k^2) lower bound. The model supplied the forbidden-configuration and fixed-cardinality insight and a short prefix-trie proof; the paper's separate diameter algorithm is human-developed downstream work.
For planar graphs at sufficiently small fixed hard-core activity, including λ≤1/1443, Guo and Zhang give a randomized polynomial-time approximation scheme for the partition function without a maximum-degree bound.
For any number of indivisible goods and agents with nonnegative additive valuations, an allocation exists with bundle sizes differing by at most one, envy-free up to one good, and no balanced fractional reallocation that Pareto improves it.
For the single-neuron modular-addition setting, Appendix A extends Memana’s human counterexample to selection-independent Clarke trajectories and dead-neuron examples under the specified ReLU, dead-zone smoothing and fixed-step full-batch gradient-descent dynamics.
For every deterministic fractional online vertex-cover algorithm and every ε>0, a finite bipartite edge-arrival instance forces a competitive ratio of at least 2−ε. The paper extends the lower bound to randomized integral or fractional algorithms against an oblivious adversary.
For the released fixed four-terminal planar acyclic gadget with coordinatewise positive cost-difference scenarios, the normalized additive upper arc-deviation supremum is (299−41√41)/32 with one scenario, 17/8 with two, 3 with three, and 4 with at least four. The two-scenario value is not attained under scenario-wise cost non-increase.
Assuming the randomized rational-weighted exact-volume Small-Set Expansion Hypothesis, no randomized polynomial-time algorithm can guarantee both additive ε accuracy in rational sparse least squares and support O(k κₛ₊ₖ^(1−γ)) for any fixed γ∈(0,1]. The hard matrices have full column rank and at most two nonzeros per row.
Feng, Fu and Guo give a fully polynomial randomized approximation scheme for s-to-t reachability probability in general directed graphs with independent edge-opening probabilities. An existing reduction extends it to undirected graphs.
For the n×n permanent over the complex numbers and n≥32, Chapter 5 gives at least n⁴/(128 log₂ n) variable leaves for division-free arithmetic formulas and n⁴/(192 log₂ n) for valid rational formulas. A separate circuit lower bound has different scope.
A direct 3SAT reduction proves NP-hardness of approximating Euclidean closest vector within n^(1/400), under the dimension and promise conventions of Chapter 7. The chapter gives separate consequences for binary nearest codeword and finite p norms.
Meiburg constructs a total Boolean function on 2,017,584 variables with block sensitivity at least 14,011 and spectral sensitivity at most 89.0162. Composition yields an exponent of at least 2.127, refuting a proposed quadratic upper bound.
Domina and Ceriotti give noncongruent atom-centered environments with identical complete correlations through seven-neighbor order, including three-dimensional lifts. For every finite correlation order and angular cutoff, they also construct indistinguishable noncongruent environments.
The Paley tournament on 43 vertices has predictability 181/301, above the five-voter threshold 3/5, yet cannot be induced by five voters. The proof uses a co-backing constraint for cyclic triangles and an exact finite shell-and-screen enumeration.
KEMENY SCORE is NP-complete for exactly three unweighted complete rankings. Peters reduces from SIMPLE MAX CUT with long even edges to a unit-margin majority tournament realizable by three voters.
Conditional on Conjecture 1.1 about large prime divisors, the authors construct an S*_m-decoding polynomial with the optimal k+1 nonzero terms for every fixed k, for suitable k-prime m and finite field. The construction gives the stated fixed-server PIR tradeoff; finite prime checks make it unconditional through 15 servers.
For synchronizing one-cluster automata of positive level, Zhu proves the relative extending-word property and reset bound (m−1)(n−1)+mℓ≤(n−1)^2. Binary examples attain his refined 3n−5 bound.
Testing whether a supplied family of meet-irreducibles is the complete canonical family of a finite implication closure system is coNP-complete, even with acyclic implications of premise size at most three and a genuine irredundant supplied family. The hard direction reduces 3-UNSAT to a missing member.
The rooted terminal-only Manhattan cost-radius spanning-tree decision problem is NP-complete under polynomial-time many-one reductions through a weak reduction from positive-integer Partition.
Assadi, Jiang and Xiang construct optimal blueprints showing that no deterministic or randomized single-pass semi-streaming algorithm using O(n polylog n) memory and constant success probability can guarantee better than a one-half approximation for maximum bipartite matching. Greedy attains one half.
For every fixed sequence of T=n vectors in [−1,1]^n chosen before the algorithm runs, the model-generated proof gives O(√n) maximum online prefix discrepancy. The authors also prove longer-horizon and sparse-column extensions, including O(√n) for T up to exp(cn/log^(2+η)(en)) with the stated high probability.
Under the stated balanced stochastic-block-model signal and community-growth assumptions, the authors give a high-probability polynomial-time pure node-private estimator attaining the prior exponential-time method’s minimax exact-recovery risk, with privacy parameter of order log n.
A polynomial-time algorithm finds a planar embedding of minimum edge-outerplanarity for every finite loopless planar graph, including graphs with parallel edges. It reduces edge-peeling depth to minimum face-depth of an auxiliary planar graph.
For every nonempty inclusion-free hypergraph on n vertices, every d≥1 and arbitrary real edge offsets, Faber and Harris prove that at least n times the sum of j^(n−1) for j=0,…,d−1 assignments in [d]^n isolate a unique minimum edge. The n-singleton hypergraph attains equality.
For every fixed-in-advance sequence of T Euclidean unit vectors in dimension d, Aden-Ali’s Gaussian triplet walk assigns signs in O(dT) time. With probability at least 1−δ, every prefix has ℓ∞ discrepancy at most sqrt(18 log(2T²/δ)).
Nikolenko constructs a 17-terminal common-point instance proving the universal cost-preserving unsplittable-flow constant is at least 1.282494797984843521. A 3/2 congestion ceiling applies only to common-point cells of complement mass two.
Nikolenko gives a six-terminal planar instance requiring congestion factor at least 58676765987259/50000000000000, or 1.17353531974518, under cost nonincrease. He also identifies exact local envelopes E(2)=1, E(3)=9/8 and E(4)=(299−41√41)/32.
Under OVH or SETH, Saha, Xu and Ye prove n^(2−o(1)) lower bounds for exact Furthest Pair, Bichromatic Closest Pair, Maximum Inner Product and Hopcroft’s problem in every efficiently constructible superconstant dimension.
For every feasible set of row and column margins on an m×n binary matrix, with m,n≥2, the lazy 2×2 swap chain has spectral gap at least 1/[binom(m,2)binom(n,2)].
For every M≥2 with N=4M−5, a nonempty open set of complex N×M measurement matrices has a noninjective phase-retrieval map. Hence an IID complex Gaussian matrix is noninjective with positive probability, so p_M<1.
A biconnected graph with nine vertices and ten edges, with s=1 and t=3, gives a longest path of six under MaxSTN and seven under MinSTN, refuting the proposed inequality for these algorithms.
A connection from relaxed thin perfect matching to a single-tree cut sparsifier helps yield an O(log n log log n)-thin perfect matching relative to any fractional perfect matching, allowing edges outside its support. The separate O(n log n) bound requires support-constrained edges.
Anari’s Theorem 4 reports that the flip–repair walk associated with any real skew-symmetric n×n matrix has spectral gap at least 2/n. In the flat 0/1 determinant case it also gives an entropy bound and nearly linear mixing; this feeds a near-uniform sampler for strongly connected directed Eulerian multigraphs with m arcs in O(m^(3/2) polylog(m/ε)) time.
For an n×n Hermitian positive semidefinite matrix A, an entropy-corrected concave Gram-factor relaxation P̂ satisfies e^(−γn)P̂≤per(A)≤P̂ and gives a deterministic polynomial-time e^((γ+ε)n)-approximation for every ε>0. The exponent is optimal up to lower-order terms under the cited NP-hardness, assuming P≠NP.
For a feasible d-dimensional Max-Cut SDP solution strengthened by triangle inequalities, a randomized polynomial-time rounding algorithm attains expected cut value at least (α_GW+2^(−O(d))) times the SDP value.
A polynomial-time algorithm finds an optimal size-k committee for any Thiele rule on voter-interval approval profiles, including voter-specific scoring sequences.
In the IID request model, the stationary expected access cost of Transposition is at most OPT+1, where OPT orders items by decreasing request probability. GPT-5 Pro proposed the central inequality sⱼ≤pⱼ and a coefficient-nonnegativity route; Christian Coester supplied the final proof.
The AlphaEvolve report gives a rank-48 decomposition of the 4×4 matrix-multiplication tensor over the complex numbers, yielding a bilinear algorithm using 48 scalar multiplications.
The authors report packing heuristics that outperform tested baselines on their online one-dimensional bin-packing experiments.
DeepMind reports AI-discovered assembly sorting routines translated to C++ and incorporated in LLVM libc++. Its tests report up to 70% improvement for length-five inputs and about 1.7% for inputs exceeding 250,000 elements.
The authors report an algorithm multiplying 4×4 matrices over the two-element field using 47 scalar multiplications, compared with 49 from two levels of Strassen’s algorithm.
Algorithms and the time, memory, queries and other resources needed to solve computational problems.
Publication dates and editorial changes.
Group constructions cannot refute the Set Cover Conjecture, including tensor powers
Approximate Counting for Mixed p-Spin Glasses in Weak External Fields via Cluster Expansion
Approximate Counting for Mixed p-Spin Glasses in Weak External Fields via Cluster Expansion
Approximate Counting for Mixed p-Spin Glasses in Weak External Fields via Cluster Expansion
Explicit circuits for the Walsh–Hadamard transform below the textbook operation count
Explicit circuits for the Walsh–Hadamard transform below the textbook operation count
The Euclidean algorithm is not optimal among uniform processes
Sum-of-squares minimizers for a left-shift binary GCD algorithm
Exact Storage-Repair-Bandwidth Tradeoffs for (6,5,5) and (7,6,6) Exact-Repair Regenerating Codes
Formally Verified Autoresearch for Analyzing the Expressivity of Multi-Head Attention
Revisiting Separators in String Graphs, Region Intersection Graphs, and Induced-Minor-Free Graphs
OpenAI formalizes an alternative proof of random k-SAT thresholds
Codex-derived rule lowers randomized metric distortion to 2.1441
OpenAI proves a 9/4 complex matrix-multiplication exponent bound
OpenAI proves variable-dimension Weisfeiler–Leman equivalence EXPTIME-complete
AI-found blueprints show greedy is optimal for one-pass matching
A 17-terminal flow instance raises a general congestion lower bound
An exact planar witness raises the unsplittable-flow lower bound
Codex-assisted proof settles the positive-level one-cluster Černý bound
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