Every OpenAI Math Result, Part 1 of 4: Goldfeld's Conjecture and 82 More

Proves Goldfeld's conjecture for quadratic twists of every elliptic curve over ℚ: analytic ranks zero and one each have density 1/2, and the mean analytic rank tends to 1/2. Both statements order signed squarefree twist parameters by absolute value.

Proves that the rational Grothendieck–Teichmüller Lie algebra, with the Ihara bracket, is freely generated by one element in each odd weight 3, 5, 7, …, resolving the Deligne–Drinfeld conjecture. (Lean)

Proves that no finite modification of the primes can be written as A+B with A,B ⊆ ℤ_{≥ 0} each containing at least two elements. This resolves Ostmann's inverse Goldbach conjecture on additive indecomposability. (Lean)

Proves the restricted geometric Langlands equivalence for connected reductive groups on smooth projective connected curves over 𝔽̅_q under the four stated Lie-theoretic characteristic hypotheses. Over arbitrary algebraically closed fields of characteristic p > 0, the same conclusion holds assuming additionally that p is very good for the group and p∤ |W_G|. Over global function fields, proves Ramanujan at every place for globally generic cuspidal representations of split adjoint absolutely simple exceptional groups, without characteristic or ramification-depth restrictions, and at every unramified place for cuspidal representations of split adjoint absolutely simple groups with a generic unramified component. Assuming the finite-level Ramanujan–Arthur decomposition, constructs global Arthur enhancements of occurring cuspidal excursion parameters for split connected semisimple groups at full finite level, recovering the given parameters by diagonal specialization on the entire Weil group, including inertia.

Proves the rational Hodge conjecture for every complex CM abelian variety, in every dimension and codimension. Through Milne's theorems, this also gives the Tate conjecture for all abelian varieties over finite fields and the Hodge standard conjecture for abelian varieties in every characteristic.

Proves the unramified de Rham quantum geometric Langlands equivalence for every connected simple complex algebraic group on every smooth projective connected complex curve, at every shifted level c ∈ ℂ∖ℚ. It identifies the full derived categories of twisted D-modules for the group and its Langlands dual, retaining all global forms and connected components.

Constructs total Boolean functions with block sensitivity bs (f) ≥ s(f)^α for a fixed α > 2, disproving the quadratic strengthening of the Sensitivity Conjecture. Here s(f) counts influential individual-bit flips, while block sensitivity allows disjoint groups of bits to change together. (Lean)

Disproves Hadwiger's conjecture even for fractional coloring: arbitrarily large finite simple graphs with independence number at most two satisfy χ_f(G) > h(G), where h(G) is the largest clique-minor order. Also disproves the fractional Colin de Verdière chromatic bound χ_f(G) ≤ μ(G)+1. In the positive direction, every finite nonempty graph satisfies χ_list(G) ≤ C h(G) for a universal constant C. (Lean)

Resolves Erdős's superexponential-growth question for van der Waerden numbers. If W_r(k) is the least interval length forcing a monochromatic k-term progression in every r-coloring, then W_r(k) > k^{ck⌊log₂ r⌋} for an absolute c > 0, all r ≥ 2 and sufficiently large k, uniformly in r. In particular, W_r(k)^1/k → ∞ for each fixed r. (Lean)

For every fixed d ≥ 3, any n ≥ 2 distinct points in ℝd determine at least c_dn^2/d distinct distances, with c_d > 0 depending only on dimension. This matches the integer-grid order and resolves the higher-dimensional Erdős distinct-distances conjecture with a constant-factor bound.

Classifies finite point configurations that occur monochromatically, at their original scale, in every finite coloring of sufficiently high-dimensional Euclidean space. The characterization is an algebraic condition over the coordinate field. It also disproves the Leader–Russell–Walters conjecture that every such configuration is a subset of a finite transitive set. (Lean)

Disproves the coefficient-free reduced Baum–Connes conjecture through a failure of rational injectivity and a separate failure of surjectivity of assembly. A finitely generated torsion-free example witnesses the injectivity failure. Separately, a torsion-free group has a nontrivial projection in its reduced group C∗-algebra, disproving the Kadison–Kaplansky conjecture.

Proves Kadison's similarity conjecture: every bounded complex-linear unital algebra homomorphism from a unital complex C∗-algebra to operators on a Hilbert space becomes a *-homomorphism after conjugation by a bounded invertible operator. (Lean)

Constructs a nonzero norm-quasinilpotent operator on every infinite-dimensional separable complex Hilbert space with no nonzero proper closed subspace invariant under every commuting operator. The construction also gives operators with no nontrivial invariant projection in the hyperfinite type II1 factor. (Lean)

Proves that every bounded Hochschild cocycle of degree at least two on a complex von Neumann algebra, with coefficients in the algebra itself, has a bounded primitive. Equivalently, all higher bounded Hochschild cohomology groups vanish, resolving the Kadison–Ringrose conjecture. (Lean)

Proves De Giorgi's conjecture at its sharp dimension-eight endpoint: every entire C2 solution u:ℝ⁸ → (-1,1) of Δ u=u³-u that is strictly increasing in one direction depends on only one linear coordinate. A stronger theorem classifies all stable entire solutions v:ℝ⁷ → [-1,1] as constant wells or planar transitions, without an energy-growth assumption.

Proves Milne's rationality conjecture for abelian varieties over ℚ̅ with good reduction: specialized Hodge classes pair rationally with complementary divisor products, independently of cohomology theory. Together with result 032, every specialized Hodge class is represented by a single rational algebraic cycle simultaneously in all prime-to-p and crystalline realizations, for every residue characteristic p.

Proves the ordinary two-point Chowla conjecture, with a bound O(X/(log X)^c) for Liouville correlation sums along fixed nonproportional affine forms, where c > 0 is absolute. More generally, proves the binary corrected Elliott conjecture for complex multiplicative functions bounded by one when one factor is uniformly nonpretentious against each fixed Dirichlet character times n^it for |t| ≤ X. (Lean)

Reconstructs function fields of transcendence degree at least two over algebraically closed constants from K^M₁/ℓ, K^M₂/ℓ, and their product. These data recover the perfect closure and constants when ℓ differs from the characteristic, and the original field and its named base in equal characteristic. Also proves Bogomolov–Pop reconstruction from abelian-by-central pro-ℓ Galois data away from the characteristic. (Lean)

Every continuous odd absolutely irreducible two-dimensional 2-adic representation of G_ℚ unramified outside finitely many primes occurs in a completed Hecke algebra at some odd tame level. Also proves classical modularity up to Tate twist for irreducible odd representations with these finiteness conditions that are de Rham at 2 with distinct Hodge–Tate weights, resolving the dyadic Fontaine–Mazur case without residual restrictions.

Proves that the normalized ordered logarithms of the prime factors of p-1, counted with multiplicity, converge jointly to the Poisson–Dirichlet law PD(1) as p ranges uniformly over primes up to x and x → ∞. This resolves the Ford–Konyagin–Luca conjecture. It also proves that infinitely many integers n have more than n^1-ε totient preimages, for every ε > 0.

Proves Haar equidistribution without escape of mass for complete volume-weighted torus packets from totally real fields: arbitrary lattices and prescribed local types in fixed prime degree at least five, arbitrary-order Picard packets in primitive quartic fields, and maximal-order ideal-class packets in primitive sextic fields. Here primitive means having no proper intermediate field; the relevant order or field discriminant tends to infinity. (Lean)

Proves the abelian Zilber–Pink conjecture over ℚ̅: every irreducible subvariety has finitely many maximal atypical subvarieties relative to its smallest containing torsion coset. It also proves the full curve case in the Siegel threefold 𝒜₂ for Hodge-generic curves defined over ℚ̅, without boundary or reduction assumptions.

Proves the Margulis–Platonov conjecture over every global field, including function fields of characteristic two. For an absolutely almost simple simply connected algebraic group G over k, every noncentral abstract normal subgroup of G(k) is the inverse image of an open normal subgroup in the finite product of its anisotropic nonarchimedean local groups.

Proves that rational points on every smooth proper geometrically connected curve of genus at least two over a finite extension of ℚp correspond bijectively to conjugacy classes of sections of its full arithmetic étale fundamental group. It also proves Grothendieck's section conjecture over ℚ for the modular curves X₀(N) and X₁(N) of genus at least two.

Proves the squarefree-values conjecture for irreducible integer quartics with no fixed prime-square divisor: squarefree values on positive integers have the predicted positive Euler-product density. More generally, establishes the (d-2)-power-free density for irreducible integer polynomials of degrees four through eight under the necessary local condition; together with Browning's higher-degree theorem, this covers every d ≥ 4. (Lean)

Answers Jacobsthal's quadratic-bound question: every interval of Ck² consecutive integers contains an integer coprime to any prescribed positive integer with at most k distinct prime divisors, for an absolute constant C. The bound is uniform over prime sets and interval positions and removes the classical logarithmic loss. (Lean)

Proves that for every real shift γ and finite-valued ψ:ℕ → [0,∞), divergence of ∑_qϕ(q)ψ(q)/q implies ‖qx-γ‖ < ψ(q) for infinitely many q, for almost every x. Here ϕ is Euler's totient and the norm is distance to the nearest integer. Numerators are unrestricted; no monotonicity or Diophantine condition on γ is needed.

Proves unconditionally the all-primary-prime form of Patterson’s first-moment asymptotic: normalized cubic Gauss sums over primary Eisenstein primes of norm at most X, including both conjugates, have an explicit positive main term of order X^5/6/log X. Every fixed nonzero prime-angle Fourier mode has smaller order. (Lean)

Proves Uchida's conjecture: every continuous open homomorphism between Galois groups of possibly infinite solvably closed Galois extensions of number fields comes from a unique equivariant field embedding in the opposite direction. No restriction on the kernel or separate cyclotomic-compatibility assumption is needed.

Proves log abundance for projective log canonical threefold pairs over algebraically closed fields of characteristic p > 3 when the effective boundary is rational and the ℚ-Cartier adjoint is nef of numerical dimension one. The adjoint is semiample, without requiring the original variety to be terminal or ℚ-factorial.

Proves the ordinary-double-point volume-gap conjecture: every singular complex algebraic klt germ of dimension n ≥ 2, with zero boundary, has normalized volume at most 2(n-1)ⁿ. Equality holds precisely for an analytic ordinary double point.

Proves Fujita's freeness conjecture at its sharp bound in every dimension: for a smooth projective complex variety X of dimension n and an ample line bundle L, the adjoint K_X+mL is globally generated for every integer m ≥ n+1.

Proves Nagata's strict inequality ∑_i m_i < d√r for every nonzero effective plane curve of degree d through r ≥ 10 very general complex points, with arbitrary multiplicities mi. It also proves maximal multipoint Seshadri constants (Lⁿ/r)^1/n for every smooth polarized projective variety of dimension n ≥ 2 and all sufficiently large r: at very general points over ℂ, and at the geometric generic tuple over any algebraically closed field of positive characteristic. (Lean)

Proves Bloch's conjecture: for every smooth connected projective complex surface S with p_g(S)=0, the Albanese map CH₀(S)⁰ → Alb(S)(ℂ) on integral degree-zero zero-cycles is an isomorphism. This combines the new p_g=q=0 theorem with the classical theorem of Bloch, Kas, and Lieberman.

Proves the strong hyperkähler SYZ conjecture: every holomorphic line bundle with nonzero nef isotropic first Chern class on a compact irreducible holomorphic symplectic Kähler manifold is semiample. It also proves that every projective Lagrangian fibration with normal projective base has projective space as its base, in every dimension and deformation type.

Proves P_k=W_2k=W_2k+1 on the full rational cohomology of smooth coprime fixed-determinant, trace-free Higgs moduli spaces and their character varieties for composite ranks over smooth projective complex curves of genus at least two. This includes variant cohomology and, together with the known prime-rank theorems, establishes the fixed-determinant P = W conjecture in every coprime rank.

Proves the equivariant cohomological Hikita correspondence for every finite quiver, including loops and multiple arrows, with arbitrary dimension and framing vectors and commuting flavor torus. When every semistable point is stable and the gauge action is free, the equivariant cohomology of the Nakajima variety is canonically the coordinate ring of the scheme-theoretic cocharacter-fixed locus of its flavor-deformed Coulomb branch.

Constructs a singular normal affine complex surface with free rank-two tangent sheaf, disproving the characteristic-zero Lipman–Zariski conjecture.

Constructs a polynomial in four complex variables that is not a coordinate but becomes one after adjoining a single variable, disproving the Stable Coordinate conjecture in four variables. Every fiber is affine three-space, yet none of its embeddings is rectifiable, also disproving the Abhyankar–Sathaye conjecture even when all fibers are affine spaces. (Lean)

Constructs ample rank-two bundles on ℙ¹×ℙ¹ with no smooth Hermitian metric of strictly Griffiths-positive curvature, disproving Griffiths' positivity conjecture already on the quadric surface. (Lean)

Every compact connected Kähler manifold of positive complex dimension with no nonconstant entire curve has ample canonical bundle and is therefore projective. This proves Kobayashi's canonical-ampleness conjecture in the smooth compact Kähler setting.

For every sufficiently large admissible Hassett discriminant, a very general smooth complex cubic fourfold is irrational despite having both an untwisted geometric K3 category and an integral Hodge-theoretic K3 association. This disproves Kuznetsov's rationality conjecture and the sufficiency of the associated-K3 criterion for rationality; the discriminant threshold is ineffective.

Proves Toda's Gepner conjecture for every smooth complex quintic threefold, constructing a numerical Bridgeland stability condition with the prescribed phase shift 2/5. Also constructs numerical Bridgeland stability conditions at every sufficiently large volume on all smooth projective complex threefolds with trivial canonical bundle, with the exact ordinary and square-root-Todd central charges.

Disproves Zariski's multiplicity conjecture by constructing reduced holomorphic hypersurface germs that are ambiently homeomorphic but have different multiplicities. The examples include hypersurfaces in ℂ4 with isolated critical points and multiplicities four and five.

Every connected minimal compact complex surface of class VII with b₂ > 0 contains a global spherical shell, proving the positive-b2 Global Spherical Shell conjecture. Such a shell is a holomorphically embedded neighborhood of the standard three-sphere in ℂ²∖{0} whose complement is connected.

Proves the LeBrun–Salamon conjecture: every closed connected positive quaternionic-Kähler manifold of real dimension at least eight is homothetic to a compact symmetric Wolf space. It also proves contact-Fano homogeneity and classifies smooth connected complex projective contact manifolds of complex dimension at least three: those with b₂=1 are adjoint varieties with their canonical contact structures, while those with b₂ ≥ 2 have underlying manifold ℙ(T^*Z) for a smooth projective variety Z.

Proves the generalized Mukai conjecture: every positive-dimensional smooth complex projective Fano manifold of dimension n, Picard number ρ and pseudoindex ι satisfies ρ(ι-1) ≤ n, with equality exactly for (ℙ^ι-1)^ρ. Here the pseudoindex is the least anticanonical degree of a rational curve.

Proves topological right-triviality for every holomorphic one-parameter family of isolated hypersurface singularities in ℂ3 with constant Milnor number, resolving the surface case of the μ-constant problem. After shrinking the parameter disk and representatives, ambient homeomorphisms vary jointly continuously, fix the origin section and preserve the defining functions.

Proves the full ordinary unreduced descendant Virasoro conjecture for smooth complete intersections in complex projective space, in every genus and curve class with arbitrary cohomology insertions. The constraints also pass from any smooth projective complex base satisfying them to the projectivization of every algebraic vector bundle of rank at least two, and hence to projective-bundle towers.

Proves the Campana–Peternell conjecture in complex dimension six: every smooth connected complex projective Fano sixfold with nef tangent bundle is rational homogeneous.

Proves Brennan's conjecture: for every conformal bijection ϕ from a simply connected plane domain onto the disk, |ϕ'|^s is area-integrable for 4/3 < s < 4. The sharp universal integral-means identity is B_𝒮(t)=|t|-1 for t ≤ −2. A strict bound B_b(-1) < 1/4 for bounded univalent functions disproves Kraetzer's prediction at that parameter. (Lean)

Resolves the Falconer distance conjecture in every dimension d ≥ 2: every compact set E ⊂ ℝ^d with Hausdorff dimension greater than d/2 determines a set of Euclidean distances of positive Lebesgue measure. (Lean)

Proves that the ordinary symmetric Fourier partial sums of every complex-valued function in L log L(𝕋) converge almost everywhere along the full sequence. This resolves the classical sufficiency conjecture at the L log L scale. (Lean)

Resolves the three-dimensional Bochner–Riesz conjecture in its strict range: the Bochner–Riesz multipliers of order δ are bounded on L^p(ℝ³) for every 1 ≤ p ≤ ∞ whenever δ > max {3|1/p-1/2|-1/2,0}.

Proves Petty's projection-volume conjecture in the remaining dimensions n ≥ 4: ellipsoids uniquely minimize projection-body volume at fixed body volume. Also establishes the full Lutwak–Petty projection inequalities. In contrast, products of simplices exceed Brannen's proposed simplex maximum for normalized projection-body volume by an exponential factor in every sufficiently large dimension. (Lean)

Resolves the planar and bounded-treewidth cases of the Gupta–Newman–Rabinovich–Sinclair conjecture. Shortest-path metrics of finite connected graphs with arbitrary positive edge lengths embed into real L1 with universal distortion for planar graphs, and distortion depending only on treewidth for bounded-treewidth graphs. The corresponding multicommodity flow–cut gaps are uniformly bounded. (Lean)

Proves that the triangular lattice minimizes the lower limit of energy per particle for every nonnegative completely monotone potential of squared distance among locally finite planar configurations of centered-disk density one. It also minimizes unit-background renormalized Riesz energies for 0 < s < 2 and Coulomb energy, resolving Sandier–Serfaty and the two-dimensional Brauchart–Hardin–Saff conjecture on the linear term of optimal spherical logarithmic energy. (Lean)

Proves the logarithmic Brunn–Minkowski inequality for origin-symmetric convex bodies in every dimension, and the scalar-dilation B-conjecture for all even log-concave Radon measures. For Lebesgue volume it also proves the additive Lp Brunn–Minkowski inequality for full-dimensional origin-symmetric convex bodies throughout 0 < p < 1. (Lean)

Determines the optimal worst-case covering density as Θ(n log n), for both lattice and unrestricted translative coverings. Every convex body in ℝn, n ≥ 2, admits a lattice covering of density at most Cn log n; centrally symmetric examples in every sufficiently large dimension require at least cn log n even without the lattice restriction, for absolute c,C > 0. (Lean)

Proves a dimension-free logarithmic Sobolev inequality for centered log-concave densities with uniformly subgaussian linear marginals, with constant bounded by a universal multiple of the squared linear subgaussian parameter.

Proves that the sum of squared Gaussian first moments of any finite measurable partition is at most 9/(8π). In dimension at least two, three planar sectors of angle 2π/3, extended orthogonally, attain the bound. Combined with the separate Unique Games theorem, this proves NP-hardness of improving the loss factor (8π/9)(1-1/k) for identity-target kernel clustering with fixed k ≥ 3 on rational centered positive semidefinite inputs. (Lean)

Proves that any finite sequence in the Euclidean unit ball of ℝd admits signs keeping every partial sum within C√d, independently of length. Consequently every zero-sum family can be reordered with the same bound on unsigned partial sums. A matching lower bound gives the optimal order S₂(d)=Θ(√d). (Lean)

Every infinite-dimensional real Banach space contains a compact doubling set that admits no bi-Lipschitz embedding into any finite-dimensional normed space. The doubling constant is universal. This answers the Lang–Plaut problem negatively, even for compact subsets of Hilbert space. (Lean)

Determines the least distortion of embedding edit distance on words of length at most d into real ℓ1: it is exp(Θ(√(log d log log d))). Insertions, deletions and substitutions have unit cost. The constants are uniform over all finite alphabets with at least two symbols, even when the alphabet grows with d; binary words already force the lower bound. (Lean)

Covers the entire closed regular tetrahedron by finitely many cylinders with compact triangular perpendicular bases whose total area is less than half its smallest orthogonal projection area. This disproves Bang's half-area cylinder-covering bound and the stronger directionwise normalized conjecture in dimension three. (Lean)

Proves that simplices uniquely maximize the isotropic constant among convex bodies in every dimension, resolving the strong isotropic constant conjecture. Also establishes the sharp entropy lower bound for log-concave probability densities, with equality precisely for invertible affine images of products of one-sided exponential laws.

Proves Khot's Unique Games Conjecture. Independent direct reductions also establish NP-hardness, on unweighted graphs, of approximation beyond the Goemans–Williamson ratio for Max-Cut, below factor two for Vertex Cover, and within any fixed constant factor for Min-UnCut and directed feedback vertex set. These direct proofs use established PCP and Label Cover hardness results. (Lean)

Proves Khot's 2-to-1 Games Conjecture with perfect completeness: for every fixed rational δ ∈ (0,1), it is NP-hard to distinguish satisfiable games from games whose optimum is at most δ, on explicit unweighted instances. The alphabet depends only on δ, and every right-hand label has exactly two preimages under each constraint map. (Lean)

It is NP-hard to color a three-colorable graph using any fixed number c ≥ 3 of colors. More strongly, for every fixed 0 < δ < 1/3, a deterministic polynomial-time reduction from 3SAT produces simple unweighted graphs that are three-colorable in the satisfiable case and have no independent set of size δ n otherwise, where n is the number of vertices. (Lean)

Proves an Ω(n³) lower bound for the border determinantal complexity of the n× n permanent over ℂ. Even coefficientwise limits of determinants of affine-linear matrices require matrix size at least cn³, for an absolute c > 0 and all sufficiently large n; the same bound therefore holds for exact representations. (Lean)

Establishes a randomized competitive ratio O(log²(k+1)) for k-server on every metric space, matching the worst-case lower-bound order. One policy serves every finite oblivious request sequence, including on infinite unbounded metrics. On finite rational metrics, a uniform implementation has polynomial preprocessing and per-request bit cost in the input length and log(t+1) at request t, with a finite instance-dependent additive movement constant. (Lean)

For every finite matroid known in advance, gives a distribution-independent online rule using one independent sample per element and earning a universal constant fraction of the expected offline optimum. Values are independent and nonnegative, with finite expected optimum. The guarantee holds even when the arrival-order adversary sees all samples, values, and the rule's entire random seed; no polynomial-time implementation is asserted. (Lean)

Constructs a single language in deterministic polynomial time whose n-bit membership function requires 2^ω(√n) total gates in unbounded-fan-in OR–AND–OR circuits, at every sufficiently large input length. This crosses the square-root-exponent threshold for explicit depth-three Boolean circuit lower bounds. (Lean)

For each characteristic, constructs in deterministic polynomial bit time a polynomial-dimensional matrix tuple detecting every nonzero division-free noncommutative formula of bounded size over any field of that characteristic. Rational formulas over ℚ also admit polynomial-size hitting lists whenever they have a defined rational-matrix evaluation. (Lean)

Disproves the modified integer round-up conjecture of Scheithauer and Terno: the integral bin-packing optimum can exceed its configuration linear-programming value by an arbitrarily large additive constant. Approximating the optimum within any fixed additive constant is also NP-hard, even when every item exceeds 1/6 and each bin holds at most five items. (Lean)

Proves the Courtade–Kumar conjecture: among Boolean functions of independent uniform bits, a single coordinate retains the most mutual information after independent bit-flip noise. A stronger theorem treats randomized binary summaries at fixed initial information. The Hellinger conjecture is also proved for every Boolean output bias and noise correlation. (Lean)

Gives a randomized algorithm finding an exact maximum-cardinality matching in any simple undirected graph in (n+m)^1+o(1) word time, with success probability at least 2/3. The time bound holds on every computation path. The same guarantees apply to finding a spanning subgraph with prescribed admissible vertex degrees, or deciding that none exists.

For every fixed rational ε ∈ (0,1), gives a randomized (1+ε) approximation to unit-cost edit distance in worst-case expected time N^1+o(1), with success probability at least 2/3. The strings have total length N and polynomially bounded integer symbols. This is an asymptotic guarantee at fixed accuracy. (Lean)

At every fixed deletion probability in (0,1), reconstructing an arbitrary length-n binary string requires n^{Ω(log log n)} independent traces, ruling out polynomial-sample reconstruction. A uniform decoder achieves quasipolynomial sample and running-time bounds for known fixed rational retention probabilities. When the deletion probability is at most n^-ε for fixed ε > 0, both bounds become polynomial in the input and parameter encoding. (Lean)

Resolves the three-processor unit-job scheduling problem of Garey and Johnson: a deterministic polynomial-time algorithm minimizes makespan for nonpreemptive unit-length jobs with arbitrary precedence constraints on three identical parallel machines. For an explicitly given precedence graph, it decides deadline feasibility exactly and constructs a feasible schedule. (Lean)

Gives a deterministic polynomial-time (1+2/e+ε)-approximation for finite rational metric k-median with specified candidate facilities, for every fixed ε > 0. Assuming P ≠ NP, the optimal infimum approximation factor is 1+2/e. (Lean)

Gives a deterministic polynomial-time algorithm constructing a common superstring of length at most twice the optimum for every finite family of explicitly represented strings. The running time is polynomial in the full encoded input length, including symbol labels. (Lean)