Every OpenAI Math Result, Part 4 of 4: L = BPL, Integer Multiplication and 82 More

Proves maximal and annular r-variation bounds, for every r > 2, from complex L³(ℝ²)× L³(ℝ²) to L^3/2(ℝ²). The maximal estimate controls both hard truncation endpoints and gives almost-everywhere and norm convergence. Pairing with a third input settles the triangular Hilbert transform estimate at the symmetric L³× L³× L³ point. (Lean)

Proves a uniform strong L2 bound for the planar Hilbert transform along any Lipschitz unit vector field, at integration lengths bounded by an absolute multiple of its reciprocal Lipschitz constant. The estimate is uniform over inner truncations and yields an L2-bounded principal-value operator, establishing Stein's weak-type conjecture at this short scale. (Lean)

Resolves the planar centered-disk case of the Hajłasz–Onninen maximal-function regularity problem. For every real f ∈ W^1,1(ℝ²), the centered disk maximal function satisfies ‖∇ Mf‖₁ ≤ C‖∇ f‖₁ with an absolute constant. It belongs locally to W^1,1 and has a globally integrable weak gradient. (Lean)

Proves that the principal-value trilinear Hilbert transform with shifts x-t, x-2t, x-3t is bounded from L³(ℝ)³ to L¹(ℝ). This resolves the L3 exponent case of the standard conjecture for slopes 1, 2, 3.

For every fixed 1 < p < ∞ and distortion D > 1, every n-point subset of real Lp embeds into ℓ_p^d with distortion at most D and dimension d=n^o(1), answering Naor's sublinear-dimension question for p ≠ 2. In contrast, exact embeddings require worst-case dimension Θ(n²) when p ≠ 2. (Lean)

Disproves the Projected Lax conjecture: some hyperbolicity cones are not spectrahedral shadows. The examples admit no exact finite affine semidefinite lift, regardless of the number of auxiliary variables or the real coefficients used. This also disproves the generalized Lax conjecture that every hyperbolicity cone is spectrahedral. (Lean)

Proves L=RL=BPL, resolving derandomization for bounded-error logarithmic-space computation. An effective compiler converts each randomized polynomial-time logarithmic-space machine deciding a language with one-sided or two-sided error into a deterministic logarithmic-space decider with explicit polynomial running-time bounds. (Lean)

Proves ω ≤ 9/4 over ℂ, giving O_ε(n^9/4+ε) arithmetic operations for square matrix multiplication. In characteristic zero, some inner dimension na with a > 0.465 permits n^2+o(1) rectangular multiplication. Further square bounds give ω < 2.258 outside finitely many positive characteristics and ω < 2.371054886006746 over every fixed field. (Lean)

Multiplies two n-bit integers exactly at every input length in deterministic worst-case time O(n(log n)^1-κ), with κ=2^-182, on one fixed finite-alphabet Turing machine with finitely many one-dimensional tapes. This disproves the Schönhage–Strassen n log n optimality conjecture in the ordinary multitape bit model.

Gives a fully polynomial randomized approximation scheme for counting perfect matchings in arbitrary finite simple graphs, with exact detection of zero counts. Also proves the perfect-matching entropy conjecture of Anari, Oveis Gharan, and Vinzant, bounding the maximum entropy of a matching law at every feasible edge-marginal vector in a loopless labelled multigraph, including boundary points. (Lean)

Gives a fully polynomial randomized approximation scheme for counting common integer bases of two integral polymatroids of equal total rank, supplied by exact rank-value oracles. Capacities are binary-encoded, each integer vector counts once, and oracle calls and bit operations outside the oracles are polynomial on every execution. For matroids presented by independence oracles, the results also cover common independent sets of prescribed, unrestricted, or maximum cardinality, even when the ranks differ. (Lean)

For nonnegative integer matrices with prescribed row and column sums, gives exact uniform sampling in expected polynomial bit time and almost-uniform sampling in worst-case polynomial bit time. The dimensions and binary-encoded margins are unrestricted. Also gives a fully polynomial randomized approximation scheme for counting such tables with arbitrary individual cell bounds, including structural zeros, with polynomial cost on every execution. (Lean)

Proves that approximating Uniform Sparsest Cut within any fixed constant factor is NP-hard, even with nonnegative rational capacities and unit demands. The Goemans–Linial semidefinite relaxation also has integrality gaps of order at least √(log n)/(log log n)³, approaching the square-root-logarithmic upper bound. (Lean)

Proves that every exact semidefinite lift of the perfect matching polytope has exponential size, answering Rothvoss's polynomial-size lift question negatively. The bound holds even for the positive semidefinite rank of its odd-cut slack matrix after any fixed shift 0 < ρ < 1, allowing arbitrary real positive semidefinite factors. (Lean)

Gives a deterministic length-n discrete Fourier transform algorithm using O(n(log n)^1-δ) operations for every n, with explicit δ=10^-13. The model uses exact complex arithmetic, unrestricted coefficients and a supplied root of unity, and counts scalar preparation and logarithmic-word indexing. (Lean)

Resolves the simple-undirected Kannan–Tetali–Vempala conjecture: the lazy edge-switch chain mixes in O(n⁸) time for every graphical labeled degree sequence. The same degree-constrained graphs can also be sampled exactly uniformly by an almost-surely terminating algorithm with expected polynomial bit running time. (Lean)

Every regular language over a finite alphabet has a generalized regular expression with at most three nested Kleene stars, allowing union, concatenation and complement over the same alphabet. This establishes an absolute bound independent of automaton size, resolving the uniform-boundedness version of the generalized star-height problem. (Lean)

Gives a uniform deterministic algorithm that completely factors every nonzero dense degree-n polynomial over a prime field 𝔽p, including multiplicities, in bit complexity polynomial in (n+1) log p. The prime is supplied in binary. No randomness, integer-factorization or primitive-root oracle, or GRH assumption is required.

Proves almost-everywhere convergence of consecutive multiple ergodic averages of every finite length for invertible mixing probability-preserving transformations. For each fixed tuple of bounded functions, the limit is the product of their integrals, along all positive averaging lengths. No mixing rate or standardness assumption on the probability space is required.

Constructs a finite translational tile in ℤ3 that tiles space but admits no fully periodic tiling, disproving the periodic tiling conjecture in the smallest possible lattice dimension. Its unit-cube thickening gives the same counterexample in ℝ3, even with arbitrary real translation vectors. (Lean)

Resolves the strong thin-tree conjecture constructively. Every finite loopless k-edge-connected multigraph on at least two vertices has a spanning tree containing at most a universal C/k fraction of the edges of every cut. Such a tree can be found deterministically in polynomial time, even with binary-encoded parallel-edge multiplicities. (Lean)

For every fixed d ≥ 3, constructs a simple d-regular nonbipartite Ramanujan graph on every sufficiently large even number n of vertices, with every nonconstant adjacency eigenvalue strictly between -2√(d-1) and 2√(d-1). A deterministic algorithm outputs the full adjacency list in polynomial bit time, with exponent depending on d.

For every fixed intersective integer polynomial h of degree k ≥ 2 with positive leading coefficient, proves that a subset of {1,…,N} avoiding nonzero values h(1),h(2),… as differences has size O_h(N^1-c_k), with c_k > 0 depending only on degree. Here intersective means having a root modulo every modulus. For prime arguments, a power saving also holds when h has a unit root modulo every modulus, with exponent allowed to depend on h. (Lean)

Proves the Alon–Krivelevich–Sudakov coloring conjecture in correspondence-coloring form: graphs avoiding any fixed subgraph F need O_F(Δ/log Δ) colors when their maximum degree Δ is sufficiently large. Also proves the Ajtai–Erdős–Komlós–Szemerédi independence conjecture: for fixed r ≥ 4, every n-vertex Kr-free graph of average degree d ≥ 2 has an independent set of size Ω_r(n log d/d). (Lean)

Disproves the unrestricted infinite matroid intersection and packing/covering conjectures in ZFC, using two self-dual partitional matroids on a countably infinite ground set. The same examples answer Joó’s partitional-matroid question negatively. They are neither finitary nor cofinitary, so Nash-Williams’ original finitary conjecture remains outside the result. (Lean)

Proves the Friedgut–Kalai threshold-width conjectures for graphs and fixed-uniformity hypergraphs. For fixed 0 < ε < 1/2, every nontrivial increasing relabeling-invariant property crosses from probability ε to 1-ε within width O((log n)^-2) for graphs and O_r((log n)^-r/(r-1)) for r-uniform hypergraphs, r ≥ 3. The hypergraph influence bound also applies to nonmonotone properties. (Lean)

Starting from the complete graph on n vertices, repeatedly delete a uniformly chosen remaining triangle. The terminal edge count is asymptotic to n^3/2/(2√2), with mean-square convergence after normalization by n3/2. This proves the triangle case of the Joos–Kühn sharp-constant conjecture. (Lean)

Disproves polynomial ordered binary matrix removal with one fixed 66×66 zero–one pattern. Matrices can require many binary-entry changes to become pattern-free while their copy density is smaller than every proposed polynomial bound in that distance. Copies preserve row and column orders and match both zeros and ones. (Lean)

Proves saturation factor one for Spin (2n), n ≥ 2: for three dominant integral weights whose sum lies in the root lattice, an invariant at any common positive integral dilation already gives an invariant at the original weights. This resolves the type-D part of the simply-laced saturation conjecture.

Some finite lattices are not congruence lattices of any finite algebra, answering the finite lattice representation problem negatively. Moreover, no algorithm decides whether a finite lattice has such a representation, or whether it is a full subgroup interval of a finite group. (Lean)

Proves that every finite symmetric tensor category over an algebraically closed field k of characteristic p > 0 admits a k-linear exact faithful strong symmetric monoidal fiber functor to a higher Verlinde category Ver_pⁿ. The level may depend on the category, and the theorem includes characteristic two, resolving the finite case of the Benson–Etingof–Ostrik conjecture.

Critical Fortuin–Kasteleyn planar maps converge to Liouville quantum gravity spheres for 0 < q ≤ 4 and to the Brownian continuum random tree for q > 4, establishing the surface-to-tree geometric transition. For FK–Ising and spanning-tree-weighted maps, stationary random walks converge to Liouville Brownian motion on the limiting sphere. The FK–Ising spectral result also gives convergence of eigenvalues and heat traces, using the stated Brownian/LQG inputs. (Lean)

Constructs the canonical continuum limit of the two-dimensional nearest-neighbor O(3) model: a non-Gaussian local relativistic theory with a unique vacuum and a positive mass gap. For the square-lattice O(4) model, determines the exact leading asymptotic of the full transfer gap, m_lat(β) ∼ 32e^π/4-1/2√β e^-πβ. The family also proves exponential spin-correlation decay for two-dimensional nearest-neighbor O(n) models with n ≥ 3 at every positive temperature. (Lean)

For every fixed degree d ≥ 3, the bulk adjacency-eigenvalue point process of a uniform simple random d-regular graph converges to the Gaussian orthogonal ensemble law, including for cubic graphs. The same fixed-energy universality persists under sufficiently weak fixed iid uniform diagonal disorder, throughout compact bands strictly inside the clean spectral edges.

Determines finite-temperature variational free energies for Gaussian Ising perceptrons with bounded Borel log-potentials and Gaussian spherical perceptrons with bounded continuous potentials, at every positive pattern density. A spherical extension treats bi-orthogonally invariant disorder with compact limiting singular-value distributions and no outliers. At margin −1, the quadratic-penalty spherical model has a sharp feasibility threshold and limiting gap and force laws, with system size, zero temperature, and critical density taken in that order. (Lean)

For zero-field Gaussian SK heat-bath dynamics with rate-one updates per spin, proves worst-start cutoff on the log n scale for fixed 0 ≤ β < 1, mixing time n^2/3+o(1) at β = 1, and stretched-exponential mixing from a Gibbs-sampled fixed starting configuration for β > 1, in probability over disorder. At criticality, rescaled stationary and quench autocorrelation processes have universal random limits for Gaussian and Rademacher disorder; the quench limit relaxes to the stationary limit. (Lean)

Constructs stable distance-dependent pair interactions for three-dimensional classical particles with a first-order phase transition: the canonical free energy has a derivative jump at one inverse temperature throughout an open density interval. One potential has a divergent repulsive core; another is bounded and continuous with an integrable power-law tail, realizing the type of transition sought in Simon's continuum problem. (Lean)

Resolves Schramm’s Hausdorff-measure question for chordal SLEκ, 0 < κ < 8. The explicit gauge r^d(log log(1/r))^(2-d)/2, d=1+κ/8, gives almost surely positive finite measure to every trace segment γ([s,t]) with 0 < s < t < ∞, and finite expected measure to the trace in every bounded disk. (Lean)

On every infinite connected locally finite simple unweighted graph, the free uniform spanning forest is a factor of independent vertex labels, by one isomorphism-equivariant rule using no root. Translation-invariant strongly Rayleigh binary processes on every countable group, including invariant determinantal processes with Hermitian positive-contraction kernels, are also factors of IID. (Lean)

For random k-SAT with independent uniformly signed proper clauses sampled with replacement, proves finite positive limiting thresholds and hitting-time variance Θ_k(n) for every fixed k ≥ 3, and computability of the 3-SAT threshold. We credit Gaia Carenini with priority for resolving the threshold-existence conjecture in her concurrent ECCC TR26-229, made public October 5, 2026; this family supplies another proof and the sharper variance and computability results. (Lean)

Proves that the Thorp shuffle randomizes N=2^d labeled cards in Θ(log N) physical shuffles, settling its optimal mixing order for power-of-two deck sizes. Convergence is in total variation from the worst initial ordering and concerns the entire permutation, not just individual card positions. (Lean)

Determines the sharp exponential singularity rate of symmetric random sign matrices with independent entries on and above the diagonal. Uniform signs give Pr(det A_n=0)=(1/2+o(1))ⁿ; for fixed bias p ∈ (0,1)∖{1/2}, the rate is (p²+(1-p)²+o(1))ⁿ. In the biased case, agreeing rows attain this rate.

Every order automorphism of the Turing degrees is the identity, resolving their rigidity problem. Thus no nontrivial relabeling of degrees preserves the ordering by relative computability. (Lean)

Confirms the Blass–Gurevich–Shelah noncapture conjecture: consistency of a linear system over 𝔽3 defines a polynomial-time query on unordered finite structures that choiceless polynomial time with counting cannot express. A separate result shows that adding witnessed symmetric choice strictly increases expressive power. Both separations hold for the full counting formalism, allowing hereditarily finite sets of arbitrary finite rank. (Lean)

Assuming ZF is consistent, constructs a model in which every surjective image of a set injects into that set, yet the axiom of choice fails. Choice for ordinal-indexed families still holds. From any countable transitive model of ZFC, a separate construction gives a transitive symmetric extension with these properties and no new countable sequences of ground-model elements. (Lean)

Proves that weak normalization implies strong normalization for every pure type system: if every legal expression in every valid context has a β-normal form, every β-reduction sequence terminates. This resolves the β-Barendregt–Geuvers–Klop conjecture, including nonfunctional rules and open contexts. (Lean)

Resolves Dixmier's problem for all discrete groups: amenability is equivalent to every uniformly bounded Hilbert-space representation being similar to a unitary representation. For countable discrete groups, amenability is also equivalent to strong Ulam stability: sufficiently accurate unitary approximate representations are uniformly close in operator norm to genuine representations on the same, possibly infinite-dimensional, Hilbert space. (Lean)

Constructs an infinite finitely presented simple amenable group, answering the longstanding question of whether these properties can occur simultaneously. (Lean)

The Salvetti complex of every finite-rank Artin group is aspherical, proving the Artin K(π,1) conjecture. Arbitrary intersections of its parabolic subgroups are parabolic, proving the Parabolic Intersection Conjecture. An explicit Artin group admits no proper cocompact isometric action on any nonempty proper CAT(0) space. (Lean)

Proves that every finite system of equations over an arbitrary group whose exponent-sum matrix has full row rank over ℚ has a simultaneous solution in an overgroup, resolving Howie's conjecture. The coefficient group embeds in the presented quotient. A companion proves Kervaire's conjecture: adjoining one generator and one relation cannot trivialize a nontrivial group. (Lean)

Answers negatively whether every word-hyperbolic group is a CAT(0) group. Constructs one with a finite classifying space but no proper cocompact isometric action on any proper complete CAT (0) space, in any dimension. (Lean)

Constructs a finitely generated group with two essentially free probability-measure-preserving actions of different costs, answering the general fixed-price problem negatively. Its Bernoulli action has cost bounded away from one, while a sequence of finite height extensions has costs tending to one.

Resolves the predicted spectral contrast for the lattice Anderson model with independent uniform site potentials. In dimension two, every positive disorder strength gives almost surely pure-point spectrum. In every fixed dimension d ≥ 3, sufficiently weak positive disorder gives purely absolutely continuous spectrum on a fixed open interval with nonzero spectral weight. (Lean)

Proves N(6)=3, resolving Zauner's dimension-six mutually unbiased bases conjecture: three such bases exist in ℂ6, but four cannot. The exclusion is a complete certified computation under the stated binary64 arithmetic and compiler conditions. An independent companion proves the Matolcsi–Ruzsa–Weiner Fourier-vanishing conjecture for order-six complex Hadamard matrices outside Tao's cubic equivalence class. (Lean)

Proves the fermionic Laughlin spectral-gap conjecture for the full V1 interaction at filling 1/3 on the round sphere. The unique ground state remains uniformly gapped under sufficiently weak bounded real scalar one-body potentials projected to the lowest Landau level. Both the gap and disorder threshold are uniform over all sufficiently large particle numbers and all normalized potential profiles. (Lean)

Constructs an entangled state on ℂ¹⁰⊗ℂ¹⁰ with zero distillable secret key for the specified local-instrument protocols that complete almost surely. These allow joint local processing and authenticated two-way public communication, with no other shared private resource and an eavesdropper holding the input purification and public record. A trace-preserving PPT channel on M₂₁(ℂ) whose square is not entanglement breaking disproves Christandl's PPT-square conjecture. (Lean)

Resolves Moore's parity conjecture in the measured-output model: constant-depth quantum circuits with arbitrary one-qubit gates, unbounded-arity Toffoli gates and polynomially many total qubits cannot compute parity with any fixed positive worst-case advantage. Ancillas start in zero, one output qubit is measured, and all other registers may be discarded. Xu–Li's reductions give the same bounded-error obstruction for strict majority. (Lean)

Determines the unassisted classical capacity of every qubit generalized amplitude-damping channel, including all damping and thermal parameters. An explicit one-variable optimization gives the capacity, attained by independent two-state signal ensembles with collective decoding. Holevo capacity, minimum output entropy and regularized classical capacity are additive when tensoring with any finite-dimensional quantum channel. (Lean)

Proves exponential threshold repetition for every finite two-player one-round game: if its entangled value is v < 1, the probability of winning at least a fraction v+δ of k independent repetitions decays exponentially in k, for 0 < δ < 1-v. Arbitrary joint finite-dimensional entangled strategies and correlated question distributions are allowed. (Lean)

Proves the strongly rational case of the strong-locality conjecture: every simple unitary strongly rational complex vertex operator algebra generates a completely rational conformal net. Its simple modules are unitarizable, and its representation category agrees with the net’s finite-index sectors as a braided unitary tensor category. (Lean)

Shows that the universal bound R(f)=O((1+Q(f))⁴) for total Boolean functions is sharp in its exponent, ruling out every smaller power and disproving the conjectured cubic relation. Here R and Q are randomized and quantum worst-case bit-query complexities with error at most 1/3; computation between queries is unrestricted.

Classifies finite-index bimodules between scalar-twisted group factors of ICC groups commensurable with property-(T) lattices over characteristic-zero local fields and arbitrary ICC group factors. Every such bimodule is a summand of finite sums of models arising from finite-index subgroup isomorphisms and finite-dimensional projective representations. The classification also recovers the group, scalar cocycle and amplification scale up to the stated stable equivalence.

Resolves the free group factor isomorphism problem: L(𝔽₂) ≅ L(𝔽₃), and hence all interpolated free group factors, including L(𝔽_∞), are isomorphic. Their common factor has fundamental group ℝ_{> 0}. (Lean)

Proves Connes' bicentralizer conjecture for every type III1 factor with separable predual and every faithful normal state. More generally, for every inclusion N ⊂ M of von Neumann algebras with separable preduals admitting a faithful normal conditional expectation, constructs an amenable expected subalgebra P ⊂ N with P'∩ c(M)=N'∩ c(M), resolving the relative bicentralizer conjecture. (Lean)

Constructs a bounded self-adjoint tuple in a tracial von Neumann algebra whose microstates and nonmicrostates free entropies satisfy -∞ < χ < χ^* < ∞. This answers Voiculescu’s finite-entropy equality question negatively: the matrix-approximation and free-Fisher-information definitions differ even when both are finite. (Lean)

For every minimal homeomorphism h of an infinite compact metrizable space X, the radius of comparison of C(X)⋊_hℤ equals 1/2mdim(X,h), including infinite values. Zero mean dimension is equivalent to the small boundary property, Jiang–Su stability and finite nuclear dimension; in this case nuclear dimension is at most one.

Constructs a uniformly discrete bounded-geometry space whose maximal coarse assembly map is not rationally injective. The example is a coarse disjoint union of finite connected graphs of uniformly bounded degree, with an infinite-order kernel class. A companion gives the analogous failure for reduced coarse assembly, disproving the rational coarse Novikov conjecture. (Lean)

Resolves the Ara–Maltsiniotis conjecture: for every n ≥ 1 and n = ω, small strict globular n-categories admit proper combinatorial Thomason model structures Quillen equivalent to simplicial sets. Thus strict higher categories model the homotopy theory of spaces in every stated dimension. (Lean)

Disproves strong chromatic splitting at height three for primes p ≥ 5, and weak splitting for the derived p-completed sphere at heights p (p ≥ 5) and p+1 (p ≥ 7). Nevertheless, for n ≥ 1 and p > n+1, the overlap L_n-1L_K(n)S_p^∧ admits a 2ⁿ-stage filtration by the predicted localized-sphere pieces. At height three and prime three, even finite assembly from such pieces fails in the category of E(2)-local modules over the derived completed sphere.

Refutes the Hahn–Wilson conjecture at chromatic height two. For every sufficiently large prime p, constructs a connective p-complete spectrum of exact fp-type two that cannot be built from completed BP⟨2⟩ by finitely many sums, shifts, cones and retracts. The examples nevertheless satisfy the finite and telescopic localization comparisons.

Constructs separable real Banach spaces that are globally bi-Lipschitz equivalent but not linearly isomorphic, resolving the separable Lipschitz-isomorphism problem negatively. Thus even the complete metric structure up to bi-Lipschitz equivalence does not determine a separable Banach space's linear isomorphism class. (Lean)

Proves that every real Banach space with Markov type p for some p > 1 admits an equivalent uniformly convex norm, answering Naor's renorming question. Together with the known converse, this characterizes superreflexivity by nontrivial Markov type. (Lean)

Resolves Kirk's reflexive-space fixed-point problem: every nonexpansive selfmap of a nonempty closed bounded convex subset of a real reflexive Banach space has a fixed point. The result uses the original norm, without assuming uniform convexity. (Lean)

Constructs a real reflexive Banach space with an asymptotically midpoint uniformly convex norm but no equivalent asymptotically uniformly convex norm, extending Baudier's separation to reflexive spaces. In the same space, depth-k countably branching diamonds require distortion at least √(1+k/12), so midpoint uniform convexity does not force uniformly bounded diamond distortion even under reflexivity. (Lean)

Every closed smooth Riemannian surface admits a smooth isometric immersion into ℝ4, resolving the closed-surface form of the four-dimensional isometric-immersion problem. This includes nonorientable surfaces and metrics of arbitrary Gaussian curvature. (Lean)

Constructs a smooth positive-definite metric on (-1,1)² for which no neighborhood of the origin admits a smooth isometric immersion into ℝ3. This answers the unrestricted smooth local isometric realization problem for surfaces negatively, even after shrinking the neighborhood. (Lean)

Every complete connected smooth boundaryless n-manifold, n ≥ 3, satisfying -4Δ+Scal ≥ 1 as a quadratic-form inequality admits a continuous map to a simplicial complex of dimension at most n-2 whose entire fibers have diameter bounded only by n in the original metric. This strengthens Gromov's width conclusion to spectral scalar curvature. Universal covers of closed positive-scalar-curvature manifolds also have continuous macroscopic dimension at most n-2 for every n ≥ 2.

Proves the first-singular-time case of tangent-flow uniqueness for smooth compact connected embedded surfaces without boundary in ℝ3. At every singular point, all fixed-center backward tangent flows agree as area measures at every negative time in the original ambient coordinates, without mean-convexity or a prescribed tangent model.

Constructs a contractible domain in ℂ3 with a complete negatively pinched Kähler metric but no bounded holomorphic coordinates, disproving bounded-domain uniformization in this setting. A higher-dimensional example has sectional curvature at most −1 and only constant bounded holomorphic functions; its curvature is not bounded below. (Lean)

On every closed connected Riemannian manifold of dimension at least two satisfying weak Ma–Trudinger–Wang curvature, all tangent injectivity domains are convex, resolving Villani’s conjecture in this setting. For squared-distance transport between measurable probability densities bounded above and away from zero, the optimal map and its inverse are Hölder continuous. (Lean)

Derives the nonlinear Boltzmann equation from three-dimensional grand-canonical Newtonian gases throughout every regular kinetic interval with uniform Gaussian decay. Stable finite-range radial potentials may have attractive wells and a singular repulsive core; initial pair exclusion and spatially summable Gaussian density and gradient bounds are assumed. A companion gives finite-dimensional hard-sphere Gaussian fluctuations, centered at the exact microscopic expectation and governed by the linear fluctuating Boltzmann equation.

Zero-frequency measurements on any nonempty open boundary patch determine a smooth metric and smooth unitary connection on a trivial Hermitian rank-two bundle over a compact connected manifold of dimension at least three, up to diffeomorphism and gauge fixed on that patch. Inputs and observations use the same patch. In contrast, distinct uniformly positive bounded measurable scalar conductivities on a three-dimensional ball can have identical full-boundary data. (Lean)

For a sufficiently large odd nonlinearity power, constructs a nonempty open set of initial data in H^k(𝕋¹²), with k > 8, whose solutions of the scalar defocusing nonlinear Schrödinger equation blow up in finite time. Thus finite-time blowup is stable under Sobolev perturbations in this supercritical regime. (Lean)

Proves that full static boundary displacement-to-traction data determine both smooth real Lamé moduli on every bounded connected smooth domain in ℝ3, provided μ > 0 and 3λ+2μ > 0 on the closure. Neither analyticity, proximity to constant coefficients nor prior knowledge near the boundary is required. (Lean)