All papers

We prove Milne's rationality conjecture for abelian varieties, including residue characteristic 2. After good reduction, the pairing of a rational Hodge class with any complementary product of divisor classes on the reduction is the same rational number in every prime-to-p realization and in crystalline cohomology. Using the Hodge theorem for CM abelian varieties, we also show that every such specialized Hodge class is represented by a single rational algebraic cycle in all these realizations.

We prove the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose full q-power Selmer group has corank zero or one at some prime q. The analytic and Mordell–Weil ranks equal that corank, and the Tate–Shafarevich group is finite. The formula includes all prime factors and requires no additional hypotheses on reduction, rational torsion, isogenies, complex multiplication, or residual Galois representations.

We prove the two-primary Birch and Swinnerton-Dyer leading-term formula for every elliptic curve over the rationals whose two-power Selmer group has corank at most one. In this range, the algebraic rank, analytic rank, and Selmer corank are equal, and the Tate–Shafarevich group is finite. Combined with the quadratic-twist Selmer distribution, this gives the exact two-primary formula for a density-one set of signed squarefree twists of each fixed curve, ordered by absolute value; the common rank is zero or one, with each value having density one half.

We prove the Selmer converse in coranks zero and one for every elliptic curve over ℚ and every prime p: if the full p-power Selmer group has ℤ_p-corank \(r\in\{0,1\}\), then the analytic and Mordell–Weil ranks both equal r, and the entire Tate–Shafarevich group is finite. As an application at the additive prime 3, we prove that for every prime \(\ell\equiv4,7,8\pmod9\), the cubic \(X^3+Y^3=\ell Z^3\) has analytic and Mordell–Weil rank one and finite Tate–Shafarevich group. In particular, every such ℓ is a sum of two rational cubes.

We establish the quasi-Riemann hypothesis by proving that every Dirichlet L-function, including Riemann's zeta function, has no zeros in the half-plane \(\mathop{\mathrm{Re}}\nolimits s\gt 11/12\). More generally, we prove the same zero-free half-plane for every finite-order Hecke L-function over \(K=\mathbb Q(\sqrt{-3})\). In particular, this rules out the existence of Landau–Siegel zeros.

We prove the uniform exclusion of Landau–Siegel zeros. There is an absolute constant c > 0 such that every real zero \(\beta\in(0,1)\) of every primitive nonprincipal real Dirichlet L-function of conductor q ≥ 3 satisfies \((1-\beta)\log q\ge c\).

We prove that all finite-order Hecke L-functions over \(\mathbb Q(\sqrt{-3})\) and all Dirichlet L-functions are zero-free in the half-plane \(\Re s\gt 7/8\), with the principal pole at s = 1 allowed. In particular, the Riemann zeta function is zero-free in this half-plane, proving the quasi-Riemann hypothesis.

We give a negative answer to Hilbert's tenth problem over the rational numbers: no algorithm decides whether a polynomial with integer coefficients has a rational zero. The number of variables is part of the input.

We prove a pointwise 2-converse for elliptic curves over \(\mathbf Q\) with nonzero rational two-torsion: if the \(2^\infty\)-Selmer corank is zero or one, then the analytic rank and Mordell–Weil rank equal that corank, and the Shafarevich–Tate group is finite. The result allows arbitrary reduction at 2.

We prove that Catalan's constant \(G=\sum_{j\geq0}(-1)^j/(2j+1)^2\) is irrational.

For every elliptic curve over ℚ, we prove that the average analytic rank of its quadratic twists tends to 1/2 when signed squarefree twist parameters are ordered by absolute value. This resolves Goldfeld's mean analytic-rank conjecture in this counting convention.

We prove Goldfeld's analytic density conjecture: for every elliptic curve E over ℚ, the quadratic twists of E with analytic rank zero and one each have density 1/2 among signed squarefree twist parameters ordered by absolute value. We also prove the low-corank 2-converse: if the \(2^\infty\)-Selmer corank of E is zero or one, then it equals the analytic and Mordell–Weil ranks, and the Tate–Shafarevich group is finite.

We prove the ordinary two-point Chowla conjecture. For every fixed pair of nonproportional affine forms, the Liouville correlation has a power-of-logarithm saving at every cutoff, with an absolute exponent. We also prove the binary corrected Elliott conjecture for ordinary averages of complex multiplicative functions of modulus at most one, under uniform nonpretentiousness of at least one original factor. This qualitative conclusion holds in fixed residue classes and for fixed nonproportional affine forms.

We prove the Deligne–Drinfeld conjecture: the rational Grothendieck–Teichmüller Lie algebra is freely generated by one element in every odd weight at least three, with the corresponding isomorphism after weight completion.

We prove a mod-ℓ Bogomolov–Pop reconstruction theorem for function fields of transcendence degree at least two over arbitrary algebraically closed fields of characteristic different from ℓ. The groups \(K^{\mathrm M}_1/\ell\) and \(K^{\mathrm M}_2/\ell\), together with their full bilinear product, determine the perfect closure and its constant field. Every compatible isomorphism of these data is induced by a field isomorphism up to a single scalar in \(\mathbb F_\ell^\times\), with only Frobenius ambiguity in the field isomorphism in positive characteristic.

Let K and L be finitely generated extensions of transcendence degree at least two over algebraically closed fields of characteristic p. We prove that every isomorphism of their first Milnor K-groups modulo p preserving the degree-two Steinberg relations is a nonzero scalar multiple of the map induced by a unique field isomorphism. The proof recovers projective lines over the subfields of pth powers by an elementary calculation with derivations.

We prove the Bogomolov–Pop reconstruction conjecture for function fields of transcendence degree at least two over arbitrary algebraically closed fields of characteristic different from ℓ. The pro-ℓ abelian-by-central datum determines the perfect closure and its constant field, with precisely the Frobenius and ℓ-adic unit ambiguities.

For every odd positive integer N, every irreducible component of the full two-adic Hecke algebra of level \(\Gamma_1(N)\) has Krull dimension four. This proves the p = 2 case of Emerton's dimension conjecture, including all residual components.

Every continuous, odd, absolutely irreducible two-dimensional 2-adic representation of \(G_{\mathbb Q}\) that is unramified outside finitely many finite primes occurs in the full completed Hecke algebra at some odd tame level. The level may contain auxiliary tame primes, and scalar and reducible residual representations are included. This is a completed-Hecke occurrence result, with no de Rham hypothesis; it does not assert classical modularity.

We prove that every continuous, irreducible, odd two-dimensional 2-adic representation of \(G_{\mathbb Q}\), unramified outside finitely many primes and de Rham at 2 with distinct Hodge–Tate weights, is modular up to Tate twist. This resolves the odd, regular two-dimensional Fontaine–Mazur conjecture over ℚ at 2, including all residual representations.

We prove Erdős's conjecture on the largest fibers of Euler's totient function: for every ε > 0, infinitely many positive integers n have more than \(n^{1-\varepsilon }\) preimages. We also show that, for every fixed δ > 0, there are at least \(x^{1-o(1)}\) primes p in \(2x\lt p\le5x\) whose predecessors have no prime factor exceeding x^δ.

For a prime p chosen uniformly from \(3\le p\le x\), list the prime factors of \(p-1\) in decreasing order, with multiplicity. As \(x\to\infty\), their logarithms, divided by \(\log(p-1)\), converge in every finite joint distribution to the Poisson–Dirichlet distribution with parameter one. This proves the conjecture of Ford, Konyagin and Luca.

We prove that there are infinitely many primes p for which \(p-1\) is squarefree and has an even number of prime factors. Equivalently, there are infinitely many primes p with \(\mu(p-1)=1\).

Let \(P^+(n)\) denote the largest prime factor of n. We prove that \(\log P^+(n)/\log n\) and \(\log P^+(n+1)/\log n\) are asymptotically independent in ordinary natural density, with Dickman marginals. This resolves the Erdős–Pomerance joint Dickman conjecture positively and implies that the ordering \(P^+(n)\lt P^+(n+1)\) has natural density 1/2.

We prove Ostmann's inverse Goldbach conjecture: no set differing from the primes by finitely many elements can be written as \(A+B\), where A and B are sets of nonnegative integers with at least two elements each.

We prove the generalized Ramanujan conjecture for globally generic cuspidal automorphic representations of split connected adjoint exceptional groups over global function fields. Every local component is tempered, without restrictions on characteristic or ramification depth. The proof extends the unramified Ramanujan theorem of a companion paper to all ramified places.

For SL_n in characteristic p > n, we construct nonzero locally constructible perverse Hecke eigensheaves with Borel level at two or more marked points on a smooth projective curve of genus at least two over an algebraic closure of a finite field. The parameter is a Zariski-dense geometric ℓ-adic PGL_n-local system with unipotent tame monodromy; its nilpotent logarithms may have any Jordan type, including zero. The eigensheaves have parabolic nilpotent singular support, and their eigenisomorphisms retain the full tensor and fusion structure.

Under the finite-level Ramanujan–Arthur decomposition stated in Theorem 1.1, we construct a global Arthur enhancement for every occurring cuspidal excursion parameter at a specified full finite level for a split connected semisimple group over a global function field. A single algebraic SL₂ and a commuting Weil centralizer map recover the given parameter by diagonal specialization on the entire Weil group, including inertia. The result does not assert ellipticity, Arthur-packet classification, or a multiplicity formula.

We construct nonzero perverse Weil Hecke eigensheaves for SL_n with Borel level at one marked point, after a finite extension of the field of constants. The parameter is a geometrically dense arithmetic PGL_n-local system with tame regular-unipotent monodromy on a once-punctured curve of genus at least two over a finite field of characteristic p > n. The full multi-leg eigenstructure is Frobenius compatible with the prescribed arithmetic eigenvalue, and the geometric sheaf has parabolic nilpotent singular support.

We construct nonzero locally constructible perverse Hecke eigensheaves with Borel level at one marked point on a smooth projective curve of genus at least two over an algebraic closure of a finite field. The parameter is a Zariski-dense geometric ℓ-adic local system with tame regular-unipotent monodromy. The group is simple and simply connected and satisfies four explicit Lie-theoretic characteristic hypotheses. The eigensheaves have parabolic nilpotent singular support, and their eigenisomorphisms are compatible with tensor products, permutations, and fusion.

We prove the \(\overline{\mathbb Q}_\ell\)-linear restricted geometric Langlands equivalence for smooth projective connected curves and connected reductive groups in characteristic p > 0, with ℓ ≠ p, in two regimes. Over \(\overline{\mathbb F}_q\), we assume the four characteristic conditions of the restricted theory stated below. Over arbitrary algebraically closed fields, we assume these conditions and additionally that p is very good and does not divide the Weyl-group order. This proves Gaitsgory–Raskin's full-support conjecture [12, Conjecture 1.3.10] in these regimes.

Under the characteristic hypotheses of restricted geometric Langlands theory, we prove that the canonical Arthur filtration on finitely supported unramified automorphic functions is defined over ℚ for split connected semisimple groups. The result includes the noncuspidal part and every closed invariant nilpotent support, establishing the rationality conjecture of Gaitsgory–Lafforgue–Raskin in this setting.

We prove rational and \(\overline{\mathbb Q}_\ell\) decompositions of cuspidal automorphic functions for split semisimple groups over global function fields into subspaces indexed by nilpotent orbits of the dual group. The decompositions hold at every full finite level, with arbitrary divisor multiplicities. Each summand is governed by the same orbit at every unramified place and every complex embedding. At trivial level this proves Conjectures 3.4.5 and 3.4.6 of Gaitsgory–Lafforgue–Raskin. For split connected adjoint absolutely simple groups, we also prove that a cuspidal automorphic representation with one generic unramified local component is tempered at every unramified place. Thus globally generic cuspidal representations in this scope satisfy the unramified part of the generalized Ramanujan conjecture.

Let K range over totally real quartic fields with no proper intermediate field, and let \(\mathcal O\) be any order in K. We prove that the packets of periodic diagonal orbits attached to invertible \(\mathcal O\)-ideal classes, weighted by orbit volume, equidistribute with no escape of mass as \(|\mathop{\mathrm{Disc}}\nolimits (\mathcal O)|\) tends to infinity. The proof combines measure rigidity with a cubic-resolvent estimate that remains uniform at primes dividing the order index.

We prove that the complete ideal-class packets of maximal orders in totally real sextic fields with no proper intermediate fields become equidistributed, with no escape of mass, in the space of unimodular lattices as their field discriminants tend to infinity. The limit is Haar probability measure. Each packet includes all coordinate-sign translates and is weighted by diagonal-orbit volume.

We prove the packet form of the higher-dimensional Duke equidistribution problem for totally real fields of any fixed prime degree at least five, allowing arbitrary local homothety types of full lattices. As the multiplier-order discriminant tends to infinity, the volume-weighted packet measures converge to Haar probability measure, with no escape of mass.

We prove the abelian Zilber–Pink conjecture over \(\overline{\mathbb Q}\). Every irreducible subvariety of an abelian variety has only finitely many maximal atypical subvarieties, where atypicality is measured inside its smallest containing torsion coset.

We prove the \(E\times\mathrm{CM}\) component of Zilber–Pink for Hodge-generic algebraic curves in \(\mathcal A_2\) over \(\overline{\mathbb Q}\). Each such curve contains only finitely many points whose abelian surface is isogenous to a product of elliptic curves with at least one factor having complex multiplication.

We prove the quaternionic-division component of Zilber–Pink for curves in \(\mathcal A_2\) over \(\overline{\mathbb Q}\). A Hodge-generic algebraic curve contains only finitely many points whose full geometric rational endomorphism algebra is an indefinite quaternion division algebra over ℚ. No boundary or reduction hypothesis is required.

We prove that every Hodge-generic algebraic curve in \(\mathcal A_2\) over \(\overline{\mathbb Q}\) contains only finitely many points whose abelian surface is isogenous to the square of an elliptic curve without complex multiplication (CM). Combining this result with the companion \(E\times\mathrm{CM}\) and quaternionic-division finiteness theorems, we prove the curve case of Zilber–Pink in \(\mathcal A_2\) over \(\overline{\mathbb Q}\), without a boundary hypothesis.

We prove the conjecture that the irrationality exponent of π is 2. As a consequence, the classical Flint–Hills series \(\sum_{n\ge1}1/(n^3\sin^2 n)\) converges, with angles in radians.

We prove the Margulis–Platonov conjecture over every global function field, including characteristic two. For an absolutely almost simple simply connected algebraic group, every noncentral abstract normal subgroup of its rational points is the inverse image of an open normal subgroup of the finite product of its anisotropic local groups.

We prove the Margulis–Platonov conjecture over number fields. For an absolutely almost simple simply connected group, every noncentral abstract normal subgroup of its rational points is the inverse image of an open normal subgroup of the product of its anisotropic nonarchimedean local groups.

Let X be a smooth proper hyperbolic curve over an algebraic closure of a p-adic local field. Given finitely many disjoint small open disks, we construct one connected finite étale cover with a sheet isomorphic to their entire exterior and with degree divisible by p on every connected component above each disk.

We prove the local p-adic section conjecture for smooth proper geometrically connected curves of genus at least two, over every finite extension of ℚ_p. Combined with established finite-descent theorems, this also yields the global section conjecture for smooth proper geometrically connected curves of genus at least two over number fields when their finite-cover descent locus equals their rational points; this includes \(X_0(N)\) and \(X_1(N)\) of genus at least two over ℚ.

We prove that every irreducible integer quartic with no fixed prime-square divisor takes squarefree values with the predicted positive Euler-product density. More generally, we obtain the corresponding \((d-2)\)-power-free density in degrees \(4\le d\le8\). The proof combines number-field factorization, determinant estimates with adaptive auxiliary primes, and explicit low-degree geometry. Together with Browning's theorem for higher degrees, this gives the \((d-2)\)-power-free density for every d ≥ 4.

Let \(h(k)\) be the least integer such that every interval of \(h(k)\) consecutive integers contains an integer coprime to any prescribed positive integer having at most k distinct prime divisors. We prove \(h(k)\ll k^2/(\log\log(3k))^2\), giving an affirmative answer to Jacobsthal's quadratic-bound question.

We prove the weak inhomogeneous Duffin–Schaeffer conjecture. For every fixed γ ∈ ℝ and finite-valued \(\psi:\mathbb N\to[0,\infty)\), divergence of \(\sum_{q\ge1}\phi(q)\psi(q)/q\) implies \(\|qx-\gamma\|\lt \psi(q)\) for infinitely many q, for almost every x. Here ϕ is Euler's totient and \(\|\cdot\|\) is distance to the nearest integer. Numerators need not be reduced, and the shift satisfies no Diophantine restriction.

We prove Patterson's first-moment conjecture for normalized cubic Gauss sums over all primary Eisenstein primes, unconditionally. The sharp-cutoff main term is \((6/5)c_*X^{5/6}/\log X\), where \(c_*=(2\pi)^{2/3}/(3\Gamma(2/3))\). For every fixed nonzero angular Fourier mode of the prime argument, we also prove cancellation at the first-moment scale.

Let \(V(x)\) count the distinct values of Euler's totient function up to x. We give an explicit asymptotic equivalent for \(V(x)\). Its coefficient is a uniform limit of functions defined from finite arithmetic data. We also prove that \(V(cx)/V(x)\to c\) as \(x\to\infty\) for every fixed c > 0, answering a question of Erdős and Hall.

We prove a conjecture of Erdős: for every sufficiently large integer b, every rational number \(a/b\) with \(1\le a\lt b\) is a sum of \(O(\log\log b)\) distinct positive unit fractions, with an absolute implied constant. This order is best possible when the numerator varies. We also show that both the number of expansions of 1 with exactly k distinct terms and the least integer at least 2 that never occurs as a denominator in such an expansion grow doubly exponentially in k: their double logarithms have order k.

For every fixed C > 0, we prove that a positive proportion of consecutive prime gaps exceed \(C\log p\), where p is the smaller prime. The proportion is bounded below for every sufficiently large initial segment of the prime sequence, with a constant depending on C. It follows that the indices at which \(p_n/n\) increases have positive lower asymptotic density, answering a question of Erdős and Prachar.

We prove potential Zariski density of integral points on SL_r-character varieties of smooth complex curves in every rank. The result allows prescribed quasi-unipotent boundary monodromy, including exact nonsemisimple conjugacy classes, and holds on every component over the full ring of integers of one finite extension. Here integrality is measured in the ambient character variety, while the exact boundary conditions are imposed on the complex representation.

We prove the Gaussian moat conjecture: no infinite walk through distinct Gaussian primes can have bounded steps. More strongly, for each fixed finite step bound, the connected components of the Gaussian-prime graph have uniformly bounded size. This bound applies to every starting prime, including primes on the coordinate axes, and is nonexplicit. The proof constructs a finite periodic sieve obstruction using geometric sampling and information-theoretic estimates.

For every fixed finite set of distinct positive primes, we prove that at least \(cx/(\log x)^2\) primes in \((x,2x)\) have every member of the set as a primitive root, for some c > 0 and all sufficiently large x. The result assumes four explicitly stated analytic and sieve inputs from the companion paper on primitive roots for admissible integer bases.

We prove the infinitude assertion in Artin's primitive root conjecture: for every integer a that is neither −1 nor a square, there are at least \(c_a x/(\log x)^2\) primes in \((x,2x)\) with primitive root a, for some \(c_a\gt 0\) and every sufficiently large x.

We prove the modularity conjecture for elliptic curves over imaginary quadratic fields: every such curve is modular, with matching local parameters at every place.

We prove Uchida's conjecture on open homomorphisms of Galois groups. Every continuous open homomorphism between Galois groups of solvably closed Galois extensions of number fields is induced by a unique equivariant field embedding in the opposite direction. No restriction on the kernel or additional compatibility hypothesis is required.

We prove the rational Hodge conjecture for every finite product of projective complex K3 surfaces: every rational Hodge class is algebraic. The factors may be distinct or repeated, with no restrictions on their Picard numbers, periods, or endomorphism fields.

We prove that the Kuga–Satake correspondence is algebraic for every smooth projective complex K3 surface. More precisely, the prescribed embedding of its transcendental cohomology into the cohomology of its Kuga–Satake abelian variety is induced by a rational algebraic cycle, with the fixed normalization and full even-Clifford target. The result also holds for isogenous Kuga–Satake models with the transported embedding.

We prove the rational Hodge conjecture in every codimension on every self-power of a complex abelian sixfold with an imaginary-quadratic action and a compatible polarization whose rational homological Hermitian form is hyperbolic of signature \((3,3)\). We also prove it in every codimension on every self-power of a complex abelian variety of dimension at most five admitting an imaginary-quadratic action. Both results include nonsimple varieties and special periods with additional endomorphisms. The proof uses the companion theorem on the rational Hodge conjecture for CM abelian varieties.

We prove the rational Hodge conjecture for complex abelian varieties with complex multiplication: every rational Hodge class on such a variety is a rational linear combination of algebraic cycle classes. As consequences, we obtain the generalized Hodge conjecture for CM abelian varieties, the Tate conjecture for abelian varieties over finite fields, and the Hodge standard conjecture for abelian varieties in arbitrary characteristic.

We prove the rational Hodge and generalized Hodge conjectures in every cohomological degree of every self-power of a projective complex K3 surface whose transcendental quadratic space, with its cup-product form, admits a rational isometric embedding in \(\mathbb U_{\mathbb Q}^{\oplus2}\perp\langle-1\rangle^4\). This includes every ample P-polarized K3 surface, including all Picard jumps, for \(P=\mathbb U\oplus D_8(-1)\oplus D_4(-1)\). On this locus we construct algebraic correspondences inducing every prescribed standard even-Clifford Kuga–Satake tensor. More generally, for any projective complex K3 surface, algebraicity of one exact standard even-Clifford Kuga–Satake tensor implies both conjectures for every self-power.

We prove the rational Hodge conjecture on every self-power of the Jacobian at each tensor Hodge-generic point of the full marked variation of a connected abelian cover of curves. This holds in every base genus and for every compatible branching pattern. For any CM abelian variety, the conclusion also holds for every self-power of its product with the Jacobian, on the same Hodge-generic locus. We also prove the conjecture on every self-power of a very general member of the full smooth labelled family of diagonal complete intersections cut out by at most two equations of a common degree, in every dimension and every degree at least two.

We prove that every rational Weil class on a split abelian eightfold of Weil type is algebraic. The result holds for every imaginary quadratic field, every compatible polarization type, and every member of the split family, including those with additional endomorphisms. Thus the full two-dimensional rational Weil space in codimension four is generated by algebraic cycle classes.

For a polarized K3 surface whose primitive cohomology has full orthogonal Hodge group, one algebraic correspondence inducing a nonzero map from that cohomology to the second cohomology of an abelian variety suffices to recover the prescribed full Kuga–Satake correspondence. We give an equivalent condition using holomorphic one-forms on a generically finite surface cover. If this input holds very generally in a polarized component, the prescribed correspondence is algebraic throughout that component, for every choice of standard data on the transcendental part. The existence of the initial correspondence remains a hypothesis.

We prove the ordinary Iitaka subadditivity conjecture for surjective projective morphisms with connected fibers between smooth connected projective varieties over algebraically closed fields of characteristic zero. If F is the geometric generic fiber of \(f:X\to Z\), then \(\kappa(X)\geq\kappa(F)+\kappa(Z)\).

We prove the reverse logarithmic Kodaira inequality for a surjective connected-fiber morphism \(f:(X,E)\to(Y,D)\) of smooth projective reduced simple-normal-crossing pairs, with \(\mathop{\mathrm{Supp}}\nolimits (f^*D)\subseteq\mathop{\mathrm{Supp}}\nolimits E\), such that X and every boundary stratum are smooth over \(Y\setminus\mathop{\mathrm{Supp}}\nolimits D\). Together with logarithmic subadditivity, the inequality gives additivity, including both negative-infinity cases. This resolves Popa's logarithmic additivity conjecture positively in the projective reduced-SNC, stratum-smooth setting.

We prove Campana's orbifold Iitaka subadditivity conjecture for rational simple normal crossing boundaries on compact manifolds in Fujiki class \(\mathcal C\), including coefficient one. Ordinary and logarithmic subadditivity follow.

We prove the logarithmic Iitaka–Viehweg inequality for projective surjective morphisms with connected fibers between smooth complex quasi-projective varieties whose base has nonnegative logarithmic Kodaira dimension. The variation measures the birational field of definition of the whole geometric generic fiber. This resolves Popa's logarithmic variation conjecture positively.

We prove the compact log-smooth Kähler case of b-semiampleness. Let \(f:Y\to X\) be a surjective holomorphic map with connected fibers between smooth compact connected Kähler manifolds, and let Δ be an effective rational divisor with simple normal crossing support and coefficients in \([0,1]\), with \(K_Y+\Delta\sim_{\mathbb Q}f^*L\) for \(L\in\mathop{\mathrm{Pic}}\nolimits (X)_{\mathbb Q}\). There is a smooth compact Kähler modification \(S\to X\) for which the threshold-moduli line satisfies \(M_{S_1}=\nu^*M_S\) in \(\mathop{\mathrm{Pic}}\nolimits (S_1)_{\mathbb Q}\) for every smooth compact Kähler modification \(\nu:S_1\to S\), and some positive multiple of M_S is represented by a holomorphic line bundle generated by global sections. Horizontal components of coefficient one are allowed; neither projectivity nor a Campana orbifold Iitaka hypothesis is assumed.

We prove a uniform index theorem for connected projective semi-log-canonical log Calabi–Yau pairs in every fixed dimension at least four over an algebraically closed field of characteristic zero. For boundary coefficients in a fixed finite rational set, a single multiple of the log canonical divisor is Cartier and linearly trivial. The multiple depends only on the dimension and coefficient set, not on the number of irreducible components. Together with the established theorem in dimensions at most three, this resolves the finite-rational-coefficient semi-log-canonical index conjecture.

Assuming orbifold Iitaka subadditivity, the specified pseudo-effective fourfold minimal model program, and abundance for nef fourfold adjoints of nonnegative Kodaira dimension, we prove the existence of good minimal models for globally strongly ℚ-factorial compact Kähler klt fourfold pairs with effective rational boundary and analytically pseudo-effective actual ℚ-Cartier adjoint. The additional step is nonvanishing. We prove it by fibration arguments and, in algebraic dimension zero, by singular metrics, holomorphic foliations, and extension from a reduced boundary. The projective abundance argument used in the proof is included in full.

For normal projective log canonical pairs over algebraically closed fields of characteristic zero, of fixed dimension d ≥ 5 and with effective boundary coefficients in a fixed finite rational set, we prove that one complete rounded pluricanonical system generates the full Iitaka field whenever the ℚ-Cartier log canonical divisor has nonnegative Kodaira dimension. Its degree depends only on the dimension and coefficient set. For the paper's normalized canonical bundle formulae over ℂ, we also bound the Cartier denominators of the moduli divisors on smooth projective determining models in terms of the dimension and coefficient set.

Assume logarithmic Iitaka subadditivity for surjective morphisms with connected fibers between smooth projective complex varieties with compatible reduced simple normal crossing boundaries. We prove log abundance for normal irreducible compact Kähler spaces in every dimension: for a log canonical pair \((X,\Delta)\) with effective rational boundary and \(K_X+\Delta\) ℚ-Cartier, analytic nefness of \(K_X+\Delta\) implies semiampleness.

We prove the effective Iitaka fibration conjecture in characteristic zero. For each dimension, one pluricanonical degree defines the Iitaka fibration of every smooth integral projective variety of that dimension and nonnegative Kodaira dimension over an algebraically closed field. The associated sections generate the full Iitaka function field.

We prove uniform denominator and effective-system bounds for log Calabi-Yau fibrations from projective log canonical complex pairs of dimension at most four onto positive-dimensional bases, with boundary coefficients in a fixed finite rational set. The bounds give a uniform trivializing degree and a uniform b-Cartier multiple of the moduli b-divisor. When the base divisor is big, a uniform complete rounded adjoint system has section ratios generating the full base function field.

Let \((H,\Theta)\) be a projective complex klt pair with effective rational boundary and nef ℚ-Cartier adjoint. On any projective log resolution, minimal semipositive metrics on the pulled-back adjoint exist and have zero Lelong numbers everywhere. On smooth projective complex varieties, we also prove an H¹-injectivity theorem for rational interior boundaries with simple-normal-crossing support when both endpoint bundles carry zero-Lelong semipositive metrics.

For a projective ℚ-factorial dlt pair with effective rational boundary over an algebraically closed field of characteristic zero, we prove that the adjoint has positive Iitaka dimension whenever a nonzero effective Cartier multiple is supported on the coefficient-one boundary and restricts to a semiample line bundle on its whole reduced support. This gives log abundance after nonvanishing in dimension at most four over ℂ.

We prove canonical nonvanishing for smooth connected projective complex fourfolds: if K_X is pseudo-effective, then \(H^0(X,mK_X)\ne0\) for some positive integer m.

We prove semiampleness of the actual ℚ-Cartier adjoint \(K_X+\Delta\) of a normal connected compact Kähler klt fourfold whenever it is analytically nef and some positive Cartier multiple has a nonzero section. The boundary is effective and rational; the fourfold need not be projective or ℚ-factorial. In Iitaka dimension zero, a positive Cartier multiple is the trivial holomorphic line bundle.

We prove the finite-rational-coefficient case of the effective log Iitaka conjecture in dimension four. For normal projective log canonical complex fourfolds with boundary coefficients in a fixed finite rational set and pseudo-effective rational Cartier adjoint, one uniform degree makes the complete rounded reflexive system nonempty and its section ratios generate the full Iitaka field. We also prove a uniform canonical index bound for projective klt complex fourfolds with rationally trivial canonical divisor.

Fix d ≥ 1 and t > 0. Let \((X,B)\) be an ordinary projective log canonical ℚ-pair of dimension d over a characteristic-zero field k, with X normal and integral, \(H^0(X,\mathcal O_X)=k\), B effective, and \(K_X+B\sim_{\mathbb Q}0\). We prove that every prime component S of B with coefficient at least t satisfies \([k_S:k]\leq N(d,t)\), where k_S is the relative algebraic closure of k in \(k(S)\). This also bounds the Stein degree of S over k, proving the contraction-to-a-point formulation of Birkar's Stein-degree conjecture for ordinary ℚ-pairs.

We prove the Campana–Peternell inequality \(\kappa(X)\geq\kappa(D)\) for a smooth connected projective complex variety X and an effective Cartier divisor D whenever \(m_0K_X-D\) is pseudo-effective for some positive integer m₀. For an algebraic fiber space \(f\colon X\to Y\) between smooth connected projective complex varieties, the hypothesis that \(m_0K_X-f^*H\) is pseudo-effective with H ample Cartier gives \(\kappa(X)=\kappa(F)+\dim Y\) for a very general smooth fiber F, as well as nonzero sections of \(mK_X-f^*H\) for all sufficiently large divisible m.

We prove the rational-boundary log abundance conjecture in every dimension over algebraically closed fields of characteristic zero: every nef ℚ-Cartier log canonical divisor on a projective log canonical pair with effective rational boundary is semiample. Over ℂ, the proof also establishes canonical nonvanishing for smooth projective varieties in every dimension.

We prove the numerical-dimension-one case of log abundance for threefolds over algebraically closed fields of characteristic p > 3. If \((X,B)\) is a projective log canonical threefold pair with effective rational boundary, and \(K_X+B\) is ℚ-Cartier, nef, and of numerical dimension one, then \(K_X+B\) is semiample. Neither terminality nor ℚ-factoriality of X is required.

Let X be a projective ℚ-factorial terminal threefold over an algebraically closed field of characteristic p > 3. We prove the numerical-dimension-one case of abundance: if K_X is nef with \(\nu(K_X)=1\), then K_X is semiample and \(\kappa(X,K_X)=1\).

Let X be a smooth connected compact Kähler manifold, let B be an effective rational simple normal crossing divisor with coefficients less than one, and let M be a nef rational holomorphic line bundle on X. If \(K_X+B\) is pseudo-effective and \(K_X+B+M\) is nef, we prove that its first Chern class in real Bott–Chern cohomology is represented by a semiample rational line bundle. This numerical statement allows a flat change of line bundle; it does not assert semiampleness of the original adjoint.

We prove the Generalised Abundance Conjecture: if \((X,B)\) is a projective klt ℚ-pair over an algebraically closed field of characteristic zero, \(K_X+B\) is pseudo-effective, M is a nef ℚ-Cartier divisor on X, and \(K_X+B+M\) is nef, then \(K_X+B+M\) is numerically equivalent to a semiample ℚ-Cartier divisor on X.

We resolve the numerical-dimension-one case of the minimal-model conjecture for smooth connected complex projective varieties of dimension at least three. If K_X is pseudo-effective and \(\kappa_\sigma(X,K_X)=1\), with κ_σ defined by section growth with a fixed ample twist, then X admits a projective ℚ-factorial terminal minimal model.

We resolve the existence form of the minimal-model conjecture for projective generalized log canonical ℚ-pairs over algebraically closed fields of characteristic zero. Such a pair admits a minimal model when its adjoint divisor is pseudo-effective, and a Mori fibre space otherwise, while keeping the nef ℚ-Cartier b-divisor fixed. Taking the nef part to be zero gives the corresponding result for ordinary log canonical ℚ-pairs.

We prove the ordinary-double-point gap conjecture for boundary-zero complex algebraic klt germs in every dimension: a singular n-dimensional germ has normalized volume at most \(2(n-1)^n\), with equality precisely at an analytic ordinary double point.

We prove that every singular complex algebraic klt fourfold germ with zero boundary has normalized volume at most 162, with equality precisely for an analytic ordinary double point. This resolves the ordinary-double-point volume-gap conjecture in dimension four.

We prove Fujita's freeness conjecture. If X is a smooth complex projective variety of dimension n and L is an ample line bundle, then \(K_X+mL\) is globally generated for every integer \(m\geq n+1\).

Let L be an ample line bundle on a smooth integral projective variety X over an algebraically closed field of positive characteristic, with \(\dim X=n\ge3\). We prove that there is a threshold \(r_0=r_0(X,L)\) such that for every integer \(r\ge r_0\), the ordinary multipoint Seshadri constant at the geometric generic tuple of r points equals the volume bound \((L^n/r)^{1/n}\). The same conclusion holds in dimension two. This establishes the positive-characteristic form of the qualitative Nagata–Biran–Szemberg assertion at geometric generic tuples.

Let L be an ample line bundle on a smooth integral complex projective variety X of dimension n ≥ 3. We prove that there is a threshold \(r_0=r_0(X,L)\) such that for every integer \(r\ge r_0\), the ordinary multipoint Seshadri constant at r very general points equals the volume bound \((L^n/r)^{1/n}\). This establishes the qualitative Nagata–Biran–Szemberg assertion in these dimensions.

We prove that a nonzero effective plane curve of degree d at r ≥ 10 very general complex points has total multiplicity strictly less than \(d\sqrt r\). The inequality holds simultaneously for all curves, including reducible and nonreduced curves, and establishes Nagata's conjecture in its strict, nonhomogeneous form.

We prove that, for every smooth integral complex projective surface S and every ample line bundle L, the multipoint Seshadri constant at r very general points equals \(\sqrt{L^2/r}\) for every sufficiently large integer r. This resolves positively the qualitative Nagata–Biran conjecture for surfaces.

We prove Bloch's conjecture for smooth connected projective complex surfaces with \(p_g=0\): the Albanese homomorphism on integral degree-zero zero-cycles is an isomorphism.

We prove 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.

We prove that the normal projective base of a projective Lagrangian fibration from a compact irreducible holomorphic symplectic Kähler manifold is projective space. This resolves the projective-space base conjecture for such fibrations in every dimension and deformation type.

We prove that every complex K3 surface X is Oka, resolving the K3 Oka conjecture. Equivalently, for every m ≥ 1, every holomorphic map to X from a neighborhood of a compact convex set in ℂ^m can be approximated uniformly on that set by entire maps \(\mathbb C^m\to X\).

We prove the P = W conjecture on the full rational cohomology of fixed-determinant, trace-free Higgs moduli spaces in composite rank and coprime degree, for smooth projective complex curves of genus at least two. Together with the established prime-rank cases, this gives the equality in every coprime rank.

We prove the equivariant cohomological Hikita conjecture for arbitrary finite quivers, including loops and multiple arrows. For any dimension and framing vectors, any commuting flavor torus, and a stability character whose semistable locus is stable and has free gauge action, the equivariant cohomology of the Nakajima variety is canonically isomorphic to the coordinate ring of the scheme-theoretic fixed locus of the stability cocharacter on the flavor-deformed Coulomb branch. This is an isomorphism of graded algebras over the common coefficient ring, retaining nilpotents and including the empty case.

We construct a smooth projective complex fourfold with large fundamental group whose universal cover is not Stein. The universal cover contains no positive-dimensional compact complex-analytic subvariety, so this disproves Shafarevich's holomorphic-convexity conjecture even under the large-fundamental-group hypothesis.

We construct a smooth connected projective complex surface whose universal cover is not holomorphically convex. This disproves the Shafarevich conjecture on holomorphic convexity in complex dimension two.

We construct an explicit integral complex affine fourfold X with \(X\times\mathbb A^1\cong\mathbb A^5\) but \(X\not\cong\mathbb A^4\). This gives a negative answer to Zariski's affine-space cancellation problem over ℂ in dimension four. The same construction disproves the Stable Coordinate Conjecture in ambient dimension five and yields smooth 𝔸³-fibrations over 𝔸¹ and 𝔸² that are not Zariski-locally trivial. These fibrations disprove the Dolgachev–Weisfeiler affine-fibration conjecture over these bases.

We construct a singular normal affine complex surface whose tangent sheaf is free of rank two. This disproves the Lipman–Zariski conjecture in characteristic zero.

We construct an explicit degree-five polynomial over \(\mathbf C\) that is not a coordinate in four variables but becomes one after adjoining a single variable. This gives a counterexample to the stable coordinate conjecture in four variables. Every fiber is isomorphic to affine three-space, yet its embedding in affine four-space is not rectifiable. Thus the example also disproves the Abhyankar–Sathaye embedding conjecture in ambient dimension four.

We construct an explicit counterexample to the Abhyankar–Sathaye conjecture: a noncoordinate polynomial in four complex variables whose zero fibre is affine three-space. Adjoining variables gives counterexamples in every ambient dimension at least four.

We give counterexamples to Griffiths' conjecture in rank two on the quadric surface \(\mathbb P^1\times\mathbb P^1\). We construct an explicit bundle G whose coordinatewise power pullbacks, tensored with \(\mathcal O(1,1)\), are ample for every positive power, but admit no smooth strictly Griffiths-positive Hermitian metric for all sufficiently large powers.

We prove that every compact connected Kähler manifold of positive complex dimension containing no nonconstant entire curve has ample canonical bundle and is projective. This resolves positively Kobayashi's canonical-ampleness conjecture in the smooth compact Kähler category.

We prove the two-summand form of Beauville's compatible splitting conjecture. If the tangent bundle of a compact connected Kähler manifold decomposes into two integrable holomorphic subbundles of positive rank, then its ordinary universal cover admits a product decomposition whose factor tangent bundles are the lifted specified summands.

We prove that both summands of every specified holomorphic splitting of the tangent bundle of a smooth rationally connected projective complex manifold into two positive-rank subbundles are integrable. This proves Höring's conjecture on rationally connected projective manifolds. Höring's product theorem then gives a product decomposition compatible with the specified splitting.

We disprove the Chow and rational-cohomological forms of Pixton's completeness conjecture for his original relation system. We construct a formal tautological class outside the original Pixton relation span whose image is zero in the Chow ring and in rational cohomology. The example has genus \(10^{60}\) and \(3\binom{10^{60}}3\) markings.

For every sufficiently large admissible Hassett discriminant, we prove that a very general cubic fourfold of that discriminant is irrational, although its Kuznetsov component is equivalent to the ordinary derived category of a projective K3 surface. The discriminant threshold is ineffective. This disproves Kuznetsov's rationality conjecture. The same cubics have associated untwisted polarized K3 surfaces in the Hodge-theoretic sense, so they also disprove the sufficiency direction of the associated-K3 rationality prediction.

Let X be a smooth projective complex threefold with trivial canonical bundle. We construct numerical Bridgeland stability conditions with the exact ordinary and square-root-Todd central charges at every sufficiently large volume. One volume threshold works on an open set of real twists and ample directions. The resulting stability conditions have the support property on the full numerical Grothendieck group and stable point sheaves. Separately, on every smooth projective complex threefold we prove a strong tilt inequality above a volume threshold uniform in the object and twist along a fixed polarization.

We prove Toda's normalized quintic Gepner conjecture. On every smooth complex quintic threefold, we construct a numerical Bridgeland stability condition for which tensoring by the hyperplane bundle, followed by the spherical twist at the structure sheaf, increases phase by 2/5.

Every sequence of generalized-canonical flips on compact Kähler fourfolds with rational boundary and fixed rational analytically nef b-divisor terminates when the small morphisms are projective. We prove this for normal globally Weil ℚ-factorial models with boundary coefficients less than one, allowing exceptional log discrepancies equal to one. No scaling rule or pseudo-effectivity assumption is required. The theorem applies to existing projective small diagrams with the specified opposite ample signs.

Every sequence of generalized log canonical flips on a normal irreducible globally Weil ℚ-factorial compact Kähler fourfold terminates, provided the flips are projective small diagrams with the stated opposite ample signs. The boundary is rational, and the nef b-divisor is fixed and represented by an analytically nef ℚ-Cartier divisor on a projective modification. No scaling rule or pseudo-effectivity assumption is required. The theorem concerns existing flip sequences; it does not assert the existence of all contractions or flips.

We prove that every maximal ordinary negative-ray program starting from a compact Kähler klt fourfold pair with effective rational boundary in the global Weil-divisor ℚ-factorial category terminates. It ends at a nef model when the adjoint is pseudo-effective and at a projective Mori fibre space otherwise.

We prove that every permitted minimal model program for a projective log canonical fourfold with rational boundary over an algebraically closed field of characteristic zero terminates. The result allows arbitrary negative extremal rays and mixed birational steps on the given models, without pseudo-effectivity or initial ℚ-factoriality.

We prove that a globally Weil-ℚ-factorial compact Kähler klt fourfold pair with effective rational boundary and canonical rational line bundle admits a finite ordinary minimal model program starting on the given pair. It ends at a nef model when the adjoint class is pseudo-effective and at a Mori fibre space otherwise.

Using the abelianity theorem for special compact Kähler manifolds in every dimension, we prove virtual abelianity of the entire orbifold fundamental group of the special order-two root orbifold associated with a pair \((X,\tfrac12D)\), where X is a smooth projective fourfold and D is a nonempty smooth connected divisor. We construct a special smooth projective eightfold whose fundamental group surjects onto the required group.

We give an independent proof that every complex linear representation of the ordinary fundamental group of a connected smooth special complex quasi-projective variety has virtually nilpotent image of class at most two. This conclusion was previously announced by Cao–Deng–Hacon–Păun. We also construct special open surfaces whose general quasi-Albanese fibres are not special.

We prove that the fundamental group of every special compact Kähler manifold is virtually abelian, resolving Campana's abelianity conjecture in all dimensions. In particular, every smooth compact connected Kähler manifold of Kodaira dimension zero has virtually abelian ordinary fundamental group.

Every nonempty connected semialgebraic bounded open subset of a complex affine variety admitting a properly discontinuous cocompact group of biholomorphisms is smooth and biholomorphic to a bounded symmetric domain. This answers the bounded-domain question of Kollár and Pardon affirmatively.

We prove the Kollár–Pardon conjecture: the universal cover of a connected normal projective complex variety is biholomorphic to a semialgebraic open subset of a projective variety if and only if it is a product of a bounded symmetric domain, a complex affine space, and a simply connected normal projective variety.

We give a negative answer to Zariski's multiplicity question in four complex variables. We construct two reduced convergent holomorphic function germs with isolated critical points and multiplicities four and five whose zero-set germs are ambiently homeomorphic.

We give a negative answer to the embedded Zariski multiplicity conjecture. We construct two reduced hypersurface germs that are ambiently homeomorphic but have multiplicities two and three. Both have isolated singularities and lie in a common complex affine space of dimension divisible by eight. Their defining function germs are also topologically right equivalent, and the same examples answer Arnold's corank problem negatively for ambient topological equivalence.

We prove the Global Spherical Shell conjecture: every connected minimal compact complex surface of class VII with positive second Betti number contains a global spherical shell. This is a holomorphically embedded neighborhood of the standard three-sphere in \(\mathbb C^2\setminus\{0\}\) whose complement is connected.

We resolve the contact-Fano homogeneity conjecture and the Riemannian LeBrun–Salamon conjecture positively. Every smooth connected complex projective contact Fano manifold of complex dimension at least three, with its given contact distribution, is contact-isomorphic to the adjoint variety of a simple complex Lie algebra. Consequently, every closed connected smooth positive quaternionic-Kähler manifold of real dimension \(4m\geq8\) is homothetic to a compact symmetric Wolf space.

We prove the generalized Mukai conjecture: every positive-dimensional smooth complex Fano manifold of dimension n, Picard number ρ, and pseudoindex ι satisfies \(\rho(\iota-1)\le n\). Equality holds precisely for the product of ρ copies of \(\mathbb P^{\iota-1}\).

We prove that every holomorphic one-parameter family of isolated hypersurface singularities in ℂ³ with constant Milnor number is topologically right-trivial. This gives a positive answer to the surface case of the μ-constant problem. The trivialization fixes the parameter and the origin section, and preserves the defining functions.

We prove that full ordinary descendant Virasoro constraints pass from a smooth projective complex base to the projectivization of any algebraic vector bundle of rank at least two. The bundle need not split and satisfies no positivity requirement. Assuming the full constraints on the base, the conclusion includes every genus, each individual integral curve class, and all cohomology insertions, including primitive and odd classes. The result also applies successively to towers of projective bundles.

We prove the Virasoro conjecture for the ordinary descendant Gromov–Witten theory of smooth complete intersections in projective space, in every genus and curve class and with arbitrary cohomology insertions. This includes primitive and odd cohomology classes, with no semisimplicity assumption.

We prove the Cartier-divisor conjecture of Birkar and Shokurov for rational boundaries in characteristic zero and for real boundaries over ℂ. For an ϵ-lc Fano type contraction with positive-dimensional base and nef negative log canonical class, a Cartier divisor through any prescribed base point can be chosen with pullback log canonical threshold bounded below in terms of the dimension and ϵ alone.

For fixed dimension d and positive rational ϵ, we prove that every ϵ-log-canonical Fano contraction over an algebraically closed field of characteristic zero admits, near each closed base point, a klt complement of index bounded only by d and ϵ. Over ℂ, we also obtain monotone klt complements for Fano type pairs with nef anti-log-canonical divisor and coefficients in a fixed finite rational set. This proves the finite-rational-coefficient form of Shokurov's bounded-klt-complement conjecture.

We prove that every smooth connected complex projective Fano sixfold with nef tangent bundle is rational homogeneous, resolving the Campana–Peternell conjecture in complex dimension six. As a consequence, a connected compact Kähler manifold X with nef holomorphic tangent bundle and \(\dim_{\mathbb C}X-\widetilde q(X)\leq6\) has ordinary universal cover \(F\times\mathbb C^{\widetilde q(X)}\), where F is a rational homogeneous manifold and \(\widetilde q(X)\) is the maximal irregularity of a connected finite étale cover.

From a smooth semipositive anticanonical metric on a smooth projective complex variety, we construct an unweighted integrable semipositive metric on a corrected anticanonical line of a smooth base, using a normal equidimensional toroidal model and full generic adjoint rank one at exponent zero. The explicit rational boundary correction pulls back to an exceptional divisor. On varieties without positive-degree holomorphic forms, two-metric transfer gives sections with prescribed boundary poles from a finite-volume log-anticanonical metric and a target line with bounded semipositive weights. In particular, a smooth projective complex variety with a smoothly semipositive anticanonical bundle and no such forms has a nonzero invariant section of a positive anticanonical multiple for every torus action with its natural linearization. On this class of varieties, generic section and adjoint-rank hypotheses give contractions of invariant fibrations that preserve both conditions unless an invariant global section already exists.

For a holomorphic action of a compact torus on a compact complex manifold, we prove that the invariant Euler characteristic of naturally linearized anticanonical powers is polynomial on a divisible progression, with its actual value at exponent zero. Independently, on a smooth projective variety with smoothly semipositive anticanonical bundle, we remove a fixed pseudoeffective error from an unbounded sequence of effective twists. These results convert invariant cohomology into twisted differential forms and prove anticanonical nonvanishing on smooth projective varieties with smoothly semipositive anticanonical bundle, by descending the forms before conversion.

Let Y be smooth projective and let C be an effective integral divisor. If \(-K_Y+C\) admits a metric with a global strict curvature lower bound for which the canonical section of C is locally square integrable, every pseudoeffective rational divisor becomes rationally effective after an effective correction supported on C. For a smooth projective X with smoothly semipositive \(L=-K_X\), sections of \(m_jL-P\) at unbounded positive exponents, with any fixed pseudoeffective Cartier error P, yield a section of a positive multiple of L.

On a smooth connected projective complex variety with smoothly semipositive anticanonical bundle, we prove that asymptotically zero valuation on the curvature-null face is attained by an effective rational anticanonical divisor. The fixed auxiliary line bundle is arbitrary.

Let X be a smooth projective complex variety with smoothly semipositive anticanonical bundle. For any torus linearization of that bundle, we show that invariant sections in unbounded degrees with one fixed negative pseudoeffective error yield an invariant section in a positive untwisted degree. Combining the natural invariant index with compact-monodromy structure, we deduce that every such X has a nonzero section of some positive anticanonical power.

Let \((W,D)\) be a projective ℚ-factorial klt pair with effective rational boundary. We prove that a positive Cartier multiple of \(-(K_W+D)\) has a nonzero section if this divisor is nef, its pullback to a resolution admits a semipositive metric with locally bounded weights, and the resolution has nonzero structure-sheaf Euler characteristic. Consequently a smooth rationally connected projective variety with smoothly semipositive anticanonical bundle has a nonzero naturally invariant anticanonical plurisection for every algebraic torus action.

Every smooth connected projective complex variety with smoothly semipositive anticanonical bundle has a nonzero section of some positive anticanonical power.

We prove the unramified quantum geometric Langlands equivalence for every connected simple complex algebraic group G and every level \(c\in\mathbb C\setminus\mathbb Q\), including non-real levels. For every smooth projective connected complex curve X, it identifies the full twisted D-module categories on \(\mathop{\mathrm{Bun}}\nolimits _G(X)\) and \(\mathop{\mathrm{Bun}}\nolimits _{G^\vee}(X)\) at the dual shifted levels c and \(-1/(rc)\), where r is the lacing number. The equivalence uses the given global forms and includes all connected components.

We prove that a circle domain with conformally removable boundary is conformally rigid, with no restriction on the number of complementary components. This establishes the removability-to-rigidity direction of the He–Schramm Conjecture.

We prove that every domain in the Riemann sphere is conformally equivalent to a circle domain, resolving Koebe's circle-domain conjecture positively.

We prove Brennan's conjecture: if a simply connected plane domain admits a conformal bijection φ onto the unit disk, then \(|\varphi'|^s\) is area-integrable for every \(4/3\lt s\lt 4\). We also prove the sharp inverse-square integral-means exponent \(B_{\mathcal S}(-2)=1\) for the normalized schlicht class \(\mathcal S\).

We prove a uniform upper bound for inverse-first-power integral means of normalized univalent disk maps with exponent strictly below 1/4. Consequently, the bounded universal integral-means spectrum satisfies \(B_b(-1)\lt 1/4\), disproving Kraetzer's conjectured spectrum at p = −1.

We resolve the Falconer distance conjecture in every dimension. For every integer d ≥ 2, a compact subset of ℝ^d with Hausdorff dimension greater than \(d/2\) determines a set of Euclidean distances of positive Lebesgue measure.

We prove the four-dimensional Hausdorff-dimension Kakeya conjecture: every subset of ℝ⁴ containing a unit line segment in every direction has Hausdorff dimension four. No compactness or regularity assumption is imposed on the set or its witnessing line family.

We prove the Kakeya maximal conjecture in three dimensions. For every ε > 0, the maximal average over unit tubes of radius δ maps \(L^3(\mathbb R^3)\) to \(L^3(S^2)\) with norm at most \(C_\varepsilon\delta^{-\varepsilon}\).

We prove that the ordinary symmetric Fourier partial sums of every complex-valued function in \(L\log L\) on the circle converge almost everywhere to the function along the full sequence. This establishes the classical \(L\log L\) sufficiency conjecture.

For every \(\varepsilon\in(0,1)\) and every sufficiently large integer N, there is a polynomial of length N with coefficients in \(\{-1,1\}\) whose modulus lies between \((1-\varepsilon)\sqrt N\) and \((1+\varepsilon)\sqrt N\) everywhere on the unit circle. Thus real Littlewood polynomials can be ultraflat through every sufficiently large integer length. The signs may be chosen separately at each length.

For every η > 0 and every sufficiently large integer N, there is a polynomial with N consecutive coefficients in \(\{-1,1\}\) whose modulus lies between \(\sqrt N/16\) and \((1+\eta)\sqrt N\) everywhere on the unit circle.

We prove that the minimum possible maximum modulus on the unit circle of a polynomial with N consecutive real coefficients in \(\{-1,1\}\) is \((1+o(1))\sqrt N\), as N tends to infinity through all integers. This disproves the real-sign analogue of Erdős's fixed relative-gap conjecture. As a consequence, the largest binary merit factor at length N tends to infinity through all integer lengths, disproving Turyn's conjecture.

We prove the bounded-data Fourier restriction conjecture for the sphere in three dimensions: the Fourier extension operator maps \(L^\infty(S^2)\) boundedly into \(L^p(\mathbb R^3)\) for every p > 3. This is the full conjectured open range, and the threshold p = 3 is sharp.

We prove the diagonal Fourier extension conjecture for compact smooth positively curved surfaces \(\Sigma\subset\mathbb R^3\), including surfaces with smooth boundary. For every \(3\lt p\lt \infty\), the extension operator maps \(L^p(\Sigma)\) boundedly into \(L^p(\mathbb R^3)\). The compact paraboloid case also yields free Schrödinger local smoothing in two spatial dimensions for every \(3\lt p\lt \infty\) and Sobolev order \(s\gt 2-6/p\).

We prove the three-dimensional Bochner–Riesz conjecture in its strict-order formulation. The main result is boundedness of the Bochner–Riesz multipliers on \(L^3(\mathbb R^3)\) for every positive order. Interpolation and duality then give the full conjectured strict range.

We prove the critical L³ local smoothing estimate for the wave equation in three spatial dimensions, with every positive Sobolev loss. This resolves Sogge's Euclidean local smoothing conjecture in dimension three.

We resolve the Sobolev endpoint of Carleson's pointwise convergence problem in every dimension n ≥ 3. For initial data in \(H^{n/(2(n+1))}(\mathbb R^n)\), the free Schrödinger evolution converges almost everywhere to the initial data as \(t\downarrow0\). The evolution is defined by removing Gaussian regularization on one full-measure spatial set, uniformly over the interval \(0\lt t\lt 1\).

We resolve the planar Sobolev endpoint of Carleson's convergence problem: for initial data in \(H^{1/3}(\mathbb R^2)\), the Schrödinger evolution converges almost everywhere to the initial data as \(t\downarrow0\). The evolution is defined by taking the Gaussian regularization limit first; on one full-measure spatial set, this limit exists for every \(0\lt t\lt 1\), and the resulting evolution is continuous at t = 0.

We prove that an n-Ahlfors–David regular Radon measure on ℝ^d is uniformly n-rectifiable if its n-dimensional Riesz transform is uniformly bounded from scalar \(L^2(\mu)\) to vector-valued \(L^2(\mu)\) over all positive hard truncations, for integers d ≥ 4 and \(2\le n\le d-2\). This gives a positive answer to the David–Semmes Riesz-transform question in its remaining higher-codimension range. The conclusion gives uniform big pieces of Lipschitz images of Euclidean balls.

We prove the annular r-variation estimate for the triangular Hilbert transform from complex \(L^3\times L^3\) to L^3/2 for every r > 2. The partitions may depend on the output point and range over all positive scales. The estimate yields the two-endpoint maximal bound and joint almost-everywhere and norm principal values, and resolves the symmetric scalar triangular Hilbert transform problem.

We prove a uniform \(L^3\times L^3\times L^3\) estimate for the dyadic triangular Hilbert form with unrestricted real inputs. The sum of the absolute local contributions over any finite set of scales is bounded by forty times the product of the input norms. In particular, the bound allows coefficients of modulus at most one to vary independently among admissible interval triples.

For arbitrary complex inputs in \(L^3(\mathbb R^2)\), we prove the pointwise maximal \(L^3\times L^3\to L^{3/2}\) estimate for the triangular Hilbert transform, with the supremum over both hard truncation endpoints. The estimate yields joint almost-everywhere and L^3/2 convergence as the lower endpoint tends to zero and the upper endpoint tends to infinity. This also proves the conjectured scalar estimate at the symmetric point.

We prove a uniform L² bound for the Hilbert transform along a Lipschitz unit vector field in the plane, with integration restricted to a fixed absolute multiple of the reciprocal Lipschitz seminorm. The bound is uniform in the inner truncation and holds for fields depending on both coordinates. This gives an affirmative answer to Stein's weak-\((2,2)\) conjecture.

We prove the geometric-progression case of the Erdős similarity conjecture. For every fixed \(q\in(0,1)\) and every \(\eta\in(0,1)\), we construct a compact set \(E_{q,\eta}\subseteq[0,1]\) of measure greater than \(1-\eta\) containing no nontrivial affine copy of \(\{q^n:n\ge1\}\), for any translation and either sign of nonzero dilation. The set may depend on q; the result makes no simultaneous assertion for different ratios.

We construct a compact subset of the unit interval, of measure arbitrarily close to one, that contains no affine copy of the dyadic sequence \(\{2^{-n}:n\ge1\}\). The conclusion holds for every translation and every nonzero real dilation, of either sign, proving the dyadic case of the Erdős similarity conjecture.

We prove the endpoint gradient bound \(\|\nabla Mf\|_1\le C\|\nabla f\|_1\) for every real-valued \(f\in W^{1,1}(\mathbb R^2)\), where \(Mf(x)\) is the supremum of the averages of \(|f|\) over disks centered at x, and C is an absolute constant. The maximal function belongs to \(W^{1,1}_{\mathrm{loc}}(\mathbb R^2)\) and has a globally integrable weak gradient. This gives a positive resolution of the planar centered-disk case of the endpoint question of Hajłasz and Onninen.

We prove that the trilinear Hilbert transform with fixed slopes 1, 2, 3 maps \(L^3(\mathbb R)\times L^3(\mathbb R)\times L^3(\mathbb R)\) to \(L^1(\mathbb R)\), resolving this case of the trilinear Hilbert transform conjecture.

We resolve the symmetric Mahler conjecture positively, including its equality classification. Every origin-symmetric convex body in ℝⁿ has volume product at least \(4^n/n!\), with equality exactly for invertible linear images of Hanner polytopes.

We resolve the Mahler conjecture for general convex bodies positively. For every convex body \(K\subset\mathbb R^n\), n ≥ 1, with Santaló point \(s(K)\), \(|K|\,|(K-s(K))^\circ|\ge (n+1)^{n+1}/(n!)^2\), with equality exactly for simplices.

For every integer n ≥ 2 and every origin-symmetric convex body \(K\subset\mathbb R^n\), we prove that the Gromov width of \(\mathop{\mathrm{int}}\nolimits K\times\mathop{\mathrm{int}}\nolimits K^\circ\) is 4. We construct smooth symplectic embeddings of standard balls of every capacity \(0\lt c\lt 4\) into this polar product. Volume preservation then resolves the symmetric Mahler conjecture positively in every dimension.

We prove that ellipsoids uniquely minimize the volume of the projection body among convex bodies of fixed volume in every dimension at least four. This proves Petty's projection-volume conjecture in these dimensions.

The product of two ten-dimensional simplices has larger normalized projection-body volume than a twenty-dimensional simplex. This gives a counterexample to Brannen's proposed simplex maximum.

We prove that every finite connected planar graph with arbitrary positive real edge lengths embeds into real L₁ with a universal distortion bound. This resolves the planar embedding conjecture positively.

For every fixed treewidth bound, the shortest-path metrics of finite connected graphs with arbitrary positive real edge lengths embed into real L₁ with uniformly bounded distortion. This resolves the bounded-treewidth case of the Gupta–Newman–Rabinovich–Sinclair conjecture positively.

We prove that the density-one triangular lattice minimizes the lower energy per particle for every nonnegative completely monotone function of squared distance, among all locally finite planar configurations of centered disk density one. The comparison includes infinite energies. The proof constructs sharp Gaussian Fourier minorants using an atomic interpolation certificate.

We prove universal energy minimality for the triangular lattice in the plane. Among locally finite configurations of centered density one, it minimizes the lower limit of centered-ball energy averages for every nonnegative completely monotone function of squared distance, including when the energy is infinite. We also prove triangular minimality for planar logarithmic and Riesz renormalized energies, with \(0\lt s\lt 2\) in the Riesz case, and the corresponding jellium minima. The proof uses sharp Gaussian Fourier bounds, positive mixtures, and heat-kernel comparison.

We prove the Sandier–Serfaty conjecture: the triangular lattice of covolume one minimizes planar Coulomb renormalized energy over all admissible curl-free fields with a unit uniform background. Combined with Bétermin and Sandier's asymptotic formula, this also proves the Brauchart–Hardin–Saff conjecture for the linear term of optimal ordered-pair logarithmic energy on the unit two-sphere. The proof uses a direct Voronoi-cell comparison with rigorous interval arithmetic.

We resolve the planar Cohn–Elkies sharpness conjecture: the two-point Fourier bound attains the optimal circle-packing density \(\pi/(2\sqrt3)\). We construct a radial Schwartz certificate and prove its global sign conditions using rigorous interval arithmetic and analytic estimates. The same certificate recovers the classical uniqueness of the triangular packing among periodic equality cases.

We prove the logarithmic Brunn–Minkowski conjecture for arbitrary origin-symmetric convex bodies in every dimension. The theorem also gives the symmetric L_p Brunn–Minkowski inequality for every \(0\lt p\lt 1\). Combined with Saroglou's transfer theorem and a support-subspace reduction, it yields the logarithmic inequality for every even log-concave Radon measure and the scalar-dilation \((B)\)-conjecture.

For every sufficiently large dimension n, we construct a centrally symmetric convex body whose translative covering density exceeds \(c n\log n\), where c > 0 is absolute. This disproves the existence of a universal linear upper bound and matches the order of Rogers' upper bound.

Every convex body in ℝⁿ, n ≥ 2, admits a covering by translates along one full-rank lattice with density at most \(Cn\log n\), for an absolute constant C. No symmetry or boundary regularity is assumed.

We prove that every centered log-concave probability measure with a Lebesgue density on ℝⁿ and linear subgaussian parameter a satisfies \(\mathop{\mathrm{Ent}}\nolimits _\mu(f^2)\le Ca^2\int|Df|^2\,d\mu\) for compactly supported smooth f, with one universal constant C. This resolves positively the dimension-free logarithmic Sobolev conjecture for subgaussian log-concave measures.

For every fixed \(1\lt p\lt \infty\) and D > 1, every n-point subset of a real L_p space embeds into \(\ell_p^d\) with distortion at most D and subpolynomial dimension \(d=n^{o(1)}\). The target has the same exponent p, and the embedding need not be linear.

We prove that not every hyperbolicity cone is a spectrahedral shadow: some closed hyperbolicity cones admit no finite affine semidefinite lift, even with arbitrary real coefficients and any finite number of auxiliary variables. This disproves the Projected Lax Conjecture and hence the generalized Lax conjecture.

We construct an exact semidefinite lift of the explicit nonspectrahedral hyperbolicity cone in twenty-three variables defined in the companion paper. The lift is a homogeneous real symmetric pencil of size 100 with 307 auxiliary variables and represents the entire closed cone, including every point with singular X. Thus, although this cone has no semidefinite representation in its original coordinates, it admits one when auxiliary variables are allowed. The stated sizes are not claimed to be minimal.

We construct a homogeneous polynomial of degree 16 in 23 real variables whose hyperbolicity cone has no representation by a finite homogeneous real symmetric linear matrix inequality. This disproves the geometric Generalized Lax conjecture.

We prove the Gaussian propeller conjecture: for every finite measurable partition of a Euclidean space, the sum of the squared lengths of its Gaussian first moments is at most \(9/(8\pi)\). For dimension at least two and at least three cells, three planar sectors of angle \(2\pi/3\), extended by an orthogonal Euclidean factor, attain the bound.

Every prescribed-order finite sequence of vectors in the Euclidean unit ball of ℝ^d has one signing for which every signed prefix has norm at most \(C\sqrt d\), with C absolute and independent of the sequence length. Consequently, every indexed zero-sum family of unit-ball vectors admits an ordering with the same bound for its unsigned partial sums. This determines the Euclidean Steinitz constant up to absolute factors, \(S_2(d)=\Theta(\sqrt d)\), and resolves the Euclidean Steinitz–Bergström conjecture.

Every infinite-dimensional real Banach space contains a compact doubling subset that admits no bi-Lipschitz embedding into any finite-dimensional real normed space. The doubling constant has a universal bound, independent of the ambient Banach space. This answers the Lang–Plaut problem negatively, even for compact subsets of Hilbert space.

We give two independent constructions of binary words of one length at most d whose ordinary edit-distance metrics require ℓ₁ distortion \(\exp(\Omega(\sqrt{\log d\,\log\log d}))\) for every sufficiently large d. We also prove a constant-distortion binary conversion for one prescribed input length. Together with the companion upper embedding theorem, these lower bounds determine the order of logarithmic distortion uniformly over finite alphabets with at least two symbols.

We give two finite-circle constructions of binary strings whose least ℓ₁ distortion is \(\exp(\Omega(\sqrt{\log d\,\log\log d}))\), where d bounds their length. Both constructions supply words of one common length for every sufficiently large cap. Two direct binary coding arguments transfer the constructions with absolute distortion and logarithmic block width. We also develop the overlapping-substring method of Ostrovsky and Rabani into a complete finite histogram embedding at the same exponential scale, uniformly over all finite alphabets and all words of length at most d, including the empty word.

We determine the exponential scale of the least ℓ₁ distortion of unit-cost edit distance on all strings of length at most d. For every sufficiently large d, uniformly over finite alphabets of size at least two, the distortion lies between \(\exp(c\sqrt{\log d\,\log\log d})\) and \(\exp(C\sqrt{\log d\,\log\log d})\) for absolute constants \(c,C\gt 0\). The lower bound already holds on binary strings of one common length. Thus the order of logarithmic distortion is sharp up to absolute constants.

The two-cylinder covering of a regular tetrahedron can be perturbed to give finite covers with total perpendicular base area strictly below half its minimum projection area. These covers give negative answers to both the half-area question and the directionwise normalized half-bound conjecture. By affine invariance, the directionwise conclusion holds for every nondegenerate tetrahedron.

We prove that radially aligned segment sweeps admit finite cylinder covers with one triangular base for each interval of any tagged partition. As the mesh tends to zero, the total perpendicular base area converges to a weighted parameter area. An explicit application covers every regular tetrahedron with total base area below half its minimum projection area, giving negative answers to the half-area question and the directionwise normalized 1-Codimensional Cylinder Covering Conjecture.

We approximate compact ruled families of segments by finitely many cylinders with square intercept tiles and perpendicular-base area at most their integral projection cost plus any positive error. The velocity field is C¹, and its differential has opposite real eigenvalues whose magnitudes are strictly below the inverse segment half-length; both eigenvalues may vanish. The result includes square-zero differentials with unrestricted shear and fields that pass between the two regimes. It holds for arbitrary compact label sets.

A regular tetrahedron admits a finite cylinder covering with compact triangular perpendicular bases whose total area is less than half its minimum orthogonal projection area. This disproves the half-area cylinder-covering conjecture. By affine invariance, the same construction gives a counterexample to the directionwise normalized half-bound for every nondegenerate tetrahedron.

We prove the strong isotropic constant conjecture: in each dimension, simplices are the unique maximizers of the isotropic constant among convex bodies. We also prove the sharp entropy bound \(h(f)\ge m+\tfrac12\log\det\mathop{\mathrm{Cov}}\nolimits (f)\) for every log-concave probability density on ℝ^m, with equality precisely for invertible affine images of products of one-sided exponential laws.

We prove the Unique Games Conjecture. For every fixed \(\varepsilon,\delta\in(0,1/2)\), we give a deterministic polynomial-time reduction from 3SAT to Unique Games over a fixed finite alphabet, with completeness at least \(1-\varepsilon\) and soundness at most δ.

We prove that minimum Vertex Cover is NP-hard to approximate within every fixed factor below two, even on simple unweighted graphs.

Approximating minimum directed feedback vertex set within any fixed constant factor is NP-hard, even on unweighted digraphs.

For every fixed C > 1, approximating Min-UnCut within factor C is NP-hard, even on simple undirected unweighted graphs.

We prove that approximating Max-Cut on simple unweighted graphs within any fixed factor greater than the Goemans–Williamson constant is NP-hard.

We prove \(\mathsf L=\mathsf{RL}=\mathsf{BPL}\), resolving the derandomization problem for polynomial-time randomized logarithmic space.

We give a uniform deterministic quasipolynomial-time algorithm for finite turn-based stochastic mean-payoff games with signed integer rewards and rational chance-transition probabilities encoded in binary. It computes exactly the vertices of nonnegative value, including value zero, for the expectation of the pathwise liminf mean payoff. The algorithm uses exact rational arithmetic and \(2^{O((\log(L+2))^2)}\) bit operations, where L is the complete binary input length.

We give a uniform deterministic quasipolynomial-time algorithm for mean-payoff parity games. It computes all vertices from which a player can enforce both nonnegative liminf mean payoff and the parity condition, with arbitrary signed binary rewards and unrestricted binary priorities. The running time is \(2^{O((\log(L+2))^2)}\) bit operations, where L is the complete input length.

We give a randomized algorithm that computes the complete zero-threshold winning set of a finite mean-payoff game with arbitrary signed integer edge weights encoded in binary. For total explicit input length L, it uses \(2^{O((\log(L+2))^2)}\) bit operations on every random tape and is correct with probability at least 7/8. A polynomial-time check certifies the winning regions and positional strategies for both players or reports failure. Independent repetition therefore gives an always-correct algorithm with the same expected quasipolynomial bit bound.

We give a deterministic algorithm that computes the complete zero-threshold winning set of a finite mean-payoff game with arbitrary signed integer edge weights encoded in binary. For total explicit input length L, it uses \(2^{O((\log(L+2))^2)}\) bit operations. A reduction also computes the exact rational value at every vertex and globally optimal positional strategies for both players within the same quasipolynomial bound.

We prove the 2-to-1 Games Conjecture with perfect completeness. For every fixed rational \(\delta\in(0,1)\), it is NP-hard to distinguish satisfiable 2-to-1 games from games of value at most δ, with a fixed alphabet and an explicitly listed unweighted multiset of constraints.

We prove that, for every fixed \(0\lt \delta\lt 1/3\), it is NP-hard to distinguish three-colorable graphs from graphs in which every independent set has fewer than δ times the number of vertices. Consequently, for every fixed integer c ≥ 3, finding a proper c-coloring of a three-colorable graph is NP-hard.

We prove that the exponent of matrix multiplication over the complex numbers is at most 9/4.

We prove that the arithmetic exponent of square matrix multiplication over every fixed field satisfies ω < 2.371054886006746. This includes every positive characteristic.

— secondary writeup Over every field of characteristic zero, we prove that the square matrix-multiplication exponent satisfies ω < 2.258, the dual exponent satisfies α > 0.465, and \(\omega(1,0.709,1)\lt 2.092\). The strict square and k = 0.709 rectangular bounds also hold over every field except possibly in one finite set of positive characteristics, in the arithmetic-operation model.

We prove that the complex border determinantal complexity of the \(m\times m\) permanent is \(\Omega(m^3)\), allowing arbitrary affine-linear determinant representations and coefficientwise limits. It also gives cubic lower bounds for exact determinantal complexity and for the numbers of vertices and edges in affine-linear algebraic branching programs, including coefficientwise limits with a fixed vertex or edge budget.

We give a deterministic algorithm that multiplies two n-bit integers in \(O(n(\lg n)^{1-\kappa})\) worst-case time, with \(\kappa=2^{-182}\), on one fixed finite-alphabet Turing machine with a fixed finite number of one-dimensional tapes. The algorithm is exact for every input length and disproves the \(n\log n\) optimality conjecture of Schönhage and Strassen in this model.

We construct a uniform randomized k-server algorithm on finite rational metrics with competitive ratio \(O(\log^2(k+1))\) against oblivious request sequences. Preprocessing is polynomial in the input length, and per-request bit complexity is polynomial in that length and the binary request-counter length. The additive movement constant is finite and instance-dependent, but may be enormous. The construction uses the companion squared-logarithmic existence theorem.

We prove that randomized k-server has competitive ratio \(O((\log(k+1))^2)\) on every metric space against oblivious request sequences, matching the known worst-case lower bound. For each metric and initial configuration, one policy works for all finite request sequences, including on infinite and unbounded spaces. When the initial server positions are distinct, no additive term is needed.

We prove that one independent sample per element suffices for a constant-competitive prophet inequality on every finite matroid. The guarantee holds even when the arrival-order adversary observes all samples, all online values, and the algorithm's entire random seed. The rule needs no description of the value distributions and achieves the absolute competitive ratio \(2^{-310}\).

We construct a language in deterministic polynomial time whose n-bit membership function requires \(2^{\omega(\sqrt n)}\) gates in an unbounded-fan-in OR–AND–OR circuit. The bound holds at every sufficiently large input length and counts all gates, including the bottom layer.

We prove the perfect-matching entropy conjecture of Anari, Oveis Gharan, and Vinzant. For every feasible vector x of perfect-matching edge marginals in a loopless labelled multigraph on \(2m\ge2\) vertices, the maximum entropy \(H(x)\) of a matching law with marginals x satisfies \(\displaystyle F(x)-(2-2/m)B(x)\le H(x)\le F(x),\) where \(F(x)=-\sum_e x_e\log x_e\) and \(B(x)=-\sum_e(1-x_e)\log(1-x_e)\). This bound holds throughout the polytope, including its boundary. We also prove the sharp bound \(|\mathop{\mathrm{supp}}\nolimits x|-\dim F_x\le3m-2\), where F_x is the minimal face of the perfect-matching polytope containing x.

We give a fully polynomial randomized approximation scheme (FPRAS) for counting perfect matchings in arbitrary finite simple undirected graphs, resolving the general-graph perfect-matching approximation problem. The algorithm returns zero with certainty when no perfect matching exists. Otherwise, it achieves relative error ε with failure probability at most δ in worst-case bit time polynomial in the input length, \(\varepsilon ^{-1}\), and \(\log\delta^{-1}\).

We give a fully polynomial randomized approximation scheme for counting common integer bases of two polymatroids with the same total rank, supplied by exact rank-value oracles. The total rank and capacities are encoded in binary, and each integer vector is counted once. On every execution, the number of oracle calls and the bit work outside the oracles are bounded by a fixed polynomial in the ground-set size, the binary input length, the inverse relative-error tolerance, and the logarithm of the inverse failure probability. The algorithm handles binary capacities directly, without expanding them into labelled copies.

We give a fully polynomial randomized approximation scheme for counting the common bases of two arbitrary matroids of the same rank, supplied by independence oracles. The algorithm requires no explicit representation of either matroid and has polynomial bounds on both oracle calls and bit operations on every execution.

We give an exact uniform sampler for nonnegative integer contingency tables with arbitrary prescribed margins. It terminates almost surely and has expected bit complexity polynomial in both dimensions and the binary length of the margins. No positivity, balance, sparsity, or fixed-dimension assumption is required.

We give a fully polynomial randomized approximation scheme for counting nonnegative integer matrices with prescribed row sums, column sums, and individual entry bounds. Both dimensions vary, all numerical data are encoded in binary, and zero bounds are allowed. The algorithm uses only unbiased random bits and has a polynomial bound on its bit operations on every execution.

We construct a single matrix substitution that detects every nonzero size-s division-free noncommutative formula in n variables over every field of a given positive characteristic. One deterministic machine, given a promised prime p in binary and n, s in unary, outputs matrices over 𝔽_p of dimension \(O(n^3s^6)\) in polynomial bit time. The same tuple works with arbitrary extension-field coefficients, including in characteristic two. The construction also applies to the stated acyclic algebraic path programs.

We construct, in deterministic polynomial bit time, a polynomial-size list of rational matrix tuples for noncommutative rational formulas over ℚ of bounded tree size. Every nonzero admissible formula has a defined, invertible value at one tuple, with no separate bounds on inverse nesting or rational constant heights. Matrix dimensions, entry bit lengths, and total output length are polynomially bounded.

We construct, in deterministic polynomial bit time, one tuple of rational matrices that detects every nonzero polynomial computed by a noncommutative division-free formula of a prescribed size. The matrices have dimension \(O(ns^2)\) for n variables and formula size s, and the same tuple works over every field of characteristic zero.

We construct uniform sparsest-cut instances whose Goemans–Linial semidefinite integrality gap is at least \(c\sqrt{\log n}/(\log\log n)^3\) along a sequence \(n\to\infty\). The demand is one between every pair of distinct vertices, and the capacities are nonnegative real numbers. This matches the Arora–Rao–Vazirani upper bound up to a power of \(\log\log n\).

We prove that, for every fixed C > 1, approximating Uniform Sparsest Cut within factor C is NP-hard. The output graphs have nonnegative rational capacities and unit demand between every pair of distinct vertices.

We disprove the Modified Integer Round-Up Conjecture for bin packing by constructing instances with arbitrarily large additive gaps between the configuration-LP value and the integral optimum. We also prove that, for every fixed nonnegative integer c, distinguishing instances that fit in B bins from those requiring more than \(B+c\) bins is NP-hard. Both results hold with rational item sizes greater than 1/6, so each bin contains at most five items.

We prove sharp contraction of the information carried by a binary channel under independent symmetric noise on a uniform discrete cube. At fixed initial information, a noisy coordinate retains the most information. The Boolean specialization resolves the Courtade–Kumar conjecture and gives an output-entropy refinement. We also establish a stronger mean-dependent entropy-production bound. The proof combines an explicit three-point optimizer for the local joining problem, two entropy capacities, and a common-output thinning inequality, followed by dimension induction and integration along the noise semigroup.

We prove the Hellinger conjecture for Boolean functions on the uniform discrete cube, with arbitrary output bias. For a Boolean function of mean m and every \(\rho\in[-1,1]\), the loss \(\sqrt{1-m^2}-\mathbb E\sqrt{1-(T_\rho f)^2}\) is at most \(1-\sqrt{1-\rho^2}\), with equality for signed coordinates. The proof combines asymmetric dimension induction, a calibrated noise-semigroup energy estimate, and finite exact arithmetic certificates. The Hellinger inequality also yields the Courtade–Kumar information bound.

We prove that maximum-cardinality matching in a simple undirected graph with n vertices and m edges can be found by one uniform randomized algorithm in \((n+m)^{1+o(1)}\) time. The bound holds on every computation path in a logarithmic-word model, and the algorithm returns an explicit maximum matching with probability at least 2/3. An explicit reduction gives the same time and probability guarantees for deciding whether a simple host graph has a spanning subgraph with prescribed valid vertex degrees, and for finding one when it exists.

We give a uniform randomized approximation scheme for unit-cost edit distance. For every fixed rational \(\varepsilon\in(0,1)\), it estimates the distance between arbitrary explicitly stored strings of total length N within a factor \(1+\varepsilon\) with probability at least 2/3, in worst-case expected time \(N^{1+o(1)}\) on a logarithmic-word RAM. The algorithm supports polynomially bounded integer alphabets and returns zero deterministically on equal strings.

We give a uniform algorithm that reconstructs every binary string from independent deletion traces when its length and rational retention probability are known. For each fixed retention probability, both the number of traces and the bit complexity are quasipolynomial in the string length. More generally, we give an explicit sample bound uniform over all rational retention probabilities, with running time polynomial in the sample budget and the binary input length. If the deletion probability is at most \(n^{-\varepsilon}\) for fixed ε > 0, the sample and running-time bounds are polynomial. Reconstruction succeeds with probability at least 2/3 for each input string.

We give an improved worst-case sample bound for reconstructing a string from independent deletion traces, with its length and retention probability known. For each fixed retention probability, the number of traces is quasipolynomial: the logarithm of the sample budget is \(O((\log n)^3(1+\log\log(2n))^6)\). If the deletion probability is at most \(n^{-\varepsilon}\) for fixed ε > 0, polynomially many traces suffice. These bounds apply to binary strings and to strings of general symbols observed exactly. They concern sample complexity and do not assert an efficient reconstruction algorithm or matching optimality.

Exact worst-case reconstruction of a binary word from independent deletion traces requires \(n^{\Omega(\log\log n)}\) samples for every fixed deletion probability \(q\in(0,1)\), even with unrestricted computation and any fixed positive success probability. This gives a negative answer to the polynomial-sample question for binary trace reconstruction. More generally, when \(q^3\log n\to\infty\), we prove a lower bound of \(n^{c\log(q^3\log n)}\) samples for every fixed \(0\lt c\lt 1/(4\log2)\). Here q is the known deletion probability, and all logarithms are natural.

We give a uniform deterministic polynomial-time algorithm for scheduling unit-length jobs with arbitrary precedence constraints on three identical parallel machines. The algorithm constructs a schedule of minimum makespan and decides exactly whether all jobs can finish by a specified deadline. The proof reorganizes feasible schedules into intervals whose job sets have descriptions of bounded size. A dynamic program searches a family containing polynomially many such descriptions. Global boundary conditions and simplification of inherited information keep the descriptions bounded throughout the decomposition.

For every fixed ε > 0, we give a deterministic polynomial-time \((1+2/e+\varepsilon)\)-approximation for finite rational metric k-median with specified candidate facilities, opening at most k facilities. Under \(P\ne NP\), the infimum approximation factor in this model is therefore \(1+2/e\).

We give an exact-budget recovery algorithm for metric k-median with single-exponential dependence on the number of comparison clusters without accurate, distinct proxies in a supplied anchor solution. On positive integral metrics of polynomially bounded diameter, a sufficiently small total proxy error and logarithmically many such clusters yield a \((1+2/e+\varepsilon)\) approximation in polynomial time with arbitrarily high success probability. We also prove bounded-price strictness for one compatible execution of the logarithmic-surplus construction. Together the recovery and payment arguments give a randomized \((2-\sigma)\) approximation, for an absolute σ > 0, on arbitrary finite rational metrics, both with high probability and in expectation, while opening at most k facilities on every output.

For every fixed \(0\lt \rho\lt 1\), the matrix indexed by odd vertex sets U and perfect matchings M of K_n, with entries \(|M\cap\delta(U)|-1+\rho\), has real positive semidefinite rank \(2^{\Omega(n)}\) as even n tends to infinity. Here \(\delta(U)\) is the edge cut of U. Consequently, every exact semidefinite lift of the perfect matching polytope has exponential size.

For every n ≥ 1 and \(1\le d\le n\), we prove that a polynomial threshold function of degree at most d on the uniform Boolean cube has average sensitivity at most \(8d\sqrt n\). This proves the asymptotic form of the Gotsman–Linial conjecture. The bound is uniform in both parameters and uses the convention \(\mathop{\mathrm{sgn}}\nolimits (0)=1\).

We give a deterministic algorithm that, for every finite family of explicitly represented ordinary strings, outputs a common superstring of length at most twice the optimum in time polynomial in the total encoded input size, including symbol labels. The guarantee applies to the algorithm constructed here, not the classical maximum-overlap Greedy procedure.

One-way liveness on h points accepts a word of binary relations when their ordered product is nonempty. For every h ≥ 2, it has a nondeterministic automaton with \(h+3\) states and no left moves, whereas every equivalent s-state two-way deterministic automaton satisfies \(4(s+2)^2\ge2^{\lfloor(h-2)/31\rfloor}\). Partial transition rules, stay moves, and nonaccepting infinite computations are allowed. The alphabets are finite and grow with h, so the result rules out an alphabet-independent polynomial state bound for deterministic two-way simulation, already for one-way nondeterministic sources.

We prove that two-way nondeterministic finite automata cannot be complemented with a polynomial number of states independent of the alphabet. For each n ≥ 4 we construct an n-state automaton over a finite alphabet whose complement requires at least \(\tfrac12 2^{\lfloor(n-4)/127\rfloor}-1\) states.

We construct exact nonuniform Fourier circuits of size \(o(n\log n)\) along an unbounded sequence of lengths, counting every addition, subtraction, and scalar multiplication. This refutes the \(\Omega(n\log n)\) lower bound in the unrestricted complex linear-circuit model. The construction uses a finite tensor saving: a tensor power of some invertible nonmonomial complex matrix can be computed with fewer matrix calls than the standard tensor-axis algorithm on the same coordinates, when invertible monomial maps are allowed freely between calls.

We give a deterministic algorithm that computes the discrete Fourier transform at every length n in \(O(n(\log n)^{1-10^{-13}})\) operations. The model uses exact complex arithmetic, unrestricted coefficients, specified Fourier roots, and unit-cost logarithmic-size indexing; scalar preparation and array organization are included.

We prove the simple-undirected form of the Kannan–Tetali–Vempala conjecture: the switch chain on simple undirected graphs mixes in polynomial time for every graphical degree sequence. For a lazy chain that proposes switches uniformly on four vertices, the total-variation mixing time at distance 1/4 is at most \(2n^8\). We also give an exactly uniform sampler for every graphical labeled degree vector. It uses unbiased random bits, terminates almost surely, and has expected polynomial bit running time.

We disprove the quadratic strengthening of the Sensitivity Conjecture by constructing nonconstant total Boolean functions whose block sensitivity grows faster than any constant multiple of sensitivity squared. In fact, for some fixed α > 2, our examples have unbounded block sensitivity and satisfy \(\mathop{\mathrm{bs}}\nolimits (f)\ge s(f)^\alpha\).

Deciding joint-update k-dimensional Weisfeiler–Leman equivalence is \(\mathsf{EXPTIME}\)-complete when the two graphs are given by explicit adjacency matrices and k ≥ 2 is encoded in binary. The result holds even for connected simple uncolored graphs of equal positive order and maximum degree at most three.

For every sufficiently large fixed k, deciding whether two n-vertex graphs are k-Weisfeiler–Leman equivalent requires \(n^{\Omega(k)}\) deterministic sequential time in the worst case. The bound holds at every sufficiently large graph order, even for simple connected uncolored graphs of diameter at most two, without a complexity assumption. Inputs are explicit adjacency matrices; the models are multitape Turing machines and sequential logarithmic-word RAMs with fixed polynomial-bit-time instructions. Both joint and separate replacement conventions are covered.

We prove that deciding whether Weisfeiler–Leman refinement of an input dimension identifies a given graph is EXPTIME-complete. The input is a nonempty finite simple uncolored graph in adjacency-matrix form and a positive binary-encoded dimension. Identification quantifies over every comparison graph.

For k ≥ 4, we construct two uncolored graphs that are k-dimensional Weisfeiler–Leman equivalent exactly when a prescribed finite-domain choice system has no compatible choice. The system has \(k+1\) domains for joint refinement and k for separate-coordinate refinement. A successful choice is detected after two joint rounds or one separate round. Applied to sparse satisfiability, the reduction gives fixed-dimension \(n^{\Omega(k)}\) time exclusions under positive-rate ETH, even for deciding equality of these early histograms.

Every regular language over a finite alphabet has generalized star height at most three, with complement taken in the same free monoid. We express finite-monoid computations using a prefix code of word pieces.

Every regular language over a finite alphabet has generalized star height at most four over that same alphabet. We give a complete construction using an affine correction that hides one interval product, a finite clock, and several scales for moving boundaries through periodic words.

Every regular language over a finite alphabet has a generalized regular expression of star height at most thirteen over that same alphabet. We prove this uniform bound by representing finite monoid computations as affine updates and recovering them through twelve successive split constructions.

Let \(\mathop{\mathrm{IMM}}\nolimits _{n,n}\) be the \((1,1)\) entry of a product of n independent \(n\times n\) matrices of variables. Over every field of characteristic zero, every syntactically homogeneous \(\Sigma\Pi\Sigma\Pi\Sigma\) circuit computing \(\mathop{\mathrm{IMM}}\nolimits _{n,n}\) has at least \(n^{\sqrt n/400}\) gates for all sufficiently large n, with an absolute threshold independent of the field. Bottom linear forms may have arbitrary support, and arbitrary finite fan-in, fan-out, and gate sharing are allowed. Over every field, a block expansion gives such circuits with at most \(n^{\sqrt n+4}\) gates for n ≥ 2. Thus the gate complexity over characteristic-zero fields is \(n^{\Theta(\sqrt n)}\).

We prove the quasilinear-size PCP-for-PPAD conjecture of Babichenko, Papadimitriou, and Rubinstein. There are fixed positive rational constants ε and δ and a deterministic polynomial-time reduction that transforms an End-of-Line instance of binary length N into a generalized circuit of total binary length \(N(\log N)^{O(1)}\). From any rational assignment of polynomial encoding length that ε-satisfies all but a δ fraction of the gates, a solution to the original End-of-Line instance can be recovered in polynomial time, regardless of which gates fail. Such assignments always exist, with one fixed polynomial bound on their encoding length.

We show that a fixed deterministic Turing machine with one writable tape and head can be simulated in \(O(T^{2/5}\mathop{\mathrm{polylog}}\nolimits (T+2))\) work-space bits when a binary time cap T ≥ 2 is supplied. The simulator computes the finite-control and halting outcome by time T; its running time is unrestricted. The result allows a fixed number of read-only input heads and requires a fixed accessor that supplies every initial writable and read-only symbol within distance T of the relevant head origin in polylogarithmic space. This improves the square-root space exponent for one-tape machines, answering Williams's question for this model.

We give a uniform randomized classical algorithm for Subset Sum with bounded error and worst-case running time \(O(2^{0.49n})\) on polynomial-bit inputs in a word-RAM model, where n is the number of input integers. The time bound holds on every random execution.

We give a uniform classical randomized decision algorithm for worst-case Subset Sum. Under every fixed polynomial bound on input-integer bit length, it uses \(\mathop{\mathrm{poly}}\nolimits (n)2^{n/2}\) time and ordinary \(O(2^{n/5})\) writable words of \(O(n+b)\) bits, where b is the largest input bit length. Both resource bounds hold on every execution. The error is one-sided: the algorithm always rejects unsolvable instances and accepts each solvable instance with probability at least 2/3.

For every fixed ε > 0, we give a sampling algorithm using at most \(C_\varepsilon d^\varepsilon\) exact first-order queries on every execution for C² potentials on ℝ^d with a known minimizer and Hessian between I_d and \(2I_d\). The output has total-variation distance at most 1/10 from the target Gibbs law. Computation between queries is unrestricted. We also prove an \(\Omega(\log d)\) query lower bound for arbitrary randomized adaptive algorithms, determining the optimal dimension exponent to be zero.

Let a finite-state streaming learner estimate a uniformly random unit vector from independent exact Gaussian linear measurements. We prove that \(o(d^2)\) bits of persistent memory and angular success probability at least 2/3 require at least \(2^{-16}d\log_2(1/\epsilon)\) samples for all sufficiently large d, uniformly for \(0\lt \epsilon\le1/10\). The proof conditions each batch on its observed projection and controls the resulting random residual subsphere.

Replacing the Gaussian rows used to select a finite message by independent rows increases the remaining conditional information by at most \(Cd\), for a uniform spherical signal, message entropy at most d², and the specified row dimensions proportional to d. As an application, we prove that learners with \(M=o(d^2)\) persistent bits need \(T=\Omega(d\log(1/\epsilon))\) exact observations to attain uniform-sphere angular success at least 3/5, for \(0\lt \epsilon\le1/10\) and a deterministic finite horizon.

We prove moment estimates for exact random projections of finite measures whose mass is controlled on Euclidean balls, and derive positive domination by countable sums of spherical cap measures. For learners with \(M=o(d^2)\) bits of memory, these estimates give three proofs that uniform-sphere average success at least 2/3 at angular accuracy \(0\lt \epsilon\le1/10\) requires \(\Omega(d\log(1/\epsilon))\) noiseless Gaussian observations. The three proofs keep their different stopping and accuracy costs explicit.

For a signal with density bounded by L relative to uniform probability on \(S^{d-1}\), we bound the information in a finite message W formed from exact Gaussian measurements, conditional on an independent projection revealed only to the analyst. For explicit row counts proportional to d, the bound is \(O(H(W)/d+d+\log(2+\log L))\). Consequently, a finite-state learner with \(o(d^2)\) persistent bits and a deterministic sample horizon needs \(\Omega(d\log(1/\epsilon))\) fresh noiseless Gaussian measurements for constant-probability angular accuracy \(0\lt \epsilon\le1/10\) under the uniform spherical prior.

For every fixed A > 0, a learner that retains at most \(Ad^2\) bits between fresh exact Gaussian linear measurements needs \(\Omega_A(d\log(1/\epsilon))\) measurements to estimate a uniformly random unit vector to angular error at most ϵ, for any \(0\lt \epsilon\le 1/10\), with probability at least 2/3. The constant is absolute for \(o(d^2)\) memory.

For the image of a uniform cube under a spherical coordinate map, we prove that finite messages from t blocks of \(\Theta(d)\) exact Gaussian measurements reveal only \(O_A(dt)\) information when each message has at most \(\exp(Ad^2)\) values, for fixed A. The same bound holds when each message is supplemented with a nested cell that restores the required geometric spread.

We prove that the existential theory of the reals lies in the counting hierarchy. More generally, we show that the truth of existential–universal sentences over the reals can be decided in a fixed level of the counting hierarchy, even when the integer polynomials are given by arithmetic circuits.

We give a uniform deterministic polynomial-time algorithm for complete factorization over prime fields. For a prime p in binary and a nonzero polynomial \(f\in\mathbf F_p[x]\) given by its dense coefficient list, the algorithm computes the irreducible factors and their multiplicities using a number of bit operations polynomial in \((\deg f+1)\log p\). The proof uses the uniform Hecke zero-free theorem from the companion paper Primitive roots for every admissible integer base.

For every degree, we prove that the number of isolated periodic orbits of a real planar polynomial vector field is bounded by a finite constant depending only on that degree. This establishes the uniform boundedness assertion in the second part of Hilbert's sixteenth problem. The proof uses separation of asymptotic expansions on nested complex domains and a finite-dimensional counting argument.

Every classical Liénard system \(\dot x=y-F(x)\), \(\dot y=-x\), with F an arbitrary real polynomial of degree at most five, has at most two geometrically distinct isolated periodic orbits, and the bound is attained. This proves the degree-five case of the Lins Neto–de Melo–Pugh conjecture.

We solve the probability-preserving form of Banach's simple Lebesgue-spectrum problem in smooth dynamics. We construct a C^∞ diffeomorphism of the three-torus that preserves standard volume and whose Koopman operator has simple Lebesgue spectrum on the entire mean-zero L² space.

Every invertible mixing probability-preserving transformation is mixing of every finite order. Thus Rokhlin's multiple-mixing problem for a single transformation has an affirmative answer.

We prove that the standard sine map of the two-dimensional torus has positive metric entropy with respect to normalized area for every sufficiently large positive parameter. This gives a full parameter tail, and hence answers Sinai's positive-parameter-measure conjecture affirmatively.

We prove that a smooth strictly convex planar billiard with positive curvature is an ellipse whenever a full neighborhood of its boundary is continuously foliated by smooth closed convex caustics. No differentiability across the leaves is assumed. This resolves the near-boundary Birkhoff conjecture in the stated smooth class.

For a smooth strictly convex planar billiard of positive curvature, a continuous foliation of a full grazing annulus by individually invariant essential curves forces an analytic boundary and a jointly analytic collar of smooth strictly convex caustics. The physical-collar rigidity theorem proved in the companion article then implies that the table is an ellipse.

We prove that the Hausdorff dimension of every finite real self-similar measure equals the minimum of one and its random-walk entropy rate divided by its Lyapunov exponent. Exact overlaps are allowed, and the contraction ratios may be unequal and negative. This resolves the entropy-rate dimension conjecture.

We prove the permanence conjecture for finite weakly reversible mass-action systems with fixed positive reaction rates. Every positive stoichiometric compatibility class admits one compact convex forward-invariant set that every positive trajectory in that class enters in finite time, even when the class is unbounded. The set depends only on the network, the rates, and the class; the entry time may depend on the initial state. Thus, after its entry time, every concentration satisfies the same positive lower and finite upper bounds throughout that class.

We prove the boundedness and persistence conjectures for finite weakly reversible mass-action systems with positive constant reaction rates. For every positive initial condition, the solution exists for all forward time, and every concentration remains bounded above and bounded away from zero. The bounds may depend on the initial condition, and no boundedness assumption is imposed on its stoichiometric compatibility class.

We prove that the unit-speed billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to π is weakly mixing for normalized Liouville measure. Equivalently, the product of the flow with itself is ergodic.

We prove that the unit-speed billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to π is ergodic for normalized area times uniform angular measure. No genericity or Diophantine condition is required. The flow is considered outside the null set of trajectories that hit a vertex.

We disprove the general C¹ self-map formulation of Shub's entropy conjecture. We construct a noninvertible C¹ self-map of a compact smooth manifold without boundary whose topological entropy is zero, while its action on second real homology has eigenvalue 2. Thus the topological entropy is strictly smaller than the logarithm of the homological spectral radius.

We construct an ergodic invertible transformation of a standard nonatomic probability space with zero Kolmogorov–Sinai entropy that has no smooth positive-volume model. More precisely, it is not measurably conjugate to any C^∞ diffeomorphism preserving a strictly positive smooth probability density on a compact finite-dimensional manifold. A single example excludes every finite dimension, including models on nonorientable manifolds and manifolds with smooth boundary.

We give an arithmetic classification of singular and absolutely continuous unbiased Bernoulli convolutions for every parameter \(\lambda\in(0,1)\). The criterion is expressed through one-sided approximation by explicitly defined finite sets of algebraic units; it is an infinite approximation condition, not a finite membership algorithm. We also establish singular examples beyond reciprocals of Pisot numbers: singularity holds at the reciprocal of every quartic Salem number in \((1,2)\) and at the reciprocal of a specified non-Pisot root of an explicit degree-31 polynomial.

For every invertible mixing probability-preserving transformation, triple ergodic averages of bounded measurable functions along any three pairwise distinct nonzero integer slopes converge almost everywhere to the product of their integrals. The slopes may be positive or negative.

We prove pointwise convergence of triple ergodic averages for every invertible mixing probability preserving transformation T of an arbitrary probability space \((X,\mathcal F,\mu)\), with measurable inverse. For every triple of bounded measurable functions \(f_1,f_2,f_3:X\to\mathbb C\), \(\displaystyle \frac1N\sum_{n=1}^N f_1(T^nx)f_2(T^{2n}x)f_3(T^{3n}x) \longrightarrow \prod_{j=1}^3\int_X f_j\,d\mu\) for μ-almost every x as \(N\to\infty\) through all positive integers.

We prove that fourfold ergodic averages along the times \(n,2n,3n,4n\) converge almost everywhere to the product of the integrals for every invertible, bimeasurable, mixing probability-preserving transformation and every fixed choice of bounded measurable inputs. Convergence holds along all positive integer averaging lengths. No quantitative mixing rate is assumed, and the probability space need not be standard.

Let T be an invertible mixing probability-preserving transformation. For every integer n ≥ 2 and every fixed tuple of bounded measurable functions, we prove that the consecutive multiple ergodic averages of length n converge almost everywhere to the product of the integrals, as the averaging length tends to infinity through all positive integers. The probability space need not be standard, and no rate of mixing is required.

We construct a finite translational tile in ℤ³ that admits tilings but no fully periodic tiling. Its unit-cube thickening has the same property in ℝ³, even when arbitrary real translations are allowed. This gives a negative resolution of the periodic tiling conjecture in dimension three.

The compact set of rank-one orthogonal projectors on ℝ⁴, with the Frobenius metric, cannot be covered by ten sets of strictly smaller diameter. It therefore gives a counterexample to Borsuk's conjecture in dimension nine.

We prove that every finite nonempty graph G satisfies \(\chi_{\mathrm{list}}(G)\le C h(G)\) for an absolute integer C, where \(h(G)\) is the largest order of a clique minor. This resolves the Linear List Hadwiger conjecture affirmatively.

We disprove the Colin de Verdière chromatic conjecture by constructing graphs whose chromatic number exceeds their Colin de Verdière invariant by more than one. The examples have independence number at most two. In fact, their ordinary fractional chromatic number also exceeds their Colin de Verdière invariant by more than one.

We disprove Hadwiger's conjecture by constructing arbitrarily large graphs whose chromatic number exceeds their Hadwiger number. The examples have independence number at most two, and even their ordinary fractional chromatic number exceeds their Hadwiger number. Thus they also disprove the fractional-coloring weakening discussed by Reed and Seymour.

We prove that every coloring of the Euclidean plane with five colors has a monochromatic unit-distance pair, with no regularity assumption on the color classes. Consequently, the chromatic number of the plane is either six or seven.

We prove Erdős's conjecture that every set of positive integers with divergent reciprocal sum contains arithmetic progressions of every finite length. More quantitatively, for every fixed k ≥ 3, we show \(\displaystyle r_k(N)\le C_kN\exp\bigl(-c_k(\log N)^{\varepsilon_k}\bigr)\) with \(C_k,c_k,\varepsilon_k\gt 0\), where \(r_k(N)\) is the largest size of a subset of \(\{1,\ldots,N\}\) with no nonconstant k-term arithmetic progression.

We prove that there are absolute constants c > 0 and K₀ such that \(W_r(k)\gt k^{ck\lfloor\log_2 r\rfloor}\) for every \(k\ge K_0\) and r ≥ 2. Consequently \(W_r(k)^{1/k}\to\infty\) for each fixed r ≥ 2, giving a quantitative positive resolution of Erdős's superexponential-growth question, including the two-color case.

We disprove Sidorenko's conjecture with a bipartite graph on 35 vertices and 66 edges: its homomorphism density in some finite simple graph is smaller than the conjectured lower bound. The same connected graph also disproves the forcing conjecture: at one fixed density, asymptotically matching the edge and pattern densities does not imply quasirandomness.

For every sufficiently large prime q, we construct a finite intersecting \((q+1)\)-partite \((q+1)\)-uniform hypergraph with covering number \(q+1\) and exactly \(q+1\) nonisolated vertices in each part. This disproves Ryser's covering conjecture, even for intersecting hypergraphs with equally sized parts.

For every sufficiently large prime \(s\equiv2\pmod3\) and every sufficiently large odd integer n, with the threshold depending on s, we construct an intersecting \((s^n+1)\)-partite \((s^n+1)\)-uniform hypergraph with covering number \(s^n+1\). This disproves Ryser's covering conjecture in its intersecting case.

We prove Hindman's finite sums and products conjecture: for every finite coloring of the positive integers and every positive integer k, there is a k-element set whose nonempty subset sums and nonempty subset products all have the same color.

We prove the Harary–Hill conjecture: for every positive integer n, the ordinary crossing number of the complete graph K_n is \(\displaystyle \frac14\left\lfloor\frac n2\right\rfloor \left\lfloor\frac{n-1}{2}\right\rfloor \left\lfloor\frac{n-2}{2}\right\rfloor \left\lfloor\frac{n-3}{2}\right\rfloor.\)

We prove the Zarankiewicz crossing-number conjecture, resolving Turán's brickyard problem. For all positive integers m, n, the ordinary crossing number of the complete bipartite graph \(K_{m,n}\) is \(\displaystyle \left\lfloor\frac m2\right\rfloor \left\lfloor\frac{m-1}{2}\right\rfloor \left\lfloor\frac n2\right\rfloor \left\lfloor\frac{n-1}{2}\right\rfloor.\)

For every fixed integer d ≥ 3, we prove that every set of n ≥ 2 distinct points in ℝ^d determines at least \(c_d n^{2/d}\) distinct distances, where \(c_d\gt 0\) depends only on d. This resolves the higher-dimensional Erdős distinct-distances conjecture positively.

We prove the weak pinned Erdős distinct-distance conjecture. For every fixed ε > 0, all but \(o(n)\) points of any n-point planar set determine at least \(n^{1-\varepsilon}\) distinct nonzero distances.

We prove a power saving for the planar unit-distance problem: for some absolute \(\beta\lt 4/3\), every set of n points in the Euclidean plane determines \(O(n^\beta)\) unordered pairs at unit distance.

We prove that an isomorphism of Bruhat intervals in arbitrary Coxeter systems preserves their equal-parameter Kazhdan–Lusztig polynomials. This resolves the full combinatorial invariance conjecture positively.

We prove that the chromatic quasisymmetric function of every natural unit interval graph is elementary-positive over \(\mathbb N[q]\). This resolves the elementary-positivity part of the Shareshian–Wachs conjecture.

We determine the sharp logarithmic exponent of the off-diagonal Ramsey number \(r(5,t)\): \(\displaystyle r(5,t)=\frac{t^4}{(\log t)^{3+o(1)}} \qquad (t\longrightarrow\infty).\)

For every fixed integer s ≥ 6, we determine the sharp logarithmic exponent of the off-diagonal Ramsey number: \(\displaystyle r(s,t)=\frac{t^{s-1}}{(\log t)^{s-2+o(1)}} \qquad (t\longrightarrow\infty).\)

We prove that the two-color Ramsey number of the n-dimensional binary cube is at most \(C2^n\), where C is an absolute constant. This resolves positively the hypercube Ramsey conjecture of Burr and Erdős.

We classify finite Euclidean Ramsey configurations by a necessary and sufficient tensor condition over their coordinate fields. The Ramsey property here concerns monochromatic congruent copies at the original scale under arbitrary finite colorings. The criterion shows that every nonempty subtransitive set and every nonempty set of at most five points on a circle is Ramsey. In particular, some Ramsey cyclic quadrilaterals are not subtransitive, disproving the necessity direction of the Leader–Russell–Walters conjectured characterization.

We prove that every nonempty finite oriented graph has a vertex with at least as many vertices at directed distance two as at directed distance one. This resolves Seymour's second neighborhood conjecture positively.

We prove that every finite loopless k-edge-connected multigraph on at least two vertices, with k ≥ 1, has a spanning tree meeting each cut in at most \(C/k\) times the size of the cut, where C is a universal constant. This resolves the strong thin tree conjecture.

We give a deterministic polynomial-time construction of strong thin trees. Given a finite k-edge-connected loopless multigraph on at least one vertex, the algorithm constructs a spanning tree meeting every cut in at most a \(C/k\) fraction of its edges, for a universal constant C. The running time is polynomial in the binary input length, including when parallel-edge multiplicities are encoded in binary.

We prove the graph-decomposition conjecture of Ascoli, He, Park, and Talagrand. Every graph admits a partition into a universally bounded number of fixed edge pieces, each having ordinary containment threshold at most a universal constant times the original graph's integral expectation threshold. The partition is chosen before sampling the random host, and the separate embeddings of the pieces need not agree on shared vertices.

We prove Talagrand's discrete-convexity conjecture. There is a universal integer k such that, whenever an arbitrary family has Bernoulli product measure at least \(1-1/k\), the sets not contained in a union of k members admit a cover of total cost at most 1/2 at the same density.

We prove Talagrand's conjecture that integral and fractional expectation thresholds are within a universal constant factor, with the same covering budget.

We prove the second Kahn–Kalai conjecture. For every finite simple graph H with h ≥ 1 edges and at most n vertices, the threshold for \(G(n,p)\) to contain an ordinary copy of H is at most \(C p_{\mathrm E}(n,H)(1+\log_2 h)\), where C is universal. Here \(p_{\mathrm E}(n,H)\) is the least density at which every subgraph of H has expected copy count at least one half.

For every integer d ≥ 3, we construct arbitrarily large finite d-dimensional simplicial complexes with uniformly bounded vertex degrees and uniform 𝔽₂ coboundary expansion in every degree below d. Together with the known graph and two-dimensional cases, this establishes the existence of bounded-degree 𝔽₂ coboundary expanders in every positive dimension.

For every fixed integer d ≥ 3, we give a deterministic algorithm that constructs a simple nonbipartite d-regular Ramanujan graph on every sufficiently large even number n of vertices. It outputs the full adjacency list in polynomially many bit operations, with an exponent that may depend on d. Every nonconstant adjacency eigenvalue lies strictly between \(-2\sqrt{d-1}\) and \(2\sqrt{d-1}\).

We prove the circulant Hadamard conjecture: a real circulant Hadamard matrix has order 1 or 4. As a consequence, Barker sequences of length greater than one exist exactly at lengths 2, 3, 4, 5, 7, 11, 13, proving the Barker-sequence conjecture.

We prove Barnette's conjecture: every finite simple cubic bipartite planar 3-vertex-connected graph has a Hamiltonian cycle.

We prove that every finite simple undirected graph on n vertices has an edge partition into at most \(Cn\) simple cycles and single edges, for an absolute constant C. This resolves the Erdős–Gallai cycle decomposition conjecture positively.

An integer polynomial is intersective if it has a root modulo every positive integer. For each degree k ≥ 2, we prove that there is an exponent \(c_k\gt 0\) such that every set \(A\subseteq\{1,\ldots,N\}\) whose differences avoid all nonzero values \(h(1),h(2),\ldots\), where h is an intersective polynomial of degree k with positive leading coefficient, satisfies \(|A|=O_h(N^{1-c_k})\). The implied constant may depend on h, but the power-saving exponent depends only on its degree.

Let h be a fixed integer polynomial of degree at least two with positive leading coefficient, having a unit root modulo every positive integer. We prove that any set \(A\subseteq\{1,\ldots,N\}\) whose differences avoid all nonzero values \(h(p)\) at primes satisfies \(|A|\le C_hN^{1-c_h}\), where \(c_h\gt 0\) and \(C_h\ge1\) depend only on h. Thus the local unit-root condition gives a fixed power saving even when polynomial arguments are restricted to primes. The proof uses the companion zero-free half-plane theorem for Dirichlet L-functions to obtain the required prime-distribution estimates.

We prove that there are absolute constants c > 0 and C < ∞ such that every set \(A\subseteq\{1,\ldots,N\}\) with no nonzero square difference satisfies \(|A|\le C N^{1-c}\). This answers the fixed-power question posed by Green and Sawhney.

There are absolute constants ε > 0 and C such that every sufficiently large even n-point set in the plane with no three collinear has at most \(Cn^{4/3-\varepsilon}\) unordered halving pairs. This gives a power saving over the classical \(O(n^{4/3})\) bound for planar halving lines. The proof is nonquantitative and does not supply explicit constants.

For every fixed integer r ≥ 4, we prove that every K_r-free graph of sufficiently large maximum degree Δ has correspondence chromatic number \(O_r(\Delta/\log\Delta)\). This resolves the Alon–Krivelevich–Sudakov coloring conjecture in the stronger correspondence-coloring form. The same bound, with a constant depending on F, holds when any fixed graph F is excluded as an ordinary subgraph. Ordinary and list coloring satisfy the same bounds.

For every fixed integer r ≥ 4, every K_r-free graph on n vertices with average degree d ≥ 2 has an independent set of size at least \(c_r n\log d/d\), where \(c_r\gt 0\) depends only on r. This proves the fixed-clique-size independence conjecture of Ajtai, Erdős, Komlós and Szemerédi.

We construct in ZFC two self-dual partitional matroids on a countably infinite common ground set that admit neither a packing/covering partition nor an intersection witness. This disproves the unrestricted infinite matroid packing/covering and intersection conjectures and answers Joó's question for two partitional matroids negatively. The examples are neither finitary nor cofinitary.

For every fixed integer r ≥ 3, we prove that every relabeling-invariant Boolean property of simple r-uniform hypergraphs on n vertices satisfies \(\mathop{\mathrm{Var}}\nolimits _p(f)\le C_r I_p(f)/(\log n)^{r/(r-1)}\). The constant depends only on r, and the bound holds uniformly for all \(0\lt p\lt 1\) without a monotonicity assumption. For increasing properties, it gives the corresponding threshold-width bound with exponent \(r/(r-1)\), proving the hypergraph threshold-width conjecture of Friedgut and Kalai.

We prove the Friedgut–Kalai sharp-threshold conjecture. For every integer n ≥ 2, every nontrivial increasing family of graphs on n vertices invariant under all vertex permutations, and every \(0\lt \varepsilon\lt 1/2\), the edge probabilities at which its probability equals ε and \(1-\varepsilon\) differ by at most \(C\log(1/(2\varepsilon))/(\log n)^2\), for a universal constant C.

We prove that Maker can achieve the Snaky hexomino within 21 actual Maker moves against arbitrary legal Breaker play on the initially empty infinite square board. The same bound holds on a \(17\times17\) square; in fact, Maker can confine its claims to a fixed 251-cell board.

Starting from the complete graph on n vertices, repeatedly remove the three edges of a uniformly chosen remaining triangle. We prove that the number of edges left at termination, divided by n^3/2, converges in L² to \(1/(2\sqrt2)\). This proves the triangle case of the sharp-constant conjecture of Joos and Kühn. In particular, the same limit holds in probability and for the normalized expectation.

We prove that \(R(C_m,K_n)=(m-1)(n-1)+1\) for every pair of integers \(m\ge n\ge3\) other than \((m,n)=(3,3)\), for which \(R(C_3,K_3)=6\). This establishes the cycle–clique conjecture of Erdős, Faudree, Rousseau and Schelp. The proof combines expansion in a minimal counterexample with a large-clique lemma and an optimization of paths joining clique vertices. These arguments reduce the remaining cases to \(3{,}099\) finite parameter-pattern instances, which are excluded by two exact implementations of proved inference rules. Complete programs and deduction traces accompany the paper.

We construct a fixed \(66\times66\) binary matrix for which ordered matrix removal has no polynomial bound. This disproves the polynomial ordered binary matrix-removal conjecture. Ordered copies preserve the separate row and column orders and match both zeros and ones; removal permits changing entries in either direction.

There are absolute constants \(\eta,c_1\gt 0\) such that, for every sufficiently large integer n, one can choose n points in the unit square so that every triangle they determine has area at least \(c_1n^{-2+\eta}\). Thus the almost n⁻² upper-bound formulation of Heilbronn's triangle problem is false. The exponent η is fixed but extremely small.

We disprove the Gopalan–Servedio square-root conjecture, even up to an arbitrary constant factor. For every real C > 0, there is a nonconstant Boolean function \(f:\{-1,1\}^n\to\{-1,1\}\) on a finite sign cube such that \(\displaystyle \sum_{i=1}^n \widehat f(\{i\})\gt C\sqrt{\deg(f)}.\) Here \(\widehat f(\{i\})\) is the linear Fourier coefficient associated with the ith input, and \(\deg(f)\) is the degree of the real multilinear polynomial representing f.

We prove Serre's positivity conjecture, including ramified mixed characteristic. If two nonzero finitely generated modules over a regular local ring have tensor product of finite length and complementary dimensions, then their intersection multiplicity is strictly positive.

We prove Lech's multiplicity conjecture: Hilbert–Samuel multiplicity cannot decrease under a flat local homomorphism of nonzero Noetherian local rings. The result holds in arbitrary dimension, with no restrictions on the residue fields or characteristics.

We construct a three-dimensional complete Noetherian normal local domain containing ℂ, with residue field ℂ, that has no nonzero finitely generated maximal Cohen–Macaulay module. This disproves the domain form of the small Cohen–Macaulay module conjecture.

We disprove Kaplansky's zero-divisor conjecture by constructing a finitely presented torsion-free group G for which \(\mathbb F_2[G]\) has nonzero zero divisors. The group admits a finite two-dimensional classifying space.

We construct a finitely presented torsion-free counterexample to Kaplansky's direct finiteness conjecture over the field of two elements. The group admits a finite two-dimensional classifying complex.

We construct a counterexample to Kaplansky's direct-finiteness conjecture in odd characteristic. For one specified odd prime p, we obtain a field K of order p⁴, a finitely generated group G containing torsion, and finite sums \(a,b\in K[G]\) with \(ab=1\) but \(ba\ne1\). The same elements define a cellular automaton on K^G that is injective but not surjective.

We disprove the unrestricted group-ring Determinant Conjecture. We construct a finitely generated group G and a square matrix over \(\mathbb Z[G]\) that is invertible over \(\mathbb Q[G]\) and has Fuglede–Kadison determinant strictly between zero and one. The logarithmic integral defining the determinant is finite.

We disprove Kaplansky's direct-finiteness conjecture by constructing a finite field K of characteristic two, a finitely presented group G, and finite sums \(a,b\in K[G]\) with \(ab=1\) but \(ba\ne1\). The group G is nonsofic. The same elements define a cellular automaton on K^G that is injective but not surjective, disproving Gottschalk's surjunctivity conjecture.

We construct a finite-dimensional complex algebra whose finite-dimensional modules have unbounded finite projective dimensions. This disproves the little finitistic-dimension conjecture.

We disprove the Auslander–Reiten conjecture for Artin algebras. We construct a finite-dimensional algebra Λ over \(k=\mathbb F_2(q,H_1,H_2)\) and a finite-dimensional nonprojective left module Z such that \(\mathop{\mathrm{Ext}}\nolimits ^i_\Lambda(Z,Z)=\mathop{\mathrm{Ext}}\nolimits ^i_\Lambda(Z,\Lambda)=0\) for every i > 0. The module is Gorenstein-projective, so the same example also disproves the Gorenstein-projective conjecture. Both counterexamples persist after every extension of k.

We disprove Tachikawa's second conjecture by constructing a finite-dimensional symmetric algebra over \(k=\mathbb F_2(q,H_1,H_2)\) with a finite-dimensional nonprojective module whose self-extension groups vanish in every positive degree. The associated endomorphism algebra also gives counterexamples to the classical, generalized, and strong Nakayama conjectures, the Auslander–Gorenstein conjecture, and the Wakamatsu tilting conjecture. These conclusions persist after every extension of k.

We prove the Eisenbud–Green–Harris and lex-plus-powers conjectures over every characteristic-zero field for homogeneous regular sequences of any positive length, with degrees at least two. For every homogeneous ideal containing such a sequence, the corresponding lex-plus-powers ideal has the same Hilbert function and at least as large a graded Betti number in every homological and internal degree.

We prove the Eisenbud–Green–Harris conjecture over every characteristic-zero field for homogeneous regular sequences of any positive length, with degrees at least two. Thus every homogeneous ideal containing such a sequence has the Hilbert function of a monomial ideal containing the corresponding pure powers.

We construct a countable division ring of characteristic zero that is algebraic over its center, generated by two elements as an algebra over that center, and infinite-dimensional over it. This gives a negative answer to the Kurosh problem for division rings.

We prove the numerical blockwise Alperin weight conjecture for every finite group and every prime p. For each p-block B, the number of irreducible Brauer characters in B equals the number of conjugacy classes of B-weights.

We prove integral Donovan finiteness: for every prime p and positive integer M, the blocks of all finite groups with defect groups of order at most M have only finitely many Morita equivalence classes over \(W(\overline{\mathbb F}_p)\). The defect groups need not be abelian, and the result includes p = 2. The same bounded-defect finiteness holds over each fixed complete discrete valuation ring of characteristic zero with algebraically closed residue field of characteristic p, including ramified rings.

We prove Donovan's conjecture over every algebraically closed field K of characteristic p > 0. For each fixed K and bound on defect-group order, blocks of finite groups represent only finitely many K-linear Morita equivalence classes. The defect groups need not be abelian, and the result includes p = 2.

We prove the saturation conjecture for \(\mathop{\mathrm{Spin}}\nolimits (2n)\), n ≥ 2. If three dominant integral weights sum to an element of the root lattice, then the existence of a nonzero tensor invariant after a positive integral dilation implies the existence of one at the original weights.

For every positive integer n other than 2, 4, and 9, we prove that some irreducible complex representation of S_n has a tensor square containing every irreducible representation. This resolves the tensor square conjecture for symmetric groups affirmatively.

For every staircase partition, we prove that the tensor square of the corresponding irreducible complex representation of the symmetric group contains every irreducible representation of that group. This proves Saxl's conjecture.

We give an explicit colored-graph characterization of the finite nonempty lattices that occur as full congruence lattices of finite algebras, and prove that deciding this representation property is undecidable. In particular, the finite lattice representation problem has a negative answer. We also prove that recognition of full subgroup intervals in finite groups is undecidable.

We give a negative solution to the finite lattice representation problem. We prove that there is a finite nonempty lattice that is not the full congruence lattice of any finite nonempty algebra of any finite signature.

We prove the ℓ¹-Bass conjecture for every discrete group. The Hattori–Stallings trace of every idempotent matrix over the complex ℓ¹ group algebra is supported on finitely many conjugacy classes of finite-order elements.

We prove the complex group-ring Bass trace conjecture for every discrete group: the Hattori–Stallings trace of a finitely generated projective module over its complex group ring is supported on conjugacy classes of finite-order elements. As a consequence, for every torsion-free group G and every commutative unital domain R of characteristic zero, the only idempotents in \(RG\) are 0 and 1. This proves Kaplansky's idempotent conjecture in characteristic zero.

We prove the finite case of the Benson–Etingof–Ostrik conjecture: every finite symmetric tensor category over an algebraically closed field k of characteristic p > 0 admits a k-linear exact faithful strong symmetric monoidal functor to a finite higher Verlinde category \(\mathop{\mathrm{Ver}}\nolimits _{p^n}(k)\). The result includes characteristic two, and the level n may depend on the category.

We construct a two-dimensional ramified regular local ring A in mixed characteristic \((0,5)\) for which the integral map \(K_3(A)\to K_3(\mathop{\mathrm{Frac}}\nolimits A)\) has nonzero kernel. This gives a negative answer to the unrestricted integral Gersten conjecture.

We construct a two-dimensional ramified regular local ring A of mixed characteristic \((0,5)\) for which \(K_5(A)\to K_5(\mathop{\mathrm{Frac}}\nolimits A)\) has a nonzero kernel. This disproves Gersten's conjecture in its unrestricted integral form.

For every finite-dimensional complex vector space V, we prove that the canonical Foulkes–Howe map \(\mathop{\mathrm{Sym}}\nolimits ^b(\mathop{\mathrm{Sym}}\nolimits ^a V)\longrightarrow\mathop{\mathrm{Sym}}\nolimits ^a(\mathop{\mathrm{Sym}}\nolimits ^b V)\) is surjective whenever a ≥ 2 and \(b\ge a(a-1)\). This gives a quadratic stabilization bound independent of \(\dim V\).

We prove the sixth-symmetric-power case of Foulkes' conjecture. For every integer b ≥ 6 and every finite-dimensional complex vector space V, there is a \(\mathop{\mathrm{GL}}\nolimits (V)\)-equivariant injection \(\mathop{\mathrm{Sym}}\nolimits ^6(\mathop{\mathrm{Sym}}\nolimits ^b V)\hookrightarrow\mathop{\mathrm{Sym}}\nolimits ^b(\mathop{\mathrm{Sym}}\nolimits ^6 V)\).

Using the conformal and metric-measure companion results and the stated Brownian/Liouville quantum gravity inputs, we prove spectral convergence for critical spherical FK–Ising maps to Liouville Brownian motion on the ordinary unit-area \(\sqrt3\)-quantum sphere. The discrete walk has total attempt rate one, uses every map edge including loops and multiplicities, and has the corner measure as its stationary law. Accelerating time by the number of map edges gives joint convergence of the metric-measure space, all ordered eigenvalues with multiplicities and padding, and the heat trace locally uniformly at strictly positive times. The conductivity and clock constant are one in these conventions.

We prove that stationary random walks on critical spherical FK–Ising planar maps converge to Liouville Brownian motion on the ordinary unit-area \(\sqrt3\)-quantum sphere, using the geometric and electrical results of the spectral companion. The walk chooses uniformly among all incident half-edges, retaining loops and multiple edges. With the corner measure as the stationary law, the deterministic time acceleration is exactly the number of map edges. For any deterministic metric scale giving the metric-measure limit, convergence retains that same surface and any fixed finite number of conditionally independent walks with their time parameters.

We prove that stationary random walk on a planar map sampled with weight equal to its number of spanning trees converges to Liouville Brownian motion on the unit-area \(\sqrt2\)-Liouville quantum sphere. The convergence retains the conditional path law jointly with the measured metric space. The walk chooses uniformly among all incident half-edges and starts from the stationary degree measure. For total attempt rate one and the continuum Dirichlet form with factor 1/2, the time acceleration is exactly the number of map edges. The result uses the companion contour and metric limits for this same ensemble.

We construct the field and area law of the unit-area critical Liouville quantum sphere as a limit of ordinary subcritical quantum spheres, and equip it with its critical intrinsic metric. We then prove that spherical Fortuin–Kasteleyn planar maps at q = 4, embedded by their equilateral flag uniformizations, converge jointly to this sphere decorated by an independent nested conformal loop ensemble CLE₄. With deterministic distance normalization, the convergence includes the area measure, the full embedded distance function, and every macroscopic interface through all positive integer edge counts.

We resolve the finite spherical cases of Gwynne and Miller's graph-metric conjecture for critical Fortuin–Kasteleyn maps at each fixed \(q\in(0,4)\) and for uniform spanning-tree-decorated maps. After deterministic rescaling of graph distances, these maps, equipped with the vertex probability measure proportional to degree, converge in Gromov–Hausdorff–Prokhorov law to their ordinary unit-area Liouville quantum gravity spheres. Distances use every primal edge, and convergence holds through all positive integer edge counts.

For every fixed \(0\lt q\lt 4\), we prove joint convergence of spherical Fortuin–Kasteleyn planar maps in their flag-triangle uniformization to the corresponding unit-area Liouville quantum gravity sphere decorated by an independent conformal loop ensemble. The convergence includes the area measure, deterministically rescaled graph distances between all vertex pairs, and the full nested interface collection, with interfaces converging uniformly up to reparameterization.

We prove the finite-volume continuum-random-tree prediction for critical Fortuin–Kasteleyn planar maps at every fixed q > 4. After rescaling graph distances by a constant times n^−1/2, an n-edge map with normalized degree measure converges to the Brownian continuum random tree in the Gromov–Hausdorff–Prokhorov topology. The convergence holds through all positive integer sizes.

We prove that the limit shape of undirected nearest-neighbor first-passage percolation on ℤ² with independent exponential edge weights is strictly convex and has a C¹ boundary. This resolves the strict convexity and differentiability conjectures for the planar exponential model. More generally, we prove differentiability of the time-constant norm for every Gamma edge-weight law with positive shape and rate.

We prove that planar first-passage percolation with independent identically distributed nonnegative nonatomic edge weights has almost surely no doubly infinite geodesic, provided the minimum of four independent weights has finite second moment. This resolves the planar no-bigeodesics conjecture under that moment assumption. The conclusion rules out all bigeodesics simultaneously, without any regularity assumption on the limit shape.

We prove that critical Bernoulli bond percolation has no infinite cluster on any infinite connected locally finite quasi-transitive graph with critical probability less than one. This resolves the bond form of the criticality conjecture of Benjamini and Schramm.

We prove that nearest-neighbor Bernoulli bond and site percolation on ℤ³ have no infinite cluster at their respective critical parameters. The proof combines a finite connection inequality for independent hyperedges with a finite-scale extension estimate and an adaptive exploration.

We prove the bond-percolation nonuniqueness conjecture of Benjamini and Schramm: every infinite connected, locally finite, nonamenable quasi-transitive graph has a nonempty interval of parameters for which Bernoulli bond percolation almost surely has infinitely many infinite clusters. We also prove Hutchcroft's stronger operator-threshold conjecture, establishing \(p_c\lt p_{2\to2}\le p_u\).

For the nearest-neighbor \(O(4)\) model on the square lattice at inverse temperature β, we prove the exact low-temperature asymptotic \(\displaystyle m_{\mathrm{lat}}(\beta)\sim 32\exp(\pi/4-1/2)\sqrt\beta\,\exp(-\pi\beta) \qquad (\beta\to\infty).\) Here the mass is the gap of the full Osterwalder–Schrader transfer operator, including rotation-invariant local-observable sectors, in units of one original lattice time step.

We construct a canonical interacting massive continuum limit of the two-dimensional nearest-neighbor \(O(3)\) model with unit-length spins, no external field, and no topological term. Normalized by susceptibility and second-moment correlation length, the limit exists as the bare coupling tends to infinity through all positive real values, without selecting subsequences. The limiting fields satisfy the Osterwalder–Schrader axioms and have a nonzero connected four-point correlation on separated time supports. Their reconstructed theory has a unique vacuum, a nonzero vacuum complement, and a positive Hamiltonian gap on that entire complement.

We consider the continuum spin field constructed from the nearest-neighbor two-dimensional \(O(3)\) model by the fixed prescription of the companion paper. We prove that its vector two-point spectral measure has a positive atom corresponding to the lowest mass, separated by a positive gap from all remaining mass support.

We prove sharp mass bounds for the two-dimensional nearest-neighbor \(O(4)\) model. For all sufficiently large inverse couplings β, the full transfer gap, including rotation-invariant local-observable sectors, is bounded above and below by positive multiples of \(\sqrt\beta e^{-\pi\beta}\). We also prove that the model has a unique periodic local limit and a positive full gap at every finite β > 0, resolving the all-temperature lattice mass-generation conjecture for this periodic state. Under the stated cutoff scaling, the mass bounds are uniform in independently fixed physical units. A continuum spectral interpretation requires separate convergence hypotheses for the transfer semigroup and the block observables.

We prove exponential decay of two-point correlations for the classical nearest-neighbor \(O(n)\) model on the square lattice, for every n ≥ 3 and every finite positive inverse temperature. The estimate is uniform over finite free-boundary subgraphs and bounded nonnegative edge strengths. This resolves positively the all-temperature exponential spin-decay conjecture for these models.

We prove the critical logarithmic correction for the nearest-neighbor cosine XY model on the square lattice. Let \(C_{b_c}(r)\) be the two-point correlation at critical inverse temperature, obtained by taking the free-box thermodynamic limit while the two sites remain r lattice steps apart along a coordinate axis. Then \(\displaystyle C_{b_c}(r)=B_{\mathrm{XY}}r^{-1/4}(\log r)^{1/8}(1+o(1)), \qquad B_{\mathrm{XY}}\in(0,\infty),\) as \(r\to\infty\).

Using the companion critical Bessel-height and pin-limit theorems, we prove that at the mass-defined critical inverse temperature of the nearest-neighbor XY model on the square lattice, the spin field on a square with boundary angles fixed to zero converges along the full sequence to the full-variance imaginary exponential of a zero-Dirichlet Gaussian free field with stiffness \(2/\pi\). The normalization uses the exact center magnetization and the lattice Green-function factor. Convergence holds in law in \(H^{-3}_{\mathrm{loc}}((-1,1)^2)\); all mixed moments and joint laws of fields smeared against smooth compactly supported test functions in \((-1,1)^2\) converge as well.

We prove the Berezinskii–Kosterlitz–Thouless essential singularity for the correlation length of the nearest-neighbor cosine XY model on the square lattice. Let \(m(b)\) be the mass obtained by taking first the free-box thermodynamic limit and then the separation limit along a coordinate axis. As the inverse temperature b approaches b_c from the massive side, \(\displaystyle \sqrt{b_c-b}\log\frac{1}{m(b)} \longrightarrow A_{\mathrm{XY}},\) where \(A_{\mathrm{XY}}\) is a finite, strictly positive model-specific constant.

Assuming the stated critical-height, local-renormalization, and spin-field inputs from the companion papers, we determine the center magnetization of the planar XY model in a square with aligned boundary spins at its mass-defined critical threshold. As \(n\to\infty\), the magnetization is \(A_{\mathrm{XY}}n^{-1/8}(\log n)^{1/16}(1+o(1))\), where \(A_{\mathrm{XY}}\) is a finite, strictly positive model-specific constant.

For the ordinary nearest-neighbor XY model on the square lattice, we prove the Berezinskii–Kosterlitz–Thouless prediction that the critical spin-correlation exponent is 1/4. More precisely, at the mass-defined critical inverse temperature, the infinite-volume correlation is \(n^{-1/4+o(1)}\). The infinite-volume limit through free square boxes is taken before the separation limit.

We prove Gaussian scaling limits for the two-dimensional discrete Gaussian height model throughout its rough phase, including the physical roughening threshold, for every finite square-symmetric interaction set containing the nearest neighbors. After the natural lattice normalization, the critical effective temperature has the universal value \(8\pi\). For the ordinary nearest-neighbor Villain and XY models at sufficiently low fixed temperatures, we also prove that their Green-function-normalized spin fields converge to the imaginary exponential of a Dirichlet Gaussian free field.

For every fixed inverse temperature β > 1, we prove that the zero-field Gaussian Ising Sherrington–Kirkpatrick free energy, centered by its expectation and divided by its standard deviation, converges in distribution to a nondegenerate law as the system size tends to infinity through all integers. We also prove that its variance divided by n^1/3 converges to a finite positive constant.

For the zero-field Gaussian Sherrington–Kirkpatrick model at every fixed inverse temperature β > 1, we prove that the standard deviation of the log partition function is \(n^{1/6+o(1)}\). The same exponent describes its typical centered absolute fluctuations, establishing the predicted one-sixth exponent in this regime.

For the planar Ising model with bonds \(J_e=1+\varepsilon\xi_e\), where the ξ_e have any fixed bounded, nondegenerate, mean-zero iid law, we prove quenched chordal SLE₃ convergence for sufficiently small ε > 0. The temperature is the spontaneous-magnetization threshold. Convergence holds in probability over environments for the full oriented curve law in deterministic Jordan-domain approximations.

We prove quenched chordal SLE₃ convergence for the planar Ising model with sufficiently weak independent symmetric two-valued ferromagnetic bonds. The temperature is the critical point defined by spontaneous magnetization. Convergence holds in probability over environments for the full oriented curve law in deterministic Jordan-domain approximations.

Assuming the stated deterministic critical-reference estimates, we prove that the relative second moment of the quenched critical spin correlation in the square-lattice Ising model with independent fair bonds \(1\pm\varepsilon\) grows as \((\log r)^{1/4+o(1)}\) for each sufficiently small fixed ε > 0. The correlation is evaluated at the physical critical temperature, with the thermodynamic limit taken before the disorder moments and the large-distance limit.

We prove that every sufficiently small, square-symmetric finite-range perturbation of the planar Ising contour energy has a domain-independent inverse temperature at which its spin interface converges to chordal SLE₃. The result allows interactions of either sign, arbitrary admissible exterior contours, and uniformly approximated Jordan domains. Convergence holds for the full oriented curve law in uniform distance modulo increasing reparametrization.

We prove conformal universality of mixed bulk spin and energy correlations for the square-lattice Ising model with sufficiently small, square-symmetric, finite-range even multispin perturbations of either sign. One critical-temperature branch and two field normalizations apply in every bounded simply connected C² Jordan domain with free, plus, or minus boundary conditions, and in the periodic thermodynamic plane state. Energies are centered in their actual states.

We prove pointwise comparison and finite stopping-band approximation theorems for the critical nearest-neighbor Ising model on the square lattice. Arbitrary common pinned spins are allowed in the comparison: the likelihood ratio is controlled by a zero-field FK connection probability on the graph with those vertices deleted. For an interface exploration across a band of fixed positive width, finitely many regular signed barriers approximate every good conditional target law in total variation, uniformly over bounded observables and sufficiently fine meshes.

For every fixed degree d ≥ 3, we prove fixed-energy GOE universality for the adjacency matrix of a uniformly random simple labelled d-regular graph with independent uniform diagonal disorder. The disorder strength is positive, sufficiently small, and fixed as the graph grows. At each fixed energy in a compact subinterval of the clean spectral band, the full microscopic eigenvalue point process converges to the GOE bulk process after rescaling by the positive density of states of the corresponding infinite-tree operator. The permitted disorder strength depends on the degree and the distance from the clean spectral edges.

We prove the fixed-degree bulk-universality conjecture for random regular graphs. For every fixed integer d ≥ 3 and every fixed energy in the open Kesten–McKay bulk, the unfolded eigenvalue point process of the adjacency matrix of a uniform simple labelled d-regular graph converges to the GOE bulk process. The convergence holds along all admissible graph sizes, without additional conditioning.

We prove a directional zero–one law for uniformly elliptic nearest-neighbor random walks in stationary, ergodic, finite-range-dependent environments on ℤ^d, d ≥ 3. For every fixed nonzero real direction, the probability of escape in that direction, averaged over the environment, is either zero or one. Finite-range dependence is imposed on the full transition rows: collections of rows at distance greater than a fixed range are independent, including collections indexed by infinite deterministic sets.

We prove that almost-sure transience in a fixed direction implies a deterministic limiting velocity with positive projection in that direction for nearest-neighbor random walks in independent and identically distributed uniformly elliptic environments on ℤ^d, d ≥ 2. This resolves the ballisticity conjecture positively.

We prove the directional zero–one conjecture for nearest-neighbor random walks in independent and identically distributed strictly elliptic environments on ℤ^d, d ≥ 3: the probability of escape in each fixed nonzero real direction is zero or one. Only strict positivity of the transition probabilities is required; no uniform lower bound or moment assumption is imposed.

We prove the Mézard–Parisi hierarchical cavity formula for diluted even-arity Ising models in the Panchenko–Talagrand class. This resolves the variational equality conjecture for that class: the limiting pressure equals the infimum of the trial functional over all finite hierarchy depths and trial laws. Only first moments of the interaction and external field are required.

We determine the limiting free energy of a spherical perceptron with a bounded continuous activation and a bi-orthogonally invariant disorder matrix. The singular values may have any compact limiting distribution, provided there are no outliers. We give an explicit variational formula for this limit.

We prove an exact variational formula for the limiting pressure of the spherical random perceptron with an arbitrary bounded continuous single-pattern potential. The formula holds at every fixed positive density and inverse temperature, with convergence in expectation and in probability.

We determine the limiting free energy of the Ising perceptron with independent Gaussian patterns for every bounded Borel log-potential, at every fixed positive temperature and pattern density. We give an explicit variational formula for the limit and prove convergence in expectation and probability.

We prove a sharp feasibility threshold and limiting gap and force laws for the spherical perceptron with margin −1 and quadratic penalty. The limits are taken successively in system size, inverse temperature, and density approaching the threshold from above. They agree for Gaussian coordinates and for the equal mixture of centered Gaussian coordinates with variances \(1-\varepsilon\) and \(1+\varepsilon\), for every sufficiently small fixed ε. The contact-removed gap cumulative law and the mean-one force cumulative law satisfy \(\displaystyle G_J(u)=u^{1-\gamma+o(1)},\qquad F_J(s)=s^{1+\theta+o(1)},\) as \(u\downarrow0\) and \(s\downarrow0\), with \(\gamma=(2+\theta)^{-1}\), \(0.4126930\lt \gamma\lt 0.4126934\), and \(0.4231063\lt \theta\lt 0.4231088\). A finite numerical certificate for these exponent intervals, together with its mathematical error bounds, is included.

We prove convergence of thermal FK–Ising interfaces, for every fixed nonzero mass of either sign, on uniformly angle-bounded isoradial lattices in bounded simply connected domains. The limit is independent of the lattice and the admissible domain approximation. For positive mass it is the unique massive SLE\(_{16/3}\) law with locally finite-energy drift prescribed by a massive boundary value problem; negative mass follows by duality and reversal. The theorem also allows Carathéodory approximations whose diameters diverge.

We prove that critical FK–Ising interfaces with sufficiently weak, symmetric, independent two-valued bond disorder converge to chordal SLE_16/3. The disorder strength is fixed as the mesh tends to zero, and convergence of the conditional curve laws holds in probability over the environment.

We prove that the rescaled counting measure of a critical square-lattice Fortuin–Kasteleyn Dobrushin interface in the unit square converges to the Minkowski-content measure of its Schramm–Loewner limit, for every fixed cluster weight \(1\le q\lt 4\). A single deterministic constant times the predicted power of the mesh gives the normalization. Convergence is joint with the ordered curve and includes the total mass, giving the scaling limit of the interface's total number of steps.

For every fixed \(1\le q\le4\), critical square-lattice random-cluster Dobrushin interfaces converge to chordal \(\mathop{\mathrm{SLE}}\nolimits _{\kappa(q)}\), where \(\kappa(q)=4\pi/\arccos(-\sqrt q/2)\), and the complete nested plane loop collections converge to whole-plane \(\mathop{\mathrm{CLE}}\nolimits _{\kappa(q)}\). The results hold under uniform marked Jordan boundary approximation and retain loop multiplicities and traversals. At q = 1 we obtain Cardy's formula for square-lattice bond percolation with free boundary edges.

For every fixed \(1\le q\lt 4\), critical square-lattice random-cluster Dobrushin interfaces converge as ordered curves to chordal \(\mathop{\mathrm{SLE}}\nolimits _{\kappa(q)}\), where \(\kappa(q)=4\pi/\arccos(-\sqrt q/2)\), confirming the Rohde–Schramm prediction in this parameter range. This holds in every bounded Jordan domain under uniform marked boundary approximation. The complete nested plane loop collections converge to whole-plane \(\mathop{\mathrm{CLE}}\nolimits _{\kappa(q)}\) in a spherical matching topology retaining multiplicities and traversals. At q = 1 we obtain Cardy's formula for square-lattice bond percolation with free boundary edges.

For every fixed \(0\lt q\lt 1\), we prove that the Dobrushin interface of the square-lattice random-cluster model at its self-dual parameter converges to chordal SLE_κ, where \(\kappa=4\pi/\arccos(-\sqrt q/2)\in(6,8)\). The approximating domains are simple closed nearest-neighbor lattice polygons with distinct marked vertices, whose marked boundary parametrizations converge uniformly to those of a bounded smooth Jordan domain. Convergence holds along the full mesh sequence in the uniform metric on oriented curves modulo increasing reparametrization. The proof combines finite connection comparisons below one, localization in irregular tiled disks, and a boundary observable that determines the limiting Loewner driver.

Taking Cardy's crossing formula for critical Poisson–Voronoi percolation as an input, we prove a universal joint limit for the near-critical crossing thresholds of rational polygonal quadrilaterals under the monotone coupling. Conditional on the Poisson tessellation, the threshold law converges in probability over tessellations to the same law as on the triangular lattice. Each model is normalized by its own expected number of color-pivotal sites for a unit-square crossing.

Taking Cardy's conformal crossing formula for critical planar Poisson–Voronoi percolation in every bounded Jordan quadrilateral as an input, we prove that the expected number of color-pivotal cells for a unit-square crossing is asymptotic to a positive constant times \(\varepsilon ^{-3/4}\), where the point intensity is \(\varepsilon ^{-2}\). No rate of convergence in Cardy's formula is required.

We prove Cardy's formula for annealed crossing probabilities in critical planar Poisson–Voronoi percolation in every bounded Jordan quadrilateral. This proves the annealed crossing-probability form of the conformal-invariance conjecture for this model.

We prove that the height function of the square-lattice six-vertex model with weights \(a=b=1\) and \(0\lt c\le2\), in the plane state obtained from balanced tori, converges to a multiple of the Gaussian free field. For unit height jumps and Green kernel \(-(2\pi)^{-1}\log|x-y|\), the squared multiplier is \(1/\arcsin(c/2)\).

We prove that the complete double-dimer loop ensemble for the Temperleyan square lattice in the upper half-plane converges to nested CLE₄. The convergence holds along the full mesh limit and matches every macroscopic loop as an unparametrized curve. This resolves the half-plane Temperleyan form of the double-dimer CLE₄ scaling-limit conjecture.

We prove joint functional convergence of the stationary and quench autocorrelations of zero-field Sherrington–Kirkpatrick heat-bath dynamics at inverse temperature β = 1, with the same random limit for Gaussian and Rademacher couplings. Each site has a rate-one clock, mean spin autocorrelations are multiplied by n^1/3, and waiting times and lags are measured in units n^2/3. The quench starts from independent fair spins, and convergence is uniform on compact sets of positive waiting times and lags. The quench limit is selected by these initial states and relaxes to the stationary limiting autocorrelation as the waiting time tends to infinity.

We prove that the stationary spin autocorrelation of the zero-field Sherrington–Kirkpatrick model at inverse temperature β = 1 has a common functional scaling limit for Gaussian and Rademacher couplings. With rate-one heat-bath clocks at every site, time is scaled by n^2/3 and the mean spin autocorrelation is multiplied by n^1/3. The functions converge in law uniformly on compact positive-time intervals. Their limiting law is not a point mass: it retains sample-to-sample randomness, and its functions decay to zero at large times.

At the critical inverse temperature β = 1, we determine the worst-start total-variation mixing exponent for single-site heat-bath dynamics in the zero-field Gaussian Sherrington–Kirkpatrick model. The mixing time is \(n^{2/3+o(1)}\) when every site has rate one, or \(n^{5/3+o(1)}\) attempted uniform-site updates, in probability over the disorder.

At every fixed inverse temperature β > 1, we prove a stretched-exponential obstruction to mixing for the zero-field Sherrington–Kirkpatrick model from typical equilibrium configurations held fixed as initial states. The Gibbs mass of states whose total-variation distance from equilibrium at time \(\exp(n^{1/10000})\) exceeds 1/4 tends to one in probability over the disorder. This holds for rate-one-per-site heat-bath dynamics and after the same stated number of discrete update attempts.

We prove worst-case total-variation cutoff for the zero-field Gaussian Sherrington–Kirkpatrick heat-bath dynamics at every fixed inverse temperature \(0\leq\beta\lt 1\). With rate-one refresh at each spin, the cutoff location is \(\log n/(2\lambda(\beta))\) for a positive deterministic rate \(\lambda(\beta)\). The location for uniformly chosen single-site update attempts is n times as large. Convergence is in probability over the disorder.

At the critical inverse temperature β = 1, we prove slow mixing for zero-field Gaussian Sherrington–Kirkpatrick heat-bath dynamics from typical equilibrium configurations held fixed as initial states. For every deterministic sequence \(t_n=o(n^{2/3})\) in rate-one-per-site time, the Gibbs mass of initial states whose time-t_n total-variation distance from equilibrium exceeds 1/4 tends to one in probability over the disorder. The same statement holds for every deterministic integer sequence \(k_n=o(n^{5/3})\) of uniform-site update attempts.

For every fixed inverse temperature β > 1, we prove subexponential mixing for rate-one-per-site heat-bath dynamics in the zero-field Gaussian Sherrington–Kirkpatrick model from a typical equilibrium configuration. If one Gibbs-sampled configuration is held fixed as the initial state, its total-variation distance from equilibrium is at most 1/4 by time \(\exp(n^{1-1/40000000})\), with joint probability tending to one over the disorder and the sampled state.

For every fixed inverse temperature \(0\lt \beta\lt 1\), we prove that the unscaled spectral gap of single-site heat-bath dynamics for the zero-field Gaussian Sherrington–Kirkpatrick model is bounded away from zero with probability tending to one over the disorder. Equivalently, the Gibbs law satisfies a dimension-free Poincaré inequality for all functions.

We construct a bounded, continuous, stable radial pair potential in three dimensions with \(|\phi(r)|\le Cr^{-3-1/32}\) for r ≥ 1. Throughout an open interval of positive densities, its canonical thermodynamic free energy is finite at every positive inverse temperature and has a strict downward derivative jump at one common finite positive inverse temperature.

We construct a stable radial pair potential in three dimensions whose canonical free energy has a strict downward derivative jump at one common finite positive inverse temperature throughout an open interval of positive densities. The potential has a divergent repulsive core, a nontrivial attractive interval, and an integrable tail satisfying \(\phi(r)=o(r^{-3})\). Its free-cube canonical free energy is finite at every positive inverse temperature and density.

We establish the exact Kesten–Stigum reconstruction threshold for the ferromagnetic four-state Potts broadcast model on every regular d-ary tree with d ≥ 2 and every Poisson Galton–Watson tree of mean d > 0. Reconstruction occurs exactly when \(d\lambda^2\gt 1\); we prove nonreconstruction at and below the threshold, including equality. In the Poisson model the whole tree is observed and the reconstruction advantage is averaged without conditioning on survival. The proof uses reproducible exact-arithmetic verification of polynomial inequalities.

For the ferromagnetic four-state broadcast model with \(0\lt \lambda\lt 1\), we prove that reconstruction on a bounded-degree deterministic rooted tree occurs exactly when its L³ capacity with edge resistances \(\lambda^{-2|e|}\) is positive. This gives an exact criterion without regularity or growth-rate assumptions on the tree, including at the exponential critical boundary.

We determine the exact reconstruction threshold for the symmetric three-state broadcast process on every regular b-ary tree, b ≥ 2, and every observed Poisson Galton–Watson tree of mean d > 1. Reconstruction occurs exactly when \(d\lambda^2\gt 1\), with d = b in the regular model; there is non-reconstruction at equality for either sign of the channel parameter. The Poisson advantage is averaged over trees and spins without conditioning on survival. This resolves the all-degree three-state regular-tree prediction. Combining the Poisson theorem with known tree-to-graph and algorithmic results gives the exact weak-recovery threshold for the symmetric three-community sparse stochastic block model with independent uniform labels, fixed within- and between-community rates \(a,b\gt 0\), and mean degree \((a+2b)/3\gt 1\): recovery is possible exactly when \((a-b)^2\gt 3(a+2b)\). Above this threshold it is achievable in \(O(n\log n)\) time; at or below it, weak recovery is information-theoretically impossible.

For each fixed \(0\lt \kappa\lt 8\), let \(d=1+\kappa/8\). The gauge \(h(r)=r^d(\log\log(1/r))^{(2-d)/2}\) at sufficiently small radii almost surely gives positive finite Hausdorff measure to every nontrivial positive-time compact segment of chordal SLE_κ. This gives an explicit solution to Schramm's Hausdorff-measure problem in this parameter range. The entire trace has finite expected measure in every bounded disk. The exponent-one iterated-logarithm gauge suggested by Schramm is not sigma-finite on any such segment.

For each \(0\lt \kappa\lt 8\), we construct a deterministic Hausdorff gauge that almost surely assigns positive finite measure to every nontrivial compact positive-time segment of chordal Schramm–Loewner evolution. This answers Schramm's Hausdorff-measure existence problem in this parameter range. The entire trace has finite expected gauge measure in each bounded box.

We prove that the free uniform spanning forest is a factor of IID on every infinite connected locally finite simple unweighted graph. One Borel rule works for all such graphs and uses no root, answering affirmatively the general factor question for unimodular random graphs. We also show that every translation-invariant strongly Rayleigh process indexed by a countable group is a factor of IID, including invariant determinantal processes with Hermitian positive-contraction kernels. This group-action conclusion requires no amenability.

We prove the Gaussian free field and SLE\(_4\) scaling limits for uniformly sampled real nearest-neighbor Lipschitz heights on the triangular lattice, resolving Schramm's Problem 2.3. On approximations of smooth simply connected domains, the centered height field converges as a random distribution to a multiple of the Dirichlet Gaussian free field. At one tuned two-arc boundary amplitude, the zero-height interface converges in uniform curve distance to chordal SLE\(_4\). We identify the relation between the field variance and the boundary height in terms of an implicit stationary tangent-flux coefficient.

We prove that the centered uniform odd integer height function on triangular-lattice approximations of a smooth simply connected domain, with neighboring differences zero or two and boundary values +1 and −1 on two arcs, converges to a universal multiple of the Dirichlet Gaussian free field. This resolves the field part of Schramm's Problem 2.2. We give an absolutely convergent finite-volume formula for the normalization. The proof combines reflection positivity, a spectral sum rule, and boundary comparison with Gaussian moment identities.

We prove a Gaussian free field scaling limit for weighted integer Lipschitz heights on the triangular lattice. With zero boundary values and a factor x for each edge on which the height changes, the field converges after division by a positive constant depending only on x to the zero-Dirichlet Gaussian free field, for every fixed \(x\in[1/\sqrt2,1]\). The convergence holds as a random distribution on every bounded C² Jordan domain under inside lattice approximations with uniformly convergent boundary parametrizations. This includes the uniform height model and the predicted critical endpoint.

For each fixed point on the critical Ashkin–Teller line, including the four-state Potts endpoint, we prove the conjectured joint scaling limit of the height and both current-cluster collections in every bounded Jordan domain, for every admissible polygonal approximation. The height converges to a Gaussian free field with the predicted coupling constant, and the clusters converge to canonical recursive two-valued local sets of that same field. The joint limit includes all nesting depths and the distinguished wired boundary cluster.

We determine the limiting pressure of an Ising spin glass with Haar orthogonal eigenvectors, a compact limiting spectral law, and no asymptotic outliers, at every fixed temperature. The pressure converges in expectation and almost surely to a variational formula. We also give the limiting pressure with deterministic external fields whose empirical laws converge in first-moment transport distance, and derive a formula for the zero-field ground-state energy by taking temperature to zero.

For random 3-SAT on n Boolean variables, with independent uniformly signed clauses on three distinct variables sampled with replacement, we prove that the first unsatisfiable prefix has variance \(\Theta(n)\). The upper bound removes the logarithmic loss in the earlier variance estimate; the matching lower bound follows from Wilson's transition-width theorem.

Let H_n be the index of the first unsatisfiable prefix in random k-SAT on n variables, with independent uniformly signed clauses using k distinct variables and sampled with replacement. For every fixed k ≥ 4, we prove \(\mathop{\mathrm{Var}}\nolimits (H_n)=\Theta_k(n)\). For k = 3, the variance is bounded below by a positive multiple of n and above by a constant multiple of \(n\log n\); the companion paper on random 3-SAT sharpens this to \(\Theta(n)\). The same upper bounds proved here hold after clipping at any fixed positive multiple of n.

The limiting satisfiability threshold of uniform random 3-SAT is a computable real. We credit Gaia Carenini [4] with priority for resolving the satisfiability conjecture, which establishes the threshold's existence. We prove that one finite deterministic machine can approximate it to any prescribed accuracy. A deletion estimate gives explicit lower certificates, while a finite hierarchical approximation of the soft pressure gives upper certificates at every larger rational density. A fair search through these certificates halts without requiring a computable rate of finite-size convergence.

For every fixed integer k ≥ 3, random k-SAT with independent uniformly signed clauses on distinct variables, sampled with replacement, has a finite positive limiting satisfiability threshold. We credit Gaia Carenini [5] with priority for resolving the satisfiability conjecture. This paper gives an alternative proof, using concentration of a capped last satisfiable index and a comparison between different system sizes.

For every integer d ≥ 3 and inverse temperature β ≥ 0, the free zero-field ferromagnetic Ising measure on the infinite d-regular tree is a factor of IID exactly when \(\tanh\beta\le(d-1)^{-1/2}\). We prove the positive implication, including equality, resolving the conjecture of Nam, Sly and Zhang; the strict converse is known. The spin factor can be chosen to commute with every tree automorphism on every label input.

We prove a uniform bound for critical honeycomb polygons through two axial marks on a periodic staircase. The bound retains an explicit power of the ratio between the period and the marked separation. We also prove sharp finite strip estimates: bridge mass of order h^−1/4, first-length mass at most \(Ch^{13/12}\), and mass at least \(ch^{-1/4}\) on bridges with length at least \(ch^{4/3}\).

At the critical activity and loop fugacity two, we prove a 1/6 partition exponent for disjoint honeycomb polygons separating opposite marks on a balanced infinite cylinder, and a 1/12 planar nesting exponent. For ordered first-exit chords in a regular hexagon of side R, with both boundary ports summed and diameter at least \(R/10\), the finite critical partition is \(R^{3/4+o(1)}\) and the normalized mean length is \(R^{4/3+o(1)}\). A separate two-bond estimate on tilted cylinders with controlled site proportions bounds the length-square mass of planar polygons of diameter at most H, modulo translations, by \(H^{2/3+o(1)}\).

For critical honeycomb self-avoiding walk, we prove spatial exponent 3/4 for the infinite irreducible-bridge law and for bridges of every sufficiently large compatible even length. We establish the corresponding thermal laws at every large discount scale, including all positive length and spatial moments. For unrestricted uniform walks, the endpoint lower law holds on a common set of lengths of natural density one. We also determine the near-critical exponential correlation scale and small-force free-energy exponent, and prove spatial local lower bounds that permit conditioning on a prescribed terminal vertex.

We prove critical diameter-tail exponents −2 for unrooted honeycomb polygons and −2/3 for length-weighted polygons, together with a truncated second-length-moment bound of exponent 2/3. Consequently, polygons conditioned to have length at least n have diameter \(n^{3/4+o(1)}\) in probability under either weight.

We prove that, under the all-length critical measure, a honeycomb bridge crossing a strip of h layers, with its initial port fixed and its terminal port free, has mean length \(h^{4/3+o(1)}\). We also obtain central visit probability \(R^{-2/3+o(1)}\) and mean length \(R^{4/3+o(1)}\) for macroscopic free-boundary chords in a regular hexagon.

For uniform self-avoiding walks of every sufficiently large integer length on the honeycomb lattice, we prove diameter exponent 3/4 and simultaneous local-mass and covering exponent 4/3, with arbitrary positive exponent slack and arbitrary polynomial failure probability.

At the critical honeycomb vertex activity, the squared-length mass of simple polygons of diameter at most H, counted modulo translations, is at most \(H^{2/3+o(1)}\). We also prove a quantitative two-mark cylinder estimate and a polynomial one-arc bound uniform even for arbitrarily unequal marked intervals.

We prove that critical honeycomb bridges crossing a strip of width R have mean length \(R^{4/3+o(1)}\), with the same exponent after confinement to a fixed multiple of the strip width. The initial boundary port is fixed, the terminal port is summed, and all finite lengths receive their critical weights. The bridge mass is comparable to R^−1/4; the half-plane arch kernel at endpoint separation m is comparable to m^−5/4.

We determine the growth exponent, at every fixed positive loop fugacity, of the critical honeycomb partition function for disjoint polygons separating two prescribed markers on a balanced cylinder. At fugacity two the cylinder exponent is 1/6; the corresponding planar nesting exponent and middle-strip nesting exponent are 1/12.

We prove a logarithmic window of critical honeycomb bridge lengths: summed through height \(2h(\log h)^{1/64}\), bridges of lengths between \(h^{4/3}(\log h)^{-1/8}\) and \(2h^{4/3}(\log h)^{1/2}\) have mass at least \(h^{3/4}(\log h)^{-C}\).

At the critical weight of the regular honeycomb lattice, the total weight of self-avoiding paths crossing a strip of height N is comparable to N^−1/4. The first horizontal-displacement moment of return paths is comparable to N^3/4.

Critical self-avoiding walks between prescribed, macroscopically separated boundary ports of a regular honeycomb hexagon have length \(R^{4/3+o(1)}\) in probability, where R is the scale of the hexagon. We prove the corresponding statements for half-plane arches, parallel cuts and nonparallel pure cuts. The half-plane law also has mean length \(R^{4/3+o(1)}\). A separate strip argument gives endpoint mean laws on one density-one set of heights and in an aligned, critically weighted mixture of all even heights in a macroscopic interval.

At the critical honeycomb fugacity, self-avoiding port-to-port chords in an equilateral lattice triangle of side R, summed over both boundary endpoints and restricted to diameter at least \(R/100\), have partition sum comparable to R^3/4 and mean length \(R^{4/3+o(1)}\). We also determine the amplitude selected by two vacuum caps on a cylinder of circumference N and prove that it grows as \(N^{1/6+o(1)}\).

We prove that a fixed number of coordinate sweeps of the Thorp shuffle on \(2^d\) cards makes the squared L² distance of the full permutation density from uniform tend to zero as \(d\to\infty\). In particular, the total-variation mixing time is \(\Theta(d)\) physical shuffles.

We prove that the Thorp shuffle on \(N=2^d\) labeled cards has full-permutation total-variation mixing time \(\Theta(\log N)\) in physical shuffles. An absolute number of coordinate sweeps suffices for the upper bound.

For \(N=2^d\) cards, we prove that an absolute number of coordinate sweeps of the Thorp shuffle brings the full permutation law to total-variation distance tending to zero from uniform. The mixing time therefore has optimal order \(\Theta(\log N)\) in physical shuffles.

We prove that an absolute number of coordinate sweeps of the Thorp shuffle on \(n=2^d\) positions brings the full permutation to total-variation distance at most \(\frac12n^{-5}\) from uniform, uniformly over the initial deck. Thus \(O(d)\) physical shuffles suffice, which is optimal in order.

For the Thorp shuffle on \(2^d\) cards, we prove that the worst-start total-variation distance after \(32800d\) physical shuffles tends to zero as \(d\to\infty\). Combining our fixed-list estimate with the companion Fourier transfer improves this bound to \(512d\) shuffles. These results follow from bounds on partially observed permutation laws in random coordinate frames.

We prove that the Thorp shuffle on \(2^d\) cards mixes in \(\Theta(d)\) complete shuffles. The full permutation law after \(1600d\) shuffles converges to uniform in total variation as \(d\to\infty\), uniformly over the initial deck, while a support count gives a lower bound of \(2d-O(1)\).

We show how information about partial permutations controls full permutation laws. Let n tend to infinity through multiples of eight. If the images of a uniformly chosen \(7n/8\) labels approach the uniform injection law in average total variation, and the sign mean tends to zero, then the product of two independent permutations with the given law converges to uniform on S_n.

Fix half the labels in a Thorp shuffle on \(2^d\) positions and reveal their complete trajectories. We prove that, after an explicit absolute number of coordinate sweeps, the conditional permutation of the remaining labels approaches uniform in expected total variation as \(d\to\infty\), uniformly in the initial layout. The resulting constructions give full-deck mixing in a constant multiple of d physical shuffles.

We bound the information remaining after paths of specified cards in the Thorp shuffle on \(n=2^d\) cards have been observed. After a fixed number of deterministic coordinate sweeps, the joint endpoint law of further cards is close to uniform on the available positions, on average over the observed paths, provided a fixed positive fraction of labels lies outside both lists. Combined with a Fourier transfer, these bounds give full-deck mixing after \(2048d\) physical shuffles.

For coordinate sweeps with near-uniform line laws and line sizes among powers of two in a suitable fixed interval, we prove a Schatten-moment bound after conditioning on any feasible collection of prescribed card trajectories. An analytic transfer to binary sweeps then shows that a fixed number of sweeps mixes the Thorp shuffle on \(2^d\) cards in \(O(d)\) physical steps, matching the order of the support lower bound.

We prove that the Thorp shuffle on \(n=2^d\) cards mixes in \(\Theta(d)\) physical shuffles. After a sufficiently large fixed number of coordinate sweeps, the total-variation distance of the full permutation from uniform tends to zero as \(d\to\infty\).

We prove a permanent inequality with exponent strictly below two for laws on S₄ sufficiently close to uniform and with exactly uniform coordinate marginals. Applied to a four-row recursion, it shows that an absolute number of coordinate sweeps brings the full permutation of the Thorp shuffle on \(N=2^d\) cards to total-variation distance tending to zero from uniform as \(N\to\infty\). Thus the mixing time has the optimal order \(O(\log N)\) in physical shuffles.

Let A_n be a symmetric random matrix whose entries on and above the diagonal are independent signs, equal to 1 with fixed probability \(p\in(0,1)\setminus\{1/2\}\). We prove that \(\mathbb P(\det A_n=0)=(p^2+(1-p)^2+o(1))^n\). The rate is attained by the event that two rows agree.

Let A_n be a symmetric \(n\times n\) matrix whose entries on and above the diagonal are independent uniform signs. We prove \(\displaystyle \Pr(\det A_n=0)=\left(\frac12+o(1)\right)^n.\)

We prove Shelah's eventual categoricity conjecture for abstract elementary classes in ZFC. For each infinite bound on the Löwenheim–Skolem number there is a uniform threshold such that categoricity in any one cardinal at or above that threshold implies categoricity in every cardinal at or above the same threshold.

Assuming the continuum hypothesis, we construct an abstract elementary class with Löwenheim–Skolem number ℵ₀ that is categorical in every sufficiently large cardinal but has at least two nonisomorphic models of cardinality \(\beth_{\omega_2}\). Thus categoricity does not transfer down to the proposed bound \(\beth_{(2^{\aleph_0})^+}\), which equals \(\beth_{\omega_2}\) under CH. Consequently, if ZFC is consistent, the prescribed-threshold form of Shelah's categoricity conjecture is not provable in ZFC.

We prove that every order automorphism of the full partial order of Turing degrees is the identity, resolving the rigidity conjecture for the Turing degrees positively.

Every recursively enumerable set of natural-number tuples has a polynomial Diophantine representation with exactly one complete auxiliary tuple for each member. This proves the single-fold conjecture and, consequently, the finite-fold conjecture.

We prove that witnessed symmetric choice strictly increases the expressive power of choiceless polynomial time with counting. A fixed sentence with one witnessed-choice occurrence defines a Boolean query on every finite input that is not definable in the original counting formalism.

We prove that choiceless polynomial time with counting does not capture polynomial time on unordered finite structures, confirming the noncapture conjecture of Blass, Gurevich and Shelah. A linear-consistency query over 𝔽₃ in a fixed binary vocabulary is decidable in polynomial time but not in the full counting formalism.

We prove that the Partition Principle does not imply the Axiom of Choice: if ZF is consistent, then so is ZF with the Partition Principle, Choice for ordinal-indexed families, and the negation of the Axiom of Choice. Separately, over every countable transitive model of ZFC, we construct a transitive symmetric model of this theory with the same ordinals and no new countable sequences of ground elements.

We prove that every weakly β-normalizing pure type system is strongly β-normalizing. Both properties quantify over all legal expressions in all valid contexts, and reduction acts inside type annotations. No functionality hypothesis is required. This resolves the β-Barendregt–Geuvers–Klop conjecture.

We prove that every hyperbolic group whose boundary is homeomorphic to the two-sphere admits a proper cocompact isometric action on hyperbolic three-space with finite kernel. This resolves Cannon's conjecture positively.

We prove that the infinite, ordinarily finitely presented periodic Steinberg group \(\Gamma=\mathop{\mathrm{St}}\nolimits _{12}(R)\) of a companion paper is residually finite. Its finite-index subgroup \(G=\ker(\Gamma\to\mathop{\mathrm{St}}\nolimits _{12}(\mathbb F_2))\) is infinite, ordinarily finitely presented, and residually finite, and every element of G has finite 2-power order. The orders of its elements are unbounded. Phases of long words in the companion's graded algebra allow finite degree truncations to detect all elements of Γ, including the central kernel.

We construct an infinite group with an ordinary finite presentation in which every element has finite order, answering the finitely presented Burnside question negatively. We also construct an infinite-dimensional finitely presented nonunital nil associative algebra over 𝔽₂ that is Jacobson radical but not nilpotent. Its unitization is finitely presented, algebraic, and infinite-dimensional. These algebras answer the finitely presented nil- and radical-algebra nilpotence questions and the finite-presentation version of Kurosh's algebraic finiteness question negatively.

We prove that Thompson's group F is nonamenable. This confirms Geoghegan's conjecture and resolves the amenability problem for F.

We construct a finitely generated residually finite group of integral cohomological dimension two and geometric dimension three, disproving the Eilenberg–Ganea conjecture.

Every finitely generated group with decidable word problem embeds in a simple group of type F_∞. This resolves the higher-finiteness strengthening of the Boone–Higman conjecture.

We prove the Boone–Higman conjecture. A finitely generated group has decidable word problem if and only if it embeds in a finitely presented simple group.

We construct a single group of type F_∞ containing every finitely presented group. Its finitely generated subgroups, up to isomorphism, are exactly the finitely generated recursively presented groups. This answers the F_∞ form of the higher-dimensional Higman embedding question.

A countable discrete group is amenable if and only if it is strongly Ulam stable: every sufficiently accurate unitary almost representation, on any complex Hilbert space, is uniformly close in operator norm to a genuine representation on the same space. We prove the converse to Kazhdan's amenable stability theorem, answering the question of Burger, Ozawa, and Thom. The inclusion of infinite-dimensional Hilbert spaces is essential.

We prove that a discrete group is amenable if and only if every uniformly bounded representation on a complex Hilbert space is similar to a unitary representation. This resolves Dixmier's unitarizability problem affirmatively for discrete groups.

We construct a torsion-free word-hyperbolic group that is not residually finite, answering the residual-finiteness question for hyperbolic groups negatively.

We construct an infinite finitely presented simple amenable group, giving a positive answer to the finite-presentation existence question for simple amenable groups.

We prove that every intersection of parabolic subgroups of any finite-rank Artin group, with arbitrary finite or infinite Coxeter labels, is parabolic. This resolves the Parabolic Intersection Conjecture affirmatively.

We prove that the standard Salvetti complex of every Artin group with finitely many standard generators is aspherical. This resolves the Artin \(K(\pi,1)\) conjecture in finite rank.

We construct an Artin group on 116 generators that admits no proper, cocompact isometric action on a nonempty proper CAT(0) space. This refutes the CAT(0) conjecture for Artin groups.

We prove that every finitely generated group quasi-isometric to a finitely generated virtually polycyclic group is virtually polycyclic. This resolves the lattice-recognition conjecture of Eskin, Fisher and Whyte, which allows the ambient solvable Lie group in the conclusion to differ from the one in the hypothesis.

Every finite nonsingular system of equations over an arbitrary group has a simultaneous solution in an overgroup. This proves Howie's conjecture on nonsingular systems.

We prove that no free product of a nontrivial group with an infinite cyclic group is normally generated by one element. This resolves the Kervaire conjecture positively.

We construct a hyperbolic group with a finite classifying space that admits no geometric action on a proper complete CAT(0) space. Consequently, a finite aspherical simplicial complex with a linear combinatorial disk-filling inequality need not have a finite locally CAT(0) homotopy model, and hence need not have a finite locally CAT(−1) model. Here metric models carry geodesic length metrics inducing the given complex topology.

We prove that every word-hyperbolic one-relator group is virtually compact special. Combined with the work of Kielak–Linton, this resolves Wise's virtual free-by-cyclic conjecture for hyperbolic one-relator groups: every such group is virtually free-by-cyclic, with the free kernel allowed to have infinite rank.

We prove Gersten's conjecture: every finitely generated one-relator group containing no subgroup isomorphic to a Baumslag–Solitar group \(\mathop{\mathrm{BS}}\nolimits (m,n)\), for nonzero integers m, n, is word-hyperbolic.

We construct a finitely generated group with two essentially free probability-measure-preserving actions of different costs. This gives a negative answer to Gaboriau's general fixed-price problem. The group is an amalgam of a free group of rank 100 with a direct product. Its Bernoulli action has cost bounded away from one, whereas finite height extensions have costs tending to one.

Under the stated energy, decay, and trapping hypotheses, we prove the sharp charged spacetime Penrose upper-area inequality for one-ended three-dimensional initial data with source-free electric and magnetic fields. The theorem allows arbitrary second fundamental form, nonzero ADM momentum, and disconnected boundary. Writing m for invariant ADM mass, Q for total charge magnitude, and r_A for the minimum-enclosing-area radius, the bound is m ≥ Q and \(r_A\le m+\sqrt{m^2-Q^2}\). The polynomial mass bound \(m\ge(r_A+Q^2/r_A)/2\) is asserted only when \(r_A\gt Q\). When m > Q, equality under the stated connected, outermost, outer-area-minimizing future-horizon hypotheses identifies the original data as a smooth spacelike slice of dyonic Reissner–Nordström, including smooth attachment at the future horizon.

We prove the spacetime Penrose Inequality for smooth three-dimensional initial-data exteriors with one spherical asymptotically anti-de Sitter end, satisfying the stated decay and integrability assumptions, the dominant energy condition, and a compact weakly future outer trapped boundary. We assume that the hyperbolic metric four-flux is future timelike; its Lorentz norm is the mass, and the area is the infimum over full enclosing cuts. On the connected outermost, outer area-minimizing horizon subclass, equality characterizes the original data as a spacelike hypersurface in Schwarzschild–anti-de Sitter spacetime with matching metric mass. No maximality, evolution, or auxiliary solvability assumption is used.

We prove the sharp Penrose inequality for three-dimensional maximal asymptotically hyperbolic initial data with spherical conformal infinity. Under the dominant energy condition, the stated decay and integrability assumptions, and a future-timelike mass covector, the invariant mass is bounded below by the Schwarzschild–anti-de Sitter mass associated with the minimum enclosing area of a weakly future outer-trapped boundary. The boundary may be disconnected, and no restriction is imposed on the compact topology.

We prove the sharp Kerr–Newman Penrose inequality \(\displaystyle m^2\ge \frac{A}{16\pi}+\frac{Q^2}{2} +\frac{\pi(Q^4+4J^2)}{A}.\) Here \(Q^2=Q_e^2+Q_b^2\), and J is the conserved total angular momentum, including its electromagnetic contribution. The result applies to smooth axisymmetric electrovacuum exteriors in the stated one-ended topological and decay class, with a connected outermost, outer-area-minimizing future marginally outer trapped boundary, Coulomb electromagnetic asymptotics, zero ADM momentum, and the explicitly assumed physical-area condition \(A\ge4\pi\sqrt{Q^4+4J^2}\). No maximality assumption is made. On the strict area branch, equality within this class characterizes the original data as an admissible spacelike exterior slice of a subextremal dyonic Kerr–Newman spacetime, with the boundary mapped smoothly to a future-horizon cross-section or the bifurcation sphere.

We prove sharp spacetime Penrose inequalities for invariant ADM mass and minimum enclosing area in three and four spatial dimensions. Under their respective designated-end conventions, the neutral numerical results allow arbitrary second fundamental form, nonzero momentum, disconnected weakly future trapped boundary, and finitely many ends. Under the stated horizon hypotheses, equality for the neutral inequalities reconstructs the original exterior data as spacelike slices of Schwarzschild spacetime in three dimensions and Schwarzschild–Tangherlini spacetime in four; the four-dimensional equality theorem concerns a connected, one-ended exterior. We also prove a three-dimensional electric–magnetic charged upper-area inequality for one-ended data with nonzero momentum, with separate purely electric rest-frame corollaries allowing finitely many ends. Finally, for each fixed transverse-traceless seed and prescribed decaying solution branch, we prove a local Schwarzschild–anti-de Sitter inequality.

We construct smooth axisymmetric electrovacuum exteriors that violate the Kerr–Newman Penrose inequality when J is the bare gravitational ADM angular momentum and the electromagnetic fields have only \(O(r^{-2})\) decay, allowing angularly varying leading tails. The examples have zero total electric and magnetic charge even though both electromagnetic fields are nonzero. They have a connected outermost, outer-area-minimizing future marginally outer trapped boundary and lie strictly on the physical area branch. We also obtain equality examples whose original data admit no Kerr–Newman spacelike realization with the corresponding mass, angular momentum, and charges. These counterexamples do not refute formulations using conserved total angular momentum with its electromagnetic correction or stronger Coulomb asymptotics.

We prove the asymptotically hyperbolic Penrose inequality for sufficiently small maximal vacuum conformal perturbations of a positive-mass Schwarzschild–anti-de Sitter exterior, for every fixed decaying transverse-traceless seed and every solution branch satisfying the stated decay and mass assumptions. The bound uses the area of the marginally outer trapped boundary itself; it requires neither outermostness nor outer area-minimization, and no bulk-versus-boundary domination condition. For sufficiently small positive parameters, equality holds for radial seeds and the inequality is strict otherwise.

Using the companion neutral spacetime Penrose theorem exactly at its stated strong-decay, future-trapped, positive-area, and future-timelike scope, we prove the sharp purely electric upper-area inequality in every spatial dimension n ≥ 4 for one-ended charged data satisfying the charged dominant energy condition and \(\mathop{\mathrm{div}}\nolimits _g E=0\), with the specified decay, integrability, and finite-flux assumptions. The data may have arbitrary interior topology and ADM momentum, with a future or past trapping sign chosen independently on each boundary component. Positive enclosing area and strict ADM timelikeness are conclusions. The polynomial mass rearrangement is asserted only when the \((n-2)\)nd power of the enclosing-area radius exceeds \(|Q|\). When \(m\gt |Q|\), equality under the stated connected, outermost, outer-area-minimizing future-horizon hypotheses recovers the original metric, second fundamental form, and electric field as a global spacelike slice of a Reissner–Nordström–Tangherlini exterior. The equality classification includes only slices whose induced magnetic two-form vanishes.

We prove the sharp spacetime Penrose inequality for smooth one-ended initial-data exteriors in every spatial dimension n ≥ 3, under the dominant energy condition, weak future trapping, the stated differentiated decay and positive enclosing area. These hypotheses force the ADM energy-momentum to be future timelike. Area is the infimum over full enclosing cuts in the original metric. In dimensions three and four, two metric derivatives and one tensor derivative suffice, and both positive enclosing area and future timelikeness follow from the hypotheses.

Equality in the spacetime Penrose inequality identifies the original initial data as a spacelike slice of a Schwarzschild–Tangherlini exterior, smoothly attached to its horizon, under the stated outermostness and decay hypotheses. We prove this in every spatial dimension n ≥ 3 for a strong-decay exterior class and for weaker decay in dimensions three and four. In dimension three the finite-component theorem forces a single spherical horizon and gives qualitative strictness for disconnected boundaries.

We give a detailed conformal-flow proof of the numerical Riemannian Penrose inequality in every dimension n ≥ 3 for complete asymptotically flat exteriors with nonnegative scalar curvature and full compact frontiers that are outer minimizing and locally perimeter minimizing. The metric extends smoothly through the possibly singular frontier; neither spin nor frontier connectedness is assumed. The proof follows the conformal-flow approach of Bray, Bray–Lee, and Bi–Zhu.

We construct graph and conformal deformations of trapped initial-data exteriors in three and four spatial dimensions. The resulting metrics have nonnegative scalar curvature, strictly negative inner mean curvature, a lower bound protecting every enclosing cut, and an arbitrarily small upper error in ADM energy. We also give a direct four-dimensional maximal vacuum construction that retains a noncompact decaying second fundamental form. These constructions give boundary routes to the corresponding numerical Penrose inequalities.

We replace a three-dimensional Cha–Khuri–Sakovich hyperboloidal end by an asymptotically flat end while preserving the dominant energy condition and each fixed compact interior. For strictly future-timelike CKS initial-data charge, the replacements lose asymptotically no enclosing area and their ADM masses tend to the invariant Bondi mass. Combining this construction with the companion numerical spacetime Penrose theorem yields the sharp Bondi bound for possibly disconnected weakly future trapped boundaries in this class. Horizon-regular Schwarzschild exteriors attain equality at every positive mass.

For each fixed positive disorder strength, we prove that the nearest-neighbor Anderson operator on the square lattice with independent uniform site potentials almost surely has pure-point spectral type throughout its spectrum. This resolves the pure-point assertion of the two-dimensional Anderson localization conjecture for the uniform single-site law.

For each fixed dimension d ≥ 3 and each fixed sufficiently small positive disorder strength, we prove that the Anderson operator on ℤ^d with independent uniform site potentials almost surely has purely absolutely continuous spectrum with nonzero weight on an open energy interval. The interval may depend on d but is independent of the disorder strength. This settles the purely absolutely continuous energy-range question in Simon's Problem 1 for this model.

We prove the sharp one-dimensional Lieb–Thirring inequality for every finite matrix size and every exponent \(1/2\lt \gamma\lt 3/2\). The optimal constant is the scalar one-bound-state constant, independently of the matrix size. The inequality bounds the full sum of negative eigenvalue moments for every measurable Hermitian positive semidefinite potential W satisfying \(\int_\mathbb R\mathop{\mathrm{tr}}\nolimits (W^{\gamma+1/2})\lt \infty\). The matrices may have arbitrary rank and need not commute at different points.

We classify all equality cases in the sharp one-dimensional Lieb–Thirring inequality for every finite matrix size and \(1/2\lt \gamma\lt 3/2\). For measurable Hermitian positive semidefinite potentials W with \(\int_\mathbb R\mathop{\mathrm{tr}}\nolimits (W^{\gamma+1/2})\lt \infty\), equality holds precisely for direct sums, in one constant unitary basis, of scalar one-bound-state solitons and zero channels. The nonzero solitons may have independent scales and centers.

We resolve affirmatively the remaining cases of the scalar one-dimensional Lieb–Thirring conjecture: for every \(\frac12\lt \gamma\lt \frac32\), the optimal constant is the one-bound-state constant. The estimate holds for every nonnegative potential in \(L^{\gamma+1/2}(\mathbb R)\), with all negative eigenvalues included.

We prove that a molecule with M fixed nuclei of real charges at least one and total nuclear charge Z strictly binds at most \(Z+CM\) electrons, where C is universal. The result holds for the full nonrelativistic Coulomb Hamiltonian with two spin states and arbitrary distinct nuclear positions. For neutral atoms with integer nuclear charge Z ≥ 1, the first ionization energy and, for every ground state, the radius outside which one half of an electron remains are each bounded above and below by positive universal constants.

We prove the radius part of the generalized ionization conjecture for neutral nonrelativistic Coulomb atoms with two spin states. For every choice of ground states, the upper and lower large-nuclear-charge limits of the radius defined by an expected exterior electron mass m are both asymptotic to \((81\pi^2/2)^{1/3}m^{-1/3}\) as m tends to infinity.

We prove the energy part of the generalized ionization conjecture for full nonrelativistic Coulomb atoms with two electron spin states. The energy needed to remove m electrons is asymptotic to \(a_{\mathrm{TF}}m^{7/3}\) whenever \(m\to\infty\) and \(Z/m\to\infty\), with the Thomas–Fermi constant in atomic units. We also obtain both conjectured iterated limits.

For every fixed rotating subextremal Kerr bridge, we prove that smooth vacuum data admitting a C² future extension of their full maximal globally hyperbolic development form a meagre set in a neighborhood defined by one finite-order seminorm. The topology allows arbitrary symbol-bounded asymptotically flat tails and imposes no symmetry.

For each fixed rotating subextremal Kerr background with mass M > 0 and rotation \(0\lt |\mathfrak a|\lt M\), we prove that the smooth vacuum data near its complete two-ended bridge whose full maximal globally hyperbolic development admits a future \(C^0\cap W^{1,2}_{\mathrm{loc}}\) extension form a meagre set in the weighted smooth topology. The extension metric is continuous and nondegenerate, and its weak connection is locally square-integrable. The neighborhood requires smallness of only the tenth weighted seminorm, while the topology tests every finite order.

For each fixed Kerr spacetime with mass M > 0 and rotation \(0\lt \mathfrak a\lt M\), we prove that a dense G_δ set of nearby smooth, complete two-ended vacuum data has a maximal globally hyperbolic development with no future C¹ extension. The neighborhood and genericity are defined in a weighted smooth topology, and ambient extensions may be nonvacuum.

We prove that the unique ground state of a uniformly gapped nearest-neighbor Hamiltonian on an \(L\times L\) square lattice admits a projected entangled-pair state approximation with bond dimension polynomial in L and global vector error at most L⁻¹ after normalization. Only the gap of the full Hamiltonian is assumed. The result is an existence theorem, with constants uniform over Hamiltonians of fixed local dimension, interaction strength, and gap.

We prove an entropy area law for the unique ground state of a finite-range Hamiltonian on any finite induced subgraph of the square lattice. A lower bound on the spectral gap of the full Hamiltonian and fixed bounds on the local dimension, interaction range, and interaction strength suffice. For every set of sites, its entanglement entropy is bounded by a constant times the number of edges crossing its boundary, independently of the size and shape of the domain.

We prove that the maximum number of mutually unbiased orthonormal bases in ℂ⁶ is three, resolving Zauner's dimension-six MUB conjecture. The upper bound is computer-assisted: under the stated binary64 arithmetic and compiler conditions, a complete execution of the documented verification pipeline excludes four arbitrary complex bases.

We prove the Fourier-vanishing conjecture of Matolcsi, Ruzsa, and Weiner: every complex Hadamard matrix of order six outside Tao's cubic equivalence class has a vanishing character sum at every permutation of \((1,1,1,-1,-1,-1)\). We also give an exact certificate excluding seven mutually unbiased bases in ℂ⁶; by Weiner's completion theorem, this yields an upper bound of five. Both certificates use only integer and rational arithmetic, and the complete verifier is included.

We prove the Bogoliubov quantum-depletion asymptotic for the ground state of a three-dimensional Bose gas with a fixed bounded, nonnegative, radial interaction of finite range and positive scattering length a. The thermodynamic limit is taken at each fixed density ρ before the dilute limit. Every thermodynamic accumulation value of the fraction outside the constant mode is \(\frac{8}{3\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt{\rho a^3})\) as \(\rho\downarrow0\). The assertion is uniform over ground-state density matrices and does not require the occupation to have a unique thermodynamic limit.

We prove the full scaled Bogoliubov momentum distribution for the depleted particles in the three-dimensional hard-sphere Bose gas at zero temperature. The thermodynamic limit is taken at each fixed density before the dilute limit, uniformly over all pure and mixed ground states. All thermodynamic accumulation values have the same dilute asymptotic: the limiting nonzero-momentum occupation measure has total mass \(8/(3\sqrt\pi)\), giving the depletion fraction \(\frac{8}{3\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt{\rho a^3})\) for density ρ and hard-sphere exclusion distance a.

We prove Bose–Einstein condensation at positive temperature in the three-dimensional dilute hard-sphere gas. For each fixed exclusion distance and every sufficiently small fixed density, there is a strictly positive temperature, independent of the volume, at which the exact canonical Gibbs state has a positive condensate fraction in the thermodynamic limit. The condensate occupies the constant orbital.

For each fixed bounded measurable, nonnegative, radial interaction of finite range in three dimensions that is not zero almost everywhere, we prove a positive lower bound on the constant-orbital condensate fraction that is uniform over all sufficiently small densities and all temperatures between zero and the square of the density. The interaction and density remain fixed in the thermodynamic limit.

We prove Bose–Einstein condensation in every ground state of a dilute three-dimensional hard-sphere Bose gas. At every sufficiently small fixed gas parameter, the constant orbital contains a positive fraction of the particles in the thermodynamic limit. The fraction can be chosen independently of the gas parameter, and the conclusion holds for arbitrary complex ground states.

We prove a uniform positive spectral gap for the pure antiferromagnetic spin-one Heisenberg chain on even periodic rings, establishing the positive even-periodic formulation of the spin-one Haldane conjecture.

We prove a uniform spectral gap for odd open spin-one antiferromagnetic Heisenberg chains with the same fixed magnetic field at both endpoints. The field \(h=3/5\) gives a gap greater than \(\log(10)/392\) on every chain of \(2L+1\) sites with L ≥ 960. Tasaki's index theorem then gives index −1 for every subsequential local limit of these boundary-selected ground states.

We prove that the fermionic Laughlin V₁ Hamiltonian at filling 1/3 on expanding round spheres retains a unique ground state and a uniform spectral gap under sufficiently weak bounded real scalar one-body potentials projected to the lowest Landau level. With coefficient one for each pair projector and Laughlin flux \(q=3(N-1)\), the gap and perturbation threshold are uniform for all sufficiently large particle numbers and all potential profiles of supremum norm at most one. The result uses the uniform unperturbed Fock-space gap and gives existential constants.

We prove the spherical fermionic Laughlin spectral-gap conjecture for the full V₁ interaction. With coefficient one for each pair projector, the gap above the Laughlin state at filling 1/3 on the round sphere is at least 1/25 for all sufficiently large systems. More generally, \(H_Q^2\ge\gamma H_Q\) for some fixed \(\gamma\gt 1/25\) on the entire lowest-Landau-level Fock space for all sufficiently large flux Q, independently of particle number.

We prove that the relative \(\mathop{\mathrm{SU}}\nolimits (2)\) BFSS Hamiltonian has infinitely many positive eigenvalues tending to infinity, with square-integrable eigenvectors. The operator is defined by closing the gauge-invariant supercharge form. This refutes, at N = 2, the exclusion of normalizable positive-energy states stated in the original BFSS paper.

For every finite N ≥ 2, we prove that undeformed \(\mathop{\mathrm{SU}}\nolimits (N)\) BFSS matrix quantum mechanics has exactly one normalizable zero-energy state after removing the center of mass. This establishes the threshold-bound-state conjecture.

We prove the spherical magnetization law for the three-dimensional nearest-neighbor isotropic quantum Heisenberg ferromagnet at every fixed positive quantum spin and every sufficiently low fixed positive temperature. As the even periodic cubes grow, the symmetric zero-field magnetization converges in moments to a uniform direction with a deterministic positive magnitude. This magnitude equals the right derivative at zero field of the infinite-volume pressure. The law is expressed through self-adjoint linear combinations of the spin components and requires no joint measurement of noncommuting observables.

We prove the first lattice correction to Bloch's law for the three-dimensional nearest-neighbor quantum Heisenberg ferromagnet at every fixed spin \(S=\tfrac12,1,\tfrac32,\ldots\). The spontaneous-magnetization deficit agrees with the full ideal-magnon density up to \(o(\beta^{-5/2})\). In addition to the leading Bloch term, this gives the correction \(3\zeta(5/2)(\beta S)^{-5/2}/(128\pi^{3/2})\). The magnetization is the right derivative at zero field of the thermodynamic pressure; the volume limit precedes the field derivative, and the low-temperature limit is taken last.

We prove Bloch's T^3/2 law for the spontaneous magnetization of three-dimensional quantum Heisenberg ferromagnets at every fixed positive quantum spin. The result holds for every nonnegative symmetric finite-range interaction whose support generates ℤ³, including spatially anisotropic couplings. The leading magnetization deficit has the exact coefficient determined by the determinant of the quadratic one-magnon dispersion. The thermodynamic limit is taken before the right field derivative at zero, and the low-temperature limit is taken last.

For every dimension d ≥ 3 and every spin \(S\in\{\frac12,1,\frac32,\ldots\}\), we prove that the nearest-neighbor isotropic quantum Heisenberg ferromagnet has a translation-invariant, spontaneously magnetized equilibrium state at every sufficiently low positive temperature. The same state satisfies the KMS condition for the zero-field dynamics and has magnetization at least \(S/4\). This resolves the low-temperature ordering problem in the spontaneous-magnetization formulation.

We construct an entangled state on \(\mathbb C^{10}\otimes\mathbb C^{10}\) with zero distillable secret key for the local-instrument protocols specified here. They allow joint processing of all copies and unlimited two-way public communication, with the input as the only shared private resource and an eavesdropper holding a purification and the complete public record. The construction also disproves the unrestricted two-map PPT-composition conjecture. A separate trace-preserving PPT channel on \(M_{21}(\mathbb C)\) has a square that is not entanglement breaking.

We prove the entropy photon-number inequality for two independent bosonic inputs with finite mean energy, for every finite number of modes. This resolves the entropy photon-number conjecture in this finite-energy setting while allowing arbitrary entanglement among the modes within either input. As a consequence, we determine the exact minimum output entropy at fixed input entropy for every finite tensor power of an identical thermal attenuator, over finite-energy inputs that may be entangled across modes. We also obtain the exact classical capacity region of the degraded two-receiver pure-loss bosonic broadcast channel under a mean photon constraint.

We prove that constant-depth quantum circuits with arbitrary one-qubit gates and unbounded-arity Toffoli gates cannot compute parity with any fixed positive worst-case advantage using polynomially many total qubits. Ancillary qubits are initialized to \(|0\rangle\), one output qubit is measured, and all other final registers may be discarded without restriction. This resolves Moore's parity conjecture in the measured-output model.

We prove that constant-depth quantum circuits with arbitrary one-qubit and unbounded-arity Toffoli gates cannot compute parity with any fixed positive worst-case advantage using polynomially many qubits. Ancillas start in zero, only one output qubit is measured, and all final garbage is unrestricted. This resolves Moore's parity conjecture in the measured-output model.

We prove that approximating the electronic ground-energy infimum for clamped unit-charge nuclei is QMA-hard on the full spinful fermionic continuum space. A deterministic classical polynomial-time reduction produces polynomially many nuclei at distinct rational positions and polynomially many electrons, with polynomial rational bit lengths and threshold separation at least one. No orbital basis, magnetic field, or additional external potential is supplied, and no binding assumption is imposed.

We prove that approximating the electronic Coulomb spectral infimum in three-dimensional space is QMA-hard when positive integer nuclear charges are encoded in binary. The nuclei have distinct rational positions, the electron number is unary, and the energy is minimized over all antisymmetric continuum states and spin sectors. A deterministic classical polynomial-time reduction produces instances with threshold separation at least one. The nuclear charges may be exponentially large, but every output has polynomial bit length.

We determine the unassisted classical capacity of every qubit generalized amplitude-damping channel, for all damping strengths and thermal occupations. It equals the one-shot Holevo capacity and is given by a one-variable maximum. A binary pure-state ensemble attains the one-use optimum, and its independent products attain the optimum at every block length. We also prove that one-shot Holevo capacity, minimum output entropy, and regularized unassisted classical capacity are additive under tensor product with every finite-dimensional completely positive trace-preserving partner channel.

For every finite two-player game with entangled value v < 1, we prove exponential decay for the probability of winning at least a \(v+\delta\) fraction of k independent repetitions, uniformly over all finite-dimensional joint strategies. The result allows arbitrary correlated question distributions and holds for every \(0\lt \delta\lt 1-v\) and k ≥ 1. Its universal rate is proportional to \(\delta^5/(1+\log(|\mathcal A||\mathcal B|))\), where \(\mathcal A,\mathcal B\) are the answer alphabets. For each fixed question distribution, we also obtain an explicit cubic rate in δ.

We construct a finite three-electron Coulomb molecule whose spin-summed ground-state density cannot be reproduced by any ground-state ensemble of noninteracting electrons in a single real, spin-independent local potential in \(L^{3/2}(\mathbf R^3)+L^\infty(\mathbf R^3)\). The molecule has two equal positive integer nuclear charges, specified by an exact finite nonnumerical formula.

We give a polynomial-time uniform quantum circuit family that outputs the complete prime factorization of every integer N ≥ 2 with probability one. A fixed finite set of bounded-arity gates suffices, and both the gate count and the number of qubits have polynomial worst-case bounds in the input length.

Every simple unitary strongly rational vertex operator algebra generates a completely rational conformal net. All its simple grading-restricted modules are unitarizable, its canonical fusion forms are positive, and the Carpi–Weiner–Xu functor gives a braided unitary tensor equivalence from its grading-restricted finite-length module category onto the finite-index sectors of the net. Gui's extension theorems then identify normalized irreducible finite-index local extensions of these nets with simple CFT-type conformal extensions of the vertex operator algebras.

We prove that every admissible integrable minimizer of the zero-temperature Parisi functional for the pure, zero-field Sherrington–Kirkpatrick model has full relative Stieltjes support on \([0,1)\). Thus its support has no gaps at any overlap scale below one. We use the covariance normalization \(\xi(t)=t^2/2\).

We prove that the Quantum Approximate Optimization Algorithm (QAOA) approaches the ground-state energy per spin of the Gaussian zero-field Sherrington–Kirkpatrick model when system size tends to infinity first and circuit depth then increases. For every accuracy, some finite depth and deterministic angles, independent of system size and disorder, achieve that accuracy in the limiting expected energy per spin using the standard cost Hamiltonian and transverse-field mixer. This proves the eventual Parisi-optimality conjecture of Basso, Farhi, Marwaha, Villalonga, and Zhou in its fixed-parameter thermodynamic formulation. We give no quantitative bound on the required depth or efficient angle-selection procedure.

We prove that scale symmetry implies local conformal symmetry for a class of four-dimensional unitary, positive-energy quantum field theories with a Poincaré- and scale-invariant vacuum. The class has a discrete, bounded-below spectrum of scaling dimensions with finite multiplicities, and each original field has finite scaling support. The original fields and their adjoints have compatible affiliated realizations in a causally commuting bounded local net, and a physical local dilatation current generates scale transformations through its local Ward identities. The operational framework admits sharply localized reconstructed fields after their joint products and compatible affiliated realizations in the same net have been established. Under these hypotheses, the stress tensor admits a symmetric, conserved, traceless improvement with unchanged translation and Lorentz charges. The corresponding conformal currents satisfy unbroken local Ward identities on the physical field algebra. The conclusion is local; integration to a global conformal action on the whole bounded net remains a separate question.

We give a positive answer to the constant-error formulation of the Aaronson–Kuperberg unitary synthesis problem. For every n, a quantum oracle circuit generated in polynomial time from n alone can approximate the channel of every n-qubit unitary to full diamond-norm error at most 1/2, after a suitable Boolean oracle is chosen. Using the fixed gates \(H,T,T^\dagger,\mathrm{CNOT}\), the circuit has polynomially many qubits, elementary gates, and oracle calls, and its oracle queries have polynomial length. The oracle may depend on the target unitary; the theorem does not give an efficient classical procedure for constructing it.

We construct total Boolean functions with a nearly quartic separation between bounded-error randomized and quantum query complexity. Writing these complexities as \(\mathrm R(f)\) and \(\mathrm Q(f)\), the examples rule out every universal bound \(\mathrm R(f)=O((1+\mathrm Q(f))^\alpha)\) with α < 4. Thus the known quartic upper bound has the optimal exponent, disproving the conjectured cubic bound. Both complexities count worst-case bit queries with error at most 1/3 on every input.

We construct a finitely generated discrete group whose reduced group C^∗-algebra contains a projection of irrational canonical trace. Its K₀-class lies outside the image of the coefficient-free reduced Baum–Connes assembly map. This disproves the coefficient-free reduced Baum–Connes conjecture for countable discrete groups.

We construct a finitely generated torsion-free discrete group whose reduced group C^∗-algebra contains a projection other than zero and the identity. This disproves the Kadison–Kaplansky projection conjecture.

We disprove rational injectivity of the coefficient-free reduced Baum–Connes assembly map for torsion-free groups. Specifically, we construct a finitely generated torsion-free discrete group whose degree-zero assembly map has a kernel class of infinite order.

We prove stable canonical recovery for scalar-twisted group factors of ICC groups commensurable with property-\((T)\) lattices over characteristic-zero local fields. Every specified stable isomorphism to a scalar-twisted factor of any countably infinite ICC group recovers the group, the cocycle up to a normalized cochain, and equal amplification scales, with an implementing partial isometry in the given matrix corners. We compute the arithmetic correspondence subcategory for arbitrary countable ICC groups, including all bounded maps, summands, conjugates, and Connes fusion, and combine it with the companion's arithmetic exhaustion theorem. Multiplication also recovers arbitrary finite-index factor neighbors, with the exact projective condition on the matched cocycle ratio.

We prove arithmetic exhaustion for bifinite correspondences between arbitrarily scalar-twisted group factors of ICC groups commensurable with property-\((T)\) lattices over characteristic-zero local fields and twisted group factors of arbitrary countable ICC groups. Every such correspondence is a closed summand of a finite direct sum of models obtained from actual finite-index subgroup isomorphisms and finite-dimensional projective representations of the matched cocycle ratio. The proof includes reducible products, mixed real and finite places, and quaternionic and Cayley rank-one factors.

We solve the free group factor isomorphism problem affirmatively by proving that \(L(\mathbb F_2)\) and \(L(\mathbb F_3)\) are isomorphic as tracial von Neumann algebras. The classical free group factor alternative then implies that all interpolated free group factors, including \(L(\mathbb F_\infty)\), are isomorphic and have fundamental group \(\mathbb R_{\gt 0}\).

We prove that every bounded complex-linear unital algebra homomorphism from a unital complex C^∗-algebra into the bounded operators on an arbitrary Hilbert space is similar to a \(*\)-homomorphism. This resolves Kadison's similarity conjecture positively. We also obtain one universal hyperreflexivity constant for all unital von Neumann algebras on arbitrary complex Hilbert spaces.

We give a negative answer to the unrestricted small-spatial-embedding problem for one-sided near inclusions of von Neumann algebras. On separable complex Hilbert spaces, we construct pairs of unital von Neumann algebras with common identity whose one-sided gaps tend to zero. Spatial embeddings of the source into the target exist, but every implementing unitary stays a fixed positive distance from the identity. The obstruction therefore concerns small implementing unitaries, rather than the existence of spatial embeddings.

We disprove the separable C^∗-algebraic spatial form of the Kadison–Kastler conjecture. For every ε > 0, we construct unital, norm-separable C^∗-algebras on a common separable complex Hilbert space, with the same identity and Kadison–Kastler distance less than ε, that are not conjugate by any unitary. The two algebras have the same von Neumann closure.

We prove that sufficiently close unital von Neumann algebras are conjugate by a unitary arbitrarily close to the identity. The tolerance depends only on the prescribed distance of that unitary from the identity, uniformly over all algebras, representations, and Hilbert spaces. This resolves the strong Kadison–Kastler conjecture.