Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More

Proves unconditional n^Ω(k) deterministic time lower bounds for joint and separate k-dimensional Weisfeiler–Leman equivalence, for sufficiently large fixed k in the specified sequential adjacency-matrix models. With dimension as input, joint equivalence is EXPTIME-complete even on subcubic graphs; deciding whether refinement identifies a graph is also EXPTIME-complete. (Lean)

Over every characteristic-zero field, the (1,1) entry of a product of n independent n× n variable matrices requires n^Θ(√n) gates in homogeneous depth-five sum–product circuits. This sharp bound allows shared gates and bottom linear forms involving all variables. (Lean)

Resolves the quasilinear PCP-for-PPAD conjecture. An End-of-Line instance of length N reduces to numerical circuit constraints of total length N(log N)^O(1) such that any polynomially encoded rational assignment satisfying all but a fixed fraction to fixed accuracy yields an endpoint solution. Such assignments always exist, giving robust local verification with only quasilinear size overhead.

Determines the halting and finite-control outcome of a fixed deterministic one-writable-tape machine up to time T using O(T^2/5 log^C(T+2)) space, improving the square-root exponent. Heads move at most one cell per step; finitely many read-only input heads are allowed. Initial contents are independent of T, and contents and input symbols have polylogarithmic-space access. Simulation time is unrestricted. (Lean)

For C2 potentials with a supplied minimizer and I ⪯ ∇²V ⪯ 2I, proves that sampling within total variation 1/10 requires only C_ε d^ε exact value-and-gradient queries for every fixed ε > 0. The bound holds on every run, with unrestricted computation between queries. A logarithmic lower bound also holds, so the optimal power-law exponent in this oracle model is zero. (Lean)

For fixed A > 0, a one-pass learner with Ad² persistent bits needs Ω_A(d log(1/ϵ)) noiseless Gaussian samples to recover a unit vector to angular error 0 < ϵ ≤ 1/10 with probability 2/3, uniformly in accuracy for large d. Computation and randomized updates are unrestricted, but output uses only the terminal state, stopping index and fresh randomness. (Lean)

Proves that the existential theory of the reals lies in the counting hierarchy. More generally, truth of existential–universal real sentences can be decided at one fixed level of that hierarchy, even when their integer polynomials are specified by arithmetic circuits.

Resolves the probability-preserving form of Banach's simple Lebesgue-spectrum problem within smooth dynamics. A smooth volume-preserving diffeomorphism of the standard-volume three-torus has simple Lebesgue spectrum on its entire complex mean-zero L2 space: the bilateral iterates of one real observable form an orthonormal basis of that space. (Lean)

Proves that every invertible mixing probability-preserving transformation is mixing of all finite orders, resolving Rokhlin's multiple-mixing problem for a single transformation. Correlations among any finite collection of measurable sets converge to the product of their measures whenever all pairwise time separations diverge. (Lean)

Resolves the near-boundary Birkhoff conjecture for smooth strictly convex planar billiards of positive curvature. Such a billiard is an ellipse whenever a full grazing annulus is continuously foliated by individually invariant essential curves. A continuous physical collar of smooth closed convex caustics also suffices.

For every self-similar measure on the line generated by finitely many contracting similarities, proves dim_Hμ=min {1,h_RW/χ}, where h_RW is the entropy rate of random composed maps and χ the average logarithmic contraction. This resolves the entropy-rate dimension conjecture without a separation assumption, allowing exact overlaps and unequal contraction ratios. (Lean)

Proves the permanence conjecture for every finite weakly reversible mass-action system with fixed positive rate constants. Every positive stoichiometric compatibility class, even an unbounded one, has a common compact convex forward-invariant absorbing set. All positive trajectories in that class therefore eventually share positive lower and finite upper concentration bounds. (Lean)

Constructs a noninvertible C1 self-map of a compact smooth manifold with zero topological entropy but eigenvalue 2 on second homology. This disproves the homological entropy lower bound for general C1 self-maps: homological growth need not force positive orbit complexity. (Lean)

Constructs a zero-entropy ergodic invertible transformation of a standard nonatomic probability space that is not measurably conjugate to any C∞ diffeomorphism preserving a strictly positive smooth probability density on a compact finite-dimensional manifold. One example rules out every finite dimension. (Lean)

Classifies singular and absolutely continuous unbiased Bernoulli convolutions for every λ ∈ (0,1) by an infinite, one-sided approximation condition using explicit finite sets of algebraic units. It also proves singularity at reciprocals of every quartic Salem number in (1,2), giving examples beyond reciprocal Pisot parameters.

Constructs a compact subset of ℝ9 that cannot be covered by ten sets of strictly smaller diameter, disproving Borsuk's covering assertion already in dimension nine. The example consists of rank-one orthogonal projectors onto lines in ℝ4, with the Frobenius metric. (Lean)

Disproves Sidorenko's conjecture with a connected bipartite pattern on 35 vertices and 66 edges that occurs less frequently than in a random graph of the same edge density. The same pattern disproves the forcing conjecture of Skokan and Thoma: matching its density and the edge density of a constant graphon need not force quasirandomness. (Lean)

Disproves Ryser's covering conjecture by constructing intersecting (q+1)-partite, (q+1)-uniform hypergraphs with covering number q+1, rather than the predicted bound q, for every sufficiently large prime q. A separate construction over extension fields also disproves Gyárfás's monochromatic tree-cover conjecture. (Lean)

Proves Hindman's finite sums and products conjecture: every finite coloring of the positive integers contains sets of any prescribed finite size whose nonempty subset sums and nonempty subset products all have one common color.

Proves the weak pinned Erdős distance conjecture: for every fixed ε > 0, all but o(n) points of any n-point planar set determine at least n^1-ε distinct nonzero distances. A complementary theorem bounds the number of unit-distance pairs by O(n^4/3-δ) for an absolute δ > 0. (Lean)

Resolves the full combinatorial invariance conjecture: isomorphic Bruhat intervals in arbitrary Coxeter systems have identical equal-parameter Kazhdan–Lusztig polynomials. Thus the abstract order of the interval determines the polynomial, even across different Coxeter systems. (Lean)

Resolves the elementary-positivity part of the Shareshian–Wachs conjecture: the chromatic quasisymmetric function of every natural unit interval graph has elementary-basis coefficients in ℕ[q]. The coefficients count explicitly described permutations, giving a combinatorial explanation of positivity. (Lean)

Proves Seymour's second-neighborhood conjecture: every nonempty finite oriented graph has a vertex with at least as many vertices at directed distance exactly two as at directed distance one. Oriented graphs may be arbitrary apart from the exclusion of loops and oppositely directed edge pairs. (Lean)

Proves that integral and fractional expectation thresholds differ by at most a universal factor, and resolves Talagrand's discrete-convexity conjecture. An application proves the Ascoli–He–Park–Talagrand graph-decomposition conjecture: every graph's edges split into a universally bounded number of fixed pieces, each with containment threshold at most a universal constant times the original graph's integral expectation threshold. The pieces' embeddings need not agree on shared vertices. (Lean)

Proves the second Kahn–Kalai conjecture: for every finite simple graph H with h ≥ 1 edges and at most n vertices, its appearance threshold in G(n,p) is at most C p_E(n,H)(1+log₂ h), with universal C. Here p_E is the least density at which every subgraph of H has expected copy count at least 1/2. (Lean)

Constructs arbitrarily large finite d-dimensional simplicial complexes, for every d ≥ 3, 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 such expanders in every positive dimension. (Lean)

Proves that real circulant Hadamard matrices exist exactly in orders 1 and 4, resolving the circulant Hadamard conjecture. Together with classical Barker-sequence results, this shows that binary sequences whose nontrivial aperiodic autocorrelations have magnitude at most 1 exist at lengths n > 1 exactly when n ∈ {2,3,4,5,7,11,13}. (Lean)

Improves the planar halving-line bound to O(n^4/3-ε) for sets with no three collinear and an absolute ε > 0. More generally, an n-point set with no three collinear has O(n(k+1)^1/3-ε₀) strictly separable k-subsets for 1 ≤ k ≤ n/2, with an absolute ε₀ > 0. The constants and positive exponents are nonquantitative. (Lean)

Proves the Erdős–Faudree–Rousseau–Schelp conjecture: R(C_m,K_n)=(m-1)(n-1)+1 for every m ≥ n ≥ 3, except R(C₃,K₃)=6. This is the exact threshold forcing a red m-cycle or a blue n-clique in every red–blue coloring of a complete graph. (Lean)

Disproves the proposed square-root bound relating a Boolean function's linear Fourier coefficients to its polynomial degree. For every C > 0, there is a sign-valued Boolean function f with ∑_if̂({i}) > C√(deg(f)). Thus its total signed correlation with individual input bits can exceed the proposed bound by an arbitrary factor. (Lean)

Proves strict positivity of Serre's intersection multiplicity χ^R(M,N) for nonzero finitely generated modules over any regular local ring, provided M⊗_R N has finite length and dim M+dim N=dim R. This resolves the positivity conjecture, including ramified mixed characteristic.

Proves e(R) ≤ e(S) for every flat local homomorphism of nonzero Noetherian local rings, where e is Hilbert–Samuel multiplicity. This resolves Lech's conjecture in every dimension and characteristic. (Lean)

Constructs a three-dimensional complete Noetherian normal local domain over ℂ with no nonzero finitely generated maximal Cohen–Macaulay module. A three-dimensional local domain essentially of finite type over ℂ has the same property, disproving the domain form of the small Cohen–Macaulay module conjecture. (Lean)

Constructs a finitely presented torsion-free group G whose group algebra 𝔽₂[G] has nonzero zero divisors, disproving Kaplansky's zero-divisor conjecture. The group has a finite two-dimensional classifying space. (Lean)

Constructs a finitely presented torsion-free nonsofic group whose group algebra over 𝔽2 is not directly finite, disproving Kaplansky's conjecture even without torsion. Companion examples give injective nonsurjective cellular automata on all configurations, refuting Gottschalk's surjunctivity conjecture. Another counterexample is an integral group-ring matrix, invertible over the rational group ring, with Fuglede–Kadison determinant strictly between zero and one, disproving the unrestricted Determinant Conjecture. (Lean)

Constructs a finite-dimensional complex algebra whose finite-dimensional modules have unbounded finite projective dimensions. This disproves the little finitistic-dimension conjecture. (Lean)

Constructs finite-dimensional algebras over a characteristic-two rational-function field that disprove the Auslander–Reiten and Gorenstein-projective conjectures, and Tachikawa's second conjecture. An associated endomorphism algebra also disproves the classical, generalized and strong Nakayama conjectures, the Auslander–Gorenstein conjecture, and the Wakamatsu tilting conjecture. The counterexamples persist under every extension of the base field. (Lean)

Proves the Eisenbud–Green–Harris and lex-plus-powers conjectures over every characteristic-zero field. Any homogeneous ideal containing a regular sequence, of arbitrary length and degrees at least two, admits a lex-plus-powers ideal with the same Hilbert function and no smaller graded Betti numbers.

Constructs a countable characteristic-zero division ring that is algebraic over its center and generated by two elements over that center, but has infinite dimension over it. This answers Kurosh's division-ring problem on local finiteness negatively.

Proves the numerical blockwise Alperin weight conjecture for every prime and every finite group: the number of irreducible Brauer characters in a block equals the number of conjugacy classes of its weights.

Proves Donovan's conjecture: over each fixed algebraically closed field of characteristic p, blocks of finite groups with bounded defect-group order have only finitely many Morita-equivalence classes, for every prime p. Also proves the integral form over each fixed complete mixed-characteristic discrete valuation ring with algebraically closed residue field.

Proves Saxl's conjecture: the tensor square of every staircase representation contains every irreducible complex representation of the corresponding symmetric group. More generally, every Sn with n ∉ {2,4,9} has an irreducible representation whose tensor square contains all irreducibles. (Lean)

Proves the ℓ1-Bass conjecture for every discrete group: Hattori–Stallings traces of idempotent matrices over ℓ¹(G) are supported on finitely many finite-order conjugacy classes. The algebraic companion proves the integral Bass trace conjecture and Kaplansky's idempotent conjecture for torsion-free groups over every commutative unital characteristic-zero domain. (Lean)

Disproves unrestricted integral Gersten injectivity in degrees 3 and 5. Two explicit two-dimensional ramified regular local rings of mixed characteristic (0,5) have nonzero integral K-theory classes that vanish over their fraction fields.

Proves the sixth case of Foulkes’ conjecture: Sym ⁶(Sym ^bV) embeds equivariantly in Sym ^b(Sym ⁶V) for every b ≥ 6 and finite-dimensional complex V. More generally, the canonical multiplication map Sym ^b(Sym ^aV) → Sym ^a(Sym ^bV) is surjective for a ≥ 2 and b ≥ a(a-1), giving dimension-independent quadratic stabilization. (Lean)

Proves that planar first-passage percolation has no doubly infinite geodesic for iid nonnegative nonatomic edge weights when the minimum of four weights has finite second moment. For exponential weights, the limit shape is strictly convex with C1 boundary. Differentiability also holds for every Gamma law with positive shape and rate. (Lean)

Proves p_c < p_u for Bernoulli bond percolation on every infinite connected locally finite nonamenable quasi-transitive graph, resolving the Benjamini–Schramm nonuniqueness conjecture. Thus there is a nonempty range of probabilities with infinitely many infinite clusters. A stronger operator bound also establishes the critical triangle condition. (Lean)

For the square-lattice nearest-neighbor cosine XY model, proves critical axis correlations C_βc(r) ∼ Ar^-1/4(log r)^1/8 and the Berezinskii–Kosterlitz–Thouless essential singularity √(β_c-β) log ξ(β) → B, with A,B > 0 after the free-box thermodynamic limit. For finite square-symmetric interactions containing nearest neighbors, discrete Gaussian heights converge to Gaussian fields throughout the rough phase, including its threshold, along geometric torus sizes. Critical center-magnetization and spin-field conclusions retain their stated height, renormalization, and field-input assumptions.

For every fixed inverse temperature β > 1, determines the fluctuation scale and limiting law of the zero-field Gaussian Sherrington–Kirkpatrick log partition function. Its variance is asymptotic to c_β n^1/3, with c_β > 0, confirming the predicted n1/6 standard-deviation scale. Exact centering and standardization give full-sequence convergence to a uniquely characterized nondegenerate law.

Weak finite-range square-symmetric even multispin perturbations of the square-lattice Ising model preserve critical bulk spin and energy limits; weak square-symmetric contour interactions also yield chordal SLE3 interface limits. With sufficiently weak iid bond disorder of any fixed bounded nondegenerate mean-zero law, critical spin interfaces converge to the same law in probability over environments. (Lean)

On ℤd, d ≥ 3, directional escape has probability zero or one for iid strictly elliptic nearest-neighbor environments, and for stationary ergodic finite-range-dependent environments under uniform ellipticity. In iid uniformly elliptic environments with d ≥ 2, almost-sure directional transience implies a deterministic limiting velocity with positive projection in that direction, resolving the ballisticity conjecture. (Lean)

Proves the Mézard–Parisi hierarchical cavity formula for Poisson-diluted even-arity Ising models satisfying the Panchenko–Talagrand factorization and positivity assumptions, with only first-moment integrability. The limiting free energy equals the infimum over finite-depth hierarchical trial laws. This includes the Viana–Bray model, symmetric diluted even-spin models, and weighted soft even-K satisfiability. (Lean)

Proves chordal SLEκ limits for critical square-lattice random-cluster interfaces for 0 < q ≤ 4, with κ=4π/arccos(-√q/2): bounded Jordan domains are allowed for q ≥ 1, and smooth Jordan domains for q < 1, under the stated marked-boundary approximations. For 1 ≤ q ≤ 4, complete nested plane loops converge to CLEκ.

Proves Cardy's formula for annealed critical Poisson–Voronoi crossing probabilities in every bounded Jordan quadrilateral. With each model normalized by its own expected unit-square pivotal count, the conditional joint near-critical crossing-threshold laws for rational polygonal quads converge in environment probability to the triangular-lattice reference law. This establishes quenched near-critical universality for crossing thresholds.

The balanced square-lattice six-vertex height field with a=b=1 and 0 < c ≤ 2 converges to a Gaussian free field, including at the endpoint c = 2. The plane state is defined by balanced-torus limits. For unit height increments and Green kernel -(2π)^-1 log|x-y|, the exact variance multiplier is 1/arcsin(c/2).

Proves the exact reconstruction threshold dλ² > 1, with nonreconstruction at equality, for three-state symmetric and four-state ferromagnetic broadcasting on regular trees (d ≥ 2) and observed Poisson trees (mean d > 1 and d > 0, respectively), with Poisson advantage averaged without conditioning on survival. The three-state theorem allows both signs of λ and gives the exact weak-recovery threshold for the symmetric three-community stochastic block model. (Lean)

Proves Gaussian free field limits on bounded smooth simply connected domains for triangular-lattice height models: uniform odd heights with increments 0,±2 and two-arc boundary values ±1, and zero-boundary integer Lipschitz heights weighted by fixed x ∈ [1/√2,1]. Uniform real Lipschitz heights also converge to a Gaussian field; at a tuned opposite-boundary amplitude, their interface converges to chordal SLE4, establishing Schramm’s real-field/interface predictions.

Identifies the joint scaling limit of Ashkin–Teller heights and both complete current-cluster collections throughout the critical line, including the four-state Potts endpoint. In bounded Jordan domains with admissible lattice approximations and wired primal/free dual boundaries, the height converges to the predicted Gaussian free field, and the clusters to canonical recursive sets of that same field, retaining every nesting depth.

Gives an exact variational formula for the limiting pressure of orthogonally invariant Ising spin glasses at every fixed temperature, both almost surely and in expectation. The coupling matrix is a Haar-random rotation of a deterministic spectrum converging to a compactly supported law, with extreme eigenvalues converging to its support edges. The zero-field ground-state energy follows as temperature tends to zero. (Lean)

Determines when the free zero-field ferromagnetic Ising state on the infinite d-regular tree is a factor of independent vertex labels: exactly when tanh β ≤ (d-1)^-1/2, including equality, for d ≥ 3 and β ≥ 0. The construction uses no root and is almost surely equivariant for each fixed tree automorphism, resolving the ferromagnetic case of Lyons's question. (Lean)

Proves Shelah's eventual categoricity conjecture in ZFC: for each bound on the Löwenheim–Skolem number, a uniform threshold makes categoricity of an abstract elementary class in one cardinal above that threshold imply categoricity throughout the same tail. Categoricity means uniqueness up to isomorphism at a given cardinality. Under the continuum hypothesis, a proposed specific Hanf threshold need not suffice. (Lean)

Every recursively enumerable set of tuples of natural numbers has a Diophantine representation with exactly one auxiliary solution for each member and none for nonmembers. This proves the single-fold conjecture and hence the finite-fold conjecture. Diophantine solvability over the nonnegative integers remains undecidable even with an at-most-one-solution promise. (Lean)

Every word-hyperbolic group with boundary homeomorphic to S2 admits a proper cocompact isometric action on hyperbolic three-space with finite kernel, proving Cannon's conjecture. Every torsion-free such group is therefore the fundamental group of a closed hyperbolic three-manifold. (Lean)

Constructs an infinite finitely presented residually finite group whose elements all have finite 2-power order, answering the finitely presented Burnside problem negatively even in this class. The construction also yields an infinite-dimensional finitely presented nil associative 𝔽2-algebra and a finitely presented infinite-dimensional algebraic unitization, giving negative answers to the corresponding nilpotence and Kurosh finiteness questions. (Lean)

Proves that Thompson's group F, the group of dyadic piecewise linear homeomorphisms of the interval, is nonamenable, resolving its longstanding amenability problem. (Lean)

Constructs a finitely generated residually finite group with integral cohomological dimension two and geometric dimension three, disproving the Eilenberg–Ganea conjecture. It has no two-dimensional classifying space, even with infinitely many cells. (Lean)

A finitely generated group has decidable word problem exactly when it embeds in a finitely presented simple group, proving the Boone–Higman conjecture. The target can have type F∞: a classifying space with finitely many cells in each dimension. A single group of type F∞ can also contain every finitely presented group. (Lean)

Constructs a torsion-free word-hyperbolic group that is not residually finite, answering the residual-finiteness question negatively. One fixed nonidentity element is killed by every finite-dimensional linear representation over every commutative field, so the group is not linear over any such field. (Lean)

Proves that every finitely generated group quasi-isometric to a finitely generated virtually polycyclic group is virtually polycyclic, resolving the Eskin–Fisher–Whyte lattice-recognition conjecture. Equivalently, a group quasi-isometric to a lattice in a connected simply connected solvable Lie group is virtually a uniform lattice in some such Lie group, possibly a different one. (Lean)

Proves Gersten's conjecture: every finitely generated one-relator group containing no Baumslag–Solitar subgroup BS(m,n), with m,n ≠ 0, is word-hyperbolic. It also proves that every word-hyperbolic one-relator group is virtually compact special.

Proves the sharp enclosing-area spacetime Penrose inequality for smooth one-ended asymptotically flat initial data in every spatial dimension n ≥ 3, under dominant energy, weak future trapping, positive enclosing area, and the stated decay assumptions. It bounds invariant ADM mass below using minimum enclosing area, with equality rigidity under additional horizon hypotheses. Charged upper-area bounds treat dyonic three-dimensional data; the higher-dimensional purely electric extension uses the matched neutral theorem. (Lean)

Proves the sharp one-dimensional Lieb–Thirring inequality for 1/2 < γ < 3/2 and arbitrary finite-matrix potentials W ≥ 0 with ∫tr (W^γ+1/2) < ∞: the optimal constant is the scalar one-bound-state value, independent of matrix size. All equality cases are direct sums, in one constant unitary basis, of scalar sech2 solitons with independent scales and centers, and zero channels. (Lean)

For the full nonrelativistic Coulomb model with two electron spin states, proves that a molecule with M fixed nuclei of charges at least one and total charge Z strictly binds at most Z+CM electrons. Neutral-atom first ionization energies and radii containing all but an expected half-electron have universal positive upper and lower bounds. The energy cost of removing m electrons has Thomas–Fermi asymptotics as m → ∞ and Z/m → ∞; neutral-atom outer radii have the corresponding iterated-limit asymptotics, taking Z → ∞ first. (Lean)

Proves local strong cosmic censorship near each fixed rotating subextremal Kerr bridge. A dense Gδ subset of a weighted smooth neighborhood of smooth complete two-ended asymptotically flat vacuum data has full maximal globally hyperbolic developments with no future continuous nondegenerate extension whose weak connection is locally square-integrable. No symmetry is imposed; extensions need not satisfy the vacuum equations.

Proves an entropy area law for unique ground states of finite-range Hamiltonians on arbitrary finite induced square-lattice domains, using only a uniform full-system spectral gap and bounds on the local interactions. On open L× L squares, uniformly gapped nearest-neighbor ground states also admit projected entangled-pair state approximations with polynomial bond dimension and global vector error at most L−1.

Proves Bose–Einstein condensation for the exact canonical Gibbs state of the three-dimensional hard-sphere gas: each fixed exclusion distance and sufficiently small fixed density admit a strictly positive temperature, independent of volume, with positive condensate fraction in the thermodynamic limit. At zero temperature, proves the Bogoliubov leading quantum-depletion law for hard spheres and fixed bounded nonnegative radial finite-range potentials of positive scattering length, taking the thermodynamic limit before the dilute limit. (Lean)

Proves the spin-one Haldane gap conjecture for the pure antiferromagnetic Heisenberg chain on even periodic rings: the spectral gap stays uniformly positive as the chain grows. A companion establishes a gap for odd open chains with endpoint field h=3/5 and gives boundary-selected infinite-volume states with topological index −1.

Proves that the undeformed relative SU(N) BFSS model has exactly one normalizable zero-energy state for every finite N ≥ 2, resolving the threshold-bound-state conjecture. For SU(2), a companion proves infinitely many normalizable positive-energy eigenstates with unbounded energies, contradicting the original BFSS paper's exclusion of additional bound states at N = 2.

Proves the entropy photon-number inequality for beam-splitter mixing of two independent finite-energy bosonic inputs in any finite number of modes: the output's entropy photon number is at least the transmissivity-weighted average of the inputs'. Arbitrary entanglement within each input is allowed, and product thermal inputs attain equality even when their entropies differ. (Lean)

Proves QMA-hardness of approximating the electronic Coulomb energy infimum in three dimensions, minimizing over the full spinful fermionic continuum space. Deterministic polynomial-time reductions work even with only unit-charge nuclei at distinct rational positions, polynomially many electrons and an energy-threshold separation of at least one. (Lean)

Constructs a three-electron Coulomb molecule with two equal positive-integer-charge nuclei whose absolute ground-state density has no noninteracting ground-state ensemble representation by a single real spin-independent local potential in L^3/2(ℝ³)+L^∞(ℝ³). This disproves Kohn–Sham ensemble representability for that potential class; the required nuclear charge is specified nonnumerically.

Proves that QAOA approaches the ground-state energy of the Gaussian zero-field Sherrington–Kirkpatrick model when system size tends to infinity before circuit depth. For every accuracy, finite depth and deterministic angles independent of size and disorder achieve the required limiting expected energy per spin. This also yields leading-order optimal expected MaxCut values on large-degree random regular graphs, with size tending to infinity before degree. (Lean)

Under the stated bounded-local-net and field-reconstruction hypotheses, proves that scale symmetry implies local conformal symmetry for four-dimensional unitary positive-energy theories with a discrete bounded-below scaling spectrum of finite multiplicity, finite scaling support and a physical local scale current. The stress tensor has a traceless improvement with unchanged spacetime charges. The conclusion concerns local Ward identities, not a global conformal action on the whole net.

Solves the constant-error Aaronson–Kuperberg unitary synthesis problem: a uniform polynomial-size quantum oracle circuit approximates every n-qubit unitary channel within diamond-norm error 1/2, after a suitable Boolean oracle is chosen. Gates, qubits, oracle calls and query length are polynomially bounded. The target-dependent oracle may have an unrestricted truth table; its efficient classical construction is not asserted.