Spacetime Penrose Inequality, Strong Cosmic Censorship Near Kerr and 8 More

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)