Artificial proofs.
Human understanding.

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 disprove the four-dimensional disc embedding conjecture without a fundamental-group hypothesis, even when no homotopy classes or output framings are prescribed. We construct a compact oriented smooth four-manifold containing finitely many disc maps with framed algebraic dual spheres whose boundary circles bound no disjoint locally flat discs. Consequently, the free group on two generators is not good in the sense of Freedman–Quinn, and neither is any group containing it as a subgroup.

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.

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.

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.

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 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 disprove the scalar-curvature extension conjecture in its unrestricted all-dimensions form. In sufficiently high dimension, we construct a Ricci flow on a closed manifold whose scalar curvature remains uniformly bounded while full curvature diverges at a finite maximal time. In one fixed sufficiently high dimension, the examples have two-sided power-law curvature blowup with arbitrarily large exponents.

The singular set of every stationary integral m-varifold in an open Euclidean set has Hausdorff dimension at most m-1, in every positive dimension and codimension. This sharp bound proves the Euclidean singular-set conjecture recorded by Brena, Decio, and De Lellis.

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.

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.

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 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.