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Mathematical physics

25 results from OpenAI’s math release. 25 explained so far. OpenAI has published Lean proofs for 17: 7 of the main theorem, 10 of part of the claim.

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• 260 Spacetime Penrose inequalities: enclosing area, charge, rotation, and anti-de Sitter extensions

  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.

  13 papers ▶Episode, 4:54 OpenAI published a Lean proof of part of the claim.

• 261 Localization and delocalization in the Anderson model

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

  2 papers ▶Episode, 4:16 OpenAI published a Lean proof of part of the claim.

• 262 Sharp finite-matrix Lieb–Thirring inequalities and all equality cases

  Proves the sharp one-dimensional Lieb–Thirring inequality for $1/2\lt \gamma\lt 3/2$ and arbitrary finite-matrix potentials W ≥ 0 with $\int\mathop{\mathrm{tr}}\nolimits (W^{\gamma+1/2})\lt \infty$: 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.

  3 papers ▶Episode, 3:47 OpenAI published a Lean proof of part of the claim.

• 263 The ionization and generalized ionization conjectures

  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 $m\to\infty$ and $Z/m\to\infty$; neutral-atom outer radii have the corresponding iterated-limit asymptotics, taking $Z\to\infty$ first.

  3 papers ▶Episode, 3:50 OpenAI published a Lean proof of part of the claim.

• 264 Strong cosmic censorship near two-ended Kerr data

  Proves local strong cosmic censorship near each fixed rotating subextremal Kerr bridge. A dense G_δ 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.

  3 papers ▶Episode, 4:03

• 265 Area laws and tensor networks for two-dimensional gapped systems

  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 $L\times L$ 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.

  2 papers ▶Episode, 3:46

• 266 Exactly three mutually unbiased bases in dimension six

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

  2 papers ▶Episode, 4:17 OpenAI published a Lean proof of part of the claim.

• 267 Positive-temperature Bose–Einstein condensation and exact quantum depletion

  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.

  5 papers ▶Episode, 4:22 OpenAI published a Lean proof of part of the claim.

• 268 The spin-one Haldane gap

  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.

  2 papers ▶Episode, 3:54

• 269 Uniform Laughlin gap and stability under bounded scalar disorder

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

  2 papers ▶Episode, 4:31 OpenAI published a Lean proof of the main theorem.

• 270 Threshold and positive-energy bound states of the BFSS matrix model

  Proves that the undeformed relative $\mathrm{SU}(N)$ BFSS model has exactly one normalizable zero-energy state for every finite N ≥ 2, resolving the threshold-bound-state conjecture. For $\mathrm{SU}(2)$, 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.

  2 papers ▶Episode, 3:41

• 271 Bloch's law, its lattice correction, and the spherical magnetization law

  Proves Bloch's T^3/2 law with its exact coefficient for three-dimensional quantum Heisenberg ferromagnets at every positive quantum spin, allowing nonnegative symmetric finite-range couplings whose support generates ℤ³. The thermodynamic limit precedes the zero-field derivative and low-temperature limit. The family also proves spontaneous magnetization for nearest-neighbor models in every dimension d ≥ 3 and determines the first lattice correction for three-dimensional nearest-neighbor couplings.

  4 papers ▶Episode, 4:09 OpenAI published a Lean proof of part of the claim.

• 272 Entanglement without distillable secret key

  Constructs an entangled state on $\mathbb C^{10}\otimes\mathbb C^{10}$ with zero distillable secret key for the specified local-instrument protocols that complete almost surely. These allow joint local processing and authenticated two-way public communication, with no other shared private resource and an eavesdropper holding the input purification and public record. A trace-preserving PPT channel on $M_{21}(\mathbb C)$ whose square is not entanglement breaking disproves Christandl's PPT-square conjecture.

  ▶Episode, 4:08 OpenAI published a Lean proof of part of the claim.

• 273 The entropy photon-number inequality

  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.

  ▶Episode, 3:28 OpenAI published a Lean proof of the main theorem.

• 274 Parity is not in QAC⁰

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

  2 papers ▶Episode, 3:32 OpenAI published a Lean proof of the main theorem.

• 275 QMA-hardness of continuum Coulomb energy

  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.

  2 papers ▶Episode, 3:42 OpenAI published a Lean proof of the main theorem.

• 276 Classical capacity of generalized amplitude damping

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

  ▶Episode, 3:21 OpenAI published a Lean proof of the main theorem.

• 277 Threshold repetition for entangled games

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

  ▶Episode, 3:25 OpenAI published a Lean proof of the main theorem.

• 278 Failure of Kohn–Sham ensemble representation

  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 $L^{3/2}(\mathbb R^3)+L^\infty(\mathbb R^3)$. This disproves Kohn–Sham ensemble representability for that potential class; the required nuclear charge is specified nonnumerically.

  ▶Episode, 3:34

• 279 Exact quantum factoring over a fixed finite gate set

  Gives a polynomial-time uniform quantum circuit family that outputs the complete prime factorization of every integer with probability one. Both gate count and qubit count are polynomial in the input length, and one fixed finite gate set suffices.

  ▶Episode, 3:16 OpenAI published a Lean proof of the main theorem.

• 280 Unitary vertex operator algebras and conformal nets

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

  ▶Episode, 3:57 OpenAI published a Lean proof of part of the claim.

• 281 QAOA attains the SK optimum in the thermodynamic-first limit

  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.

  2 papers ▶Episode, 3:39 OpenAI published a Lean proof of part of the claim.

• 282 From scale symmetry to local conformal symmetry in four-dimensional QFT

  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.

  ▶Episode, 4:12

• 283 Polynomial-time unitary synthesis from a Boolean oracle

  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.

  ▶Episode, 3:23

• 284 The optimal quartic separation between randomized and quantum queries

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

  ▶Episode, 3:27

Summary and paper titles from OpenAI’s catalog, used under the Apache License 2.0. Last checked October 19, 2026.