Mathematical physics

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.