Average the four correlated local Pauli operations to remove coherences in the Bell states basis. A one-party Pauli operation can permute the largest of the four resulting weights into the singlet sector; that weight is at least $1/4$. Correlated common local unitaries then permute the three triplet projectors, while collective-unitary covariance of the two-qubit singlet keeps the singlet projector unchanged. Averaging all six permutations makes the triplet weights equal, producing a two-qubit Werner state. Shared randomness or local operations and classical communication coordinates these local operations. Common unitaries alone cannot increase an initially small singlet weight, because that weight is invariant under them.

For any normalized pure two-qubit vector, its squared overlap with a maximally entangled state is at most $(1+C)/2$: in a Schmidt decomposition the overlap is bounded by $(\sqrt{\lambda }+\sqrt{1-\lambda })/\sqrt{2}$. Averaging gives $C(W_F)\ge 2F-1$. Equality follows from the six-state ensemble $\sqrt{F}|{\Psi }_{-}⟩\pm \sqrt{1-F}|t_j⟩$, with equal weights, where $t_1={\Psi }_{+}$, $t_2={\Phi }_{-}$ and $t_3=i{\Phi }_{+}$. The coherences cancel and each coefficient determinant gives pure concurrence $2F-1$.

Write $W_F=(I\otimes I-w{\sum }_j{\sigma }_j\otimes {\sigma }_j)/4$, with $w=(4F-1)/3\in (1/3,1]$. For positive matrices $X=A^{†}A=a_{0}I+a\cdot \sigma$ and $Y=B^{†}B=b_{0}I+b\cdot \sigma$, the success probability is $p=a_0{b}_0-wa\cdot b$. It is at least $a_0{b}_0-|a||b|\ge \sqrt{(a_0^2-|a{|}^2)(b_0^2-|b{|}^2)}=|\det A\det B|$. Squaring the middle comparison leaves $(a_0|b|-b_0|a|{)}^2\ge 0$.

A geodesic on the embedded two-dimensional de Sitter spacetime is the intersection with a two-plane through the ambient origin. For initial point $P$ and tangent $V$ satisfying $P^2=-1$, $P\cdot V=0$, $V^2=\epsilon$, its affine parametrization is $P\cosh s+V\sinh s$ when timelike, $P+sV$ when null and $P\cos s+V\sin s$ when spacelike. These solve $X^{\prime \prime }=\epsilon X$. The spacelike geodesics are closed; timelike geodesics and null geodesics extend to both conformal infinities with infinite affine length.

The conformal time $T=2\arctan (e^t)-\pi /2$ satisfies $\sin T=\tanh t$ and $\cos T=\mathrm{sech}t$. Multiplying the physical metric by ${\cos }^{2}T$ gives a finite conformal compactification with periodic spatial coordinate. Its past and future conformal boundaries are spacelike circles. Null geodesics have slopes $d\chi /dT=\pm 1$.

The unrestricted spatial coordinate unwraps the spatial circle of two-dimensional de Sitter spacetime. The conformal time $\eta =\arcsin (\tanh t)$ lies in $(-\pi /2,\pi /2)$, and null geodesics have slopes $d\xi /d\eta =\pm 1$. Constant-$\eta$ lines are Cauchy hypersurfaces: along any causal curve $|d\xi /d\eta |\le 1$, so a curve cannot escape to spatial infinity in a bounded conformal-time interval. The covering spacetime is therefore globally hyperbolic, despite its observer-dependent cosmological horizons. Identifying $\xi$ modulo $2\pi$ gives the de Sitter cylinder, also globally hyperbolic.

On the universal cover, $-\pi /2<\psi <\pi /2$ and $t\in \mathbb{R}$. The two conformal boundaries are timelike. Null geodesics are $t\pm \psi =\text{constant}$ and can travel from either boundary to the center in finite coordinate time, despite taking infinite physical affine parameter to reach the boundary. There is no event horizon for the complete central static observer. The spacetime is not globally hyperbolic, unlike the global conformal cylinder of two-dimensional de Sitter spacetime; well-posed field evolution requires boundary conditions as well as initial data.

A geodesic of the unit embedded hyperboloid lies in the plane spanned by its initial point and tangent. The constraint $X^2=1$ forces the displayed ambient acceleration. Timelike geodesics are $X=P\cos s+V\sin s$, null ones are $X=P+sV$, and spacelike ones are $X=P\cosh s+V\sinh s$. Timelike geodesics are closed on the hyperboloid but lift to complete nonclosed curves on its universal cover. In static coordinates, conserved $E={\cosh }^{2}rdt/ds$ gives $(dr/ds{)}^2=E^2/{\cosh }^{2}r-1$ and $\sinh r=\sqrt{E^2-1}\sin (s-s_0)$. Null affine parameters obey $\lambda =\pm \sinh r/E+\text{constant}$, proving completeness at both spatial ends.

Put each successive item into a currently lighter bin. Before receiving weight $w_j$, that bin contains at most half the total weight already assigned, hence at most $(s-w_j)/2$, where $s$ is the overall total. Apply this to the last item in a finally heaviest bin to get $\mathrm{ALG}\le s/2+w_{\max }/2$. Since $\mathrm{OPT}\ge \max (s/2,w_{\max })$, the approximation ratio is at most $3/2$. Ordered weights $1,1,2$ give loads $3,1$ rather than the optimal $2,2$, proving the bound is sharp. Its relative-error guarantee is $\epsilon =1/2$.

For integer weights, subset-sum dynamic programming finds the greatest achievable total at most $s/2$ in time $O(ns)$, where $s$ is the total weight. For rational weights and $0<\epsilon <1$, scale by $K=\epsilon s/(2n)$ and round down to $q_i=\lfloor w_i/K\rfloor$. The scaled total is at most $2n/\epsilon$, so exact dynamic programming costs $O(n^2/\epsilon )$ arithmetic operations. Lifting its partition to the original weights increases the maximum load by at most $nK\le \epsilon \mathrm{OPT}$. Thus it is a fully polynomial-time approximation scheme for this two-bin problem. Bit complexity also accounts for the rational input and epsilon encodings.

The projective space of one-dimensional complex subspaces of twistor space. A point corresponds to an alpha-plane in complexified compactified Minkowski space; a projective line corresponds to a spacetime point.

For a fixed affine spacetime point $x$, this linear relation cuts out a two-dimensional subspace $W_x=\{(ix\pi ,\pi )\}$ of twistor space. Projectivizing gives its twistor line. For a fixed projective twistor with $\pi \ne 0$, its solutions are $x=x_0+u^A{\pi }^{A^{\prime }}$, a totally null affine alpha-plane. The compact description replaces affine solutions by the two-planes containing the fixed twistor line in the vector space.

For $Z=(\omega ,\pi )$ and $W=(\eta ,\rho )$ the displayed Hermitian form has signature $(2,2)$. It is zero on every pair of vectors in $W_x=\{(ix\pi ,\pi )\}$ when $x$ is Hermitian, that is, Lorentzian real. Conversely a transverse totally isotropic two-plane is such a graph with $x$ Hermitian. This gives the Lorentzian reality structure on twistor space.

Choose an antilinear quaternionic structure $J$ on ${\mathbb{C}}^4$. It induces a fixed-point-free involution on projective twistor space. A real Euclidean spacetime point is a $J$-invariant complex two-plane, equivalently a quaternionic line in ${\mathbb{H}}^2$. These form ${\mathbb{HP}}^1\cong S^4$, the conformal compactification of Euclidean space. Every projective twistor lies on the unique invariant line $\mathbb{P}{\mathrm{span}}_{\mathbb{C}}(Z,JZ)$, giving a fibration ${\mathbb{CP}}^3\to S^4$ with fiber ${\mathbb{CP}}^1$.

The four-complex-dimensional Grassmannian of two-planes in twistor space. Its Plücker embedding is the Klein quadric. An affine chart consists of graphs $W_x=\{(ix\pi ,\pi )\}$. A Lorentzian reality structure on twistor space or an Euclidean reality structure on twistor space selects different four-real-dimensional compact spacetimes inside this same complex manifold.

The projective line associated to a two-plane $W_x$ representing a point of complexified compactified Minkowski space. Two distinct twistor lines intersect precisely when their spacetime points are null separated. On the Lorentzian real slice, the line lies entirely in projective null twistor space and parametrizes the null geodesics through the point.

An alpha-plane in the compact twistor correspondence is the set of two-planes containing a fixed line $\ell$ in twistor space. Its affine portion has tangent vectors $u^A{\pi }^{A^{\prime }}$ for fixed primed spinor $\pi$, so any two tangent vectors are orthogonal and null. Two distinct compact alpha-planes meet at the unique two-plane spanned by their defining lines; that point can lie at conformal infinity.

A holomorphic vector bundle on an appropriate twistor domain, trivial on its space-time incidence lines, determines a complex anti-self-dual gauge gauge connection. Split its patching matrix on each line and apply the tangent vector fields ${\pi }^{A^{\prime }}{\partial }_{A{A}^{\prime }}$. They annihilate the unsplit twistor patching matrix, so the two logarithmic derivatives of the factors agree and define a global degree-one function of $\pi$. It is consequently ${\pi }^{A^{\prime }}{A}_{A{A}^{\prime }}$. The inverse splitting bundle frame is parallel along each alpha-plane; commuting these parallel equations proves that the primed gauge curvature two-component spinor vanishes. Reality conditions recover unitary Euclidean gauge fields; Euclidean instanton bundles need triviality on the real twistor lines, not on every possible complex line.

Modulo the three quadric relations of the twisted cubic, replace $x_1^2$ by $x_0{x}_2$, $x_2^2$ by $x_1{x}_3$, and $x_1{x}_2$ by $x_0{x}_3$. Each replacement reduces the total exponent of $x_1,x_2$. The resulting monomials are $x_0^a{x}_3^b$, $x_0^a{x}_1{x}_3^b$, and $x_0^a{x}_2{x}_3^b$. In degree $m$ their images under $x_i\mapsto s^{3-i}{t}^i$ are all $3m+1$ distinct monomials of degree $3m$. They are linearly independent, so the three quadrics generate the full homogeneous ideal, not merely its radical. The quotient is the third Veronese subring of $k[s,t]$.

The sum of reciprocals of twin primes converges, whether or not there are infinitely many twin primes. The twin-prime upper bound from a quadratic sieve and reciprocal-sum convergence from a counting bound prove it. This contrasts with the Euler proof that the sum of reciprocals of primes diverges.