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For $X\in L$, define the Adjoint representation of a Lie algebra by ${\mathrm{ad}}_X(Y)=[X,Y]$. The Jacobi identity gives, for every $Z$,Thus $\mathrm{ad}$ preserves brackets and is a Lie algebra representation. Its Killing form isUsing the representation identity and the cyclic property of the trace,The Killing form is also symmetric by cyclicity.

In a basis $T_a$, write $[T_a,T_b]=c_{ab}{}^d{T}_d$. The lowered structure constants satisfyThe first expression is antisymmetric in $a,b$ and the second in $b,c$. These transpositions generate all permutations, so $c_{abc}$ is totally antisymmetric.

An invariant subspace of $L$ under its Adjoint representation of a Lie algebra is exactly an ideal of a Lie algebra: $[L,I]\subseteq I$. A simple Lie algebra has no nonzero proper ideal, proving the adjoint representation is irreducible.

Put $D_X=d(X)$. The same trace calculation proves invariance of the Trace form of a Lie algebra representation:Because the $D_X$ are anti-Hermitian, $H$ is a real symmetric bilinear form. The Killing form of a simple compact Lie algebra is negative definite. Define a real linear map $T$ by $H(X,Y)=\kappa (TX,Y)$. Invariance of both forms implies that $T$ commutes with every ${\mathrm{ad}}_Z$, while symmetry makes $T$ self-adjoint for the positive inner product $-\kappa$. By the spectral theorem, its real eigenspaces are invariant under the Adjoint representation of a Lie algebra. Irreducibility forces a single eigenvalue, so $T=\mu I$ and $H=\mu \kappa$. This also avoids the extra care needed when applying the complex form of the Schur lemma to a real vector space.

The kernel of $d$ is an ideal of a Lie algebra. It cannot be all of $L$: a trivial representation of dimension greater than one is reducible. Simplicity therefore makes $d$ faithful. For every nonzero $X$,Hence the positive trace index for a compact simple Lie algebra givesin the adapted basis.

For the cubic trace tensor of a Lie algebra representation, the identityexpands toSet $Y=T_d$, $X=T_a$, $Z=T_b$, $W=T_c$. Cyclicity of $B$ converts $B_{\ell bc}$ to $B_{bc\ell }$ and $B_{a\ell c}$ to $B_{ca\ell }$. Thus the invariance identity for a cubic trace tensor is

For the final contraction, it is important to keep the specified index-raising convention. In the adapted basis write $C_{ab}{}^c=c_{ab}{}^c$. Then $c_{abc}=-C_{ab}{}^c$, while raising the last two indices with ${\kappa }^{ab}=-{\delta }^{ab}$ gives $c_a{}^{mn}=-C_{am}{}^n$. Total antisymmetry and the definition of the Killing form implyOnly the antisymmetric part of $B$ in $m,n$ contributes, soConsequently the Killing-normalized contraction of a cubic trace tensor isThe negative sign printed in the last requested identity is incompatible with raising indices by the inverse Killing form. The opposite sign is obtained if one instead defines the lowered structure constants using the positive Euclidean metric, a different convention from the one specified.

For a concrete check, take the two-dimensional representation of $\mathfrak{su}(2)$ with $T_a=-i{\sigma }_a/(2\sqrt{2})$, where the ${\sigma }_a$ are Pauli matrices. ThenTheir contraction is $+{\delta }_{ab}/8$, confirming the positive sign with $\mu =1/4$ and providing a counterexample to the printed sign.

Use ${\eta }_{\mu \nu }=\mathrm{diag}(1,-1,-1,-1)$, ${\sigma }^{\mu }=(I,\mathbf{\sigma })$ and ${\bar{\sigma }}^{\mu }=(I,-\mathbf{\sigma })$, where $\mathbf{\sigma }$ are the Pauli matrices. The translation and Lorentz generators obey the Poincare algebraOne can derive the additional relations by realizing the supercharges as differential operators on superspace. With left Grassmann derivatives and $P_{\mu }=i{\partial }_{\mu }$, takeUsing $\{{\partial }_{{\theta }^{\alpha }},{\theta }^{\beta }\}={\delta }_{\alpha }{}^{\beta }$, the two mixed derivative-coordinate terms each give $i{\sigma }_{\alpha \dot{\beta }}^{\mu }{\partial }_{\mu }$. The remaining terms cancel by Grassmann anticommutation. Equal-chirality terms cancel in the same way. HenceAll coefficients are independent of $x$, so direct differentiation gives $[P_{\mu },Q_{\alpha }]=[P_{\mu },{\bar{Q}}_{\dot{\alpha }}]=0$. This also shows that supercharges commute with translations and preserve the four-momentum of a state.

For the Lorentz generators, define the left-handed spinor matricesThe identity ${\sigma }^{\mu }{\bar{\sigma }}^{\nu }+{\sigma }^{\nu }{\bar{\sigma }}^{\mu }=2{\eta }^{\mu \nu }I$ gives their Lorentz commutation relations. Acting on the spinor superspace coordinates and their dual derivatives then gives, in a column-component convention,Here dotted index values are identified with the conjugate undotted values, and ${\bar{Q}}_{\dot{\alpha }}=Q_{\alpha }^{†}$. The second formula also follows directly by taking the adjoint of the first, using Hermitian $M_{\mu \nu }$. For example, $S_{12}={\sigma }^3/2$, so $[J_3,Q_1]=-Q_1/2$ and $[J_3,Q_2]=Q_2/2$ for $J_3=M_{12}$. This explicit component convention avoids reversing signs by mixing state matrices with operator-component matrices. Covariance of the mixed anticommutator follows by applying these matrices to ${\sigma }^{\rho }{P}_{\rho }$.

These are the minimal Super-Poincaré algebra relations. In the ordinary particle algebra a possible scalar term in $\{Q_{\alpha },Q_{\beta }\}$ would be proportional to ${ϵ}_{\alpha \beta }$, but an anticommutator is symmetric in its two labels; with only one supersymmetry such a central term vanishes. Extended algebras can instead use an antisymmetric internal index tensor. Tensor charges associated with extended objects are not part of this minimal particle algebra.

For a positive-energy massless supermultiplet, choose $p^{\mu }=(E,0,0,E)$ with $E>0$, so $p_{\mu }=(E,0,0,-E)$ andPositivity of the Hilbert-space norm implies that both $Q_1$ and $Q_1^{†}$ annihilate every state: their squared norms sum to zero. SetFor momentum along the positive third axis, helicity is $h=J_3$. The Lorentz commutators give $[h,a]=a/2$ and $[h,a^{†}]=-a^{†}/2$. Choose a highest-helicity Clifford vacuum $|\lambda ⟩$ with $a|\lambda ⟩=0$. Its partner $a^{†}|\lambda ⟩$ has unit norm if the first state does and has helicity $\lambda -1/2$. No further state is generated because $(a^{†}{)}^2=0$. ThereforeThe two states have opposite fermion parity. A CPT completion of a supermultiplet adds helicities $(-\lambda ,-\lambda +1/2)$ and conjugate internal charges when necessary. Taking $\lambda =1/2$ produces a complex scalar and a Weyl spinor after CPT completion; taking $\lambda =1$ produces the massless vector multiplet. This construction assumes ordinary finite-helicity particle representations, rather than continuous-spin representations.

Represent the pure states by density matrices ${\rho }_u=(I+u\cdot \sigma )/2$ and ${\rho }_v=(I+v\cdot \sigma )/2$. The Pauli matrices obey $\mathrm{Tr}{\sigma }_i=0$ and $\mathrm{Tr}({\sigma }_i{\sigma }_j)=2{\delta }_{ij}$. ConsequentlyOn the other hand, ${\rho }_u=|u⟩⟨u|$ and ${\rho }_v=|v⟩⟨v|$ make that trace equal to $⟨u|v⟩⟨v|u⟩$. Thus the pure-qubit overlap identity isFor mixed states the trace formula still holds, but its left side is then a Hilbert–Schmidt overlap rather than this squared pure-state overlap.

For each hidden variable $\lambda$, a deterministic local hidden-variable model assigns values $a_i(\lambda ),b_j(\lambda )\in \{0,\dots ,d-1\}$ to all settings. The responses depend only on the local setting; their averaging distribution is setting independent, as required by measurement independence.

Let $T(\lambda )$ be the sum of the nonnegative modular representatives in the chained modular Bell inequality. Before reduction modulo $d$, the chain telescopes:Reduction changes this integer only by multiples of $d$. Since all reduced terms are nonnegative,The smallest possible nonnegative value in this residue class is $d-1$. Averaging over the hidden variable provesAdding local random seeds to $\lambda$ gives the same result for stochastic local hidden-variable theories.

For the coplanar qubit realization of the chained Bell inequality, shareAt angle $\theta$ in the $xz$ plane, use the projective measurement of the observableHere ${\sigma }_x,{\sigma }_z$ are Pauli matrices, and the classical bit $a$ records eigenvalue $(-1{)}^a$. Since $⟨{\sigma }_z\otimes {\sigma }_z⟩=⟨{\sigma }_x\otimes {\sigma }_x⟩=1$ and both crossed expected values vanish in this Bell state,The one-party expected values vanish, so the complete joint probabilities areFor binary outcomes, $[a-b]$ is one exactly when $a\ne b$, while $[b-a-1]$ is one exactly when $a=b$. Consequently

For $N\ge 2$, set $\delta =\pi /(2N)$ and choose Alice's angle ${\theta }_i=2(i-1)\delta$ and Bob's angle ${\varphi }_i=(2i-1)\delta$. Every adjacent pair $A_i,B_i$ or $B_i,A_{i+1}$ differs by $\delta$, giving expectation ${\sin }^2(\delta /2)$. The closing pair has angle difference $(2N-1)\delta =\pi -\delta$; its agreement probability is also ${\sin }^2(\delta /2)$. Thus every one of the $2N$ terms contributes equally:Indeed $\sin x<x$ for $x>0$, so this value is below ${\pi }^2/(8N)\le {\pi }^2/16<1$. For an explicit two-setting instance, Alice measures ${\sigma }_z,{\sigma }_x$ and Bob measures $({\sigma }_z+{\sigma }_x)/\sqrt{2},(-{\sigma }_z+{\sigma }_x)/\sqrt{2}$. The four relevant expected values sum toviolating the local bound one. At $N=1$ there is no violation; the two terms always sum to one. As $N\to \infty$, the displayed quantum value is asymptotic to ${\pi }^2/(8N)$ and approaches the algebraic lower bound zero.

The violation rules out a local hidden-variable theory with setting-independent preassigned local responses reproducing these quantum correlations. It does not permit faster-than-light communication: the joint formula gives each local outcome probability $1/2$, independent of the remote setting, as required by the no-signalling theorem. The contradiction concerns the joint correlations and the Bell assumptions, not the ability to send a signal.

The PDF's first Pauli term contains an undefined ${\rho }_x$. For the stated depolarizing channel, use ${\sigma }_x\rho {\sigma }_x$, consistently with the other two terms. This is the minimal typographical correction needed to define the intended Pauli channel.

Write $\rho =\frac{1}{2}(I+\mathbf{s}\cdot \mathbf{\sigma })$ using its Bloch vector. The Pauli matrices satisfy ${\sigma }_j{\sigma }_k{\sigma }_j={\sigma }_k$ for $j=k$ and $-{\sigma }_k$ for $j\ne k$. Thus conjugating by each nonidentity Pauli matrix leaves one Bloch component unchanged and reverses the other two. Summing the three conjugates givesSubstitution yieldsso the Pauli-mixture parametrization of qubit depolarization, with $p$ here the total error probability, givesThere is also a genuine range error in the printed claim. For its full $0<p<1$ assumption, $0<q<4/3$, and $0<q<1$ holds exactly when $0<p<3/4$. For example $p=9/10$ gives $q=6/5$; on $|0⟩⟨0|$ the output is $\mathrm{diag}(2/5,3/5)$, with negative $z$ Bloch component. No $q\in (0,1)$ could produce this output from that input, since its component would be $1-q>0$.

The corrected channel is still completely positive in the larger range: its original Pauli probabilities are $1-p,p/3,p/3,p/3$. This is depolarizing noise weight above one; the formula with $q>1$ is not a convex mixture with weights $1-q,q$, but it has a valid random unitary channel representation. The subsequent parts can therefore be solved both in the printed $q\in (0,1)$ regime and in the actual parameter range.

The Pauli matrices act by $X|0⟩=|1⟩$, $X|1⟩=|0⟩$, $Z|0⟩=|0⟩$, and $Z|1⟩=-|1⟩$. Consequently $X\otimes X$ swaps $00$ with $11$ and $01$ with $10$. The plus combinations are its $+1$ eigenvectors, and the minus combinations its $-1$ eigenvectors. Meanwhile $Z\otimes Z$ is $+1$ on the even-parity basis vectors and $-1$ on the odd-parity ones. Thus, with the paper's labels,This verifies all four Bell states directly. The two observables commute: $XZ=-ZX$ on each factor, so the two minus signs cancel in their tensor products. Their joint eigenvalues distinguish the four states, as used in a Bell-basis measurement.

Write $|0⟩=|\uparrow ⟩$, $|1⟩=|\downarrow ⟩$, and use $X_j,Y_j$ for the Pauli matrices on particle $j$. Their actions on the computational basis areSet $P_1=X_1{Y}_2{Y}_3$, $P_2=Y_1{X}_2{Y}_3$, $P_3=Y_1{Y}_2{X}_3$, and $P_4=X_1{X}_2{X}_3$. Any two distinct operators in this list differ by an $X$ versus $Y$ at exactly two sites. At each such site $XY=-YX$, while factors on different particles commute. Swapping the two full products therefore introduces two minus signs, givingThis is the Pauli-string commutation parity rule. Each product is Hermitian and squares to the identity, so its only eigenvalues are $\pm 1$.

For the negative-phase Greenberger–Horne–Zeilinger state, put $|G_{-}⟩=(|000⟩-|111⟩)/\sqrt{2}$. The two $Y$ factors in each of $P_1,P_2,P_3$ contribute $i^2=-1$ on $|000⟩$ and $(-i{)}^2=-1$ on $|111⟩$. Thus these operators take $|000⟩$ to $-|111⟩$ and $|111⟩$ to $-|000⟩$. By contrast, $P_4$ flips these two basis states without a phase. ConsequentlyThe GHZ state is therefore a simultaneous eigenstate of the four commuting products. As a sign check, direct multiplication givesThis is compatible with the displayed quantum eigenvalues.

Now apply the EPR criterion of reality together with locality for the spacelike separated particles. In the context with local settings $X_1,Y_2,Y_3$, the product of the three outcomes is certainly $+1$. Therefore the two remote $Y$ outcomes predict the first particle's $X$ outcome with certainty, without disturbing it under the locality assumption. The other perfect-correlation contexts similarly predict each particle's $X$ or $Y$ outcome from measurements performed only on the other particles. The EPR criterion of reality then assigns predetermined local values $x_j,y_j\in \{+1,-1\}$ to these six observables. Locality makes the value assigned at one site independent of which measurements were chosen at the other sites.

To reproduce the three certain mixed-setting quantum predictions, every such assignment must satisfyMultiplying these scalar equations cancels each squared $y_j$ and gives $x_1{x}_2{x}_3=+1$. But the all-$X$ quantum prediction, from the $P_4$ eigenvalue, is $x_1{x}_2{x}_3=-1$ with certainty. HenceThe perfect GHZ correlations contradict the simultaneous local elements of reality inferred by the EPR argument. No statistical Bell inequality is needed. The locality assumption is essential to that inference; the certainty criterion without the premise that distant choices cannot disturb the local element does not by itself imply a local hidden-variable model. Nor does the argument require simultaneous measurement of $X$ and $Y$ on a single particle: different experimental contexts establish the perfect predictions, and it is the EPR/locality interpretation that attempts to assign all their values at once.

The individual rates describe population relaxation: ${\gamma }_{12}$ transfers population from level one to level two, and ${\gamma }_{21}$ transfers it back. With $p_1+p_2=1$ and the conventional qubit coordinate $r_z=p_1-p_2$,Thus $T_1^{-1}={\gamma }_{12}+{\gamma }_{21}$ is the longitudinal relaxation rate. The parameter $\Gamma =T_2^{-1}$ is the transverse relaxation rate: it damps the off-diagonal coherence, or the $x,y$ components of the Bloch vector. In the usual completely positive two-level model,so $\Gamma$ includes both the coherence loss caused by population relaxation and additional pure dephasing.

There is a normalization switch in the displayed qubit Affine Bloch equation. Its constant term is appropriate to the conventional Pauli matrices and unit-radius coordinate $r_k=\mathrm{Tr}(\rho {\sigma }_k^{\mathrm{Pauli}})$. With the orthonormal generators used earlier, $s=r/\sqrt{2}$, the same physical rates would instead give a constant term $({\gamma }_{21}-{\gamma }_{12})/\sqrt{2}$. The drift matrix is unchanged. We interpret the supplied qubit equation in its conventional $r$ coordinates.

With zero controls and $g={\gamma }_{12}+{\gamma }_{21}>0$, $\Gamma >0$, the steady state is unique:The earlier normalized vector is $s_{\ast }=r_{\ast }/\sqrt{2}$. Indeed $r_x,r_y$ decay as $e^{-\Gamma t}$ and $r_z-r_{\ast ,z}$ as $e^{-gt}$.

For vanishing rates, the full equilibrium conditions of the Affine Bloch equation are $\Gamma r_x=\Gamma r_y=0$ and $g{r}_z={\gamma }_{21}-{\gamma }_{12}$, together with $\Vert r\Vert \le 1$. In particular, for nonnegative rates, $g=0$ means both population rates vanish. If then $\Gamma >0$, all diagonal density operators are stationary; if also $\Gamma =0$, every density operator is stationary. The formal case $g>0$, $\Gamma =0$ permits extra transverse equilibrium coordinates, but is excluded by the completely positive rate bound above.

The Pauli matrices obey $(\mathbf{n}\cdot \mathbf{\sigma }{)}^2=I$. Expanding the exponential into even and odd powers gives the time-evolution operatorFor orthogonal states the identity contribution vanishes, so the transition probability is $|⟨{\chi }^{\prime }|\mathbf{n}\cdot \mathbf{\sigma }|\chi ⟩{|}^2{\sin }^2(\gamma Bt/2)$. For spin up to spin down in a static field $(B_x,0,B_z)$, only ${\sigma }_x$ has the required matrix element. Hence

For the weak rotating transverse field, take $H_0=-\hslash \gamma B_z{\sigma }_z/2$. The unperturbed energies give $E_{\downarrow }-E_{\uparrow }=\hslash \gamma B_z$. The perturbation matrix element is $⟨\downarrow |V(t)|\uparrow ⟩=-\hslash \gamma A{e}^{i\alpha t}/2$. Thus first-order perturbation theory givesSquaring its magnitude yieldsThe hypothesis is $|\gamma A|\ll |\Omega |$, excluding resonance. At $\alpha =0$ this is the leading weak-$A$ expansion of the exact static-field formula with $B_x=A$. For extremely long times, a small higher-order frequency correction can accumulate phase, so first-order perturbation is not a uniform approximation on arbitrarily long time scales.

Use metric $g_{\mu \nu }=\mathrm{diag}(1,-1,-1,-1)$ and the hypercharge convention $Q=T^3+Y$. Then the Higgs doublet has $Y=1/2$ and transforms under electroweak gauge invariance asThe covariant derivative sign consistent with the requested expression isA convention that instead calls the doublet hypercharge one puts $Y/2$ in the charge generator and derivative; the physical coupling is the same. Here $B_{\mu }\mapsto B_{\mu }-{\partial }_{\mu }\beta /g^{\prime }$ and the matrix-valued $SU(2)$ connection transforms as $A_{\mu }\mapsto U{A}_{\mu }{U}^{-1}+(i/g)({\partial }_{\mu }U)U^{-1}$, making $D_{\mu }\varphi$ transform like $\varphi$.

In unitary gauge, the upper component of the derivative comes from the off-diagonal Pauli matrices:The lower component isWriting $c_W=\cos {\theta }_W$, $s_W=\sin {\theta }_W$, inversion of the neutral rotation gives $A^3=c_{W}Z+s_{W}A$ and $B=-s_{W}Z+c_{W}A$. Since $g^{\prime }=g\tan {\theta }_W$, the coefficient of the photon vanishes, whereas $-g{c}_W-g^{\prime }{s}_W=-g/c_W$. ThusThe zero photon coupling expresses the neutrality of the surviving radial Higgs boson.

The scalar Lagrangian density is ${\mathcal{L}}_{\varphi }=(D_{\mu }\varphi {)}^{†}{D}^{\mu }\varphi -V$. Adding $-F_{\mu \nu }^a{F}^{a\mu \nu }/4-B_{\mu \nu }{B}^{\mu \nu }/4$ gives the complete bosonic electroweak Lagrangian density. Its scalar and vector mass terms, after electroweak symmetry breaking, are obtained directly asIndeed, the shifted quartic Higgs potential normalization gives ${\varphi }^{†}\varphi -v^2/2=vH+H^2/2$. Comparing with a canonically normalized real scalar mass term, a complex vector mass term and a real vector mass term yieldsThe extra factor one half in this paper's Higgs potential is why $M_H^2$ is $\lambda v^2$, rather than $2\lambda v^2$ in another common convention. The tree-level electroweak gauge-boson masses give $\cos {\theta }_W=M_W/M_Z$: measure the two vector masses and take their ratio. Beyond tree level this extraction requires a specified renormalization convention and radiative corrections.

Expanding the vector terms gives the cubic and quartic Higgs couplings to electroweak gauge bosons:The four elementary Feynman diagrams and their all-incoming Feynman rules are shown below. Differentiating with respect to two identical $Z$ fields supplies a factor two; differentiating with respect to two identical Higgs fields supplies another factor two in the quartic rules.

There are no elementary $H\gamma \gamma$, $HZ\gamma$, $HH\gamma \gamma$ or $HHZ\gamma$ vertices in this tree-level Lagrangian density. Loop-induced photon interactions are a different order. The scalar self-interactions, if included, have rules $HHH:-3i{M}_H^2/v$ and $HHHH:-3i{M}_H^2/v^2$.

The prime LEP2 search used the Higgs boson coupling to Z bosons through Higgsstrahlung, $e^{+}{e}^{-}\to Z^{\ast }\to ZH$, shown in the first panel of the process figure in Question 3. This identifies an associated-production interaction, rather than an on-shell decay of a single $Z$ into a heavier Higgs. The channel's historical role is documented in the LEP2 Higgs-search account.

For the free charged W boson, take $W_{\mu }(x)=\int d^{4}p{e}^{-ipx}{W}_{\mu }(p)/(2\pi {)}^4$, with the conjugate field carrying $e^{ipx}$. In the action, multiplying the two field strengths gives $2[p^2{W}_{\mu }^{†}{W}^{\mu }-(p\cdot W^{†})(p\cdot W)]$. ThereforeTo invert it, separate the transverse and longitudinal projectors $P_T=g-pp/p^2$, $P_L=pp/p^2$. The operator is $(p^2-M_W^2)P_T-M_W^2{P}_L$, soSince the quadratic action kernel is $-O$, its Proca propagator isFor all momentum components small relative to $M_W$, the low-momentum limit is $D_{\mu \nu }\simeq i{g}_{\mu \nu }/M_W^2$, with corrections of relative order $p^2/M_W^2$ or $p_{\mu }{p}_{\nu }/M_W^2$. This local limit leads to the Fermi interaction, with $G_F/\sqrt{2}=g^2/(8M_W^2)$ for the charged-current normalization used here.

The Pauli matrices obey $X|0⟩=|1⟩$, $X|1⟩=|0⟩$, $Y|0⟩=i|1⟩$, and $Y|1⟩=-i|0⟩$. Thus $X\otimes X$ and $Y\otimes Y$ contribute opposite amplitudes on $|00⟩,|11⟩$ and equal amplitudes on $|01⟩,|10⟩$. For the XY exchange interaction $E=(X\otimes X+Y\otimes Y)/2$,Equivalently $E=|01⟩⟨10|+|10⟩⟨01|$. It transfers an excitation across a link without creating or destroying excitations, so a sum of such links preserves the single-excitation subspace.

The EPR criterion of reality proposes that a quantity has an element of physical reality if its value can be predicted with certainty without disturbing the system. To use remote predictions this way, one additionally assumes locality: the choice and execution of a measurement in quantum mechanics on separated particles cannot disturb the local physical properties of the remaining particle. The contradiction below is with this combination of the criterion and locality, rather than with a definition of reality alone.

Use the computational basis $|0⟩=|\uparrow ⟩$, $|1⟩=|\downarrow ⟩$ and the conventional Pauli matricesFor each site $j$, let $O_j$ be the tensor product of $X$ at that site and $Y$ at all other sites. For $N=4M+3$, the number of $Y$ factors is $N-1=4M+2$, soThe minus sign in the given GHZ state then impliesHence every one-$X$, remaining-$Y$ product has measurement outcome $+1$ with certainty, while the all-$X$ product has outcome $-1$ with certainty. These are experimentally separate local Pauli measurements on identically prepared copies, not joint measurements of $X$ and $Y$ on the same particle.

For a chosen site, the other $N-1$ sites can determine its $X$ outcome with certainty by measuring $Y$ and using the corresponding $O_j$ relation. They can determine its $Y$ outcome with certainty by selecting an $O_{\ell }$ with $\ell \ne j$, measuring $X$ at $\ell$ and $Y$ at all other remote sites. Since $N\ge 3$, both choices are available. Locality and the EPR criterion of reality therefore assign each site definite values $x_j,y_j\in \{+1,-1\}$, independent of which remote choice is made, even though quantum theory does not jointly measure the local $X$ and $Y$.