2 As S 1 , 2 acquire vacuum expectation values, U (1)$_{X}$ is broken. ( x$_{e}$, x$_{µ}$, x$_{τ}$ ) as well as x$_{S}$ will be chosen in such a way that a residual symmetry of B − x$_{i}$L will remain which is exactly R parity, and that realistic neutrino masses and mixing are obtained via the canonical see- saw mechanism. We note that since the Higgs doublets do not transform under U (1)$_{X}$ and S 1 , 2 do not transform under SU (3)$_{C}$ × SU (2)$_{L}$ × U (1)$_{Y}$ , the resulting Z and Z ′ X bosons do not mix. This avoids the stringent con- straint from precision electroweak measurements at the Z resonance. (See Ref. [7] and references therein.) 〈 S 1 〉 . In model (II), the N c $_{e}$N c e entry is zero, but all others are nonzero: N c $_{µ,τ}$N c µ,τ come from 〈 S 2 〉 , and N c $_{e}$N c µ,τ from 〈 S 1 〉 . Both are general enough for obtaining a realistic neutrino mass matrix, with mixing among all three lepton families. To obtain R parity (equivalently, matter parity) as a total residual symmetry from the breaking of B − x$_{i}$L using S 1 , 2 , these new singlet superfields (with L = ± 2 and B = 0) should satisfy | x$_{S}$ | = 1 / 3. Since B = 1 / 3 for quarks and L = 1 for leptons, we obtain the usual definition of R parity for all particles if 3 x$_{e,µ,τ}$ are odd integers. For a detailed discussion about conditions to get a Z$_{N}$ out of a U (1) gauge symmetry, see Ref. [8] and references therein. Flavor-dependent U (1) models have been widely stud- ied to address many issues (e.g., see Ref. [9]). Our idea of having a particular discrete symmetry out of a flavor- dependent U (1) gauge symmetry can be viewed as a use- ful guide in constraining such models. As discussed above, the requirement of a realistic neu- trino mass matrix using only S 1 , 2 demands | x$_{S}$ | = 3 in- stead of | x$_{S}$ | = 1 / 3. This means that the total discrete symmetry from B − x$_{i}$L is not just R parity any more. It has been extended to a larger symmetry. Following the general arguments in Ref. [8], we find that our total discrete symmetry is now Z 6 , which is a direct product of R parity and baryon triality ( Z 6 = R 2 × B 3 ). Un- der baryon triality ( B 3 = Z 3 ) [10], baryons transform as ω = exp(2 πi/ 3), so that the proton is absolutely stable, being the lightest particle with that charge. This result came as a pleasant surprise, because it was not our inten- tion to construct a model which contains B 3 . It points to a possible deep connection between neutrino mass and proton stability. In regard to baryon stability, there are other relevant anomaly-free U (1) gauge symmetry models [8, 11, 12], as well as discrete gauge symmetry models [10, 13, 14]. Especially, we note that when model (I) is shifted by some hypercharge, it can reach the equivalent form of a model in Ref. [11]. The general requirements discussed above would still allow an infinite number of possible models, until realistic neutrino masses and mixing are considered. If all x$_{i}$ are different for the three families of leptons, the charged lepton mass matrix and the Dirac mass ma- trix linking ν$_{i}$ with N c j are both constrained to be diag- onal. To obtain mixing among the three neutrinos, the 3 × 3 Majorana matrix spanning N c j must have enough nonzero entries. However, the only sources of such terms are S 1 , 2 N c $_{j}$N c k and N c $_{j}$N c k if x$_{j}$ + x$_{k}$ = 0. With three dif- ferent x$_{e,µ,τ}$ and just ± x$_{S}$ to work with, this is clearly impossible. We now assume that x$_{µ}$ = x$_{τ}$ , then some simple alge- bra will show that there are only two solutions (recalling that S 1 , 2 have L = ± 2): The salient prediction of our proposal is the existence of a new neutral gauge boson Z ′ $_{X}$. It does not mix with the electroweak Z at tree level, but it couples to all quarks and leptons in a specified way. In particular, it breaks e − µ − τ universality. Thus it may be important as a one-loop effect [4] in the precision measurements of Z → ℓ $^{+}$ℓ $^{−}$. However, this effect is proportional to x 2 $_{i}$, and its contribution to nonuniversality is zero for model (I). As for model (II), the prediction is that Γ( Z → e $^{+}$e $^{−}$) should be bigger than Γ( Z → µ $^{+}$µ $^{−}$) and Γ( Z → τ $^{+}$τ $^{−}$), and it is proportional to (81 − 9) g 2 $_{X}$. The present world averages are [15] Together with Eq. (3), this means that In model (I), the 3 × 3 Majorana mass matrix spanning N c j has all nonzero entries: N c $_{e}$N c µ,τ are invariant mass terms, N c $_{e}$N c e comes from 〈 S 2 〉 , and N c $_{µ,τ}$N c µ,τ come from After adding a kinematical correction of 0.19 MeV to Γ$_{τ}$, we find the deviation of Γ$_{e}$ from the average of Γ$_{µ}$ and Γ$_{τ}$ to be bounded at 95% C.L. by Neutrino sector Baryon triality e − µ − τ nonuniversality (I) x$_{S}$ = − x$_{e}$ = x$_{µ}$, (4) (II) x$_{S}$ = − x$_{µ}$ = ( x$_{e}$ + x$_{µ}$ ) / 2 . (5) (I) x$_{e,µ,τ}$ = ( − 3 , 3 , 3) , x$_{S}$ = 3 , (6) (II) x$_{e,µ,τ}$ = (9 , − 3 , − 3) , x$_{S}$ = 3 . (7) Γ$_{e}$ = 83 . 91 ± 0 . 12 MeV , (8) Γ$_{µ}$ = 83 . 99 ± 0 . 18 MeV , (9) Γ$_{τ}$ = 84 . 08 ± 0 . 22 MeV . (10) ΔΓ$_{e}$ / Γ$_{µ,τ}$ < 0 . 002 . (11)