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Left-right symmetry breaking in $E_6^L\times E_6^R$ occurs only in spacetime -- with possible implications for strong $CP$

High Energy Physics - Phenomenology 2026-07-05 v1 High Energy Physics - Theory

Abstract

Two-sided octonionic E6L×E6RE_6^L\times E_6^R unification carries a nominal second colour SU(3)cSU(3)_{c'} which, gauged, would colour the charged lepton and must be suppressed by hand. We argue no such device is needed: the left-right symmetry acts in spacetime, not internally -- in the gravi-weak reading SU(2)RSU(2)_R is the gravitational frame group and the exchange is ordinary parity on the Dirac field; a spacetime operation cannot double the internal colour, so SU(3)c=SU(3)cSU(3)_{c'}=SU(3)_c, one vector-like colour with a colour-singlet electron. The right sector contributes one colour-blind datum, m\sqrt m (values: the exceptional-Jordan trace split, an input; NRN_R fixes only the colour representation). An anomaly no-go proven here shows every anomaly-free U(1)U(1) on visible fermions lies in span{Q,BL}\{Q,B-L\}, which the m\sqrt m pattern does not, in any sign or chirality assignment. The gauged dark-electromagnetic U(1)demU(1)_{dem} -- the mirror electroweak chain's unbroken remnant -- therefore carries the parity-mirror of electric charge on the dark sector: the visible fermions are dem-neutral (kinetic mixing the sole portal; no visible fifth force), and m\sqrt m enters visible physics only as the spectral label of the Jordan mass operator. The same recognition recovers hypercharge as the consistency relation Y=QTL3Y=Q-T^3_L, Q=N/3Q=N/3, with no right-sector generator. As an application, the spontaneous parity forbids the QCD vacuum angle (θQCD=0\theta_{QCD}=0, conditional on the gravi-weak identification), while the flavour rotors have real determinant, so argdetM=0\arg\det M=0 at tree level (exact for the Cabibbo rung, texture-contingent in full), coexisting with a nonzero CKM phase. Loop stability -- where minimal gauged-SU(2)RSU(2)_R parity solutions fail -- is plausibly evaded (no gauged WRW_R) but open, pending the Higgs-bridge matrix elements. We tag throughout what is derived, what is input, and what is open.

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Cite

@article{arxiv.2607.04132,
  title  = {Left-right symmetry breaking in $E_6^L\times E_6^R$ occurs only in spacetime -- with possible implications for strong $CP$},
  author = {Tejinder P. Singh},
  journal= {arXiv preprint arXiv:2607.04132},
  year   = {2026}
}

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47 pages