Left-right symmetry breaking in $E_6^L\times E_6^R$ occurs only in spacetime -- with possible implications for strong $CP$
Abstract
Two-sided octonionic unification carries a nominal second colour 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 is the gravitational frame group and the exchange is ordinary parity on the Dirac field; a spacetime operation cannot double the internal colour, so , one vector-like colour with a colour-singlet electron. The right sector contributes one colour-blind datum, (values: the exceptional-Jordan trace split, an input; fixes only the colour representation). An anomaly no-go proven here shows every anomaly-free on visible fermions lies in span, which the pattern does not, in any sign or chirality assignment. The gauged dark-electromagnetic -- 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 enters visible physics only as the spectral label of the Jordan mass operator. The same recognition recovers hypercharge as the consistency relation , , with no right-sector generator. As an application, the spontaneous parity forbids the QCD vacuum angle (, conditional on the gravi-weak identification), while the flavour rotors have real determinant, so at tree level (exact for the Cabibbo rung, texture-contingent in full), coexisting with a nonzero CKM phase. Loop stability -- where minimal gauged- parity solutions fail -- is plausibly evaded (no gauged ) 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