The reverse paradigm of the electroweak sector: the two reversals of the hyperquaternion
Abstract
The reverse paradigm of the eleven-parameter special relativity reads the algebra of a body from the other end, by the Weyl reflection that exchanges the position and momentum blocks of u(2, 2); the reflection is forced up to the transformations that keep the blocks, its fixed point is the Planck mass, and the reading has content inside horizons. This paper does the same for the electroweak sector of the unified theory, whose algebra is the hyperquaternion H ⊗ H ⊗ H = M 4 (H), the space-time tetraquaternion times the weak isospin. The sector has two reversals where mechanics has one, because its algebra has two factors that can be turned around. The first is the Hodge reversal of the space-time factor, multiplication by the pseudoscalar ω: it sends each Clifford grade to its complement, turns the potentials into the currents and the currents into the potentials, reads every field strength as its dual, exchanges the two invariants, and sends the Higgs quaternion into the pseudoscalar slots that the ordinary reading leaves empty; it is electric-magnetic duality read on the sixty-four slots, exact on the field strength and a relabelling of the potential and the current, since a map that exchanges the scalar with the pseudoscalar is no automorphism; its reciprocity is Dirac's condition written as r Q (e) r Q (g) = n 2 L 2 P , and the fixed point of the coupling map that accompanies it is the self-dual charge α = n 2 , of charge length L P n/2. The second is the conjugation of the isospin quaternion adapted to the vacuum, q → nq * n -1 : an anti-involution that exchanges the left and right multiplications by the complex unit, hence the third isospin with the hypercharge on the Higgs quaternion, fixes the electric charge Q = T 3 + Y , fixes the vacuum, and exchanges the two Hopf fibrations of the vacuum 3-sphere, by the hypercharge and by the third isospin. Read on the couplings as the renaming (T 3 , g, W 3 ) ↔ (Y, g ′ , B), it leaves e, m Z , the Fermi constant, the photon and the neutral-current interaction fixed, reverses the Z, and exchanges sin 2 θ W with cos 2 θ W and m W = gv/2 with g ′ v/2 = m Z sin θ W = 43.1 GeV, the mass a W would have with the hypercharge coupling and the mass of no field, with
Neither fixed point is where the world is: the hypercharge coupling reaches 1 2 only at 10 42 GeV, the electromagnetic one, bounded by the weak coupling above m W , never, and sin 2 θ W is 0.223. Where each reading holds is decided as it was for mechanics. The magnetic reading of the abelian sectors never holds below the Planck mass; the magnetic reading of the weak isospin would hold beyond 5 × 10 7 m, where the one-loop coupling diverges, and is pre-empted by the vacuum at v, as the singularity is pre-empted by the bounce in the gravitational theory; what survives of it in the broken phase is the electroweak monopole of Cho and Maison, of magnetic charge 4π/e, whose charge length is L P / √ α. The isospin reversal is the parity of the left-right symmetric models restricted to the Higgs quaternion, their bidoublet, where the hypercharge is the third component of a right isospin; its fixed point there is g L = g R , and the Standard Model is the reading in which the right action is broken to its U (1). On the Standard Model's fermions the exchange T 3 ↔ Y is not realisable, their hypercharges not being isospins: the reverse reading is exact on the Higgs, a renaming on the neutral gauge sector, and a different theory beyond.
Related
- The reverse paradigm of the colour sector: the two reversals of Cl(9, 1) and confinement as the other reading
- Condensed matter at eleven parameters: the reverse paradigm on a lattice
- The reverse paradigm: special relativity at eleven parameters with momentum as the base
- The inverse of the hyperquaternion, M 4 (H)
- Lifetime reversals of neutral and charged scalar clouds around nonminimal Einstein-Yang-Mills black holes
BibTeX
@misc{reynaud2026reverse,
title = {{The reverse paradigm of the electroweak sector: the two reversals of the hyperquaternion}},
author = {Patrick .B Reynaud},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05749763v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The reverse paradigm of the electroweak sector: the two reversals of the hyperquaternion
AU - Patrick .B Reynaud
PY - 2026
DA - 2026/09/14
PB - HAL
UR - https://hal.science/hal-05749763v2
AB - The reverse paradigm of the eleven-parameter special relativity reads the algebra of a body from the other end, by the Weyl reflection that exchanges the position and momentum blocks of u(2, 2); the reflection is forced up to the transformations that keep the blocks, its fixed point is the Planck mass, and the reading has content inside horizons. This paper does the same for the electroweak sector of the unified theory, whose algebra is the hyperquaternion H ⊗ H ⊗ H = M 4 (H), the space-time tetraquaternion times the weak isospin. The sector has two reversals where mechanics has one, because its algebra has two factors that can be turned around. The first is the Hodge reversal of the space-time factor, multiplication by the pseudoscalar ω: it sends each Clifford grade to its complement, turns the potentials into the currents and the currents into the potentials, reads every field strength as its dual, exchanges the two invariants, and sends the Higgs quaternion into the pseudoscalar slots that the ordinary reading leaves empty; it is electric-magnetic duality read on the sixty-four slots, exact on the field strength and a relabelling of the potential and the current, since a map that exchanges the scalar with the pseudoscalar is no automorphism; its reciprocity is Dirac's condition written as r Q (e) r Q (g) = n 2 L 2 P , and the fixed point of the coupling map that accompanies it is the self-dual charge α = n 2 , of charge length L P n/2. The second is the conjugation of the isospin quaternion adapted to the vacuum, q → nq * n -1 : an anti-involution that exchanges the left and right multiplications by the complex unit, hence the third isospin with the hypercharge on the Higgs quaternion, fixes the electric charge Q = T 3 + Y , fixes the vacuum, and exchanges the two Hopf fibrations of the vacuum 3-sphere, by the hypercharge and by the third isospin. Read on the couplings as the renaming (T 3 , g, W 3 ) ↔ (Y, g ′ , B), it leaves e, m Z , the Fermi constant, the photon and the neutral-current interaction fixed, reverses the Z, and exchanges sin 2 θ W with cos 2 θ W and m W = gv/2 with g ′ v/2 = m Z sin θ W = 43.1 GeV, the mass a W would have with the hypercharge coupling and the mass of no field, with Neither fixed point is where the world is: the hypercharge coupling reaches 1 2 only at 10 42 GeV, the electromagnetic one, bounded by the weak coupling above m W , never, and sin 2 θ W is 0.223. Where each reading holds is decided as it was for mechanics. The magnetic reading of the abelian sectors never holds below the Planck mass; the magnetic reading of the weak isospin would hold beyond 5 × 10 7 m, where the one-loop coupling diverges, and is pre-empted by the vacuum at v, as the singularity is pre-empted by the bounce in the gravitational theory; what survives of it in the broken phase is the electroweak monopole of Cho and Maison, of magnetic charge 4π/e, whose charge length is L P / √ α. The isospin reversal is the parity of the left-right symmetric models restricted to the Higgs quaternion, their bidoublet, where the hypercharge is the third component of a right isospin; its fixed point there is g L = g R , and the Standard Model is the reading in which the right action is broken to its U (1). On the Standard Model's fermions the exchange T 3 ↔ Y is not realisable, their hypercharges not being isospins: the reverse reading is exact on the Higgs, a renaming on the neutral gauge sector, and a different theory beyond.
ER -