The reverse paradigm of the colour sector: the two reversals of Cl(9, 1) and confinement as the other reading
Abstract
The reverse paradigm reads an algebra from its other end by an involution and asks for the fixed point of the reading and for the domain in which the other reading is the right one; it was done for the eleven quantities of a body, where the Weyl reflection of u(2, 2) exchanges position and momentum, the Planck mass is the fixed point and the reverse reading holds inside horizons, and for the electroweak sector, where the hyperquaternion has two reversals whose fixed points, the self-dual charge and sin 2 θ W = 1 2 , the world does not reach. This paper does it for the colour sector of the unified theory, whose algebra is T ⊗ Cl(0, 6) = Cl(9, 1) = M 32 (R), the space-time tetraquaternion times the Clifford algebra of the octonion units, with colour the stabiliser of the seventh unit e 7 in G 2 . The sector has two reversals. The first is the Hodge reversal of the space-time factor, which exchanges the slots of the gluon potentials with those of the colour currents and reads every field strength as its dual: it is the electric-magnetic duality of chromodynamics, whose magnetic reading is the dual superconductor of 't Hooft and Mandelstam, exact on the field strength and a relabelling of the potentials and the currents as in the electroweak sector; and it renames the slots that the Wilson loop and the 't Hooft loop read, the two loops whose area and perimeter laws distinguish, without fundamental matter, the Higgs phase from the confining one. The second is a Cayley-Dickson involution of the octonions, an automorphism that reverses e 7 : it reverses the complex structure of colour, acts on su(3) by its outer automorphism 3 ↔ 3, fixes a quaternion subalgebra H ⊂ O and three of the eight gluons and reverses the other five; it is the colour part of charge conjugation. The two Clifford pseudoscalars of the sector, ω and L 7 , each of order four with no real fixed vector, multiply, up to sign, to the pseudoscalar of Cl(9, 1), an honest involution whose fixed and reversed spaces are the two sixteen-dimensional Majorana-Weyl spinors of Spin(9, 1), each tying one sign of the space-time chirality to one sign of the colour structure; the colour reversal is a grade-three element of Cl(0, 6), anticommutes with that pseudoscalar, and exchanges the two spinors, which is the exchange of 3 with 3 at fixed four-dimensional chirality, a flip of the ten-dimensional one. What is new against the two earlier sectors is that both reverse readings are realised. The fixed point of the coupling map, α s = 2 for the fundamental charge and 1 for the adjoint, whose counterpart the hypercharge reaches only beyond the Planck mass and the electromagnetic coupling never, is crossed by the colour coupling at the hadronic scale, between 0.3 and 0.7 GeV at one loop, between Λ and the proton mass. Below that scale the magnetic reading is the right one, confinement being the condensation of the reverse reading's charges, the flux tube its body and the string tension σ = (0.44 GeV) 2 its invariant, with the lattice placing the vacuum deep on the type-I side of the marginal vortex; the reverse reading of colour holds outside a fraction of a fermi where that of mechanics held inside a gravitational radius, the ultraviolet and infrared exchanged with the charge that is a length. And the second reversal is an exact symmetry of the sector's dynamics, chromodynamics being vector-like, so that nothing in the law of the sector selects the quark reading over the antiquark one; the state of the universe does, by its baryon number, the one selection in the series that is a fact about the state and not about the law of the sector.
Related
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- Condensed matter at eleven parameters: the reverse paradigm on a lattice
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BibTeX
@misc{reynaud2026reverse,
title = {{The reverse paradigm of the colour sector: the two reversals of Cl(9, 1) and confinement as the other reading}},
author = {Patrick .B Reynaud},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05749761v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The reverse paradigm of the colour sector: the two reversals of Cl(9, 1) and confinement as the other reading
AU - Patrick .B Reynaud
PY - 2026
DA - 2026/09/14
PB - HAL
UR - https://hal.science/hal-05749761v2
AB - The reverse paradigm reads an algebra from its other end by an involution and asks for the fixed point of the reading and for the domain in which the other reading is the right one; it was done for the eleven quantities of a body, where the Weyl reflection of u(2, 2) exchanges position and momentum, the Planck mass is the fixed point and the reverse reading holds inside horizons, and for the electroweak sector, where the hyperquaternion has two reversals whose fixed points, the self-dual charge and sin 2 θ W = 1 2 , the world does not reach. This paper does it for the colour sector of the unified theory, whose algebra is T ⊗ Cl(0, 6) = Cl(9, 1) = M 32 (R), the space-time tetraquaternion times the Clifford algebra of the octonion units, with colour the stabiliser of the seventh unit e 7 in G 2 . The sector has two reversals. The first is the Hodge reversal of the space-time factor, which exchanges the slots of the gluon potentials with those of the colour currents and reads every field strength as its dual: it is the electric-magnetic duality of chromodynamics, whose magnetic reading is the dual superconductor of 't Hooft and Mandelstam, exact on the field strength and a relabelling of the potentials and the currents as in the electroweak sector; and it renames the slots that the Wilson loop and the 't Hooft loop read, the two loops whose area and perimeter laws distinguish, without fundamental matter, the Higgs phase from the confining one. The second is a Cayley-Dickson involution of the octonions, an automorphism that reverses e 7 : it reverses the complex structure of colour, acts on su(3) by its outer automorphism 3 ↔ 3, fixes a quaternion subalgebra H ⊂ O and three of the eight gluons and reverses the other five; it is the colour part of charge conjugation. The two Clifford pseudoscalars of the sector, ω and L 7 , each of order four with no real fixed vector, multiply, up to sign, to the pseudoscalar of Cl(9, 1), an honest involution whose fixed and reversed spaces are the two sixteen-dimensional Majorana-Weyl spinors of Spin(9, 1), each tying one sign of the space-time chirality to one sign of the colour structure; the colour reversal is a grade-three element of Cl(0, 6), anticommutes with that pseudoscalar, and exchanges the two spinors, which is the exchange of 3 with 3 at fixed four-dimensional chirality, a flip of the ten-dimensional one. What is new against the two earlier sectors is that both reverse readings are realised. The fixed point of the coupling map, α s = 2 for the fundamental charge and 1 for the adjoint, whose counterpart the hypercharge reaches only beyond the Planck mass and the electromagnetic coupling never, is crossed by the colour coupling at the hadronic scale, between 0.3 and 0.7 GeV at one loop, between Λ and the proton mass. Below that scale the magnetic reading is the right one, confinement being the condensation of the reverse reading's charges, the flux tube its body and the string tension σ = (0.44 GeV) 2 its invariant, with the lattice placing the vacuum deep on the type-I side of the marginal vortex; the reverse reading of colour holds outside a fraction of a fermi where that of mechanics held inside a gravitational radius, the ultraviolet and infrared exchanged with the charge that is a length. And the second reversal is an exact symmetry of the sector's dynamics, chromodynamics being vector-like, so that nothing in the law of the sector selects the quark reading over the antiquark one; the state of the universe does, by its baryon number, the one selection in the series that is a fact about the state and not about the law of the sector.
ER -