The four reversals and the three faces: the readings of the unified theory as phases
Abstract
The unified theory of gravity and the Standard Model at eleven parameters has been read from the other end sector by sector: the Weyl reflection of u(2, 2) exchanges position and momentum, the Hodge map of the space-time tetraquaternion exchanges potentials and currents in the electroweak and in the colour sector, the conjugation of the isospin quaternion exchanges the third isospin with the hypercharge, and a Cayley-Dickson involution of the octonions exchanges colour with anticolour. Three sectors with an ordinary and a reverse reading each suggest eight combinations, and this paper asks whether they are equivalent descriptions or distinct physical states. The named reversals are five, one for mechanics and two for each gauge sector, and they are four once the Hodge map, the pseudoscalar of the tetraquaternion that the two gauge sectors share, is taken as one map, which the Clifford container of the theory forces; the four generate, modulo the sign, a group (Z 2 ) 4 of sixteen readings, and the readings are of two kinds. The three that are automorphisms or anti-automorphisms give one kinematics in two sets of variables, and the reading that holds is chosen by a state or fixed by the law, never by a symmetry: the spectral condition bounds the momentum in one reading and the event in the other, the vacuum selects the ordinary isospin reading on the Higgs and the chiral content of the fermions stops the reversal there, the baryon number of the universe selects quarks over antiquarks. The Hodge map is no automorphism; its two readings are two regimes of one sector, exchanged by Dirac's condition with the self-dual charge as fixed point, and the reading that holds is decided by the side of the fixed point on which a running quantity sits, so that the switch from one to the other is a crossing governed by an equation of the theory: the colour coupling runs through its self-dual value between 0.3 and 0.7 GeV, and the mechanical reversal, which has both mechanisms, crosses its fixed point M P at the gravitational radius of a body heavier than the Planck mass; the electroweak sector never crosses inside the domain of the theory, its weak coupling pre-empted by the vacuum and its abelian couplings below the fixed point throughout. Of the sixteen readings four are realised, as domains the world passes through by crossing and not as states a transformation switches: a hadron at rest, the quark-gluon plasma, a star fallen inside the gravitational radius of a large hole, and matter at the bounce. The readings are phases. The three faces of the theory locate the statement: every reversal is a choice on face A, the symmetry and its units; on the abelian Hertz chain of face B the Hodge map is exact, exchanging d with δ, closedness with co-closedness, and leaving the single equation invariant; every selection is made on face C, the matter. The container that holds the three sectors on one level exists, the Clifford algebra Cl(6, 8) = M 64 (H) of the unified theory, in which the four reversals are Clifford objects; what it cannot hold is a scale. The two constants that connect the three sets of dimensionless quantities, v/M P = 2.0 × 10 -17 and Λ 3 /M P = 1.7 × 10 -20 , are the ratios of the vacuum values of the three normalisers of face A, which have the same Weyl weight; they are the two order parameters of the two crossings the electroweak and colour sectors do or do not make, constant only in the broken and confined vacuum, and they remain inputs.
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BibTeX
@misc{reynaud2026four,
title = {{The four reversals and the three faces: the readings of the unified theory as phases}},
author = {Patrick .B Reynaud},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05749775v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The four reversals and the three faces: the readings of the unified theory as phases
AU - Patrick .B Reynaud
PY - 2026
DA - 2026/09/14
PB - HAL
UR - https://hal.science/hal-05749775v2
AB - The unified theory of gravity and the Standard Model at eleven parameters has been read from the other end sector by sector: the Weyl reflection of u(2, 2) exchanges position and momentum, the Hodge map of the space-time tetraquaternion exchanges potentials and currents in the electroweak and in the colour sector, the conjugation of the isospin quaternion exchanges the third isospin with the hypercharge, and a Cayley-Dickson involution of the octonions exchanges colour with anticolour. Three sectors with an ordinary and a reverse reading each suggest eight combinations, and this paper asks whether they are equivalent descriptions or distinct physical states. The named reversals are five, one for mechanics and two for each gauge sector, and they are four once the Hodge map, the pseudoscalar of the tetraquaternion that the two gauge sectors share, is taken as one map, which the Clifford container of the theory forces; the four generate, modulo the sign, a group (Z 2 ) 4 of sixteen readings, and the readings are of two kinds. The three that are automorphisms or anti-automorphisms give one kinematics in two sets of variables, and the reading that holds is chosen by a state or fixed by the law, never by a symmetry: the spectral condition bounds the momentum in one reading and the event in the other, the vacuum selects the ordinary isospin reading on the Higgs and the chiral content of the fermions stops the reversal there, the baryon number of the universe selects quarks over antiquarks. The Hodge map is no automorphism; its two readings are two regimes of one sector, exchanged by Dirac's condition with the self-dual charge as fixed point, and the reading that holds is decided by the side of the fixed point on which a running quantity sits, so that the switch from one to the other is a crossing governed by an equation of the theory: the colour coupling runs through its self-dual value between 0.3 and 0.7 GeV, and the mechanical reversal, which has both mechanisms, crosses its fixed point M P at the gravitational radius of a body heavier than the Planck mass; the electroweak sector never crosses inside the domain of the theory, its weak coupling pre-empted by the vacuum and its abelian couplings below the fixed point throughout. Of the sixteen readings four are realised, as domains the world passes through by crossing and not as states a transformation switches: a hadron at rest, the quark-gluon plasma, a star fallen inside the gravitational radius of a large hole, and matter at the bounce. The readings are phases. The three faces of the theory locate the statement: every reversal is a choice on face A, the symmetry and its units; on the abelian Hertz chain of face B the Hodge map is exact, exchanging d with δ, closedness with co-closedness, and leaving the single equation invariant; every selection is made on face C, the matter. The container that holds the three sectors on one level exists, the Clifford algebra Cl(6, 8) = M 64 (H) of the unified theory, in which the four reversals are Clifford objects; what it cannot hold is a scale. The two constants that connect the three sets of dimensionless quantities, v/M P = 2.0 × 10 -17 and Λ 3 /M P = 1.7 × 10 -20 , are the ratios of the vacuum values of the three normalisers of face A, which have the same Weyl weight; they are the two order parameters of the two crossings the electroweak and colour sectors do or do not make, constant only in the broken and confined vacuum, and they remain inputs.
ER -