The mass as a vector at eleven parameters: the spin's pure quaternion in the two readings, and the vector of a family

@misc{reynaud2026mass,
  title = {{The mass as a vector at eleven parameters: the spin's pure quaternion in the two readings, and the vector of a family}},
  author = {Patrick .B Reynaud},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  url = {https://hal.science/hal-05750050v2},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - The mass as a vector at eleven parameters: the spin's pure quaternion in the two readings, and the vector of a family
AU  - Patrick .B Reynaud
PY  - 2026
DA  - 2026/09/14
PB  - HAL
UR  - https://hal.science/hal-05750050v2
AB  - In four-dimensional relativity the mass of a body is a Casimir, a number with no geometry. At eleven parameters it acquires a direction and, in the reverse reading, a length in time, and this paper states them. The Pauli-Lubański vector of a body at rest is mc times its spin, so that in the rest frame W/(c|s|) is a pure quaternion of norm m along the spin axis. The reverse reading of the same algebra, the Weyl reflection that exchanges position and momentum through the Planck scale, builds from the position the mirror vector W = -1 2 ϵ(M ′ , x), equal, at the spatial origin of the frame, to the time times the angular momentum; on the free worldline of the body's own frame W = (τ /m) W , and the two vectors coincide, in Planck units, at the proper time t g = Gm/c 3 of the event hyperboloid. The mass is then a vector with a geometric reading, the gravitational time of the body carried along its spin axis, invisible at four parameters, where neither the scale that exchanges x with p nor the spin is among the data of an event; no new symmetry is involved, the little group of the position on the worldline being that of the momentum. The rest frame of a spinning body has three Euler angles, of which the vector uses two; the third, the rotation about the spin axis, is conjugate to the Casimir s, and is a phase. A three-space in which three angles are all observable exists for a family, the three generations of one charge: the vector m = k √ m k e k , whose squared norm is the mass of the family, has a polar angle from the diagonal and an azimuth about it, and each particle is defined by those two angles and its Z 3 angle 2πk/3. The symmetric group of the generations is the dihedral S 3 inside the octahedral group; Koide's relation for the charged leptons is the statement that the polar angle is 45 • , the equipartition of the vector between the singlet and the doublet of S 3 , exact to 10 -4 degree with the 2024 masses, and the azimuth is 2/9 to 2.5 × 10 -6 rad; the lightest member of a family is massless at the azimuth π/12, and the electron is light because a rotation of 2.3 • would annul its component. The frame of the diagonal is the tribimaximal mixing matrix, so that the lepton mixing is the composition of the chargedlepton frame with the frame in which the azimuth is measured. The angles carry Weyl weight zero; only the norms carry the scale. Run backwards, Koide's relation gives the tau mass in closed form, 1776.969 MeV, 0.4 standard deviations from the measured value. The bosons sort themselves by the same two data: the W ± and the Z carry the vector and the Weinberg angle, the Higgs a norm without a direction, on which the reflection has nothing to move, and the massless bosons live on the light cone, the one shell that is its own image.  The reflection keeps the spin and moves the mass: the two readings are one coin, the spin its edge. In the electroweak and colour sectors the same pair is the norm and direction of the coupling vector, the electric and magnetic charge lengths of a charge and the two Casimirs of the colour charge, and the Regge trajectory is the colour sector's Kerr bound.
ER  -