The reverse paradigm: special relativity at eleven parameters with momentum as the base

@misc{reynaud2026reverse,
  title = {{The reverse paradigm: special relativity at eleven parameters with momentum as the base}},
  author = {Patrick .B Reynaud},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  url = {https://hal.science/hal-05749776v2},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - The reverse paradigm: special relativity at eleven parameters with momentum as the base
AU  - Patrick .B Reynaud
PY  - 2026
DA  - 2026/09/14
PB  - HAL
UR  - https://hal.science/hal-05749776v2
AB  - Special relativity at eleven parameters (SR11) describes a body by three 4-vectors with one constraint, time-position, energy-momentum and Pauli-Lubański spin, and admits the Planck constants by making the eleven dimensionless with one scale, four of them at order ε 0 and seven at order ε 1 . In that description the four coordinates lead and the seven follow, expressed as lengths through ℏ and G; the world of the four is compact once conformally completed, and the world of the seven is hyperbolic. We reverse the roles. The reversal is not a choice but a theorem of the algebra: the sixteen quantities of a body form one element X of u(2, 2), the exchange of leaders and followers is the Weyl reflection P ↔ K of so(2, 4), and it is realised inside SU (2, 2) by a single element, the (2, 2) form η itself, X → ηXη, which exchanges the position and momentum blocks exactly, keeps the angular momentum and the helicity, and reverses the mass moment and the dilatation charge. The reverse theory is the module of the reciprocal Weyl algebra K ⊕ M ⊕ D, which has the structure constants of P ⊕ M ⊕ D: the same algebra in the other representation. What the reversal does not preserve is the place of the constraints: the mass shell p2 = (m/M P ) 2 becomes the hyperboloid x2 = (m/M P ) 2 of events at proper time Gm/c 3 from the reference event, the Compton length and the gravitational radius are exchanged, their product is L 2 P , and the Planck mass is the fixed point. The reverse postulates follow by the same map: two frames are related by a relative 4-position on a hyperboloid X • X = c 2 T 2 , T = GM/c 3 the gravitational time of the reference body, the boost is the square root of the position, L(X) = (cT + X)/ 2cT (X 0 + cT ), which is the boost to the frame of the inertial worldline joining the two events, the composition of positions is linear, and the light cone, |x|/(ct) = tanh ξ, is a range and not a law, as v < c was. The geometry of the two worlds is then made explicit. The four close into the conformal compactification of Minkowski space, where time is a circle; the seven of a massive body form the spin sphere over the mass-shell hyperboloid, five-dimensional and hyperbolic; for a massless body the seven are a twistor modulo its phase, the orbit is the seven-dimensional quadric SU (2, 2)/SU (2, 1), and its compact form is the 7-sphere S 7 = SU (4)/SU (3) = Sp(2)/Sp(1) with its two Hopf fibrations, over projective twistor space CP 3 by the phase that the helicity generates, and over the compact 4-sphere S 4 = HP 1 by the spin S 3 = SU (2): space-time is attached to the seven by the second fibration and, in the Minkowski face, by Penrose's incidence relation, which needs ℏ and attaches the event sharply only for a spinless body, the blur for helicity H being exactly ℏ|H|/(2E). The hyperbolic, quaternionic and compact faces are the real forms su(2, 2), sl(2, H), su(4) of one complex algebra, and they are the Clifford faces Cl(2, 4) = Cl(5, 1) = Cl(6, 0) of the hyperquaternion M 4 (H) of the companion paper on the 64D inverse. The two Weyl groups decide the two arrows: translations freeze the momentum block and special conformal transformations freeze the position block, so that in SR11 the sign of the energy is absolute and the chronology of an event relative, while in the reverse paradigm the chronology relative to the reference event is absolute and the sign of the energy is not. What flows in both readings is the phase, generated by the helicity, the centre of u(2, 2) and the one quantity invariant under η. The reverse of the elevenparameter general relativity is a curved momentum space in the sense of Snyder and of relative locality, with Snyder's commutators [x µ , x ν ] = -η 44 L 2 P M µν and discrete positions, 1 and with the non-linear composition of momenta back; its scale is bounded below by the Planck mass. The reverse reading is the right one where the two hyperboloids cross, inside the gravitational radius of a body heavier than M P , the domain of the GR11 bounce, where chronology is absolute and the sign of the conserved energy is not; on the Schwarzschild interior this is verified literally, the Killing energy of a future-directed body taking every sign as a spacelike momentum component, the fall being the one absolute direction, and the interior's maximal proper lifetime being π GM/c 3 , π times the reverse invariant. Finally the homogeneous worlds of the sixteen are listed with their stabilisers, from the two fourdimensional worlds, whose stabilisers are the two Weyl algebras (4 = 15 -11), through AdS 5 , twistor space and the twistor quadric to the eight-dimensional future tube, the phase space of a body, and the twelve-dimensional adjoint orbit: the two paradigms are the two Lagrangian polarisations of the eight-dimensional world, exchanged by η, and four dimensions is the only world in which the eleven are the symmetries fixing a point.
ER  -