Temperature at eleven parameters: the thermal element of a gas of bodies, its two ceilings and the history it reads
Abstract
The theory at eleven parameters is a theory of bodies, and a body has no temperature. Temperature belongs to ensembles, and this paper states what the ensembles of the eleven are. Their phase space contains the sphere of spin directions, whose holonomy is the sign of exchange, so that the two statistics are properties of a body's phase space. Their equilibrium is labelled by a thermal element, one element of the dual of the algebra u(2, 2) in which the sixteen quantities of a body close. Its parts are an inverse-temperature fourvector, a thermal vorticity whose electric part is the Unruh bath and whose magnetic part polarises spins, a dilatation that exists for radiation only, four conformal accelerations, and a helicity potential. The multipliers a kind of matter can carry are fixed by one proposition. Temperature has Weyl weight -1 and entropy weight zero, so that the invariant thermometer is the temperature in units of the compensator, and the Tolman-Ehrenfest and Klein laws hold in every gauge. The acceleration bound of the eleven gives two ceilings, mc 2 /(2πk B ) for the vacuum seen by a body of mass m and T P /(2π) for any body, which is Sakharov's maximal temperature with its coefficient. In Euclidean form, the radius of the thermal circle cannot be smaller than the reduced Compton wavelength, nor than the Planck length. Below the ceiling the thermal field theory of the unified Lagrangian is the Standard Model's, term by term, with g * = 106.75, because the compensator is pure gauge, the Weyl vector and the torsion are algebraic, and the gravitons never reach equilibrium. Every thermal number is then a ratio to the compensator, and the hierarchy number of the unified paper is, at one loop, the electroweak crossover temperature in Planck units, (k B T EW /M P c 2 ) 2 = ζ/c T . The QCD crossover falls between the ceiling of the nucleon, 149 MeV, and the Unruh temperature of the string, 168 MeV, a relation that the lattice can test along the quark-mass axis, and in which the paper reads the body's mass as a proxy for the string tension. The thermal history is read from the ceiling down: every estimate of the bounce temperature lies above it, and the ceiling selects the relic M P /4 and a bounce at half the Planck density. Below it the history is the Standard Model's, the microwave background staying Planckian as the conformal equilibrium that radiation alone can carry. The polarisation of hyperons is the magnetic half of the Lorentz multiplier, with a relaxation time of 14 to 15 fm/c from the Σ * cycle, or 4 to 7 in the full kinetics, that makes it a memory of hadronisation. Newton's constant is fundamental, the fields' share of the horizon entropy being a regulator's number. Every statement is marked with its kind, and three measurements would test the paper's own numbers.
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BibTeX
@misc{reynaud2026temperature,
title = {{Temperature at eleven parameters: the thermal element of a gas of bodies, its two ceilings and the history it reads}},
author = {Patrick B. Reynaud},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05765513v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Temperature at eleven parameters: the thermal element of a gas of bodies, its two ceilings and the history it reads
AU - Patrick B. Reynaud
PY - 2026
DA - 2026/09/26
PB - HAL
UR - https://hal.science/hal-05765513v2
AB - The theory at eleven parameters is a theory of bodies, and a body has no temperature. Temperature belongs to ensembles, and this paper states what the ensembles of the eleven are. Their phase space contains the sphere of spin directions, whose holonomy is the sign of exchange, so that the two statistics are properties of a body's phase space. Their equilibrium is labelled by a thermal element, one element of the dual of the algebra u(2, 2) in which the sixteen quantities of a body close. Its parts are an inverse-temperature fourvector, a thermal vorticity whose electric part is the Unruh bath and whose magnetic part polarises spins, a dilatation that exists for radiation only, four conformal accelerations, and a helicity potential. The multipliers a kind of matter can carry are fixed by one proposition. Temperature has Weyl weight -1 and entropy weight zero, so that the invariant thermometer is the temperature in units of the compensator, and the Tolman-Ehrenfest and Klein laws hold in every gauge. The acceleration bound of the eleven gives two ceilings, mc 2 /(2πk B ) for the vacuum seen by a body of mass m and T P /(2π) for any body, which is Sakharov's maximal temperature with its coefficient. In Euclidean form, the radius of the thermal circle cannot be smaller than the reduced Compton wavelength, nor than the Planck length. Below the ceiling the thermal field theory of the unified Lagrangian is the Standard Model's, term by term, with g * = 106.75, because the compensator is pure gauge, the Weyl vector and the torsion are algebraic, and the gravitons never reach equilibrium. Every thermal number is then a ratio to the compensator, and the hierarchy number of the unified paper is, at one loop, the electroweak crossover temperature in Planck units, (k B T EW /M P c 2 ) 2 = ζ/c T . The QCD crossover falls between the ceiling of the nucleon, 149 MeV, and the Unruh temperature of the string, 168 MeV, a relation that the lattice can test along the quark-mass axis, and in which the paper reads the body's mass as a proxy for the string tension. The thermal history is read from the ceiling down: every estimate of the bounce temperature lies above it, and the ceiling selects the relic M P /4 and a bounce at half the Planck density. Below it the history is the Standard Model's, the microwave background staying Planckian as the conformal equilibrium that radiation alone can carry. The polarisation of hyperons is the magnetic half of the Lorentz multiplier, with a relaxation time of 14 to 15 fm/c from the Σ * cycle, or 4 to 7 in the full kinetics, that makes it a memory of hadronisation. Newton's constant is fundamental, the fields' share of the horizon entropy being a regulator's number. Every statement is marked with its kind, and three measurements would test the paper's own numbers.
ER -