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A torsion-regulated gravitational bound sets a maximum density of 9.72e+57 kg per cubic metre and a critical shear rate of 8.56e+32 per second. This paper asks a deliberately narrow question: by what factor do the midplane densities and Keplerian shear rates of an ordinary protoplanetary disc fall below those two ceilings? For a Hayashi-type disc around a solar-mass star, the midplane density at 1 AU is 1.36e-06 kg per cubic metre, a factor of 1.40e-64 of the density ceiling, and the Keplerian shear there is 2.99e-07 per second, a factor of 3.49e-40 of the critical shear. Even at 0.1 AU the density reaches only 7.86e-62 of the ceiling. Keplerian shear would equal the critical rate only at 4.96e-27 AU from the star, deep inside the star itself, and the midplane density would reach the ceiling only at 6.03e-24 AU. The negative result is plain: the bound does not constrain protoplanetary disc structure at any radius, and every operative constraint on such discs comes from disc physics and observation, not from the gravitational ceiling. All figures are computed from stated assumptions, not measured.
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William Alexander Aldred (2026) studied this question.
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