Galactic Shear Against Self-Gravity: The Density a Cloud Needs to Survive Differential Rotation, and Its Distance from the Shear Bound

This paper addresses a single question: How dense must a cloud in the solar neighbourhood be before its self-gravity defeats the stretching imposed by Galactic differential rotation, and how does the actual Galactic shear compare with the maximum shear permitted by the torsion bound? Using published molecular-cloud and young-stellar-object measurements as inputs, and carrying the spin-torsion density bound of 9.72 x 10^57 kilograms per cubic metre and its critical shear of 8.56 x 10^32 per second as fixed comparison scales, the calculation finds that a cloud needs a mean density of only about one particle per cubic centimetre to hold itself against the local tidal field, so ordinary clouds clear the requirement by two to three orders of magnitude and even diffuse envelopes clear it by about two orders of magnitude; the Galactic shear itself, about 5 x 10^-16 per second, lies roughly 48 orders of magnitude below the critical shear of 8.56 x 10^32 per second, so the shear bound imposes no restriction on cloud survival - a plain negative result. The headline values are: rho tide kg m3 = 3.38e-21; A per s = 4.86e-16; shear ratio to bound = 5.68e-49; margin typical = 346.0. All values are computed from the stated formulae and inputs; this is a simulated and computed study, not a measurement. Where the comparison scales supplied by the torsion framework turn out to be too remote to influence the process, that outcome is reported as a negative result rather than passed over.