The 1-arc-second likely originates from this turntable vendor whitepaper - Oswalds Mill Audio - K3 Whitepaper | OMA of which MF owns (or owned). There are a number of errors in the white paper:
"...our device needs to actually physically trace modulations in a vinyl groove to the order of .005 microns." The inherent surface roughness of the record reported by a couple sources is 0.01 to 0.005-microns (100 to 50 angstroms). The RIAA curve is specifically intended to avoid the noise inherent to the record's inherent surface roughness, so the realistic smallest modulation is 0.1-microns, at least 10X the surface roughness. And just to put the record's surface roughness into perspective, its surface finish (inherent to the material and pressing process) is finer than mirror polished stainless steel.
As far as the arc-sec, read Minute and second of arc - Wikipedia, it's an angle measurement. To apply that to a record of 12" diameter spinning at 33.333-rpm with outside groove tangential velocity of 51-cm/sec and inside groove velocity of as low as 20-cm/sec. Converting 51-cm/s to radians/sec Angular Velocity Calculator = 3.3456 rads/sec which Angular Velocity Conversion Calculator - Radians, Degrees, Revolutions, and Grads = 191.74 deg/sec Degrees to Seconds Converter = 687,600 arc-secs. Like the OP, the ability to detect 1-arc-sec is not possible. However, angle encoders appear to have accuracies of 5-arc-sec and better accuracy_of_angle_encoders.pdf where "Angular measurement error (arc seconds) = bearing wander (μm) x 412.5/D" where D=mm. If it's assumed a bearing radial wander of 5-microns and diameter of 13-inch, the angular measurement error in arc-sec is 1.2492. But this does not take into account circuit noise, feedback error, etc.
Like the 0.005 micron claim, the white paper claim of 1-arc-sec appears (being diplomatic) to be very 'optimistic'. Otherwise, keep in mind that MF is not a scientist or mathematician. He would help himself if he referenced the information source that is well out of his swim-lane, versus making claims that appear to be attributed to him.