S. L. U.@unitedWeStreamU

Oct 28

Axiom

@axiommathai

Oct 27

We invite group theorists and the Lean community to validate it: rerun the proof check, and confirm that the Lean statement says what CFSG says.

cfsg.axiommath.ai

Jared Duker Lichtman

@jdlichtman

Oct 27

Axiom

@axiommathai

Oct 27

The largest math proof in human history is now computer-verified.

Today, Axiom announces a complete Lean auto-formalization of the Classification of Finite Simple Groups, the "periodic table of symmetry."

About 15,000 pages, built by over 100 mathematicians across generations.

S. L. U.@unitedWeStreamU

Oct 22

Philosophy Of Physics

@PhilosophyOfPhy

Oct 22

The Hafele–Keating experiment is one of the most direct tests of relativity ever made. It showed that time is not universal: two identical clocks, carried along different paths, disagree when they meet again.

In October 1971, Joseph Hafele and Richard Keating flew four caesium e world on ordinary commercial flights, once eastward and once westward. They then compared them with reference clocks at the US Naval Observatory that had stayed on the ground.

Two effects compete. Special relativity says a moving clock runs slow. In a frame fixed to the centre of the Earth, the ground is already moving east at up to 465 m/s because of the Earth's rotation. A plane flying east adds its speed to that motion, so its clock runs slower than the ground clock. A plane flying west moves more slowly than the ground, so its clock runs faster. General relativity adds a second effect: a clock higher in a gravitational field runs faster. At about 10 km altitude, both flying clocks gain time.

For a clock at height h moving at ground speed v, the fractional rate difference is approximately

Δτ/τ ≈ gh/c² − (v² + 2vωR)/2c²

where ω is the Earth's rotation rate and R its radius. The first term is gravitational, the second kinematic. The eastward trip lasted 65.4 hours.

Integrating along the actual flight paths gave clear predictions. Eastward: the kinematic effect causes a loss of 184 ns and gravity a gain of 144 ns, so a net loss of 40 ± 23 ns. Westward: the kinematic effect gives a gain of 96 ns and gravity a gain of 179 ns, so a net gain of 275 ± 21 ns. The clocks measured a loss of 59 ± 10 ns going east and a gain of 273 ± 7 ns going west. Both results agree with relativity within the experimental uncertainty.

There are clear limits. The precision was only about 10%, and the predictions depended on flight data such as speed, altitude and route. Some critics later questioned how the clock data were combined. Later experiments settled the matter with far greater precision. Gravity Probe A in 1976 confirmed the gravitational effect to better than one part in ten thousand. Today, optical clocks detect the change in the rate of time over a height difference of a single millimetre.

Relativity is also part of everyday technology. GPS satellite clocks run about 38 microseconds per day fast relative to the ground: gravity speeds them up by about 45, and their orbital speed slows them down by about 7. Without that correction, positions would drift by roughly 10 km per day.