Flyby anomaly and Earth’s Schwarzschild radii are both millimeter order. It’s because they both come from Earth’s gravitational potential. SO it should never have been considered an anomaly in the first place. Call it a puzzle. Earth’s S-radii is 9 mm and anomalous speed ranges upto 13 mm. reasonable. (My detailed analysis gives exactly why S-radii determines this, S-radii=2mu, at surface speed anomaly is of order S-radii/actual-Radii) Note that all it means is since we are using light speed in our theory we must make an error at the order of S-radii since by definition S-radii is the distance over which the relevant mass does not let even light escape. SO thats the minimum error or speed anomaly. That is where the singularity of theory of Relativity exists. Sun’s S-radii is 3 kms hence 3kms/s order error on satelites is permitted by theory of Relativity, at the surface or away from it which gradually drops down as we go far from the effect of gravity, asymptotically to zero. Which is what we are seeing. Except we do not go close to it’s surface hence we do not see 3kms/s. WE are far away from it and seeing 1.27 mm/s. (400 kms/year is not small, millimeter accuracy is inbuilt into theory of Relativity, or it would not explain cosmology.)

Schwarzschild radius (S-radii) is like the edge of a hole, if Gravity makes a hole in space time it would make an edge of the order of S-radii. So earth makes 9 mm edge in a spacetime hole which gradually drops down as we go farther from earth. Sun makes an edge of 3 km and what dwarfs make even bigger edes. (call it the choco-pie edge of Universe) A proton makes an edge of ~10^-30 milimeter.

If a proton will shine off a photon (or even an electron) you would not know the trajectory of the light or electron by that amount, 10^-30 mm. But when the light or electron has escaped to far away distance the deflection would be much larger. eg in 10^30 years the light would have deflected by 365*3600 milimeters = 1.3 million milimeters = 1.3 kms. A proton lives for ~10^30 years. So in its life time it has mis led zillions of photons.

What is the surface potential of a neutrino?

That would be ~2X10^-48. which is 10^-9 smaller than even the protons. So a photon can be misled by a neutrino, ~1.3 micrometers in 10^30 years.

How to prove theory of Relativity wrong for neutrinos?

Earth would deflect photons by ~9 mm in one second, 0.22 micrometers in 1 OPERA tof=2.43 milisecs and 540 pico-meters in 60 nanosecs. neutrinos will be deflected more if theory of relativity is correct for neutrinos, which are much more massive than light hence suffer more eccentricity. SO to prove theory of relativity incorrect you need nanometer accuracy in nanosecond time accuracy. GPS provides milimeter accuracy. It registers the excess does not prove the theory to be wrong.