Neutrons don't act at distance with the electric force, because they don't have electric charge.
They act at distance with the gravitational force, because they have mass.
For example, you can fill a glass with heavy water that has Deuterium instead of Hydrogen. https://en.wikipedia.org/wiki/Heavy_water
The number of molecules is almost equal, because the "size" of the molecule does not depend on the number of neutrons (almost). But the mass and the weigh will increase like a 10% because now there are more neutrons in the glass. The neutrons are acting at distance with the Earth, so the weight measure with an usual balance is greater.
At one time we didn't know how the sun could burn for so long. Even made of anthracite coal, it would only last a few hundred years.
As for your example, the neutrons in the glass are also attracted to the protons in the earth.
1) Inertial mass M: It's the number that appears in the equation F = a * M, where F is the force and a is the acceleration of the ball.
2) Gravitational mass m: It's the number that appears in the equation F = g * m, where F is the force (weight) and g is the gravity constant on Earth surface that is approximately 9.8m/s^2 or 32ft/s^2.
All the experiments and General Relativity say both number are equal, so we usually use the same letter and name for both.
The idea is that if the ball moves and the only force is the gravity, then the mass get cancel in the equation and then if you fix the initial direction and velocity, all balls have the same trajectory that is the same parabole.
It's easy to repeat this experiment with balls of different materials, and verify this result experimentally. If the ball is very light, you need to make the experiment in vacuum, but for a big heavy ball that's not a problem.
Each material has a different proportion of neutrons and protons. A ball of iron has approximately 47% protons and 53% neutrons. A ball of lead has approximately 40% protons and 60% neutrons.
IIUC in your theory, the attraction is:
neutron - neutron : yes
neutron - proton : yes
proton - neutron : yes
proton - proton : yes
So the neutron in the ball will be attracted by all the neutron and protons of the Earth, but the protons on the ball will be attracted only the neutrons of the Earth.
So the idea is to build two balls of the same inertial mass m, one is made of iron and te other is made of lead. Then you put both in a launcher aimed at the same direction and make a video of the trajectory of the ball.
The current most popular theory says that both ball will follow the same path, and that's what you get if you make the experiment in the lab.
With your theory, the force(weight) on the iron ball will be smaller than the force(weight) on the lead ball, so the jump of the iron ball will be higher and land more far away. It would be like 3% higher and land like 3% more far away, that is easy to measure in the lab, so this would have been discovered a long time ago.
It is worth rereading this comment. Start with a cloud of ordinary hydrogen–1 with no neutrons at all. It can still collapse under gravity to form stars. All of the hydrogen atoms are attracted to all the other hydrogen atoms, even though they none of them have neutrons. In the early stages of the universe, all stars would have been of this type. These stars were the forges that created most of the neutrons that exist today. If protons did not attract each other gravitationally then this would never have happened.
It would be an interesting DIY project to liquify some hydrogen and measure its properties. There would, however, be considerable difficulty in replicating the original experiment directly using liquid hydrogen, because it has to be kept quite cold in order to stay a liquid. It also has the disadvantage of being quite a bit less dense than lead or iron, so your instruments would need to be correspondingly more precise. You might have better luck by measuring the deflection due to gravity of a beam of protons coming out of a particle accelerator. Be sure you make some video recordings of your experiments.
How much dark matter is in the universe according to the experts?
https://spaceplace.nasa.gov/dark-matter/en/ "Dark matter is stuff in space that has gravity, but it is unlike anything scientists have ever seen before. Together, dark matter and dark energy make up 95% of the universe. That only leaves a small 5% for all the matter and energy we know and understand."
From what we see in the sky it is a fairly logical assumption that 95% of the baryonic matter up there are protons and 5% of it is neutrons. There is your missing mass. We only think there is missing mass because of the strange motion of the cosmos doesn't line up with our laws. As a programmer I abhor fudge factors because they don't always work.
Think about black holes sucking in matter. Mass down the the center loses it's electric charge for the most part and becomes like neutrons. Quasars are supposed to be quite massive but it appears mass is leaving them. Perhaps quasars are the exit points of black holes. The mass gets converted from neutral to non neutral somewhere at the edges of the universe.
If I'm right, then the sun and most stars are a heck of a lot more massive than we think they are. Also this would explain why the atmosphere at earth's beginning was so much hotter because the sun was considerably larger due to ongoing neutron-genesis.
Thank you for your time. I'd do the experiment myself if I was able to.
So I don't see how a universe where protons didn't create gravity could develop stars.
Cavendish experiment with Hydrogen <--> Hydrogen: I agree, probably no one has done it.
But your proposal must change the result of experiments with balls of Iron instead of balls of Plumber. That's much easier, and I'm sure someone has done it, but I can't find a good example in Google. It's a tricky experiment, so it would be nice if you try, but beware it's difficult.
After thinking for a while, I notice your proposal would change the time of oscillation of a pendulum. The standard formula is
T = 2π sqrt(L/g)
but it can be rewritten as
T = 2π sqrt(L/(W/M))
where W is the weight and M is the inertial mass.
The standard theory says that W/M has the same value for all materials and is usually called g. Your theory predicts a 3% difference for iron and plumber IIUC.
The usual pendulum with a ideal string and a ideal ball has some problems, because it's difficult to measure the "center" and get a high accuracy. But there are better solutions that use a solid big rod and measure the different times hanging the rod from a few points instead of one, for example https://en.wikipedia.org/wiki/Kater%27s_pendulum . I used a similar device in the past, but we used more points and it was just a rod instead of a profesiona device. It's not difficult to build one of this pendulums with different materials and measure the W/M of different materials yourself.