It would mean black holes have different gravitational properties. Normal matter's field falls off at 1/r^2, but a black hole would be more like 1/r^2 + x(1/r). The 1/r term would make the outer edges spin faster than expected.
Your theory is that at far distances black holes act like normal gravity wells, and at near distances the gravity is lower (or higher?) than would be expected for the real mass.
This is directly contradicted by evidence specifically for the case of galactic-scale supermassive black holes, and broadly by observations of black holes.
We have directly observed accretion disks, which would look very different if the near field of a black hole was different from the far field. We have directly observed discrete objects falling into black holes with minimal accretion disks, ie with no other forces except for gravity, and have found that redshift evidence very strongly agrees with general relativity. There is just no way you are correct for most black holes. They behave the same close up as they do from far away.
If supermassive black holes acted differently then we would have seen evidence of your theory with Sagittarius A*. It's our galaxy's central supermassive black hole. We have measured the orbits of objects light years (cluster GCIRS 13E) away and light hours (star S2) away from its center. Gravity changes as expected.
A near-field linear effect is not sufficient to cause us to think that black holes are less massive than they actually are, which would be necessary for your theory to explain why stars in the near parts of galaxies don't orbit as quickly as we'd expect based on the outer stars. If gravity stopped increasing as quickly below a certain radius, we would have seen it.
> Your theory is that at far distances black holes act like normal gravity wells, and at near distances the gravity is lower (or higher?) than would be expected for the real mass.
No that's not right. It's the inverse. At far distances gravity is higher than expected for black holes compared to regular stars.
> At far distances gravity is higher than expected for black holes compared to regular stars.
If it doesn't follow the square-cube law in the far field, it violates conservation of energy and creates a whole bunch of situations that don't make any sense. The farther away from something you are, the heavier it would appear to be. The event horizon of a black hole would shrink as you got closer, because light coming from far away would be pulled more over the longer distance. Neutron stars would appear to be black holes if you were far enough away.
Gauss' law for gravity says the total flux through any enclosing surface is proportional to the enclosed mass. If the gravitational potential decreases at 1/d and the area increases at 1/d^2 the mass must increase. That fact also makes it irrelevant if this is something that only happens for black holes- if you're far enough away, the enclosed mass will make it look like a black hole. That means that this effect would be apparent at measurable scales. Tungsten motes would look like miniscule black holes and asteroids would not emit light once you were far enough away.
The Triangulum galaxy is ~2.7 Mly away and ~2.6 kly wide. At 5 kly the rotation curve is totally unexplainable with conventional mass and gravity. If the strength of gravity falls off with 1/r past that distance, the flux enclosed by a 2.6 Mly sphere would be 1000x higher (d^2 area times 1/d flux density) and the apparent mass would be 1000x higher.
General relativity would have quite a few things to explain if galaxies caused more gravitational lensing the farther away they were despite not having any more stars. The fact that we are seeing significant, unignorable effects at the edges of galaxies means that those effects should be FAR larger at even fractions of the distances between galaxies.
