I think your argument involves reflection somewhere. The desk calculator agrees that 2+2=4, and it's not reflective. Putting two pebbles next to two pebbles also agrees.

Agreement with statements such as 2+2=4 is not a function that desk calculators perform. It is not the function performed when you place two pebbles next to two pebbles.

Agreement is an evaluation performed by your mind from its unique position in the universe.

... this implies there is something to be converged upon.

The conclusion that convergence has occurred must be made from a context of evaluation. You make observations and derive a conclusion of convergence from them. Convergence is a state of your map, not a state of the territory.

Mathematical realism also explains my observations and operates entirely within the mathematical universe; ...

Mathematical realism appears to confuse the map for the territory -- as does scientific realism, as does physical realism.

When I refer to physical reality or existence I am only referring to a convenient level of abstraction. Space, time, electrons, arithmetic, these all are interpretations formed from different contexts of evaluation. We form networks of maps to describe our universe, but these maps are not the territory.

Gottlob Frege coined the term context principle in his Foundations of Arithmetic, 1884 (translated). He stated it as "We must never try to define the meaning of a word in isolation, but only as it is used in the context of a proposition."

I am saying that we must never try to identify meaning or existence in isolation, but only as they are formed by a context of evaluation.

When you state:

Putting two pebbles next to two pebbles also agrees.

I look for the context of evaluation that produces this result -- and I recognize that the pebbles and agreement are states formed within your mind as you interact with the universe. To believe that these states exist in the universe you are interacting with is a mind projection fallacy.