What replaces the singularities in the Big Bang and black holes?
A singularity signals that a particular spacetime description has reached a limit, such as incomplete paths or unbounded curvature. It need not be a physical object containing an actual infinity. OPH proposes finite observer records beneath the smooth geometry, so a continuum singularity can mark the failure of that effective description. That is a possible route around the problem, not a completed replacement history. A finite construction must still say what observers record, how clocks and signals continue, and whether the relevant quantum evolution is well defined. Simply capping a number in a simulation proves none of those facts. The cosmological and black-hole cases require their own source-generated continuations and comparison with the surrounding physical geometry.
See also What was the Big Bang?, What happens inside a black hole?, Do infinities exist in reality or only in our descriptions?
Sources: Observer spacetime and the conditional Einstein reconstruction
Also in Chapter 33: Why Was Every Open Problem the Same Problem?, Chapter 35: What Is Anything Made Of?.
This answer comes from OPH, Observer Patch Holography. OPH is a rigorous, consistent and complete description of the unified, self-creating computational Universe we inhabit. Its consistency is machine-checked: the fixed point at its core is constructed explicitly, and a theory with a model contains no contradiction.
To understand OPH and check every step yourself, start with the free book Reverse Engineering Reality. It builds the theory from the ground up, and the OPH Notebook answers your questions while you read.
read the free bookask the notebookor ask sage:read chapter 25: Why Does Gravity Look Like Geometry?watch a universe make its own particles