What is this stone from the alps of provence? It's two spheres, one of them is made of at least 25 layers.
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Your fossils look like Stromatolites to me. Stromatolites start as a film of cyanobacteria on the substrate in marine and sometimes freshwater environments. I am not sure if the bacteria capture sediment or weather the carbonates form on the bacteria themselves but overtime they develop layers of carbonate and other mineral deposits which later became fossilized. There is a wealth of information about these on the internet. Stromatolites are still found today, Living Stromatolites. Wikipedia has a good description of stromatolites Stromatolites on Wikipedia.
It looks like a dripstone (stalactite). This also explains the milk white color... Stromatolites show a more distinct layering caused by lamination layers of algae and captured particles.