Those proposing Neo-Darwinian Evolution as the mechanism that developed past and present living species have yet to offer an actual proof. Of course, "proof" of past events is always difficult because the only method is the disproof of alternatives and the relative plausibility of those explanations that remain. Additionally, new information may reveal a possibility that had not been noticed before. So questions about the past are often never settled.
The Neo-Darwinian Evolution proposes that the living species we see today and in fossil evidence developed over time, each from the previous by accidental mutation, step by step, guided by the success of each new form in the existing or changing environment. It is assumed that an enormous span of time allowed successes despite the poor likelihood of survivable mutations. This theory is an extension of Darwin's original Theory of Adaptation where existing variations within a species allow adaptation to changing conditions by the success of those variations under different conditions.
There are many tantalizing and suggestive clues that led us to take the Theory seriously, but none of these rise to the level of proof. In particular, these are the fossil evidence, the DNA evidence, various DNA mechanisms in cell reproduction, and the success of the Theory of Adaptation.
The fossil evidence shows a sequence of forms generally from simplest to more complex over time. However, these forms do not approach completeness due to the limited conditions which are conducive to fossil formation, and our failure to dig up the entire planet down to magma. If the current Theory is correct, we would expect to find at least some instances of successive, fine, refinement moving from one form to the next. Instead we find major changes with no step-by-step intermediates. Also there are examples of fossils out of sequence, which might or might not result from incorrect interpretation of the conditions used to establish the time sequence. Whatever the case, the fossil evidence shows only sequence and gives no clue as to mechanism.
The analysis of differences in the DNA of existing organisms into time sequences is also suggestive of development over time. Some ancient DNA has also been reconstructed. As we learn more about DNA and cell mechanisms, we may find clues to mechanism. But nonetheless, like the fossil evidence, DNA sequences merely imply sequence not mechanism.
Much research into DNA and the various mechanisms within the cell give us plausible mechanisms for alteration of DNA and consequently change in an organism. This research is very sophisticated but is still in a very early stage. We are a long way from a comprehensive understanding of DNA, cell function, and differentiation in organism development, and hence the biological basis to substantiate the Theory. It may be that DNA facilitates evolution as yet another adaptive mechanism for survival.
The Theory of Adaptation has been successful in showing that existing genetic variation within a species can allow survival under changing conditions. This has been subjected to experiment in the present and hence rises to the level of proved. The origin of the variations is presumed to be mutation, and because most such variations are are simple changes, has high plausibility. It is proposed that Evolution is just a stepwise extension of Adaptation. However there is an enormous leap of complexity to a new species, and in many cases there is doubt that stepwise refinement can explain a large or novel change. Also recent experiments have shown that mutations that adapt an organism are all destructive. That is, they destroy an existing genetic function to help the organism survive but provides no path to novel improvement.
So what would constitute proof? Proof of the possibility would be a complete description, consistent with the Theory, of the transition from a predecessor to a successor form including a complete description of each mutation, how it manages to survive long enough and in sufficient quantity for the next mutation, and of course complete, molecular level details. The entire process must then be reproduced in the laboratory. Subsequent to the first successful proof, a wide range of other life forms would have to be similarly proved. Clearly, we are a hundred years away from this effort because we barely understand DNA and how it functions within the cell, let alone how it controls the development of complex living organisms.
An inductive proof is possible but unlikely. It would involve detailed monitoring of a population for a very long time, awaiting the development of new species. Perhaps this could be accelerated, but the time involved would exceed our patience and resources. Any mistake or break in the chain of custody would invalidate the result.
Since it is obvious that no reasonable proof of the Neo-Darwinism exists and that much work needs to be done, we should stop lying to ourselves and our children. Instead we should tell the truth, the Theory of Neo-Darwinian Evolution is what we have, so get yourselves into Science and help us figure it out!