Proof that a neutron event changes carbon dating is needed.
Carbon 14 is normally generated high in the atmosphere as a loose neutron crashes into an atom of Nitrogen 14, ejecting a proton. The number of atomic particles remains the same (14), but by losing a positively charged proton and replacing it with a neutrally charged neutron, the charge on the nucleus goes down (from 7 to 6), which is why we call the new atom Carbon rather than Nitrogen. After the atomic bomb tests of the 1960s, the amount of Carbon 14 (radiocarbon) in the atmosphere almost doubled, so we know that proton radiation from atomic disintegration certainly can affect the proportion of radiocarbon in a material.
The factors affecting how much material is affected can be modelled very precisely, and Bob Rucker, who has been doing this kind of modelling professionally for many years, such that the safety of nuclear power stations has literally depended on his work, has calculated quite precisely the number of neutrons that would have had to be emitted from a disintegrating body in order to produce the exact gradient that we find in the radiocarbon dating results from 1988. So far, so uncontroversial.
Is there any evidence that this can actually happen?
No. This is where the miracle comes in. Experimentally, a recognised radioactive source is required, which dead bodies are not. The only evidence that such an event could have occurred lies in the chronological gradient of the results of the 1988 dating tests, which must be assumed, for the purposes of this speculation, to be accurate and quite precise.
that data does not pass the mathematical tests needed to confirm that the evidence in question actually is indicative of a date.
I think this needs a bit of unpacking. Noticing that their results were statistically anomalous in terms of an entirely uniform bit of cloth, the authors of the Nature paper assumed that the errors quoted were too small, as I mentioned to Jan10000 above, and proceeded to calculate an overall date of 1260-1390, as is well known. More precisely, the authors took the 12 individual results from the 12 little pieces into which the samples had been subdivided (4 at Tucson, 3 at Oxford and 5 at Zurich), and combined them into 3 means, one for each laboratory, and then combined those to produce the final date.
Disputing this assumption (that the errors were too small), Riani and Atkinson took the 12 individual results as correct and accurate measurements, and deduced the arrangement of the little pieces which best fitted the chronological sequence of the dates.
Rucker and Antonacci have taken the middle path, and used the three laboratory means as indicative of the gradient derived from neutron radiation. The statistics used to achieve these dates have been assumed to be reliable.