The three C14 labs were accurate on what they tested... a medieval repair. Essentially, because of sampling error and the breaking of the agreed protocols for taking the sample from the Shroud, the 1988 C14 test has been invalidated. This has now been conclusively proved, chemically, physically, and photographically, in peer-reviewed work published by the late Raymond N. Rogers in Thermochimica Acta, Vol. 425, Issues 1-2, 20 January 2005, Pages 189-194 in 2005. Independent researchers under Dr. John .L Brown, using different techniques from Rogers chemical analysis approach, including electron microscopy, confirmed Rogers findings in 2005, Microscopical Investigation of Selected Raes Threads From the Shroud of Turin. In mid 2008, Dr. Robert Villarreal, et al, using entirely different approaches, confirmed the finding that the C14 sample used in 2008 was a melange of mixed original material and skillfully re-woven material probably from the 16th Century "Los Alamos National Laboratory team of scientists prove carbon 14 dating of the Shroud of Turin wrong".
Rogers work, attempting to FALSIFY the repair theory, instead confirmed it. Rogers now confirmed findings showed that the 1988 test sample was approximately 40-60% dyed COTTON, which contains approximately 2% aluminum from an alum mordant used to adhere the dye to the cotton, intermixed with between 60-40% original Shroud material (depending on location on the sample)... the main body of the Shroud is Linen, which is made of Flax, is not dyed, contains no aluminum, and contains no cotton.
The test proposed by Dr. John Jackson in your linked article is not related to the proof that the 1988 C14 sample has been proved, with independent confirmation, to have been falsified because of sampling error. The Jackson theory, that Ramsey is going to test on NON-Shroud material, has to do with an atom substitution theory. This theory is moot because of the prior work proving that what was tested was not exemplar of the Shroud.
Thanks for the links. However, at best the research you link dates the Shroud to be somewhere between 1300 and 3000 years old.
Perhaps more research is needed to better date the Shroud to determine if it is authentic or a hoax.
BTTT