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(One gram of modern carbon contains about 6 x 10 ¹⁰ ¹⁴ C atoms, and 43.6 half-lives should reduce that number by a factor of 7.3 x 10 ^-14 .)

It's a quibble, but that's a very tiny reduction factor where a great one is needed. (You just cut the original amount of C-14 in half 43 times.) I would expect a scientific paper to be proofread.

... organic samples from every portion of the Phanerozoic record show detectable amounts of C! 14 C/C ratios from all but the youngest Phanerozoic samples appear to be clustered in the range 0.1-0.5 pmc (percent modern carbon), regardless of geological ‘age.’

The short answer to what is going on here is "Abuse the instrument, measure noise." There's a preferred instrument for every date range, one for which the element half-life makes sense. C-14 is only useful for very recent objects because of the short half-life. With any physical measurement, as the thing being measured shrinks, the spike of signal tends to go down not to zero but into a fringe of noise. These guys appear to be lawyering on the noise.

"The short answer to what is going on here is "Abuse the instrument, measure noise." There's a preferred instrument for every date range, one for which the element half-life makes sense. C-14 is only useful for very recent objects because of the short half-life. With any physical measurement, as the thing being measured shrinks, the spike of signal tends to go down not to zero but into a fringe of noise. These guys appear to be lawyering on the noise."

No. The lower limit of detection (LLD) for the AMS instrument is 0.002 pmc (percent of the modern ratio of C-14). In Figure 3 in their paper, the mean pmc for 10 samples of coal was pmc = 0.247, which is about two orders of magnitude higher than the pmc.

No, they are not "measuring noise".

I just wrote a document dealing with LLD determination for radioactive dose measurements. (I am the lead researcher on a new instrument.) They are not measuring noise.

The tables have turned. The statues are falling. Many scientists are being exposed as being not only fallible but biased, deceiving and dishonest on the question of evolution.