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What justification is there for the article listing the ISO 4 abbreviation as "Phil. Trans. R. Soc. A"? This does seem to be an abbreviation that the journal itself uses, but not the ISO 4 abbreviation. (Perhaps the question, then, becomes what justification is there for the Royal Society using it?)
Consulting the ISO 4 List of Title Word Abbreviations, "Philosophical", for example, should be abbreviated "Philos." (not "Phil."). The US NLM lists the ISO abbreviation for "Philosophical Transactions of the Royal Society of London. A" as "Philos Trans R Soc Lond A". By this logic (and by the LTWA above) "Philosophical Transactions of the Royal Society A" should be "Philos. Trans. R. Soc. A" (assuming that 'R' is right for Royal. I've trouble finding it in the LTWA).