Bullwort

Bullwort

Observation - Bullwort - UK and Ireland. Description: An umbellifer I haven't been able to identify - I've considered Fool's Parsley, which feels appropri

An umbellifer I haven't been able to identify - I've considered Fool's Parsley, which feels appropriate given how long I have spent mulling it over, but have started to wonder if it might be Bullwort. However, my "Atlas of the Kent Flora" from 1982 describes the neophyte Bullwort as "An occasional plant of cultivated and waste ground, either as a wool-alien or from bird-seed" and gives only three locations across the county where it had been found during the atlas survey.
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There were several plants here, growing in an arable field on a chalky slope which has been set aside and was full of Welted Thistle, Field Madder and Wild Mignonette. I only photographed one plant. The bracts and bracteoles are distinctive, I think, also the stem leaves; the lower leaves, while similar, were already dying back (although this might possibly have been a consequence of the very dry warm weather of the last two weeks). In context, it didn't seem likely to be a wool-alien or a bird-seed incidental; I would have expected it to have been a calcicole and a plant typical of southern chalk downland now emerging in an era of set-aside. I might add that the beech and yew woodland adjoining the field is an SSSI.
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Any thoughts much appreciated.
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EDITED 24 May 2020: title changed to reflect ID confirmed by JoP and lavateraguy