What is the danger in Thomas' writing down the instruction for thread dying in Gathering Blue?
The answer to this question can be found in Chapter Nine. Having met Annabella for the first time, Kira feels completely bewildered by all that she has to remember and very overwhelmed by what is expected of her. When she confesses her feelings to Thomas, he comes up with a suggestion. He proposes that he writes the information down for her so that she can have a record that he can read to her to help her remember. Writing and reading are presented as important skills that only a few boys are taught. Thomas, being one of these select few, is able to thus help Kira.
Note the way that knowledge is presented as being carefully controlled by the Council. It is "not permitted" for girls to learn how to read and write, and thus it is potentially very dangerous for Thomas to record this information and help Kira as Kira might learn how to read herself. Note Kira's response to seeing Thomas write the words down:
When he read the word hollyhock aloud with his finger on the word, she saw that it was long, with many lines like tall stems. She turned her eyes away quickly so that she would not learn it, would not be guilty of something clearly forbidden to her. But it made her smile, to see it, to see how the pen formed the shapes and the shapes told a story of a name.
Thus the danger lies in the way that women are not allowed to learn how to read or write or have anything to do with it. Discovery of what Thomas and Kira are doing could jeapordise Kira's position.
In the book Gathering Blue Kira and Thomas live in a society where if a person is disabled in any way they are usually put into a graveyard in the forest and left for the beasts to get them or to die. KIra has a disfigured and relatively useless leg. In truth there are no beasts but the community does not know that.
Kira has been spared because she has special talent. She is able to weave and mend the coat worn by the singer each year. The coat is supposed to guide the people towards their future and reveal and document the past.
If the information about dying is in writing someone else may be able to learn how to do it an then the powers might decide that Kira is no longer needed.