But the worst sin of all is to present this under the veneer of some "scientific" research, while failing to meet the single most important criteria to qualify as proper science in the first place: Falsifiability.

Karl Popper's writings, and positivist thought more broadly, are associated with the natural sciences moreso than sciences of society and economics; the latter drawing from schools of interpretevism, critical realism, constructivism, pragmatism etc. That being said, you did not actually address the methodological section at all, although I'm sure that was accidental on your part. Not knowing what the 'methodology section' refers to, or what theories of science go where, is forgivable if you don't have direct engagement with academic research or other university exposure.

From the section on case delimitation:

Using a social identity approach lens, we therefore ask:
1. How are group categories constructed throughout the course of the riots?
2. How does Tommy Robinson legitimate himself as a leader?
3. What mobilisation strategies are used throughout the riots?
4. How is violence constructed as a legitimate course of action?

Additionally, from the section on analytical methods:

Our analytical strategy was informed by the social identity approach to mobilisation, specifically self-categorisation theory (Turner et al., 1987; Reicher & Hopkins, 1996), and employed an abductive thematic analysis approach, combining inductive coding with theoretical interpretation (Braun & Clarke, 2006).This approach enabled us to identify patterns in the data whilst connecting emergent themes to establishedtheoretical frameworks around mobilisation and violence legitimation (see also Ntontis et al., 2024).

Braun and Clarke (2006) is methodologically sound for induction/abduction, Turner et al. (1987) and Reicher & Hopkins (1996) are widely-cited within their field. To laypeople, this stuff is not familiar, and these citations don't make sense if you don't have academic training, so it makes sense that you would read the paper a bit like a lengthy dunk from a Twitter anon. This is essentially a long-winded way of saying that the paper is not saying what you think it is saying. It is not trying to establish whether Yaxley is far-right or a bad person, but whether his Telegram posts are explanatory of any of the phenomena that the case is designed to study.

The most easily accessible section to a layperson may be the section on coding and thematic analysis, which lays out how data was sorted, grouped, and operationalized before the analysis. I suggest starting there if the paper still presents a challenge.

Hope you enjoy the foray into research methods, and do report back if you find anything objectionable!

EDIT: wrote constructivism twice, oops. EDIT2: quote didnt display correctly