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prooftree is designed to set a proof in a centred, display environment, with appropriate vertical spacing. You don't want that so don't use the environment. Instead, use \DisplayProof to display the proofs in-line. This can be used to set proofs within text or side-by-side. I've used a regular center environment to display the two proofs centred and with ...

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You are specifying [display] which typesets the label of the chapter in a separate paragraph. However, you are also specifying an empty label. So you get an empty paragraph for each \chapter. Since unnumbered chapters don't get labels anyway, there is no such paragraph in those cases. If you do not want numbered chapters, you should use \chapter*. Or, if, ...

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The example has lots of strange markup. I removed the \, (letterspacing should be applied elsewhere) and all the page styles (why change pagestyle each chapter???). I then added \showoutput so that TeX leaves a representation of the page in the log file. % [[- LaTeX prelude \documentclass[11pt,letterpaper,oneside]{book} %\usepackage{fontspec} ...

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As clearly shown in your error message $\decayarrow} \mu + \num$ You have a } with no { in that math expression. The markup is hmm strange, you really shouldn't have any \hspace or \newline within the document markup. Hard to say from a fragment but this should presumably be using align. in addition \hbox{\big|}\ from the definition you show will ...

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article by default uses \raggedbottom so the format adds stretch glue at the bottom of each page, your \vfil balances that glue so they both stretch the same anount, centring the text in the remaining space. If you use ll glue then it expands to the full amount and the l glue from \raggedbottom is restricted to its natural size of 0pt.

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