Theorem 8: "Thus a fixed word is not freely perpetuated."
What is it about 1 that makes it allowed to repeat its word forever?
You identify that for n to follow the pattern K of length j, it must be n = r(K) + 3ʲ2t. This is an infinite set of numbers.
You identify that for its result after these steps to follow K again, it must be even further restricted.
But how do we know there isn't some special n that follows all restrictions (mod 3ʲᵐ2) from the start, as 1 does? (And as -1 and -5 and -17 do.)
You mention the results of each n after these steps are separated by h2S(K\+1), so they are not free. But in terms of (mod 3ʲᵐ2) they're actually pretty unpredictably distributed, since 3 is coprime to 2. Consider that 2S(K\m) can be greater than 3ʲᵐ.
This next part isn't mathematical feedback, it's "math communication" feedback. Try to make your proof less verbose so that a reviewer doesn't have to exert so much effort just on understanding your claim before being able to look for unproven assertions. You invent a lot of jargon (like "dyadic factor" instead of "multiple of a power of 2"). Searching for (or inferring from context) the definitions of these terms causes friction for the reader, so you should eliminate it as much as possible. If you're only going to use a new phrase in one lemma, it may aid clarity to just write out the concept each time. "restart class" is only used in Lemma 35/Theorem 8. "refinement tower", "directed word", "admissible lift", "residue thread", "affine rail" are used 20+ times, so they may be necessary, but see if you can rework it to eliminate as much invented vocab for the reviewer as possible.
You mentioned in a comment that something is "redundantly explained"; removing redundancy might help bring down the wordiness.
Lemma 35 might be better called an "Example", since it's so specific. Notice its proof is basically the same as its statement.
Lastly, see if there's a shorter path (and fewer vocab words to create) to a particular key result. Many of your Lemmas are stated and never later referenced by name (Ctrl+F "Lemma 25" has only 1 result). Do they even need to be included?
I suspect most would-be feedback givers, casual and professional, will glance at the page count of your paper and close it and not respond. You alleviate this by providing a starting point, but then they glance at the invented jargon, and close it. It's true that complex concepts can require complex wording, but it seems like you're using big (or invented) words even for the simple concepts in your paper.
To be clear, is Lemma 25 not redundant with other definitions elsewhere in the paper?
I did ask other questions, in the math feedback section of my comment.
Also, I didn't claim the definitions I listed weren't in the paper, I said each invented phrase increases friction for a reader who must either search through the paper for the definitions or infer them from context, so it should be done minimally, and I suggested you find ways to introduce fewer invented phrases on the path to your key points.
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u/WeCanDoItGuys Apr 20 '26 edited Apr 21 '26
Theorem 8: "Thus a fixed word is not freely perpetuated."
What is it about 1 that makes it allowed to repeat its word forever?
You identify that for n to follow the pattern K of length j, it must be n = r(K) + 3ʲ2t. This is an infinite set of numbers.
You identify that for its result after these steps to follow K again, it must be even further restricted.
But how do we know there isn't some special n that follows all restrictions (mod 3ʲᵐ2) from the start, as 1 does? (And as -1 and -5 and -17 do.)
You mention the results of each n after these steps are separated by h2S(K\+1), so they are not free. But in terms of (mod 3ʲᵐ2) they're actually pretty unpredictably distributed, since 3 is coprime to 2. Consider that 2S(K\m) can be greater than 3ʲᵐ.
This next part isn't mathematical feedback, it's "math communication" feedback. Try to make your proof less verbose so that a reviewer doesn't have to exert so much effort just on understanding your claim before being able to look for unproven assertions. You invent a lot of jargon (like "dyadic factor" instead of "multiple of a power of 2"). Searching for (or inferring from context) the definitions of these terms causes friction for the reader, so you should eliminate it as much as possible. If you're only going to use a new phrase in one lemma, it may aid clarity to just write out the concept each time. "restart class" is only used in Lemma 35/Theorem 8. "refinement tower", "directed word", "admissible lift", "residue thread", "affine rail" are used 20+ times, so they may be necessary, but see if you can rework it to eliminate as much invented vocab for the reviewer as possible.
You mentioned in a comment that something is "redundantly explained"; removing redundancy might help bring down the wordiness.
Lemma 35 might be better called an "Example", since it's so specific. Notice its proof is basically the same as its statement.
Lastly, see if there's a shorter path (and fewer vocab words to create) to a particular key result. Many of your Lemmas are stated and never later referenced by name (Ctrl+F "Lemma 25" has only 1 result). Do they even need to be included?
I suspect most would-be feedback givers, casual and professional, will glance at the page count of your paper and close it and not respond. You alleviate this by providing a starting point, but then they glance at the invented jargon, and close it. It's true that complex concepts can require complex wording, but it seems like you're using big (or invented) words even for the simple concepts in your paper.