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I'm familiar with the construct:

sentence + に違いない
surely/must be/no doubt that sentence

So in the sentence

中身はケーキか焼き菓子のたぐいに間違いない。
The contents must surely be of the cake or baked sweet kind (my translation).

I'm assuming that に違いない plays the same role.

What is the difference between these constructs? My gut tells me that に間違いない shows more certainty that に違いない, but my gut is very unreliable when it comes to Japanese.

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According to this thread it seems to me that the majority of the answers say that に間違いない is a stronger version of に違いない, while に違いない is closer to a personal "sureness". I'll paraphrase some of the answers for convenience.

According to the top answer, に違いない means that the speaker came to a conclusion and is very sure that it is correct, but it is ultimately their own opinion. There is a little uncertainty, but the speaker is very confident in what they are saying.

に間違いない is described as a 100% certainty that there is nothing wrong with the statement. It seems to describe a personal conclusion as well, but is a bit more objective.

One of the other answers explains it like this:

「?に違いない」は、100%では無いがそれに近い場合の言い方です。言い換えると、『?にほぼ間違いない』となります。

So, に違いない = にほぼ間違いない.
I don't know if there is a maybe more nuanced difference between the two, but this seems to be the general agreement on the internet. Some other sites like this HiNative page say similar things.

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(One difference that I can think of right now is...)

「~に違いない」は、確信の高い推測・推定を表すことができます。「~はずだ」と似た意味があります。

◎「雨が降るに違いない。」(✕「雨が降るに間違いない。」)
◎「犯人はこの男に違いありません。」
◎「中身はケーキに違いない。」(≂ 中身はきっとケーキだ。)

「~に間違いない」は、証拠があって確信をもって断言・断定する表現です。証言するときにも使われます。

◎「覚醒剤は私の物に間違いありません。」(警察に捕まった時に言う)
◎「犯人はこの男に間違いありません。」(「面通し」などで)

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    「覚醒剤は私の物に間違いありません。」(警察に捕まった時に言う) <-- You must have some very honest criminals in Japan :-) – user3856370 Jan 9 at 9:42

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