Ok so you’re saying it never happened, but then in the very next sentence you acknowledge that you know it is happening with TI today

You asked me what I do if my students show me 2 different answers what do I tell them, and I told you that has never happened. None of my students have ever had one of the calculators which does it wrong.

that both behaviors are intentional and documented

Correct. I already noted earlier (maybe with someone else) that the TI calculator manual says that they obey the Primary School order of operations, which doesn’t work with High School order of operations. i.e. when the brackets have a coefficient. The TI calculator will give a correct answer for 6/(1+2) and 6/2x(1+2), but gives a wrong answer for 6/2(1+2), and it’s in their manual why. I saw one Youtuber who was showing the manual scroll right past it! It was right there on screen why it does it wrong and she just scrolled down from there without even looking at it!

none of these calculators is “wrong”.

Any calculator which fails to obey The Distributive Law is wrong. It is disobeying a rule of Maths.

there is no ambiguity where there actually is.

There actually isn’t. We use decimal points (not commas like some European countries), the obelus (not colon like some European countries), etc., so no, there is never any ambiguity. And the expression in question here follows those same notations (it has an obelus, not a colon), so still no ambiguity.

i recommend reading this one instead: The PEMDAS Paradox

Yes, I’ve read that one before. Makes the exact same mistakes. Claims it’s ambiguous while at the same time completely ignoring The Distributive Law and Terms. I’ll even point out a specific thing (of many) where they miss the point…

So the disagreement distills down to this: Does it feel like a(b) should always be interchangeable with axb? Or does it feel like a(b) should always be interchangeable with (ab)? You can’t say both.

ab=(axb) by definition. It’s in Cajori, it’s in today’s Maths textbooks. So a(b) isn’t interchangeable with axb, it’s only interchangeable with (axb) (or (ab) or ab). That’s one of the most common mistakes I see. You can’t remove brackets if there’s still more than 1 term left inside, but many people do and end up with a wrong answer.

By “we” do you mean high school teachers, or Australian society beyond high school?

I said “In Australia” (not in Australian high school), so I mean all of Australia.

Because, I’m pretty sure the latter isn’t true

Definitely is. I have never seen anyone here ever use a colon to mean divide. It’s only ever used for a ratio.

Do you have textbooks where the fraction bar is used concurrently with the obelus (÷) division symbol?

All my textbooks use both. Did you read my thread? If you use a fraction bar then that is a single term. If you use an obelus (or colon if you’re in a country which uses colon for division) then that is 2 terms. I covered all of that in my thread.

EDITED TO ADD: If you don’t use both then how do you write to divide by a fraction?