asiddham bahiraGgam antaraGge

असिद्धं बहिरङ्गमन्तरङ्गे

A /bahiraGga is /asiddha to an /antaraGga.

This is a /paribhASA. It says that inword rules work first, and when we have all our words made, we join them using outword rules.

Example. Ordinarily we

first make the word **pacAma पचाम using several inword rules,
then the word **idam इदम् with other inword rules,
and then we join them and the outword rule AdguNaH applies,
and thusly we get the sentence pacAmedam पचामेदम् "let's cook this".

. Booring. We have always done that in class. That's how the grammar works. What's the need of making it into a rule?

Historical reasons. It appears that /pANini did not bother to make this into a sUtra because he took for granted that students would gather this general principle for the teaching, just like you did.

Yet, some centuries afterward, some smartie student told his teacher this —

Rules ekaHpUrvap... and antAdivacca teach that after making pacAmedam पचामेदम् by AdguNaH, the pacAme पचामे must be considered to be a replacement of pacAma पचाम. And nothing in the /pANinIya prevents rule eta::ai from applying. It follows that either we should say pacamaidam, or /pANini screwed up.

Both options were equally unacceptable. So the teachers got together and agreed to this answer —

The problem with your argument is that eta::ai is an inword rule, and inword rules can never work after outword rules have worked — /pANini intended us to first apply inword rules, and outword rules only afterwards. We know that this is so because blablabla...

And that is how the saying asiddham bahiraGgam antaraGge असिद्धं बहिरङ्गमन्तरङ्गे got added to the grammatical tradition.