The path example is extremely fertile ground for this kind of discussion! It is definitely true that both paths encode the information that one's house is connected to the local store. But is that all they encode? Homotopy theory is all about the different paths between two points, and it tells us some quite interesting things! In particular, if you have two paths from point A to point B, you can ask: can you smoothly animate an image of the first path into an image of the second, such that every still frame in-between is also a legitimate path? (If you can't, that tells you that there's some form of hole in between them!)

In the house/store example, a path is also a witness to the fact that, if you perform a road closure anywhere not on the path, then connectivity is preserved. Simply stating that the two points are connected doesn't tell you whether it's safe to close a road! Moreover, taking the two paths together tells you that performing a single road closure that only affects one of the paths will still leave a route you can take.

In both examples, if the paths were logically interchangeable, you wouldn't be able to get more information out of the both of them than you could from just one. But because they aren't equivalent -- because each contains some information that the other does not -- we can deduce more from both together than from either individually.

> all theorems about a certain statement being true must be reducible to a single irreducible logical chain of argument.

Why is this necessarily true? We know that true statements in topology (for example) don't all reduce down to being equivalent (eg if I have a loop that goes through the ring of a donut/toroid it doesn't reduce the same as if I have a loop on the surface of the donut/toroid so establishing facts about one wouldn't tell me facts about the other). So how do we know that statements in logic reduce? Could the space of logical statements not have topological characteristics like that?

You are being too literal -- I was providing an analogy, not an example.

Also:

> ... all theorems about a certain statement being true must be reducible to a single irreducible logical chain of argument.

Citation needed ... I have no reason to believe this is true.

But here's an example of two proofs.

Proving sqrt(2) is irrational.

Proof 1: Every rational number has a finite Continued Fraction representation. But the normalised Continued Fraction representation of sqrt(2) is [1;2,2,2,...], which is infinite. Since this is infinite, sqrt(2) is irrational.

Proof 2: Consider integers a and b, and suppose 2(b²)=a². Consider the prime decompositions of a and b, and count how many times "2" turns up on each side. It's odd on the left, it's even on the right, so this cannot happen. Therefore we can never have integers a and b with 2(b²)=a². Therefore we can't have 2=(a²)/(b²)=(a/b)². So any fraction when squared cannot equal 2, so sqrt(2) is irrational.

Do these really feel "the same" to you?

> But if you consider deeply, both of these proofs are implicitly encoding same information about the space between your house and the local store

That is only _some_ of the informations that they encode, and particularly informations shared by both proofs, but that it not the only information they encode! The exact way to reach the local food store is also some information, and they encode different ways, hence different informations.

> What I wanted to opine was merely the fact that since all proofs use logic

Note that there's no single logic! There are at least two big logics, classical and constructive/intuitionistic, each with their own variants.

For example a proof by contradiction is valid in classical logic but not in constructive one. It would give you a proof that there must be a way to reach the local store without giving you the way to reach it. Would it still count as the same proof as the other two for you? It doesn't encode how to reach it, so for some it's not even a valid proof.