6.1262 Proof in logic is only a mechanical means to make perception of a tautology easier in cases where it is complicated.
Hence the quotation marks earlier (6.126) around “proof”. Proof is relative to a system. Proof of guilt in law is proof of guilt beyond a reasonable doubt. Proof in mathematics depends on the laws/rules in play. For Frege, proof means logical proof, i.e. demonstration that a sentence/formula is equivalent to an axiom or derivable from some other part of the system. Derivable, presumably, according to the rules of derivation in the system. But presumably something, whether axioms or processes of derivation, must be simply given or accepted as intuitively obvious. Experiments are particular. Manipulation of actual rods might be an experiment in physics, but there are no experiments in mathematics. If one rod breaks, this can prove nothing mathematical. Similarly if the paper tears when drawing a diagram. A diagram might prove something in geometry, but only if taken a certain way, so that physical features of the diagram are ignored, for instance.
McManus p. 85: “ontological distinctions are now [with the right mechanical means, i.e. Begriffsschrift] shown simply in that ‘they’ are shown up for what they are: namely, the confused product of word-play.”