Has anyone thought about the possibility of breaking quantum codes via an inspection of the quantum error corrections that are always needed to make a fault-tolerant quantum computation? Error corrections deal with the static or noise on stored quantum information, bad quantum gates, errors in preparation, and correctable or corrected measurements.

You are way ahead of yourself. Quantum error correction does not employ redundancy. We have not discovered how to clone quantum data (information). Shor’s method of unfolding the qubits into nine fold parts works to identify if the data has been previously scanned, but little else.

Simple quantum computers are already being built, but they are not close to being as powerful as the most powerful PCs. The problem is the energy source, and that is again not a technical problem but a money problem.

“Quantum computers are already being built”?? I don’t think so! It is not simply a money problem. The essence of the quantum computer is molecular structure. Measuring plus and minus charges has nothing to do with the structure. Saying it does is like saying someone pulling the shades up and down on a jumbo jet is intrinsic to the jet. It is not.

No one has seriously claimed to be able to prove the full Thurston Geometrization Conjecture. Are there any takers? Thurston’s Conjecture is much more far-reaching than the PoincarĂ© Conjecture. It actually includes PoincarĂ© as a special case. Takers?

Caroline’s suggestion does not work. Take the sum of the digits of Pi, 3.14159….n.

Someplace in that sum of the list of digits there will be the solution to both the Thurston Conjecture and Poincare’s Conjecture. But there is no relation between the sets. It is easier to prove that than it is to prove that the sets will appear necessarily in the 3.14159 series.

The question was resolved by Cantor over a century ago. There are an infinity of infinities which means clearly that regardless of what magical properties you attribute to a number such as pi, the same properties occur in all other infinities. This is the same sort of thinking that metaphysical philosophers used to prove that you cannot conceive of a being greater than which you can conceive. It all reduces to Wittgenstein’s dictum that the limit of our logic is the limit of our language and the limit of our language is the limit of our logic. Numbers are part of the language we call mathematics.