In 1916, Hausdorff proved that the Baire hierarchy on $\mathbb{R}$, starting with the continuous functions, generates all Borel functions on $\mathbb{R}$. It remained open whether, starting with the a.e. continuous functions, the corresponding hierarchy generates all Lebesgue measurable functions on $\mathbb{R}$. We prove that, assuming the Axiom of Choice, the answer is negative.

We consider the enumeration of $N$-ary convex structures on finite sets. Our main result shows that the total number of convexities is expressed as the binomial transform of the sequence of numbers of grounded convexities. We present the exact numbers of all $N$-ary and grounded $N$-ary convexities for $|X|\leqslant 5$. The obtained integer sequences have been cross-referenced with the OEIS. As a result, we identified both known sequences (such as A000798) and new ones not previously listed.

Using a sequence of high-school level mathematics questions that were not available on the Internet, we compare the performance of the popular AI software Claude with that of my friends and fellow human beings Pierre and Catherine. Pierre had solid scientific training as a young man, while Catherine studied literature. All three were subjected to a simulated pre-calculus oral exam with main questions and follow-up questions. Their performances are compared and the ones with the best and worst performances are identified. The outcome is that the current version of Claude, even though it is an extremely useful tool that has probably recorded the solution to nearly all calculus questions that are available on the Internet, {\em exhibits only a very limited understanding of the subject} and {\em does not exhibit the ability to make intelligent connections} between different features of a pre-calculus mathematics problem that it has never seen before.

This paper bridges combinatorial geometry and music theory to solve the fundamental challenge of embedding classical Western harmony into the microtonal 19-tone equal temperament (19-TET). Inspired by Roger Penrose's observations on the mathematical elegance of 19-TET, we provide the theoretical foundation for a physical 19-TET acoustic piano currently under construction. However, playing classical 12-TET music on such an instrument poses a topological problem: emvedding the classical Euler-Riemann Tonnetz into the 19-TET universe inevitably distorts structural chords, creating dissonant ``wolves.'' By formalizing these harmonic spaces as incidence configurations (the 12_3 and 19_3 graphs) and utilizing integer cuts in our optimization model, we exhaustively prove that exactly 32 out of 36 Neo-Riemannian harmonic connections can be preserved. We demonstrate a strict 5-fold degeneracy of this optimum: there exist exactly 5 mathematically equivalent local packings for the wolf chords. Among these, we identify a unique canonical realization in which the 14 excised vertices form a perfectly contiguous geometric void along the primary Hamiltonian cycle. We reveal that the 4 inevitably broken edges represent the exact topological scars of the historical enharmonic diesis, and we formulate the Vicentino Hypothesis regarding 16th-century microtonal composition. Finally, to make this theoretical geometry physically playable, we design a novel 19-TET split-key keyboard, formalized through a biomechanical cost function that optimizes the performer's hand span. This work provides the complete theoretical, historical, and ergonomic blueprint for the next generation of microtonal acoustic instruments.