Why must observables be self-adjoint (not merely symmetric/hermitian)?

Derive the uncertainty relation’s operator core in one line.

What is the parity of $Y_{\ell m}$, and which dipole transitions does it allow?

Two states of a qubit have $\langle\sigma_z\rangle = 0$: give one pure and one maximally mixed, and how to tell them apart.

Why is the hydrogen degeneracy in $\ell$ “accidental,” and what resolves the scare quotes?

Fermi’s golden rule: state it and name its two validity conditions.

Why must a manifold be Hausdorff and second countable? Give one failure each condition prevents.

Compactness of $[0,1]$: which two properties of continuous maps does it immediately deliver?

$\pi_1$ of the figure eight, the torus, $\mathbb{RP}^2$, and $S^2$?

What data classifies connected coverings of a (nice) space $X$?

Why does $SU(2) \to SO(3)$ being a double cover matter physically?

True or false: a continuous bijection is a homeomorphism.

Why do quantum systems represent $SU(2)$ rather than $SO(3)$?

Compute $\dim SU(n)$ and $\dim SO(n)$.

What exactly does the Lie algebra fail to know about the group?

Why must roots come in pairs $\pm\alpha$, and what does the Weyl group do?

State Schur’s lemma and its physical consequence in one line each.

Decompose $\mathbf 3 \otimes \bar{\mathbf 3}$ of $\mathfrak{su}(3)$ and name the physics.

State the product’s universal property without mentioning elements.

Every monad arises from an adjunction — name the two canonical ways.

Your qubit shows $T_2 = 2T_1$ exactly. What does that tell you?

Why is “natural” in ‘natural transformation’ the right word for basis-independence?

What is a coalgebra for a functor, and one CS example?

RAPL: which functor in free $\dashv$ forgetful preserves products, and what does that predict?

In $(\mathbf{Hilb}, \otimes)$, what plays the role of $\otimes$’s unit, and what are ‘scalars’ categorically?

Why does entanglement not allow faster-than-light signaling?

Schmidt rank of $\tfrac{1}{\sqrt3}(|00\rangle + |11\rangle + |22\rangle)$, and its entanglement entropy?

What makes GHZ and W inequivalent?

An entanglement witness $W$ reports $\operatorname{tr}(W\rho) = +0.2$. Conclusion?

Why is the singlet the same in every basis?

Distillable entanglement vs. entanglement cost — which inequality always holds and when is it strict?

Why can’t you define the derivative of a vector field along another vector field without extra structure — and what are the two standard fixes?

State the regular level set theorem and use it in one line on $SL(n,\mathbb{R})$.

What does $d^2 = 0$ buy, in one sentence?

Why does integration on manifolds need an orientation?

Frobenius: when is a 2-plane field on $\mathbb{R}^3$ tangent to a family of surfaces?

State Stokes and name three classical theorems it contains.

Homology vs. homotopy groups: which is computable and why?

Compute $H_*(\text{Klein bottle};\mathbb{Z})$ from the word $abab^{-1}$.

Why does $H_n$ of a closed orientable $n$-manifold equal $\mathbb{Z}$, and what breaks without orientability?

State Poincaré duality with its pairing interpretation.

Give the Brouwer fixed point proof in two sentences.

Two spaces with the same homology groups but different cohomology rings?

Why does relativistic QM force fields (infinitely many degrees of freedom)?

Why must the Dirac field anticommute?

What, in renormalization, is actually being ‘hidden’ — and what is measurably left behind?

What breaks the symmetry in spontaneous symmetry breaking?

Count the Higgs mechanism’s bookkeeping in electroweak breaking.

Why is Wigner’s little group the right definition of spin?

Why must ‘only connectivity matters’ be proved rather than assumed?

In diagram terms: what is a state, an effect, a number, and the Born rule?

Why does a compact category with cartesian $\otimes$ collapse?

What distinguishes $\mathbf{FHilb}$ from $\mathbf{Rel}$ despite both being dagger compact?

Two spiders of the same color touch. What happens, and when is it forbidden?

State the CP construction’s move in one sentence.

Why ‘completely’ positive — what goes wrong with merely positive maps?

State Stinespring in one sentence and its philosophical reading.

Von Neumann entropy is invariant under which operations and concave in what sense?

What does negative conditional entropy $S(A|B) < 0$ signal, and can classical systems do it?

Why does the Heisenberg limit not violate anything fundamental?

Sound speed in air: why $\sqrt{\gamma p/\rho}$ and not $\sqrt{p/\rho}$?

What is 0 dB SPL anchored to, and how many pascals is a painful 120 dB?

Double one wall’s absorption coefficient in a room. What happens to $T_{60}$, roughly?

Why do wind and temperature gradients bend sound, and which way at night?

A duct is driven below its first transverse cutoff. What propagates?

Where does linear acoustics fail first, and what is the telltale waveform?