= Solution
For <fractal geometry>, let $C_n$ be the $n$th middle-third construction of the <Cantor set>. It contains $2^n$ intervals of length $3^{-n}$, with total length $(2/3)^n$. Each finite $C_n$ contains intervals, so its <Hausdorff dimension> is one. The limiting set $C=\bigcap_nC_n$ has zero length and
$$
\boxed{\dim_H C=\frac{\log2}{\log3}.}
$$
Indeed, $2^n$ intervals of scale $3^{-n}$ cover it, giving the upper dimension bound from $2^n3^{-ns}\to0$ whenever $s>\log2/\log3$. For the lower bound, assign mass $2^{-n}$ to each level-$n$ interval. Any sufficiently small interval of length $r$ meets only a bounded number of construction intervals at the comparable scale, giving a bound $\mu(I)\le Kr^{\log2/\log3}$. Covering $C$ by such intervals then forces their dimension-weighted sizes to sum to at least $1/K$, yielding the matching lower bound. The noninteger dimension is deducible from a simple iterative rule even though it is absent from every finite-stage set at arbitrarily fine resolution. A finite construction nevertheless exhibits the same scaling over intermediate resolutions. Thus the order of idealization and probing ever smaller scales matters: an actual finite-resolution pattern can have useful fractal behaviour without possessing the exact limiting dimension at all scales.
For a <phase transition>, finite-spin <partition functions> are finite sums of positive exponentials at real finite temperature and field. Their <free energies> are therefore analytic there. Genuine thermodynamic nonanalyticity can appear only after a suitable infinite-system limit. The <Curie–Weiss model> gives an explicit demonstration. With $s_i=\pm1$, take
$$
H_N=-\frac{J}{2N}\left(\sum_i s_i\right)^2-h\sum_i s_i,\qquad J>0,\qquad\beta=(k_BT)^{-1}.
$$
Grouping configurations by $m=N^{-1}\sum_i s_i$ and using their binomial multiplicities gives the limiting <free energy> as the minimum of
$$
f(m)=-\frac J2m^2-hm-\beta^{-1}s(m),\qquad s(m)=-\frac{1+m}{2}\log\frac{1+m}{2}-\frac{1-m}{2}\log\frac{1-m}{2}.
$$
The number of possible magnetizations grows only linearly with $N$, so their largest exponential contribution determines the limiting <free energy>. Differentiating the variational expression gives
$$
m=\tanh\{\beta(Jm+h)\}.
$$
At $h=0$ and $\beta J>1$, its two stable minima have magnetizations $\pm m_*\ne0$. Every finite zero-field system instead has exactly zero mean magnetization by spin-inversion <symmetry>. Consequently
$$
\boxed{\lim_{h\downarrow0}\lim_{N\to\infty}\langle m\rangle_{N,h}=m_*,\qquad\lim_{N\to\infty}\lim_{h\downarrow0}\langle m\rangle_{N,h}=0.}
$$
This noncommutation of limits explains how <spontaneous symmetry breaking> emerges. Below the critical temperature, the limiting <free energy> has a cusp in $h$ at zero, while finite systems have a sharp but smooth crossover. Away from the critical point, the relative weights of the two phases behave approximately as $e^{2\beta Nhm_*}$; the rounded magnetization is correspondingly about $m_*\tanh(\beta Nhm_*)$. Exact singularity belongs to the idealized limit, whereas the narrow crossover and long-lived phases can be physically useful at finite size. The <theoretical reduction> is the derivation and its approximation control, not a claim that a finite analytic <partition function> is already nonanalytic.
For <superselection>, consider an infinite spin lattice with <quasi-local observable algebra> $\mathcal A$. These are zero-temperature equilibrium phases of a ferromagnetic Ising spin Hamiltonian with nearest-neighbour interactions $-J\sigma_z^{(i)}\sigma_z^{(j)}$. At finite $N$, the all-up and all-down vectors belong to one Hilbert space, and the coherent state
$$
|\Psi_N\rangle=\frac{|\uparrow\cdots\uparrow\rangle+e^{i\theta}|\downarrow\cdots\downarrow\rangle}{\sqrt2}
$$
is distinguishable from the corresponding mixture by a suitably chosen global <observable>. But if $A_R$ acts on only $R<N$ spins, its off-diagonal matrix element between these vectors vanishes: the untouched spins contribute $\langle\uparrow|\downarrow\rangle=0$. Hence its expectations already equal those of the equal-weight mixture, independent of $\theta$. Increasingly global <observables> can recover the phase at finite $N$, but the infinite product of spin flips is not a norm limit of finite-support <observables> and is absent from $\mathcal A$.
In the infinite-volume limit the up and down phases have disjoint state representations. The averaged magnetization $M_N=N^{-1}\sum_i\sigma_z^{(i)}$ commutes asymptotically with any fixed local <observable>: $\|[M_N,A_R]\|\le2R\|A_R\|/N$. In the phase representations its limiting values are $+1$ and $-1$, distinguishing sectors. Finite spin-flip vectors form a dense set on which these averages tend to their respective scalar values, so boundedness extends the strong limits to the phase Hilbert spaces. An intertwiner between the representations would obey $(-I)T=T(+I)$ and must therefore vanish, explaining their disjointness. In the combined representation the <observable> algebra is block-diagonal between them, so no quasilocal measurement detects their relative phase. This yields an emergent <superselection rule> relative to the chosen <observable> algebra. It is not a derivation of physical wave-function collapse. \b[Fractal dimension, a thermodynamic singularity and sector separation each exhibit new limiting structure while retaining an explicit microscopic construction.]
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