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Using the diameter normalization, define ${\mathcal{H}}_{\delta }^d(E)=\inf \{{\sum }_j(\mathrm{diam}{U}_j{)}^d:E\subseteq {\bigcup }_j{U}_j,\mathrm{diam}{U}_j\le \delta \}$ and the Hausdorff measure ${\mathcal{H}}^d(E)={\lim }_{\delta \downarrow 0}{\mathcal{H}}_{\delta }^d(E)$. The Hausdorff dimension is $\inf \{d:{\mathcal{H}}^d(E)=0\}$, equivalently $\sup \{d:{\mathcal{H}}^d(E)=\infty \}$.

The Cantor set is the intersection of the sets left by successively removing open middle thirds from $[0,1]$. At level $k$ it has $2^k$ Cantor cylinders of diameter $3^{-k}$. For $s=\log 2/\log 3$, their total $s$-cost is $2^k{3}^{-ks}=1$, so ${\mathcal{H}}^s(C)\le 1$.

The random series $\xi ={\sum }_{n\ge 1}{X}_n{3}^{-n}$ converges absolutely, and its ternary digits are all zero or two. Thus $\xi \in C$ and its law is the Cantor Bernoulli measure. Each level-$k$ cylinder has probability $2^{-k}$. Distinct such intervals have separation at least $3^{-k}$; a set $U$ of smaller diameter meets at most one. If $3^{-(k+1)}\le \mathrm{diam}U<3^{-k}$, thenSets of diameter zero have probability zero, since the measure has no measure atoms; diameters at least one give the bound trivially. For arbitrary nonmeasurable sets use outer probability. Any countable cover of $C$ therefore satisfies $1\le {\sum }_j{\mathbb{P}}^{\ast }(\xi \in U_j)\le 2{\sum }_j(\mathrm{diam}{U}_j{)}^s$. HenceThe last implication follows by multiplying the $s$-costs by powers of the cover diameter: measures above $s$ vanish and those below $s$ are infinite.

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 $C={\bigcap }_n{C}_n$ has zero length andIndeed, $2^n$ intervals of scale $3^{-n}$ cover it, giving the upper dimension bound from $2^n{3}^{-ns}\to 0$ whenever $s>\log 2/\log 3$. 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 $\mu (I)\le K{r}^{\log 2/\log 3}$. 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 $s_i=\pm 1$, takeGrouping configurations by $m=N^{-1}{\sum }_i{s}_i$ and using their binomial multiplicities gives the limiting free energy as the minimum ofThe number of possible magnetizations grows only linearly with $N$, so their largest exponential contribution determines the limiting free energy. Differentiating the variational expression givesAt $h=0$ and $\beta J>1$, its two stable minima have magnetizations $\pm m_{\ast }\ne 0$. Every finite zero-field system instead has exactly zero mean magnetization by spin-inversion symmetry. ConsequentlyThis 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 $e^{2\beta Nh{m}_{\ast }}$; the rounded magnetization is correspondingly about $m_{\ast }\tanh (\beta Nh{m}_{\ast })$. 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 stateis 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 $⟨\uparrow |\downarrow ⟩=0$. 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: $\Vert [M_N,A_R]\Vert \le 2R\Vert A_R\Vert /N$. 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. Fractal dimension, a thermodynamic singularity and sector separation each exhibit new limiting structure while retaining an explicit microscopic construction.

Suppose a connected subset $C\subseteq F$ contains distinct points $a,b$. The distance function $d_a(x)=|x-a|$ is a Lipschitz map with constant one, by the reverse triangle inequality. Its image $d_a(C)$ is connected and contains both $0$ and $|b-a|$, so it contains the whole interval $[0,|b-a|]$. Monotonicity and the Lipschitz inequality for Hausdorff dimension implya contradiction. Thus every nonempty connected subset of $F$ is a single point: $F$ is totally disconnected. The argument is the Hausdorff dimension of a connected set bound, and does not require $F$ to be closed.

Let $C$ be the middle-third Cantor set and choose a positive integer $d$ with $d\log 2/\log 3\ge k$; this is possible for every real $k$, including negative $k$. Set $X=C^d\subset {\mathbb{R}}^d$. It is closed and bounded since each factor is compact. If a subset of $X$ is connected, each coordinate projection is a connected subset of $C$ and hence a singleton; therefore the subset itself is a singleton. Thus $X$ is totally disconnected.

For completeness, the dimension of these Cartesian powers of the Cantor set can be checked directly. At stage $n$ there are $2^{dn}$ cubes of side $3^{-n}$, giving vanishing $s$-dimensional cover cost whenever $s>d\log 2/\log 3$. Conversely put independent equiprobable zero-or-two ternary digits on every coordinate. Each stage-$n$ cube has probability $2^{-dn}$. If $3^{-n}\le r<3^{-n+1}$, a Euclidean ball of radius $r$ meets at most a constant depending on $d$ such cubes. Hence its probability is at most $C_d{r}^{d\log 2/\log 3}$. For any sufficiently fine cover, summing this bound over covering sets shows that its cost at exponent $d\log 2/\log 3$ is bounded below by a positive constant, since total probability is one. These upper and lower Hausdorff measure bounds proveThe construction does not claim arbitrarily large Hausdorff dimension in one fixed ambient Euclidean dimension.

A Tangent-disk Schottky group uses side-pairing disks with disjoint interiors but permits boundaries to meet. This convention must be distinguished from the strictly separated, purely loxodromic Schottky group convention. For example, the maps $A(z)=z+2$ and $B(z)=z/(iz+1)$ pair the half-plane disks $\Re z<-1,\Re z>1$ and the disks $|z-i|<1,|z+i|<1$. The ping-pong lemma gives a free discrete group. With $C=AB{A}^{-1}{B}^{-1}$, the subgroup generated by $B,C$ is conjugate by $M(z)=iz/(2-z)$ to the principal level-two modular group, generated by $\left(\begin{array}{cc}1& 0\\ 2& 1\end{array}\right)$ and $\left(\begin{array}{cc}1& 2\\ 0& 1\end{array}\right)$. Thus its Kleinian limit set contains the circle $|z-1|=1$ and its translate $|z-3|=1$. Its Hausdorff dimension is at least one and it cannot be homeomorphic to a single circle.