= Solution
With lattice spacing set to one and a smoothly varying auxiliary field, the discrete action becomes
$$
S[m]=\int_0^{\mathcal L}dx\left[\frac M2\left(\frac{dm}{dx}\right)^2+U(m)\right],\qquad
M=\frac1{J\sinh\kappa}.
$$
This is precisely the <Euclidean path integral> of a quantum particle with $\hbar=1$, mass $M$ and potential $U$. With fixed endpoints it gives the transition amplitude $\langle m_f|e^{-\mathcal L\widehat H_Q}|m_i\rangle$, while periodic classical <boundary conditions> give its trace. The corresponding operator is
$$
\boxed{\widehat H_Q=-\frac{J\sinh\kappa}{2}\frac{d^2}{dm^2}+U(m),\qquad
\mathcal Z_{\rm periodic}\propto\operatorname{Tr}e^{-\mathcal L\widehat H_Q}.}
$$
This is <classical Ising chain as a quantum particle>. It is the continuum approximation to the exact positive transfer kernel, so its numerical spectrum is accurate when the relevant auxiliary configurations vary slowly. The transfer-kernel reasoning and the domain-wall argument below also support the qualitative conclusion without treating a continuum approximation as an exact finite-lattice identity.
At zero field write $a=\tanh(\kappa/2)/J>0$. Expansion gives
$$
U(m)=-\log2+(a-2)m^2+\frac43m^4+O(m^6).
$$
A double well exists when $a<2$; the stationary points satisfy $am=\tanh(2m)$. For $a\geq2$ there is a single minimum at zero, with the quadratic curvature marginal at equality. Thus a double well is the low-effective-temperature regime of the model, not an unconditional property for every $J$. The exact potential remains confining because $U(m)=am^2-2|m|+O(1)$ at large $|m|$.
Even when two wells exist, a quantum particle with finite mass and a finite barrier tunnels between them. Its one-dimensional confining <Schrödinger operator> has a unique, strictly positive ground-state wavefunction. At $h=0$ reflection <symmetry (physics)> makes that state even, so every odd observable has zero expectation. An antisymmetric first excited state is split from it by a positive gap. A spectral expansion of an odd-observable correlation has factors $e^{-x(E_n-E_0)}$; the smallest contributing gap therefore gives a finite classical <correlation length>. The original spin observable, after decoupling, is $\tanh(2m+h)$, which is odd at zero field. Hence
$$
\boxed{\langle\sigma\rangle=0\ \text{at }h=0\text{ for finite }J\text{ and fixed }\kappa>0,\quad
\langle\sigma_0\sigma_r\rangle\to0.}
$$
A deep well can give an exponentially small <quantum tunnelling> gap and an extremely long <correlation length>, which can mimic order in a finite sample, but it does not produce true long-range order.
The classical counterpart is the finite cost of a <domain wall>. Because the pair sum is doubled, a sharp wall has dimensionless energy
$$
E_{\rm wall}=4J\sum_{r\geq1}r e^{-\kappa r}=\frac{4Je^{-\kappa}}{(1-e^{-\kappa})^2}<\infty.
$$
At any finite temperature such walls have nonzero fugacity, divide a long chain into domains and destroy spontaneous magnetization. The range is large for $\kappa\ll1$ but finite. The zero-temperature limit and an infinite-range limit are singular and must not be interchanged casually with the <thermodynamic limit>; an infinite-range interaction also requires an extensive normalization before comparing models.
A nonzero $h$ tilts the potential: to first order $U_h(m)=U_0(m)-h\tanh(2m)+O(h^2)$. A positive field favors the positive well and produces positive magnetization. For fixed finite parameters the <ground state> and classical <free energy> vary smoothly, so there is no zero-field discontinuity due merely to a double-well bare potential. A discrete broken-symmetry state appears only in an appropriate singular limit, not by ignoring <quantum tunnelling>.
More generally a local $d$-dimensional classical statistical model can be sliced along one coordinate and represented by a transfer operator. That coordinate becomes imaginary time; the other $d-1$ coordinates become the spatial coordinates of a quantum system. Infinite extent in the transfer direction corresponds to its ground-state, or zero-quantum-temperature, limit. The classical <Ising model> is thus related to a quantum Ising system in $d-1$ spatial dimensions, with transverse dynamics generated by the transfer operator. For $d=1$ this is quantum mechanics of finitely many local variables, where <quantum tunnelling> restores <symmetry (physics)>. For $d\geq2$ the quantum system has infinitely many spatial degrees of freedom and can have distinct symmetry-broken <ground states>, allowing a classical finite-temperature ordering transition. Locality and anisotropy of the mapping determine the quantum couplings; the dimension correspondence alone does not fix them.
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