A Finite-Lattice Model from a Reciprocal Cost Action: Spectral and Reflection-Positivity Properties

Abstract

We study the finite-lattice statistical-mechanical model whose nearest-neighbor bond potential is the reciprocal cost $J(e^{\epsilon })=\cosh \epsilon -1$, selected by the d'Alembert functional equation under the stated regularity and calibration assumptions. The structural inputs are stated explicitly; once they are fixed, the analysis is rigorous mathematics about the bond action $V(\Delta \varphi )=\cosh (\Delta \varphi )-1$ on finite boxes in ${\mathbb{Z}}^3\times \mathbb{Z}/8\mathbb{Z}$. Our main result pairs a negative and a positive statement about reflection positivity. For the continuous noncompact model the natural temporal kernel $K(u)=\exp [-(\cosh u-1)]$ fails the Bochner positive-definiteness test: an interval-certified quadrature gives $\tilde{K}(3)<0$. Thus the standard Bochner route to Osterwalder-Schrader reflection positivity is obstructed. For a finite-alphabet variant, with field values restricted to a finite symmetric set $\Phi =v_0\{-N,\dots ,N\}$, reflection positivity holds whenever the finite crossing-bond Toeplitz matrix $(K_{\Phi (v_0,N)}{)}_{a,b}:=K(b-a),a,b\in \Phi$, is positive semidefinite. For $v_0\in \{1.2,1.5,2.5\}$, this is discharged by a rigorous diagonal-dominance certificate uniform in $N$, and the associated one-step transfer operator is then positive and self-adjoint in an explicit reflection-positivity inner product. These finite-volume results do not provide a continuum Wightman theory, Osterwalder-Schrader reconstruction, LSZ scattering, or a continuum mass gap.

45 pages, 3 figures, no tables. Ancillary files include an 8-page supplementary PDF, Python verification scripts, saved numerical output, requirements file, and figure alternative text