arXiv:2610.39815

High-temperature expansion of the inverse-square Ising chain in a magnetic field: exact coefficients and finite-size reconstruction

M. A. Rajabpour

Published Oct 29, 2026• 0 citations

Abstract

The one-dimensional Ising chain with inverse-square interactions is a paradigmatic long-range system: it exhibits the unusual Thouless transition, with a discontinuity in the spontaneous magnetization, yet its full field-dependent thermodynamics remains unknown in closed form. We develop the high-temperature expansion of its free energy while retaining the dependence on a uniform magnetic field at every order. We prove that, after a natural rescaling of the coupling, each coefficient is a rational polynomial in the squared hyperbolic tangent of the reduced field. Remarkably, at each order $n$ the field-dependent free-energy coefficient of a finite, periodically repeated ring is an exact polynomial of degree at most $n$ in $L^{-2}$. The infinite-chain coefficient is its constant term and can therefore be reconstructed from $n+1$ finite rings without numerical extrapolation. Using this reconstruction, we determine the full field-dependent expansion through order 26: 26 polynomials containing 377 rational coefficient entries. We further derive exact formulas, valid at every order, for the coefficients of the two highest powers of the field variable using matching determinants and a cycle-Laplacian reduction. These results reveal an unexpected algebraic structure in the microscopic thermodynamics of the inverse-square chain and provide high-order information about its magnetic response.

DOI: 10.48550/arxiv.2610.39815

Related papers

Finite Gluing of the Two-Dimensional Ising Free Energy

Yoshiki Ueoka

Sep 2, 20260 citations

On Ising model in magnetic field on the lattice

Raghav G. Jha

arXiv:2504.18744Apr 25, 20250 citations

Low-temperature behavior of the finite-size one-dimensional Ising model and the partition function zeros

Julian Lee

arXiv:1407.6919Journal of the Korean Physical SocietySep 1, 201410 citations

Microscopic Origin of the Shannon Renyi Boundary Phase in the Critical Ising Chain

M. A. Rajabpour

quant-pharXiv:2609.27909Aug 24, 20260 citations

Low-temperature series expansion of the square-lattice Ising model: A study based on Fisher zeros

De-Zhang Li, Xin Wang, Xiao‐Bao Yang

arXiv:2412.07328Physical review. B./Physical review. BApr 21, 20253 citations

Finite Gluing of the Three-Dimensional Ising Free Energy

Yoshiki Ueoka

Sep 2, 20260 citations

arXiv:2610.39815

High-temperature expansion of the inverse-square Ising chain in a magnetic field: exact coefficients and finite-size reconstruction

M. A. Rajabpour

Published Oct 29, 2026• 0 citations

arXiv HTML PDF

Abstract

n

the field-dependent free-energy coefficient of a finite, periodically repeated ring is an exact polynomial of degree at most

n

L^{-2}

. The infinite-chain coefficient is its constant term and can therefore be reconstructed from

n+1

finite rings without numerical extrapolation. Using this reconstruction, we determine the full field-dependent expansion through order 26: 26 polynomials containing 377 rational coefficient entries. We further derive exact formulas, valid at every order, for the coefficients of the two highest powers of the field variable using matching determinants and a cycle-Laplacian reduction. These results reveal an unexpected algebraic structure in the microscopic thermodynamics of the inverse-square chain and provide high-order information about its magnetic response.

DOI: 10.48550/arxiv.2610.39815

Related papers

Finite Gluing of the Two-Dimensional Ising Free Energy

Yoshiki Ueoka

We test the Gluing Method on the square-lattice two-dimensional Ising model by reconstructing global free-energy generators from complementary local relations. The emphasis is kept explicit throughout: the inputs used as \emph{gluing material} are separated from quantities obtained \emph{after gluing} and from data reserved for validation. At zero magnetic field, Kramers--Wannier duality is absorbed into the coordinate $\tau=\log\sinh 2K$ , so that the reduced free energy decomposes as $\Phi(\tau)=\tau/2+G(|\tau|)$ . The gluing materials are then only a critical logarithmic local expansion and a low-order high-temperature outer expansion; the low-temperature sector is generated by duality rather than fitted independently. Positive flat weights preserve the prescribed critical asymptotics to every finite algebraic order. With one crossover parameter fixed from three scalar anchors, the resulting bulk free energy agrees with the exact Onsager benchmark at a maximum relative error of $1.915\times10^{-5}$ on the tested interval. Internal energy and specific heat are derived from the same glued generator. Zero-field finite-size corrections are organized separately as one-variable scaling functions multiplying $L^{-2},L^{-4},\ldots$ . At nonzero field in the thermodynamic limit, the gluing materials are the established high-temperature, critical, and low-temperature local expansions of the universal Ising magnetic scaling free energy $\Phi_{\rm mag}(\eta)$ , $\eta=m/|h|^{8/15}$ . The low-temperature input contains the $|h|$ cusp associated with spontaneous symmetry breaking. Mapping the glued scaling free energy back to the square lattice through nonlinear scaling fields yields a single generator for the magnetic free-energy increment, magnetization, and susceptibility. The construction reproduces high-temperature linear response, the critical isotherm $M\propto |H|^{1/15}$ , and the two one-sided limits $M(H\to0^\pm)=\pm M_0$ without separately fitting magnetization branches. Finally, finite size and finite field are combined in the scaling variables $x_t=Lm$ and $x_h=L^{15/8}h$ . The two protected one-dimensional axes, a central mixed-response expansion, a low-temperature spectral-rounding relation, and a massive outer description are glued into a two-variable scaling surface $\mathcal Y_{\rm glue}(x_t,x_h)$ . The resulting lattice free energy closes in the compact form $\Phi_L^{glue}(K,H)=\Phi_{lat}^{glue}(K,H)+\frac{\mathcal{Y}_{glue}(Lm,L^{15/8}h)- Y_{bulk}(Lm,L^{15/8}h)}{L^2}+O(L^{-4})$ where $m=m(r,H)$ and $h=h(r,H)$ are the nonlinear scaling fields and $Y_{\rm bulk}$ is the thermodynamic singular scaling surface expressed in the same variables. Thus all finite-size/finite-field information is carried by the excess $\mathcal Y_{\rm glue}-Y_{\rm bulk}$ , while the thermodynamic lattice free energy remains the previously constructed $\Phi_{\rm lat}^{\rm glue}$ . On eleven transfer-matrix points withheld from the construction, the blind free-energy test gives RMS absolute error $3.90\times10^{-4}$ and maximum absolute error $9.55\times10^{-4}$ in $\mathcal Y$ . Differentiation of the same free energy produces finite-size magnetization and susceptibility; magnetization is especially stable on the low-temperature side, while the second derivative is more sensitive in crossover regions.We test the Gluing Method on the square-lattice two-dimensional Ising model by reconstructing global free-energy generators from complementary local relations. The emphasis is kept explicit throughout: the inputs used as \emph{gluing material} are separated from quantities obtained \emph{after gluing} and from data reserved for validation. At zero magnetic field, Kramers--Wannier duality is absorbed into the coordinate $\tau=\log\sinh 2K$, so that the reduced free energy decomposes as $\Phi(\tau)=\tau/2+G(|\tau|)$. The gluing materials are then only a critical logarithmic local expansion and a low-order high-temperature outer expansion; the low-temperature sector is generated by duality rather than fitted independently. Positive flat weights preserve the prescribed critical asymptotics to every finite algebraic order. With one crossover parameter fixed from three scalar anchors, the resulting bulk free energy agrees with the exact Onsager benchmark at a maximum relative error of $1.915\times10^{-5}$ on the tested interval. Internal energy and specific heat are derived from the same glued generator. Zero-field finite-size corrections are organized separately as one-variable scaling functions multiplying $L^{-2},L^{-4},\ldots$. At nonzero field in the thermodynamic limit, the gluing materials are the established high-temperature, critical, and low-temperature local expansions of the universal Ising magnetic scaling free energy $\Phi_{\rm mag}(\eta)$, $\eta=m/|h|^{8/15}$. The low-temperature input contains the $|h|$ cusp associated with spontaneous symmetry breaking. Mapping the glued scaling free energy back to the square lattice through nonlinear scaling fields yields a single generator for the magnetic free-energy increment, magnetization, and susceptibility. The construction reproduces high-temperature linear response, the critical isotherm $M\propto |H|^{1/15}$, and the two one-sided limits $M(H\to0^\pm)=\pm M_0$ without separately fitting magnetization branches. Finally, finite size and finite field are combined in the scaling variables $x_t=Lm$ and $x_h=L^{15/8}h$. The two protected one-dimensional axes, a central mixed-response expansion, a low-temperature spectral-rounding relation, and a massive outer description are glued into a two-variable scaling surface $\mathcal Y_{\rm glue}(x_t,x_h)$. The resulting lattice free energy closes in the compact form $\Phi_L^{glue}(K,H)=\Phi_{lat}^{glue}(K,H)+\frac{\mathcal{Y}_{glue}(Lm,L^{15/8}h)- Y_{bulk}(Lm,L^{15/8}h)}{L^2}+O(L^{-4})$ where $m=m(r,H)$ and $h=h(r,H)$ are the nonlinear scaling fields and $Y_{\rm bulk}$ is the thermodynamic singular scaling surface expressed in the same variables. Thus all finite-size/finite-field information is carried by the excess $\mathcal Y_{\rm glue}-Y_{\rm bulk}$, while the thermodynamic lattice free energy remains the previously constructed $\Phi_{\rm lat}^{\rm glue}$. On eleven transfer-matrix points withheld from the construction, the blind free-energy test gives RMS absolute error $3.90\times10^{-4}$ and maximum absolute error $9.55\times10^{-4}$ in $\mathcal Y$. Differentiation of the same free energy produces finite-size magnetization and susceptibility; magnetization is especially stable on the low-temperature side, while the second derivative is more sensitive in crossover regions.

Sep 2, 20260 citations

On Ising model in magnetic field on the lattice

Raghav G. Jha

We study the free energy of the classical square-lattice Ising model in a uniform real magnetic field $h$ , for which no exact solution is known. We first conjecture a closed-form expression built from the one-dimensional Ising chain and Onsager's hypergeometric solution. Comparing with high-precision tensor renormalization group (TRG) data, we find that it deviates by up to $2.6\%$ , has no spontaneous magnetization, and gives an effective critical-isotherm exponent $δ\to 1$ . However, using the Lee--Yang circle theorem, we show that the exact free energy is a superposition of one-dimensional Ising chains in the same field, weighted by a temperature-dependent (normalized, but not always positive) distribution of squared Lee--Yang gaps, of which our first expression is a single-chain approximation. Exact results (Onsager, Yang, the strong-field expansion and the Lee--Yang solution at $βh = iπ/2$ ) become sum rules for this distribution. We propose a simple form for it, fixed by Onsager's free energy and the exact strong-field limit without any fit to numerical data. This \emph{improved} conjecture agrees with TRG to $0.15\%$ for all $β$ and $h$ , reproduces Yang's spontaneous magnetization exactly, has a positive specific heat, and yields the critical isotherm $M\propto h^{1/δ}$ with $δ= 15.2$ and amplitude $1.055$ , compared with the exact $δ=15$ and $1.059$ . A refined form, which builds in additional input from Ising field theory (including the exact critical isotherm, $δ=15$ ) and a few constants adjusted to transfer-matrix and TRG data, matches the TRG free energy to $4\times 10^{-5}$ .We study the free energy of the classical square-lattice Ising model in a uniform real magnetic field $h$, for which no exact solution is known. We first conjecture a closed-form expression built from the one-dimensional Ising chain and Onsager's hypergeometric solution. Comparing with high-precision tensor renormalization group (TRG) data, we find that it deviates by up to $2.6\%$, has no spontaneous magnetization, and gives an effective critical-isotherm exponent $δ\to 1$. However, using the Lee--Yang circle theorem, we show that the exact free energy is a superposition of one-dimensional Ising chains in the same field, weighted by a temperature-dependent (normalized, but not always positive) distribution of squared Lee--Yang gaps, of which our first expression is a single-chain approximation. Exact results (Onsager, Yang, the strong-field expansion and the Lee--Yang solution at $βh = iπ/2$) become sum rules for this distribution. We propose a simple form for it, fixed by Onsager's free energy and the exact strong-field limit without any fit to numerical data. This \emph{improved} conjecture agrees with TRG to $0.15\%$ for all $β$ and $h$, reproduces Yang's spontaneous magnetization exactly, has a positive specific heat, and yields the critical isotherm $M\propto h^{1/δ}$ with $δ= 15.2$ and amplitude $1.055$, compared with the exact $δ=15$ and $1.059$. A refined form, which builds in additional input from Ising field theory (including the exact critical isotherm, $δ=15$) and a few constants adjusted to transfer-matrix and TRG data, matches the TRG free energy to $4\times 10^{-5}$.

arXiv:2504.18744Apr 25, 20250 citations

Low-temperature behavior of the finite-size one-dimensional Ising model and the partition function zeros

Julian Lee

In contrast to an infinite chain, the low-temperature expansion of a one-dimensional free-field Ising model has a strong dependence on boundary conditions. I derive an explicit formula for the leading term of the expansion both under open and periodic boundary conditions and show that they are related to different distributions of the partition function zeros on the complex temperature plane. In particular, when a periodic boundary condition is imposed, the leading coefficient of the expansion grows with increasing size of the chain, due to the zeros approaching the origin.In contrast to an infinite chain, the low-temperature expansion of a one-dimensional free-field Ising model has a strong dependence on boundary conditions. I derive an explicit formula for the leading term of the expansion both under open and periodic boundary conditions and show that they are related to different distributions of the partition function zeros on the complex temperature plane. In particular, when a periodic boundary condition is imposed, the leading coefficient of the expansion grows with increasing size of the chain, due to the zeros approaching the origin.

arXiv:1407.6919Journal of the Korean Physical SocietySep 1, 201410 citations

Microscopic Origin of the Shannon Renyi Boundary Phase in the Critical Ising Chain

M. A. Rajabpour

Shannon--Rényi entropies probe the full measurement distribution of a quantum many-body state, but their microscopic evaluation remains difficult even in free-fermion systems. We show that the Born distribution of the critical transverse-field Ising chain is exactly the parity boundary of a weak inverse-square classical Ising model, a structure revealed by the Cauchy form of its probability kernel. The construction is exact at finite size for every real $n>0$ and isolates the long-distance sectors controlling the $n>1$ thermodynamics. Cutting a finite interval generates the universal $c\,n/[8(n-1)]$ logarithm directly from the missing $1/r^2$ interactions, with $c=1/2$ the Ising central charge. On the ring, the explicit finite-size factor has no constant term, while any additional noninteger singularity is confined to a residual cycle-only free energy. All explicit long-distance contributions become marginal only at $n=1$ .Shannon--Rényi entropies probe the full measurement distribution of a quantum many-body state, but their microscopic evaluation remains difficult even in free-fermion systems. We show that the Born distribution of the critical transverse-field Ising chain is exactly the parity boundary of a weak inverse-square classical Ising model, a structure revealed by the Cauchy form of its probability kernel. The construction is exact at finite size for every real $n>0$ and isolates the long-distance sectors controlling the $n>1$ thermodynamics. Cutting a finite interval generates the universal $c\,n/[8(n-1)]$ logarithm directly from the missing $1/r^2$ interactions, with $c=1/2$ the Ising central charge. On the ring, the explicit finite-size factor has no constant term, while any additional noninteger singularity is confined to a residual cycle-only free energy. All explicit long-distance contributions become marginal only at $n=1$.

quant-pharXiv:2609.27909Aug 24, 20260 citations

Low-temperature series expansion of the square-lattice Ising model: A study based on Fisher zeros

De-Zhang Li, Xin Wang, Xiao‐Bao Yang

Low-temperature expansion of Ising model has long been a topic of significant interest in condensed matter and statistical physics. In this paper we present new results of the coefficients in the low-temperature series of the Ising partition function on the square lattice, in the cases of a zero field and of an imaginary field $i(\ensuremath{\pi}/2){k}_{B}T$. The coefficients in the low-temperature series of the free energy in the thermodynamic limit are represented using the explicit expression of the density function of the Fisher zeros. The asymptotic behavior of the sequence of the coefficients when the order goes to infinity is determined exactly, for both the series of the free energy and of the partition function. Our analytic and numerical results demonstrate that the convergence radius of the sequence is dependent on the accumulation points of the Fisher zeros, which have the smallest modulus. In the zero field case this accumulation point is the physical critical point, while in the imaginary field case it corresponds to a nonphysical singularity. We further discuss the relation between the series coefficients and the energy state degeneracies, using the combinatorial expression of the coefficients and the subgraph expansion.Low-temperature expansion of Ising model has long been a topic of significant interest in condensed matter and statistical physics. In this paper we present new results of the coefficients in the low-temperature series of the Ising partition function on the square lattice, in the cases of a zero field and of an imaginary field $i(\ensuremath{\pi}/2){k}_{B}T$. The coefficients in the low-temperature series of the free energy in the thermodynamic limit are represented using the explicit expression of the density function of the Fisher zeros. The asymptotic behavior of the sequence of the coefficients when the order goes to infinity is determined exactly, for both the series of the free energy and of the partition function. Our analytic and numerical results demonstrate that the convergence radius of the sequence is dependent on the accumulation points of the Fisher zeros, which have the smallest modulus. In the zero field case this accumulation point is the physical critical point, while in the imaginary field case it corresponds to a nonphysical singularity. We further discuss the relation between the series coefficients and the energy state degeneracies, using the combinatorial expression of the coefficients and the subgraph expansion.

arXiv:2412.07328Physical review. B./Physical review. BApr 21, 20253 citations

Finite Gluing of the Three-Dimensional Ising Free Energy

Yoshiki Ueoka

We construct a single reduced free-energy generator for the simple-cubic three-dimensional Ising model by finite gluing of complementary local relations. With the convention $\Phi=+\log Z/V$ , the final object is $\Phi_3(K,H,L)=\Phi_0(K)+\Delta\Phi_H(K,H)+\Phi_{\rm FSS}(K,H,L),$ from which $M=\partial_H\Phi_3$ , $\chi=\partial_H^2\Phi_3$ , and $U_K=\partial_K\Phi_3$ are generated. The construction covers the high- and low-temperature regimes, the critical region, finite magnetic field, strong-field saturation, periodic-film finite size, and simultaneous finite $H$ and finite $L$ . The materials and operations are separated explicitly. A diagnostic dimensional stage uses $d=1,2,4$ anchors in $\mu_c=-\log\tanh K_c$ to transport critical coordinates and positive film amplitudes toward $d=3$ . The practical 3D generator then protects established 3D values of $K_c$ , critical exponents, selected amplitude ratios, and scalar zero-field critical data. The zero-field term joins truncated simple-cubic high- and low-temperature series through a singular critical patch with $C^2$ matching. The inner magnetic increment joins an $O(K^2)$ high-temperature field expansion, a minimal 3D parametric critical free energy fixed only by the basic 3D anchors, and a low-temperature/cluster continuation using fixed quintic switches in $K$ . A recovered $z^4$ strong-field expansion is integrated to an absolute free energy. The final inner-to-strong crossover is performed by constructing a monotone beta-CDF magnetization whose endpoint values and integral reproduce the two endpoint free energies, and then integrating that derivative; thus convexity is enforced at the free-energy level rather than by clipping observables. The periodic-film excess is built from dimensionally weighted 2D and leading 4D film surfaces in common correlation-length coordinates, with an $H$ -independent residual stabilizer that preserves the $H=0$ axis. A previously reconstructed full 3D parametric equation-of-state benchmark and the full 3D finite-size scaling function are withheld from construction. The pure dimensional stage gives $K_c(3)=0.22301549$ and sub-percent estimates of $\beta$ , $\delta$ , and $\nu$ . After 3D calibration, the final implementation passes 29/29 stated structural and numerical checks. In the withheld low-field window $|K-K_c|\le0.01$ , $H\le0.1$ , the magnetization has RMS relative error $1.21\%$ and maximum error $4.29\%$ . The result is not an exact lattice solution; it is a finite, auditable reconstruction in which the input anchors, switch intervals, matching conditions, numerical tolerances, and validation data are kept distinct.We construct a single reduced free-energy generator for the simple-cubic three-dimensional Ising model by finite gluing of complementary local relations. With the convention $\Phi=+\log Z/V$, the final object is $\Phi_3(K,H,L)=\Phi_0(K)+\Delta\Phi_H(K,H)+\Phi_{\rm FSS}(K,H,L),$ from which $M=\partial_H\Phi_3$, $\chi=\partial_H^2\Phi_3$, and $U_K=\partial_K\Phi_3$ are generated. The construction covers the high- and low-temperature regimes, the critical region, finite magnetic field, strong-field saturation, periodic-film finite size, and simultaneous finite $H$ and finite $L$. The materials and operations are separated explicitly. A diagnostic dimensional stage uses $d=1,2,4$ anchors in $\mu_c=-\log\tanh K_c$ to transport critical coordinates and positive film amplitudes toward $d=3$. The practical 3D generator then protects established 3D values of $K_c$, critical exponents, selected amplitude ratios, and scalar zero-field critical data. The zero-field term joins truncated simple-cubic high- and low-temperature series through a singular critical patch with $C^2$ matching. The inner magnetic increment joins an $O(K^2)$ high-temperature field expansion, a minimal 3D parametric critical free energy fixed only by the basic 3D anchors, and a low-temperature/cluster continuation using fixed quintic switches in $K$. A recovered $z^4$ strong-field expansion is integrated to an absolute free energy. The final inner-to-strong crossover is performed by constructing a monotone beta-CDF magnetization whose endpoint values and integral reproduce the two endpoint free energies, and then integrating that derivative; thus convexity is enforced at the free-energy level rather than by clipping observables. The periodic-film excess is built from dimensionally weighted 2D and leading 4D film surfaces in common correlation-length coordinates, with an $H$-independent residual stabilizer that preserves the $H=0$ axis. A previously reconstructed full 3D parametric equation-of-state benchmark and the full 3D finite-size scaling function are withheld from construction. The pure dimensional stage gives $K_c(3)=0.22301549$ and sub-percent estimates of $\beta$, $\delta$, and $\nu$. After 3D calibration, the final implementation passes 29/29 stated structural and numerical checks. In the withheld low-field window $|K-K_c|\le0.01$, $H\le0.1$, the magnetization has RMS relative error $1.21\%$ and maximum error $4.29\%$. The result is not an exact lattice solution; it is a finite, auditable reconstruction in which the input anchors, switch intervals, matching conditions, numerical tolerances, and validation data are kept distinct.

Sep 2, 20260 citations

n

the field-dependent free-energy coefficient of a finite, periodically repeated ring is an exact polynomial of degree at most

n

L^{-2}

. The infinite-chain coefficient is its constant term and can therefore be reconstructed from

n+1

\tau=\log\sinh 2K

, so that the reduced free energy decomposes as

\Phi(\tau)=\tau/2+G(|\tau|)

. The gluing materials are then only a critical logarithmic local expansion and a low-order high-temperature outer expansion; the low-temperature sector is generated by duality rather than fitted independently. Positive flat weights preserve the prescribed critical asymptotics to every finite algebraic order. With one crossover parameter fixed from three scalar anchors, the resulting bulk free energy agrees with the exact Onsager benchmark at a maximum relative error of

1.915\times10^{-5}

on the tested interval. Internal energy and specific heat are derived from the same glued generator. Zero-field finite-size corrections are organized separately as one-variable scaling functions multiplying

L^{-2},L^{-4},\ldots

. At nonzero field in the thermodynamic limit, the gluing materials are the established high-temperature, critical, and low-temperature local expansions of the universal Ising magnetic scaling free energy

\Phi_{\rm mag}(\eta)

\eta=m/|h|^{8/15}

. The low-temperature input contains the

|h|

cusp associated with spontaneous symmetry breaking. Mapping the glued scaling free energy back to the square lattice through nonlinear scaling fields yields a single generator for the magnetic free-energy increment, magnetization, and susceptibility. The construction reproduces high-temperature linear response, the critical isotherm

M\propto |H|^{1/15}

, and the two one-sided limits

M(H\to0^\pm)=\pm M_0

without separately fitting magnetization branches. Finally, finite size and finite field are combined in the scaling variables

x_t=Lm

and

x_h=L^{15/8}h

. The two protected one-dimensional axes, a central mixed-response expansion, a low-temperature spectral-rounding relation, and a massive outer description are glued into a two-variable scaling surface

\mathcal Y_{\rm glue}(x_t,x_h)

. The resulting lattice free energy closes in the compact form

\Phi_L^{glue}(K,H)=\Phi_{lat}^{glue}(K,H)+\frac{\mathcal{Y}_{glue}(Lm,L^{15/8}h)- Y_{bulk}(Lm,L^{15/8}h)}{L^2}+O(L^{-4})

where

m=m(r,H)

and

h=h(r,H)

are the nonlinear scaling fields and

Y_{\rm bulk}

is the thermodynamic singular scaling surface expressed in the same variables. Thus all finite-size/finite-field information is carried by the excess

\mathcal Y_{\rm glue}-Y_{\rm bulk}

, while the thermodynamic lattice free energy remains the previously constructed

\Phi_{\rm lat}^{\rm glue}

. On eleven transfer-matrix points withheld from the construction, the blind free-energy test gives RMS absolute error

3.90\times10^{-4}

and maximum absolute error

9.55\times10^{-4}

\mathcal Y

. Differentiation of the same free energy produces finite-size magnetization and susceptibility; magnetization is especially stable on the low-temperature side, while the second derivative is more sensitive in crossover regions.

We study the free energy of the classical square-lattice Ising model in a uniform real magnetic field

h

, for which no exact solution is known. We first conjecture a closed-form expression built from the one-dimensional Ising chain and Onsager's hypergeometric solution. Comparing with high-precision tensor renormalization group (TRG) data, we find that it deviates by up to

2.6\%

, has no spontaneous magnetization, and gives an effective critical-isotherm exponent

δ\to 1

. However, using the Lee--Yang circle theorem, we show that the exact free energy is a superposition of one-dimensional Ising chains in the same field, weighted by a temperature-dependent (normalized, but not always positive) distribution of squared Lee--Yang gaps, of which our first expression is a single-chain approximation. Exact results (Onsager, Yang, the strong-field expansion and the Lee--Yang solution at

βh = iπ/2

) become sum rules for this distribution. We propose a simple form for it, fixed by Onsager's free energy and the exact strong-field limit without any fit to numerical data. This \emph{improved} conjecture agrees with TRG to

0.15\%

for all

β

and

h

, reproduces Yang's spontaneous magnetization exactly, has a positive specific heat, and yields the critical isotherm

M\propto h^{1/δ}

with

δ= 15.2

and amplitude

1.055

, compared with the exact

δ=15

and

1.059

. A refined form, which builds in additional input from Ising field theory (including the exact critical isotherm,

δ=15

) and a few constants adjusted to transfer-matrix and TRG data, matches the TRG free energy to

4\times 10^{-5}

n>0

and isolates the long-distance sectors controlling the

n>1

thermodynamics. Cutting a finite interval generates the universal

c\,n/[8(n-1)]

logarithm directly from the missing

1/r^2

interactions, with

c=1/2

the Ising central charge. On the ring, the explicit finite-size factor has no constant term, while any additional noninteger singularity is confined to a residual cycle-only free energy. All explicit long-distance contributions become marginal only at

n=1

We construct a single reduced free-energy generator for the simple-cubic three-dimensional Ising model by finite gluing of complementary local relations. With the convention

\Phi=+\log Z/V

, the final object is

\Phi_3(K,H,L)=\Phi_0(K)+\Delta\Phi_H(K,H)+\Phi_{\rm FSS}(K,H,L),

from which

M=\partial_H\Phi_3

\chi=\partial_H^2\Phi_3

, and

U_K=\partial_K\Phi_3

are generated. The construction covers the high- and low-temperature regimes, the critical region, finite magnetic field, strong-field saturation, periodic-film finite size, and simultaneous finite

H

and finite

L

. The materials and operations are separated explicitly. A diagnostic dimensional stage uses

d=1,2,4

anchors in

\mu_c=-\log\tanh K_c

to transport critical coordinates and positive film amplitudes toward

d=3

. The practical 3D generator then protects established 3D values of

K_c

, critical exponents, selected amplitude ratios, and scalar zero-field critical data. The zero-field term joins truncated simple-cubic high- and low-temperature series through a singular critical patch with

C^2

matching. The inner magnetic increment joins an

O(K^2)

high-temperature field expansion, a minimal 3D parametric critical free energy fixed only by the basic 3D anchors, and a low-temperature/cluster continuation using fixed quintic switches in

K

. A recovered

z^4

strong-field expansion is integrated to an absolute free energy. The final inner-to-strong crossover is performed by constructing a monotone beta-CDF magnetization whose endpoint values and integral reproduce the two endpoint free energies, and then integrating that derivative; thus convexity is enforced at the free-energy level rather than by clipping observables. The periodic-film excess is built from dimensionally weighted 2D and leading 4D film surfaces in common correlation-length coordinates, with an

H

-independent residual stabilizer that preserves the

H=0

axis. A previously reconstructed full 3D parametric equation-of-state benchmark and the full 3D finite-size scaling function are withheld from construction. The pure dimensional stage gives

K_c(3)=0.22301549

and sub-percent estimates of

\beta

\delta

, and

\nu

. After 3D calibration, the final implementation passes 29/29 stated structural and numerical checks. In the withheld low-field window

|K-K_c|\le0.01

H\le0.1

, the magnetization has RMS relative error

1.21\%

and maximum error

4.29\%

. The result is not an exact lattice solution; it is a finite, auditable reconstruction in which the input anchors, switch intervals, matching conditions, numerical tolerances, and validation data are kept distinct.