We derive the critical exponents of the three-dimensional Ising universality classas exact rational numbers from the SkinnerTorrado curvature coecients C(n) =Γ(1 + n)2/Γ(1 + 2n) established in Articles 1a1b [1, 2].Three independent predictions are obtained algebraically from the four knownenrollment subspaces of M:η3D =C(3)^3/(C(1)^3 · C(2)^2)=9/250= 0.0360, (I)ν3D =C(3)^3/(C(1) · C(2)^2 · C(4)) =7!/20^3=63/100= 0.63, (II)β3D =C(4)^2/(C(1)^2 · C(3)^2)=16/49= 0.326531 . . . (III)These agree with the state-of-the-art conformal bootstrap determinations [9] to0.005% (ν), 0.034% (β), and 0.82% (η). All remaining exponents follow by standardscaling relations and are likewise exact rationals: γ = 30933/25000, α = 11/100,δ = 1241/259.We show that the exact critical exponents of the two-dimensional Ising model(Onsager solution, 1944) are recovered as powers of C(1) = 1/2: η2D = C(1)2 = 1/4,ν2D = C(0) = 1, β2D = C(1)3 = 1/8. Mean-eld exponents (d ≥ 4) are similarlyrecovered: νMF = βMF = C(1) = 1/2, ηMF = 0.We then construct the complete thermodynamic framework of GOD Theory:phase transitions as enrollment transitions of M; the Carnot cycle as a closed trajectory in enrollment space; adiabatic compression as topological compacticationof a fold; and the connection between tokamak disruptions (Article 3, [3]) and classical phase transitions via the common structure of the SkinnerTorrado algebra atenrollment thresholds.
We derive the critical exponents of the three-dimensional Ising universality classas exact rational numbers from the SkinnerTorrado curvature coecients C(n) =Γ(1 + n)2/Γ(1 + 2n) established in Articles 1a1b [1, 2].Three independent predictions are obtained algebraically from the four knownenrollment subspaces of M:η3D =C(3)^3/(C(1)^3 · C(2)^2)=9/250= 0.0360, (I)ν3D =C(3)^3/(C(1) · C(2)^2 · C(4)) =7!/20^3=63/100= 0.63, (II)β3D =C(4)^2/(C(1)^2 · C(3)^2)=16/49= 0.326531 . . . (III)These agree with the state-of-the-art conformal bootstrap determinations [9] to0.005% (ν), 0.034% (β), and 0.82% (η). All remaining exponents follow by standardscaling relations and are likewise exact rationals: γ = 30933/25000, α = 11/100,δ = 1241/259.We show that the exact critical exponents of the two-dimensional Ising model(Onsager solution, 1944) are recovered as powers of C(1) = 1/2: η2D = C(1)2 = 1/4,ν2D = C(0) = 1, β2D = C(1)3 = 1/8. Mean-eld exponents (d ≥ 4) are similarlyrecovered: νMF = βMF = C(1) = 1/2, ηMF = 0.We then construct the complete thermodynamic framework of GOD Theory:phase transitions as enrollment transitions of M; the Carnot cycle as a closed trajectory in enrollment space; adiabatic compression as topological compacticationof a fold; and the connection between tokamak disruptions (Article 3, [3]) and classical phase transitions via the common structure of the SkinnerTorrado algebra atenrollment thresholds.
One axiom. One operation. Zero free parameters. At a phase transition, physical quantities follow power laws whose critical exponents are universal across utterly different materials. This paper forces the exponents to be fold rationals. The order-parameter exponent β = (m−1)/m at m=2 = 1/2 (the half-One); the correlation-length ν = 1/2; the susceptibility γ = 1 (the One); the critical isotherm δ = 3 (the colour count); and the exponents that would be zero (η, α) sit at the floor, because the fold has no zero. These satisfy the scaling relations exactly: Widom γ = β(δ−1) = ½·2 = 1, and Rushbrooke α + 2β + γ = 2 gives α at the floor. The quantum-Hall conductance plateaus are the same fold rational counts. Machine-checked; reproduces from one command. A standalone result within the Smithian Fold Theory of Everything (SFTOE). Full corpus, code, and the run-it-yourself VERIFY.md protocol: https://github.com/MettaMazza/Smithian-Fold-Theory
For pt. II, see abstr. A64455 of 1973. The critical-point exponents of two Ising models with pure three-spin coupling terms are examined using low-temperature (T<T c ) series expansions for the thermodynamic functions. In particular the critical isotherm exponent delta is examined by applying numerical extrapolation methods to the critical isotherm. The two models are (a) a two-dimensional triangular lattice and (b) a three-dimensional face-centred cubic lattice. Direct numerical estimates of alpha ', beta and delta for (b) show that the scaling relations hold for this model and it is found that alpha '=0.655, beta =0.0538, gamma =1.24, nu =0.448, eta =-0.76, and delta =24 on the basis of the scaling relations. The numerical accuracy of the exponents beta and delta for model (a) does not permit an independent verification of the scaling relations; assuming scaling to hold here also, and using the exact result alpha '= 2 / 3 obtained by Baxter and Wu, it is found that gamma '=1.187+or-0.006, nu = 2 / 3 , eta =0.219+or-0.009, beta =0.073+or-0.003, and delta =17.3+or-0.8.
A careful Monte Carlo investigation of the phase transition very close to the critical point (T -> Tc, H -> 0) in relatively large d = 3, s = 1/2 Ising lattices did produce critical exponents beta = 0.3126(4) =~ 5/16, delta^{-1} = 0.1997(4) =~ 1/5 and gamma_{3D} = 1.253(4) =~ 5/4. Our results indicate that, within experimental error, they are given by simple fractions corresponding to the linear interpolations between the respective two-dimensional (Onsager) and four-dimensional (mean field) critical exponents. An analysis of our inverse susceptibility data chi^{-1}(T) vs. /T - Tc/ shows that these data lead to a value of gamma_{3D} compatible with gamma' = gamma and Tc = 4.51152(12), while gamma values obtained recently by high and low temperature series expansions and renormalization group methods are not.
In extensive Monte Carlo simulations the phase transition of the random-field Ising model in three dimensions is investigated. The values of the critical exponents are determined via finite-size scaling. For a Gaussian distribution of the random fields it is found that the correlation length \ensuremath{\xi} diverges with an exponent \ensuremath{\nu}=1.1\ifmmode\pm\else\textpm\fi{}0.2 at the critical temperature and that \ensuremath{\chi}\ensuremath{\sim}${\ensuremath{\xi}}^{2\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}}$ with \ensuremath{\eta}=0.50\ifmmode\pm\else\textpm\fi{}0.05 for the connected susceptibility and ${\mathrm{\ensuremath{\chi}}}_{\mathrm{dis}}$\ensuremath{\sim}${\ensuremath{\xi}}^{4\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}$ with \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=1.03\ifmmode\pm\else\textpm\fi{}0.05 for the disconnected susceptibility. Together with the amplitude ratio A=${\mathrm{lim}}_{\mathit{T}\ensuremath{\rightarrow}\mathit{T}\mathit{c}}$${\mathrm{\ensuremath{\chi}}}_{\mathrm{dis}}$/${\mathrm{\ensuremath{\chi}}}^{2}$(hr/T)2 being close to one this gives further support for a two-exponent scaling scenario implying \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\eta}. The magnetization behaves discontinuously at the transition, i.e., \ensuremath{\beta}=0. However, no divergence for the specific heat and in particular no latent heat is found. Also the probability distribution of the magnetization does not show a multipeak structure that would be characteristic for the phase-coexistence at first-order phase-transition points.
FINDING: The 2D Ising model on a square lattice with D4 point group symmetry exhibits Z2 symmetry breaking with critical exponents that are exact rational numbers, linked to the A1 root system and conformal field theory. MATH: - Critical exponents (2D Ising exact): α = 0 (log), β = 1/8, γ = 7/4, δ = 15, ν = 1, η = 1/4. - These satisfy scaling relations: α + 2β + γ = 2, γ = ν(2 - η), etc. - Central charge c = 1/2 (minimal model M(4,3)). - Partition function zeroes lie on unit circle (Fisher zeros) with angular spacing related to modular invariance. - D4 point group: order 8, dihedral symmetry of square; irreducible representations correspond to Z2 × Z2. - A1 root system: simple Lie algebra su(2); the Ising model's order parameter (magnetization) transforms under the 1D representation of Z2, and the critical theory is a free Majorana fermion (spin-1/2 representation of A1). - Exact ratios: β/ν = 1/8, γ/ν = 7/4, δ = 15 = (5×3) — note 15 = 1.618^2 × 5.73? No, but 15 appears Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: 2D Ising model critical exponents are rational fractions derived from exact solution, with √2 appearing in the spontaneous magnetization exponent. MATH: - Critical exponents: α = 0 (log divergence), β = 1/8, γ = 7/4, δ = 15, ν = 1, η = 1/4 - Spontaneous magnetization: M ∝ (T_c - T)^β, β = 1/8 = 0.125 - Correlation length exponent: ν = 1 - Susceptibility exponent: γ = 7/4 = 1.75 - Specific heat exponent: α = 0 (logarithmic) - Exact solution via Onsager (1944): free energy expressed in terms of elliptic integrals, critical temperature T_c = 2/ln(1+√2) ≈ 2.269 J/k_B - √2 appears in T_c formula and in β = 1/8 (since 1/8 = (1/2)^3, but deeper: β = 1/8 emerges from algebraic order parameter scaling involving √2 in the lattice combinatorics) CONNECTION: - β = 1/8 = 0.125 is a rational fraction, not directly a golden ratio, but the critical temperature T_c = 2/ln(1+√2) involves √2, linking to square lattice symmetry (crystallographic point group D4). - The exponent Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com