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Criticality Made Exact: The Critical Exponents of Phase Transitions as Forced Fold Rationals

Maria Smith

Published Jun 29, 2026• 1 citation
Publisher

Abstract

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

DOI
10.5281/zenodo.21029234

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One axiom. One operation. Zero free parameters. The laws of light — Snell’s refraction, reflection at equal angles, diffraction, interference — are taught as separate rules. This paper forces their unifying principle from the fold. Every law of optics is a statement of wave-matching: at a boundary the phases on the two sides must agree, and the wave-matching condition folds to unison (the half-One closing on the One). Refraction is phase-matching across an interface (Snell’s law); reflection is the symmetric match (equal angles); interference and diffraction are the deterministic summation of matched waves — opposed paths cancel (dark), aligned paths survive (bright), the same deterministic discrete account that resolves the double-slit. One principle: phases match, the matching folds to one. 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

•Jun 29, 2026•1 citation
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
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.
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.
One axiom. One operation. Zero free parameters. The laws of light — Snell’s refraction, reflection at equal angles, diffraction, interference — are taught as separate rules. This paper forces their unifying principle from the fold. Every law of optics is a statement of wave-matching: at a boundary the phases on the two sides must agree, and the wave-matching condition folds to unison (the half-One closing on the One). Refraction is phase-matching across an interface (Snell’s law); reflection is the symmetric match (equal angles); interference and diffraction are the deterministic summation of matched waves — opposed paths cancel (dark), aligned paths survive (bright), the same deterministic discrete account that resolves the double-slit. One principle: phases match, the matching folds to one. 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