Scaling Laws for Ising Models Near T_c

The paper-local cell-variable construction is the root of the main argument. A separate scaling transformation consumes its checked output before free-energy homogeneity may use checked sector scaling. Correlation and exponent relations consume both checked scaling and homogeneity premises. The effective-coupling regularity route has no dependencies and is not used by the main exponent result.

Let L denote the selected cell scale and ξ the coherence length named by the model. The setup does not assign numerical values; it requires the qualitative scale separation

Within such a cell, spins predominantly align and the cell magnetization is represented by a two-valued variable. Between cells, a nearest-neighbor description is assumed sufficient and no additional long-range cell interaction is needed. The setup also requires regular finite-cell free energy, transformed parameters that remain critical, long-range correlations in the transformed problem, and controlled block and scale-restoration operations.

The formal route does not stop at the word “two-valued.” It selects the reduced temperature, weak external magnetic field, cell site count, cell average spin, and cell-spin variable, then requires a supplied calculation of the cell-spin sum, effective field, nearest-neighbor effective Ising description, free-energy decomposition, and long-distance correlation replacement.

cell_spin_construction_retains_effective_ising_data establishes this paper-local result. established_cell_construction_retains_equations_four_through_eleven projects the value-level bundle before the shared scaling adapter is used, while established_cell_construction_retains_free_energy_and_correlation_equations still exposes the paper-local comparison predicates. The finite countermodel two_valued_cell_spin_does_not_supply_effective_scaling_equations blocks treating a bare binary cell label as if it already supplied the effective field and scaling comparisons; two_valued_spin_does_not_supply_equation_four_spin_sum separately blocks a two-valued spin from manufacturing equation (4)'s numeric equality.

The scaling route starts only after EstablishedCellVariableConstruction is available. It maps the original Ising description to a reduced block description at scale L, then restores the comparison scale and produces an effective description. Observables and values are rescaled in parallel. The selected sector contains exactly the singular free energy and long-distance correlation.

Once the singular free-energy prediction is known to scale across the block-and-restoration step, the supplied homogeneity calculation compares the singular terms in the cell relation. The Lean calculation now retains the nonlogarithmic source formulas themselves before returning the paper-local homogeneous-free-energy proposition.

When the above-critical specific heat is logarithmic, the supplied calculation retains the displayed above-critical singular term, the extra scale logarithm generated by equation (13), and its regular-cell compensation. The below-critical logarithmic formula and amplitude equality are not available until the second logarithmic branch is supplied.

The correlation calculation consumes the scaled long-distance-correlation prediction and retains scoped equations (27)-(29) and (31). Its main outputs carry the actual correlation and zero-field coherence formulas. The exponent calculation consumes those scoped outputs with homogeneous singular free energy and retains equations (22)-(26) and (32). The printed field power (30) remains a separate assertion outside this derivation.

Admissible comparison points and actual coordinate agreement

Equation (27) inherits the cell hierarchy and the original and transformed near-critical parameters from equation (11). Equations (28)-(29) are restricted to their corresponding critical windows. Equation (31) uses a long-distance window without assigning a finite coherence length to the exact critical point.

selected_cell_equations_give_the_scoped_correlation_reexpression combines equations (4), (9), (11), (13), and (17) at the selected point. The separate SelectedCorrelationCoordinateAgreementAt identifies the actual temperature coordinate and the original and blocked correlation readouts. A shared real-valued codomain alone does not establish these identifications, and one selected comparison does not prove a universal scaling law.

The printed field power is recorded and separately tested

The original page 269 prints the exponent -(s-psi), not its reciprocal. The formalization preserves that statement in EquationThirtyCoherenceLengthFieldFormulaAt. Its compatibility with length rescaling is a separate conditional question, not a replacement formula attributed to the source.

With L = 2, theta = 1/2, delta = 1/4, and field weight y = 2, both test profiles admit h = 1/16. The printed profile xi = |h|^(-2) fails covariance; the reciprocal profile xi = 1/sqrt(|h|) satisfies it but fails the printed power. At h = 1/8 the original field remains in the window while its transformed value h' = 1/2 leaves it. These are inhabited function-valued diagnostics, not constructed microscopic Ising measures.

The exponent route consumes the scoped correlation formulas and the zero-field coherence law (29). Neither the printed field power nor the independent covariance diagnostic is supplied as a premise of that route.

The paper describes a correlation range without specifying this exponential convention. The Lean refinement computes the connected channel from the raw spin product and the average spin, then uses an eventually nonzero logarithmic decay rate. Raw-tail scaling and consistent mean-square scaling together supply the connected-tail identity.

A profile with mean spin 1/2 has a nonzero raw plateau but a finite connected length. Two exponentially suppressed profiles can also agree asymptotically in absolute error while having different lengths. These profile tests do not construct microscopic critical Ising measures. Critical-isotherm raw scaling, mean-square scaling, and finite-length meaning remain additional inputs.

Two cell comparisons need the intermediate window

The shared numerical transport interface composes actual distance maps and weights, while the Kadanoff adapter retains both parameter transformations. The equality and residual routes separately identify their intermediate correlation readout; neither is an assertion of microscopic reconstruction or automatic effective-model closure.

For the separately declared exponential-length convention, the relevant error is multiplicative in the connected tail. A correction may grow or shrink algebraically and need not approach a nonzero constant; its logarithm must instead be sublinear in distance.

The proof uses actual logarithmic rates, not an interpretation label or an assumed equality of lengths. Absolute tail error is insufficient, as before; even relative error is insufficient if it is measured against the uncentered observable.

The final route assumes that the effective coupling of the selected cell model is regular in temperature. The source map classifies this premise and the resulting equalities with author-expectation status. A supplied calculation then retains the exact equation-(15) relation nu = nu' and yields equality of the correlation-length exponents above and below T_c, and likewise for the specific-heat exponents.

The coarse-grained model stores microscopic, reduced, and restored effective descriptions separately. Its setup certifies the selected critical regime and control of both block transformation and scale restoration. The scaling bridge transfers only the selected-sector predictions.

The actual-law refinement separates an exact statistical projection from the physical model assumptions used by the scaling route. A measurable map q assigns actual two-valued configurations to microscopic states. Normalization gives a genuine retained probability law, not an empty prediction relation.

This theorem concerns cell-pair observables pulled back through q. It does not identify them with original microscopic-spin correlations, determine the cell-average magnitude, or derive the nearest-neighbor ansatz. Those source assumptions and calculations remain in the preceding routes.

Here nonvacuity means an actual normalized observable law; it does not require a nonzero mean. Integrability is a separate hypothesis for the expectation identity.