Source
Machta, Chachra, Transtrum and Sethna (2013), Science 342:604, LIT-1299, arXiv:1303.6738v1, read in full in NOTE-1018: main text Fig. 3, supplement §IV (the model), §V (the metric before coarsening, Fig. 4), §VI (after coarsening, Eqs. 32–41) and §VII (the compatible Monte Carlo).
What was actually shown
The computation. H = Σ θ_αβ Φ_αβ + θ_h Φ_h with Φ_αβ = Σ s_ij s_{i+α, j+β} over nearby offsets and Φ_h = Σ s_ij, evaluated on the ordinary Ising line (θ₀₁ = θ₁₀ = J, the rest 0, h = 0) at J_c. The data at level n are the spins on the n-th checkerboard sublattice, still drawn from the full-lattice ensemble. The coarse metric is g^n = Cov({Φ_μ}, {Φ_ν}), with {Φ} the mean of Φ conditional on the observed sublattice (Eq. 34; the same identity as THEORY-742, step 2). It is estimated from M = 10⁴–10⁵ Wolff configurations and, for each, a sub-ensemble drawn by compatible Monte Carlo with the observed spins frozen, using products over distinct replicas to remove the bias (Eqs. 36, 46). At level 1 the conditional means are exact, because the free spins are conditionally independent (Eq. 42).
The result, as plotted. The field eigenvalue sits near 10⁵ at every level. The ∇T_c eigenvalue sits near 4 × 10³ and falls to about 2 × 10³ by step 6. The other eigenvalues start clustered between about 3 and 40 and are coloured into three groups, with shaded bands. Read from the rendered figure, the bands fall by about 0.27, 0.87 and 1.35 decades per step, against 0.30, 0.90 and 1.51 for 2^{y−1} with y = 0, −2, −4. The lowest group leaves the plot after step 4, near 10⁻⁵.
The explanation offered. Supplement Eqs. 29–41. Split F into singular and analytic parts in scaling variables u_α with u′ = b^{y_α} u. The singular part of the metric is ∼ A ξ^{2y−d}, the analytic part ∼ A, so near criticality only directions with 2y ≥ d are dominated by the singular part (the field, y = 15/8, and temperature, y = 1 = d/2, logarithmically). After n steps the singular part is unchanged and the analytic part is ∼ A b^{n(2y−d)}, because the matrix it multiplies “should be smoothly varying” and its eigenvalues “should continue to take a characteristic value”. This is a scaling argument. It also assumes (Eqs. 37–40) that the coarse metric in the bare coordinates is the Hessian of a coarse free energy, which needs the closure and linear parameter map that THEORY-742 states.
What could have come out otherwise. The field and temperature eigenvalues could have fallen with the rest. The others could have fallen at one common rate (b^{−d}, the rate of losing sites) rather than at separated ones. Neither happens in the figure.
What this does not say
- Not a measured rate. No exponent is fitted and no error is reported. The rates are those of bands drawn by the authors, and the assignment of each eigenvalue to a band is by colour. The agreement is “quantitative”, in the supplement’s word, only by eye.
- Not the 2D Ising irrelevant spectrum, as such. The bands are labelled y = 0, −2, −4, “assuming that our variables project onto irrelevant and marginal scaling variables” with those dimensions, with the authors citing Caselle et al. (2002) for −2 and −4. The 2D Ising model has no marginal operator; a direction with no singular part and no RG scaling loses information at b^{−d} per step simply because half the sites are dropped, and the y = 0 band is that rate.
- Not more than one size or temperature. At T_c on a 64×64 lattice, ξ is the system size, and six steps reach an 8×8 lattice. The slight fall of the ∇T_c eigenvalue at steps 5 and 6 is consistent with the logarithmic case and with b^n nearing L. Nothing is shown off criticality.
- Not a noise-free spectrum. Eigenvalues ten decades below the largest are reported from Monte Carlo covariances with no estimate of the estimator’s floor, and the sub-ensemble sizes are not given.
- Not that relevant directions are exactly those that do not contract. The main text says so. The supplement says that relevant directions with 0 < 2y < d contract like irrelevant ones. The 2D Ising model has none, so the figure cannot test the line.
- The caption’s rate is wrong. “Further eigenvalues shrink by a factor of 2^{−d−y_i} in each step” gives 1/4, 1 and 4 for y = 0, −2, −4. The bands and the supplement’s Eq. 41 give b^{2y−d} = 2^{y−d/2}, that is 1/2, 1/8 and 1/32.
Connections
This is the numerical support that LIT-1254 cites for its Eq. 21 (“our confidence in the result comes from the numerical work in Ref. [15]”), whose derivation it repeats in Fisher-metric language. The proved part of that later paper is THEORY-742; this entry is the empirical part it leans on. The same paper’s diffusion half is THEORY-774, which is proved. On QUESTION-217, the contraction order here is an order of relevance to the bare couplings along a refinement chain of sublattices. It says nothing about acquisition.
Note of 2026-10-30: what holds it back
The eigenvalues reported as falling by up to six decades come from Monte Carlo covariances with no estimate of the estimator’s noise floor and no per-step error, and the rates are bands the authors drew, read off the figure by eye. Whether the lowest groups’ decay is signal is a question about the study’s internal validity (ADR-065 §7), so the condition stays. Nothing rests on this entry, so narrowing it to what Machta et al. report would free nothing.