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LIT-1299: Parameter Space Compression Underlies Emergent Theories and Predictive Models

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StatusActive — read 2026-10-20 (NOTE-1018) in full, from arXiv:1303.6738v1 with its nine-page supplement; worth reading for the original numerics behind the claim that coarse-graining contracts irrelevant directions of the Fisher metric (Fig. 3: a 64×64 square-lattice Ising model at its critical point, 13 couplings and a field, up to six checkerboard decimations, in which the field and temperature eigenvalues stay put and the rest fall in bands drawn at b^(2y−d) per step for y = 0, −2, −4), for the supplement's scaling argument for that rate (Eqs. 28–41), which already states that only directions with 2y ≥ d are carried by the singular part, and for the hopping model of diffusion, whose density at time t carries Fisher eigenvalues scaling as t^(3/2−k) (THEORY-774). Read it knowing that the Ising rates are read off a figure with no error bars, no fitted exponents and no second lattice size, and that the eigenvalues are assigned to y by drawn bands, with y = 0 not an operator of the 2D Ising model (THEORY-777, Proposed); that the derivation of the rate is the same scaling argument Raju, Machta and Sethna (2018) later repeat, and assumes the coarse metric is the Hessian of a coarse free energy in the bare couplings; that the Fig. 3 caption's rate 2^(−d−y) contradicts the supplement and the figure, whose bands fall as 2^(y−d/2); that the diffusion model is not a coarse-graining of one data set, so its leading eigenvalues grow with t rather than staying put; that the main text's "'relevant' directions in the RG are exactly those whose FIM eigenvalues do not contract" is contradicted by its own supplement for relevant directions with 0 < 2y < d; and that the closing claim that emergent theories are "due to" the sloppy hierarchy is not shown, since the paper traces both to the same separation of scales.
Version2
Tagsinformation-geometry, natural-sciences, complex-systems
Cited as “paper” byNOTE-1018 — Machta, Chachra, Transtrum and Sethna — Parameter space compression
Cited as “source” byTHEORY-774 — In a one-dimensional lattice hopping model with 2N+1 free hopping probabilities, all particles starting at the origin and the density after t steps observed with fixed Gaussian error, the Fisher metric is exactly t² times the distribution of the difference of two independent (t−1)-step walks, evaluated at the lag between the two hops, so it does not depend on the drift; when the walk is aperiodic with nonzero variance, its k-th largest eigenvalue is of order t^(3/2−k) for k = 0, …, 2N, so successive eigenvalues separate by a factor of order t, THEORY-777 — In Monte Carlo on a 64×64 periodic square-lattice Ising model at its critical point, parametrized by a field and 13 nearby couplings and observed only through the spins left after n checkerboard decimations (b = √2 per step, n ≤ 6), the two Fisher eigenvalues along the field and the gradient of T_c stay within a factor of about two of their microscopic values, while the others fall by up to about six decades, in three groups whose decay is consistent with b^(2y−d) per step for y = 0, −2 and −4

Graph View

Machta, B. B., Chachra, R., Transtrum, M. K. & Sethna, J. P. (2013). Parameter Space Compression Underlies Emergent Theories and Predictive Models. Science 342(6158): 604–607. doi:10.1126/science.1238723. arXiv:1303.6738.

Read in full in NOTE-1018, from the arXiv copy and its supplement.

Key takeaways

  • In a 1-D hopping model whose density at time t is observed with fixed Gaussian error, the Fisher metric is exactly t² times the distribution of the difference of two (t−1)-step walks, and its k-th eigenvalue scales as t^(3/2−k) (THEORY-774). The leading directions measure particle creation, drift and the diffusion constant.
  • In a decimated critical Ising model, the two eigenvalues along the field and the gradient of T_c stay roughly constant over six steps of b = √2, and the others fall by many decades, in bands the authors draw at b^(2y−d) per step for y = 0, −2, −4 (THEORY-777, Proposed). These are the numerics on which LIT-1254 rests its Eq. 21 (THEORY-742).
  • The supplement’s scaling argument (Eqs. 31 and 41) states that the singular part dominates only when 2y ≥ d, the dividing line NOTE-972 found on Raju, Machta and Sethna’s scalings.

Standing in the record

Here as the source of the numerics behind THEORY-742’s excluded scaling claim. It does not change that claim’s standing: the paper’s own derivation is the same scaling argument, and its Monte Carlo is one lattice size at one temperature, without error bars.

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