Asymmetric Scaling Formula

• Asymmetric scaling formula is a framework where different variables or regimes are scaled non-uniformly with distinct exponents and prefactors.
• It appears in diverse applications such as knowledge distillation, sparse-feature modeling, plasma turbulence, and gravitational thermodynamics.
• The method captures intricate system behaviors by encoding directional, channel-dependent, or regime-specific asymmetries in its scaling relations.

An asymmetric scaling formula is a scaling relation in which distinct variables, channels, geometrical sectors, or fluctuation regimes are not rescaled uniformly. In the arXiv literature, the term appears in several unrelated technical settings, including knowledge distillation, sparse-feature neural scaling laws, tokamak momentum transport, charged Lifshitz black-hole thermodynamics, finite-size percolation, asymmetric nuclear response modeling, and large-deviation theory. Across these settings, asymmetry typically means that one side of a relation is controlled by a different exponent, prefactor, temperature, or charge coefficient than another, or that different phases obey genuinely different asymptotic laws (Li et al., 2022, Sous et al., 22 May 2026, Ball et al., 2016, Bravo-Gaete et al., 2015, Žeželj et al., 2011, Martinez-Consentino et al., 10 Sep 2025, Monthus, 2019).

1. General forms of asymmetric scaling

The literature exhibits several recurrent mathematical forms. One form assigns different scaling parameters to different components of the same object. In knowledge distillation, asymmetric temperature scaling uses

$$\mathbf{p}_c(\tau_1,\tau_2) = \frac{\exp(\mathbf{f}_c/\tau_c)} {\sum_{j\in[C]}\exp(\mathbf{f}_j/\tau_j)}, \qquad \tau_i=\mathcal{I}\{i=y\}\tau_1+\mathcal{I}\{i\neq y\}\tau_2,$$

with the recommendation $\tau_1 > \tau_2 > 0$, so the correct class and wrong classes are softened differently (Li et al., 2022). A second form assigns different exponents to different asymptotic regimes. In sparse random-feature models, the summary law

$$\ell_{\mathrm{Bayes}}(N,D)\asymp N^{-(\alpha_1+\alpha_2+1)}+ D^{-(\alpha_1+\alpha_2+1)/(\alpha_1+1)}$$

gives one exponent in the underparameterized regime and another in the overparameterized regime (Sous et al., 22 May 2026). A third form distinguishes symmetry classes or channels. In tokamaks, non-mirror symmetric shaping yields a power law, whereas mirror-symmetric shaping yields exponential suppression (Ball et al., 2016). In asymmetric nuclear 2p2h response modeling, the scaling is channel dependent and uses separate proton and neutron Fermi momenta (Martinez-Consentino et al., 10 Sep 2025). In finite-size percolation, asymmetry enters through aspect-ratio-dependent prefactors $a_i(r)$ and $b_i(r)$ in

$$\langle n \rangle_{L,r} = n_c + L^{-1/\nu}\sum_{i=0}^\infty a_i(r) L^{-\theta_i}, \qquad \Delta^2_{L,r} = L^{-2/\nu}\sum_{i=0}^\infty b_i(r) L^{-\theta_i}.$$

(Žeželj et al., 2011)

Domain	Representative formula	Asymmetry source
Knowledge distillation	$\mathbf{p}_c(\tau_1,\tau_2)$	Correct vs wrong classes
Sparse-feature scaling laws	$$\ell_{\mathrm{Bayes}}(N,D)\asymp N^{-\alpha_N}+D^{-\alpha_D}$$	Model-limited vs data-limited regimes
Tokamak momentum flux	power law vs $\Pi_s \sim -\beta e^{-\gamma m}$	Non-mirror symmetric vs mirror symmetric shaping
2p2h MEC response	$$R^K(X)\approx R^K({}^{12}\mathrm{C})\times(\text{factor})\times\mathcal C(X)$$	Proton-neutron imbalance and channel dependence
Finite-size percolation	$\tau_1 > \tau_2 > 0$0 in $\tau_1 > \tau_2 > 0$1	Aspect-ratio asymmetry

These patterns show that “asymmetric scaling formula” is not a single canonical equation. It is a family resemblance across formulas in which the scaling map itself encodes a directional, structural, or regime-dependent inequivalence.

2. Learning-theoretic asymmetric scaling

In knowledge distillation, the relevant asymmetry is class selective. The KD term is decomposed into Correct Guidance, Smooth Regularization, and Class Discriminability, with the last term governed by variance among wrong-class probabilities. The paper defines

$\tau_1 > \tau_2 > 0$2

and proves the factorization

$\tau_1 > \tau_2 > 0$3

The stated motivation is that complex teachers tend to be over-confident and traditional temperature scaling limits the efficacy of class discriminability, so ATS uses a larger $\tau_1 > \tau_2 > 0$4 on the correct class and a smaller $\tau_1 > \tau_2 > 0$5 on wrong classes to enlarge the variance of wrong-class probabilities (Li et al., 2022).

A distinct learning-theoretic use of asymmetric scaling appears in sparse-feature neural scaling laws. There, the data are sparse coordinates with

$\tau_1 > \tau_2 > 0$6

and the expected number of coordinates active at least once across $\tau_1 > \tau_2 > 0$7 samples is

$\tau_1 > \tau_2 > 0$8

This creates an asymmetric bottleneck: in the data-limited regime, rare coordinates are never observed, while in the parameter-limited regime the bottleneck is the number of learnable directions. The resulting exponents are

$\tau_1 > \tau_2 > 0$9

with $$\ell_{\mathrm{Bayes}}(N,D)\asymp N^{-(\alpha_1+\alpha_2+1)}+ D^{-(\alpha_1+\alpha_2+1)/(\alpha_1+1)}$$0 for $$\ell_{\mathrm{Bayes}}(N,D)\asymp N^{-(\alpha_1+\alpha_2+1)}+ D^{-(\alpha_1+\alpha_2+1)/(\alpha_1+1)}$$1, and the interpolation threshold is near

$$\ell_{\mathrm{Bayes}}(N,D)\asymp N^{-(\alpha_1+\alpha_2+1)}+ D^{-(\alpha_1+\alpha_2+1)/(\alpha_1+1)}$$2

The same paper derives a compute-optimal frontier under

$$\ell_{\mathrm{Bayes}}(N,D)\asymp N^{-(\alpha_1+\alpha_2+1)}+ D^{-(\alpha_1+\alpha_2+1)/(\alpha_1+1)}$$3

finding

$$\ell_{\mathrm{Bayes}}(N,D)\asymp N^{-(\alpha_1+\alpha_2+1)}+ D^{-(\alpha_1+\alpha_2+1)/(\alpha_1+1)}$$4

so the compute-optimal frontier favors more data than parameters (Sous et al., 22 May 2026).

These two cases use asymmetry differently. ATS imposes asymmetry directly in the softmax map. Sparse-feature scaling laws derive asymmetry from a latent observation bottleneck. In both cases, the asymmetry is mechanistic rather than merely empirical.

3. Geometry-, channel-, and field-dependent formulas in plasma and scattering

In tokamak turbulence, the scaling law depends on whether the flux surface shaping is non-mirror symmetric or mirror symmetric. For non-mirror symmetric up-down asymmetric shaping, the turbulent toroidal momentum flux obeys

$$\ell_{\mathrm{Bayes}}(N,D)\asymp N^{-(\alpha_1+\alpha_2+1)}+ D^{-(\alpha_1+\alpha_2+1)/(\alpha_1+1)}$$5

for sufficiently weak shaping, specifically when

$$\ell_{\mathrm{Bayes}}(N,D)\asymp N^{-(\alpha_1+\alpha_2+1)}+ D^{-(\alpha_1+\alpha_2+1)/(\alpha_1+1)}$$6

In the explicit two-mode example with $$\ell_{\mathrm{Bayes}}(N,D)\asymp N^{-(\alpha_1+\alpha_2+1)}+ D^{-(\alpha_1+\alpha_2+1)/(\alpha_1+1)}$$7, this becomes

$$\ell_{\mathrm{Bayes}}(N,D)\asymp N^{-(\alpha_1+\alpha_2+1)}+ D^{-(\alpha_1+\alpha_2+1)/(\alpha_1+1)}$$8

By contrast, for mirror-symmetric shaping,

$$\ell_{\mathrm{Bayes}}(N,D)\asymp N^{-(\alpha_1+\alpha_2+1)}+ D^{-(\alpha_1+\alpha_2+1)/(\alpha_1+1)}$$9

with $a_i(r)$0 and $a_i(r)$1 independent of $a_i(r)$2. The asymmetry therefore lies not only in up-down parity, but in whether different shaping harmonics can beat together and survive averaging over the fast coordinate (Ball et al., 2016).

Asymmetric magnetic reconnection with in-plane flow shear uses another class of formulas. The X-line convection speed is

$a_i(r)$3

the reconnection rate is

$a_i(r)$4

and steady reconnection is suppressed above

$a_i(r)$5

Here asymmetry is encoded in the combinations $a_i(r)$6 and $a_i(r)$7, which weight the contributions of the two upstream regions (Doss et al., 2016).

In asymmetric nuclei, the 2p2h MEC response is scaled from $a_i(r)$8C by channel-specific multiplicative laws. For electron scattering, the paper proposes

$a_i(r)$9

$b_i(r)$0

$b_i(r)$1

For charged-current neutrino scattering, the formulas are

$b_i(r)$2

$b_i(r)$3

The stated motivation is phase-space dominance with distinct proton and neutron Fermi seas, and the paper reports that using $b_i(r)$4Ca as a proxy for $b_i(r)$5Ar leads to a systematic error of approximately $b_i(r)$6 (Martinez-Consentino et al., 10 Sep 2025).

4. Finite-size, anisotropic, and probabilistic asymmetric scaling

Finite-size scaling in percolating stick systems uses a generalized scaling function

$b_i(r)$7

The complementarity relation

$b_i(r)$8

induces parity constraints on the prefactors of the moment expansions. For the mean threshold density, $b_i(r)$9 is odd in $$\langle n \rangle_{L,r} = n_c + L^{-1/\nu}\sum_{i=0}^\infty a_i(r) L^{-\theta_i}, \qquad \Delta^2_{L,r} = L^{-2/\nu}\sum_{i=0}^\infty b_i(r) L^{-\theta_i}.$$0 and $$\langle n \rangle_{L,r} = n_c + L^{-1/\nu}\sum_{i=0}^\infty a_i(r) L^{-\theta_i}, \qquad \Delta^2_{L,r} = L^{-2/\nu}\sum_{i=0}^\infty b_i(r) L^{-\theta_i}.$$1 is even; for the variance, $$\langle n \rangle_{L,r} = n_c + L^{-1/\nu}\sum_{i=0}^\infty a_i(r) L^{-\theta_i}, \qquad \Delta^2_{L,r} = L^{-2/\nu}\sum_{i=0}^\infty b_i(r) L^{-\theta_i}.$$2 is even and $$\langle n \rangle_{L,r} = n_c + L^{-1/\nu}\sum_{i=0}^\infty a_i(r) L^{-\theta_i}, \qquad \Delta^2_{L,r} = L^{-2/\nu}\sum_{i=0}^\infty b_i(r) L^{-\theta_i}.$$3 is odd. The paper also identifies a characteristic aspect ratio $$\langle n \rangle_{L,r} = n_c + L^{-1/\nu}\sum_{i=0}^\infty a_i(r) L^{-\theta_i}, \qquad \Delta^2_{L,r} = L^{-2/\nu}\sum_{i=0}^\infty b_i(r) L^{-\theta_i}.$$4 at which the threshold probability becomes scale invariant (Žeželj et al., 2011).

Anisotropic random fields on $$\langle n \rangle_{L,r} = n_c + L^{-1/\nu}\sum_{i=0}^\infty a_i(r) L^{-\theta_i}, \qquad \Delta^2_{L,r} = L^{-2/\nu}\sum_{i=0}^\infty b_i(r) L^{-\theta_i}.$$5 provide another meaning of asymmetry. Partial sums are taken over rectangles of side lengths $$\langle n \rangle_{L,r} = n_c + L^{-1/\nu}\sum_{i=0}^\infty a_i(r) L^{-\theta_i}, \qquad \Delta^2_{L,r} = L^{-2/\nu}\sum_{i=0}^\infty b_i(r) L^{-\theta_i}.$$6 and $$\langle n \rangle_{L,r} = n_c + L^{-1/\nu}\sum_{i=0}^\infty a_i(r) L^{-\theta_i}, \qquad \Delta^2_{L,r} = L^{-2/\nu}\sum_{i=0}^\infty b_i(r) L^{-\theta_i}.$$7, and the scaling transition is described by a critical exponent $$\langle n \rangle_{L,r} = n_c + L^{-1/\nu}\sum_{i=0}^\infty a_i(r) L^{-\theta_i}, \qquad \Delta^2_{L,r} = L^{-2/\nu}\sum_{i=0}^\infty b_i(r) L^{-\theta_i}.$$8. For congruous scaling, the critical exponent is generally $$\langle n \rangle_{L,r} = n_c + L^{-1/\nu}\sum_{i=0}^\infty a_i(r) L^{-\theta_i}, \qquad \Delta^2_{L,r} = L^{-2/\nu}\sum_{i=0}^\infty b_i(r) L^{-\theta_i}.$$9 or its reciprocal. For incongruous or oblique dependence axis, the paper proves the universal result

$\mathbf{p}_c(\tau_1,\tau_2)$0

This sharply separates axis-aligned and obliquely oriented dependence structures (Pilipauskaitė et al., 2020).

Weakly asymmetric bridges classify asymptotic behavior by the size of the asymmetry

$\mathbf{p}_c(\tau_1,\tau_2)$1

The paper identifies three thresholds: $\mathbf{p}_c(\tau_1,\tau_2)$2 for the hydrodynamic transition between the heat equation and nonlinear Hamilton–Jacobi/Burgers behavior, $\mathbf{p}_c(\tau_1,\tau_2)$3 for the comparison between mean shape and fluctuations in equilibrium, and $\mathbf{p}_c(\tau_1,\tau_2)$4 for the KPZ window (Labbé, 2016). Closely related weakly asymmetric interfaces use the critical scale

$\mathbf{p}_c(\tau_1,\tau_2)$5

which yields invariant measures tilted by continuum area functionals and dynamical limits given by stochastic heat equations or reflected stochastic heat equations with additive drift $\mathbf{p}_c(\tau_1,\tau_2)$6 (Etheridge et al., 2014).

Asymmetric trap models exhibit a different scaling mechanism. For the finite complete graph, the rescaled trap-depth process is

$\mathbf{p}_c(\tau_1,\tau_2)$7

while the small-time scaling for the K-process is

$\mathbf{p}_c(\tau_1,\tau_2)$8

valid only in the regime $\mathbf{p}_c(\tau_1,\tau_2)$9. Here the asymmetry parameter $$\ell_{\mathrm{Bayes}}(N,D)\asymp N^{-\alpha_N}+D^{-\alpha_D}$$0 modifies the effective stable index of the limit process (Bezerra et al., 2012).

Large-deviation theory supplies perhaps the clearest statistical meaning of asymmetric scaling. For the empirical average with stretched exponential tails and $$\ell_{\mathrm{Bayes}}(N,D)\asymp N^{-\alpha_N}+D^{-\alpha_D}$$1,

$$\ell_{\mathrm{Bayes}}(N,D)\asymp N^{-\alpha_N}+D^{-\alpha_D}$$2

Below the typical value, the cost is collective and extensive; above it, the cost is a one-big-jump mechanism. The same logic extends to non-integer empirical moments (Monthus, 2019).

5. Anisotropic gravity and parity-dependent renormalization

In gravitational thermodynamics, anisotropic scaling means

$$\ell_{\mathrm{Bayes}}(N,D)\asymp N^{-\alpha_N}+D^{-\alpha_D}$$3

For charged Lifshitz black holes, the entropy is organized by a generalized Cardy formula involving the black-hole energy $$\ell_{\mathrm{Bayes}}(N,D)\asymp N^{-\alpha_N}+D^{-\alpha_D}$$4, the ground-state energy $$\ell_{\mathrm{Bayes}}(N,D)\asymp N^{-\alpha_N}+D^{-\alpha_D}$$5, the electric and magnetic charge sectors, and a theory-dependent coefficient $$\ell_{\mathrm{Bayes}}(N,D)\asymp N^{-\alpha_N}+D^{-\alpha_D}$$6. The same $$\ell_{\mathrm{Bayes}}(N,D)\asymp N^{-\alpha_N}+D^{-\alpha_D}$$7 appears in the Smarr relation

$$\ell_{\mathrm{Bayes}}(N,D)\asymp N^{-\alpha_N}+D^{-\alpha_D}$$8

which in the three-dimensional Lifshitz case becomes

$$\ell_{\mathrm{Bayes}}(N,D)\asymp N^{-\alpha_N}+D^{-\alpha_D}$$9

For hyperscaling violation, the effective dimensionality is

$\Pi_s \sim -\beta e^{-\gamma m}$0

the entropy scales as

$\Pi_s \sim -\beta e^{-\gamma m}$1

and the same structural formula survives with $\Pi_s \sim -\beta e^{-\gamma m}$2 replacing the naive spatial dimension (Bravo-Gaete et al., 2015).

A very different parity-sensitive asymmetric scaling law appears in strongly asymmetric unimodal maps. Near the critical point,

$\Pi_s \sim -\beta e^{-\gamma m}$3

$\Pi_s \sim -\beta e^{-\gamma m}$4

The renormalization intervals $\Pi_s \sim -\beta e^{-\gamma m}$5 satisfy

$\Pi_s \sim -\beta e^{-\gamma m}$6

and the scaling alternates with parity. For large even $\Pi_s \sim -\beta e^{-\gamma m}$7,

$\Pi_s \sim -\beta e^{-\gamma m}$8

whereas for large odd $\Pi_s \sim -\beta e^{-\gamma m}$9,

$$R^K(X)\approx R^K({}^{12}\mathrm{C})\times(\text{factor})\times\mathcal C(X)$$0

The lengths of the renormalization intervals decay super-exponentially, with

$$R^K(X)\approx R^K({}^{12}\mathrm{C})\times(\text{factor})\times\mathcal C(X)$$1

The paper emphasizes that this scaling is non-universal, because $$R^K(X)\approx R^K({}^{12}\mathrm{C})\times(\text{factor})\times\mathcal C(X)$$2 depends on the map (Kozlovski et al., 2019).

6. Interpretation, scope, and non-universality

The cited literature does not support treating asymmetric scaling as a single phenomenon. In some works, asymmetry is imposed at the level of the scaling map itself, as in ATS. In others, it emerges from hidden coverage constraints, as in sparse features, from symmetry-breaking geometry, as in tokamaks, from proton-neutron imbalance and channel counting, as in 2p2h MEC scaling, or from different fluctuation mechanisms above and below a typical value, as in large deviations (Li et al., 2022, Sous et al., 22 May 2026, Ball et al., 2016, Martinez-Consentino et al., 10 Sep 2025, Monthus, 2019).

The literature also distinguishes sharply between universal and non-universal asymmetry. Oblique dependence axes in linear random fields force the universal critical exponent $$R^K(X)\approx R^K({}^{12}\mathrm{C})\times(\text{factor})\times\mathcal C(X)$$3 (Pilipauskaitė et al., 2020). By contrast, the charge-sector coefficient $$R^K(X)\approx R^K({}^{12}\mathrm{C})\times(\text{factor})\times\mathcal C(X)$$4 in charged Lifshitz black holes depends on the electromagnetic theory, the reduced coefficients $$R^K(X)\approx R^K({}^{12}\mathrm{C})\times(\text{factor})\times\mathcal C(X)$$5 in asymmetric nuclear scaling are model dependent, and the super-exponential rate $$R^K(X)\approx R^K({}^{12}\mathrm{C})\times(\text{factor})\times\mathcal C(X)$$6 in asymmetric unimodal maps depends on the map (Bravo-Gaete et al., 2015, Martinez-Consentino et al., 10 Sep 2025, Kozlovski et al., 2019).

A further point is that asymmetry need not coincide with the mere breaking of an obvious discrete symmetry. The tokamak result is explicit: mirror-symmetric flux surface shaping can be up-down asymmetric and yet produce momentum flux that is exponentially small in large shaping mode number, while non-mirror symmetric shaping yields a power law (Ball et al., 2016). This suggests that the decisive issue is often which terms survive averaging, normalization, or coarse graining, not simply whether a configuration looks asymmetric in a geometric sense.

Across fields, asymmetric scaling formulas therefore function as compact summaries of inequivalent mechanisms. They mark cases in which a single symmetric exponent or prefactor is insufficient, and in which the scaling description must remember direction, channel, parity, resource regime, or fluctuation side.