Accuracy Law: Empirical, Computational, Legal Insights

• Accuracy law is a class of formulations that models accuracy, error, or relative correctness as a systematic function of a controlling variable across various domains.
• It encompasses empirical power-law decay in machine translation and neural networks, as well as exponential error bounds in deep time series forecasting and probabilistic laws in finite element analysis.
• In legal and computational contexts, accuracy law underpins normative standards for data protection and guides performance predictions by relating complexity or class separability to measurable error rates.

“Accuracy law” denotes a class of law-like formulations in which accuracy, error, or relative correctness is expressed as a systematic function of a controlling variable. In current arXiv usage, the term appears in several distinct senses: an empirical power law for semantic degradation in sequential machine translation, a power-law relation between class separability and classification loss in physical neural networks, an exponential relation between forecasting error and window-wise pattern complexity in deep time series forecasting, probabilistic laws for the relative accuracy of finite element schemes, and a legal principle requiring personal data to be “accurate and, where necessary, kept up to date” (Sequeira et al., 2020, Ermolaev et al., 30 Jun 2026, Wang et al., 3 Oct 2025, Chaskalovic et al., 2020, Hallinan et al., 28 Sep 2025).

1. Range of meanings

In the cited literature, “accuracy law” is not a single theorem but a recurring form of formalization. Accuracy may be modeled as a decaying observable, as a lower-bounded function of task complexity, as a probability of one numerical method outperforming another, or as a normative obligation in data protection law.

Domain	Law or principle	Accuracy quantity
Sequential machine translation	$AEL(t) = (t+1)^{-\alpha}$	accumulated GLEU after $t$ hops
Physical neural networks	$\mathrm{MSE} \propto \mathrm{HTC}^{\zeta}$	classification loss as a function of HTC
Deep time series forecasting	$$\mathrm{MSE} \approx \exp(\alpha \cdot \operatorname{Complexity}(\mathbf{x})) - 1$$	minimum forecasting MSE
Finite elements	$\mathbb{P}(X^{(m)}(h) \leq X^{(k)}(h))$ under two-step or sigmoid laws	relative accuracy of $P_m$ versus $P_k$
Data protection law	GDPR Article 5(1)(d)	accuracy of personal data

These formulations differ in ontology. Some are empirical scaling laws fitted to observed curves, some are probabilistic laws derived from error bounds, and some are legal standards whose content is purpose-relative rather than statistical. This suggests that the unifying feature is not a common equation, but the attempt to render accuracy predictable, comparable, or governable.

2. Empirical power laws in machine translation and physical neural networks

A canonical technical use of the term is the Accuracy Empirical Law (AEL) for sequential machine translation. For Google Translate, translation accuracy after $t$ sequential hops is modeled by

$AEL(t) = (t+1)^{-\alpha},$

where $AEL(t)$ is the accumulated translation accuracy, measured by accumulated GLEU score, and $t$0 is the semantic divergence factor (Sequeira et al., 2020). The empirical program used 71 of the 103 available Google Translate languages, four source texts, and translation chains of up to 284 hops. Accuracy was measured by GLEU, computed as the minimum of recall and precision over $t$1-gram overlap, and the law was fitted by minimizing RMSE. The reported $t$2 values were $t$3 for randomized chains, approximately $t$4 for “common” chains of closely related languages, and approximately $t$5 for “mixed” chains using distant language families. Figures 4–7 showed monotonic decay, RMSEs in the range 0.02–0.04, slower degradation for closely related languages, and higher stepwise GLEU between related language pairs.

A distinct but structurally similar law appears in physical neural networks (PNNs). There the controlling variable is not hop count but the Hotelling Trace Criterion (HTC),

$t$6

a task-dependent scalar measure of class separability in PNN state space (Ermolaev et al., 30 Jun 2026). The paper reports a universal power-law relation between HTC and classification loss:

$t$7

For MNIST, the exponent was $t$8; for Fashion-MNIST, $t$9. Pearson correlation coefficients were reported as $\mathrm{MSE} \propto \mathrm{HTC}^{\zeta}$0 and $\mathrm{MSE} \propto \mathrm{HTC}^{\zeta}$1, and the abstract summarizes them as exceeding 0.99 for MNIST and approximately 0.97 for Fashion-MNIST. Experimental and simulated data from highly nonlinear optical fibres, large-area vertical cavity surface-emitting lasers, and coupled nonlinear oscillator networks collapsed onto a single task-specific scaling curve, and once the exponent was calibrated from a small number of trained systems, later performance predictions required no training.

In both cases, the law compresses a high-dimensional system into a low-dimensional predictor. In sequential machine translation the predictor is semantic divergence accumulated over hops; in PNNs it is class separability measured before output-layer training.

3. Forecasting bounds and adjacent scaling laws

In deep time series forecasting, the term “accuracy law” is attached to an exponential relation between irreducible forecasting error and window-wise pattern complexity (Wang et al., 3 Oct 2025). The construction begins by segmenting a series into overlapping windows of length $\mathrm{MSE} \propto \mathrm{HTC}^{\zeta}$2, computing each window’s amplitude spectrum via FFT, and defining pattern complexity as the trace of the covariance matrix of the amplitude spectra:

$\mathrm{MSE} \propto \mathrm{HTC}^{\zeta}$3

The reported law is

$\mathrm{MSE} \propto \mathrm{HTC}^{\zeta}$4

with empirical $\mathrm{MSE} \propto \mathrm{HTC}^{\zeta}$5, over a complexity interval from $\mathrm{MSE} \propto \mathrm{HTC}^{\zeta}$6 to $\mathrm{MSE} \propto \mathrm{HTC}^{\zeta}$7. The evidence was based on 2,820 newly trained deep forecasters over 940 univariate time series from LOTSA, using PatchTST, TimeMixer, and DLinear. The paper reports a Pearson correlation between window-wise pattern complexity and log MSE of 0.7767; a supplementary summary reports Pearson 0.8025 for LogMSE with RESET $\mathrm{MSE} \propto \mathrm{HTC}^{\zeta}$8. By contrast, ADF, ForeCA, and ACF showed weak or no correlation. The law was then used to argue that benchmarks such as ETT, Electricity, Weather, and Exchange-Rate are saturated because the best observed errors already approach the law’s lower bound.

Adjacent literatures do not always use the phrase “accuracy law,” but they adopt closely related predictive structures. CLIPA and CLIPA-v2 report an inverse scaling law for CLIP training: larger image/text encoders can be effectively trained and finetuned with shorter sequence lengths, and when using only 50% of input tokens during finetuning the reported accuracy drops were 0.4% for H/14, 0.8% for L/16, 1.1% for B/16, and 1.8% for S/16 (Li et al., 2023). The same work reports 81.1% zero-shot ImageNet accuracy for CLIPA-v2 H/14 at a training cost of $\mathrm{MSE} \propto \mathrm{HTC}^{\zeta}$$9366,105 for <a href="https://www.emergentmind.com/topics/openclip" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">OpenCLIP</a> G/14, as well as 81.8$$4,000.

CAMEL generalizes the same predictive ambition to data-mixture optimization for LLMs by modeling validation loss as a joint nonlinear function of model size and mixture, and then mapping validation losses to benchmark accuracy through a generalized logistic loss-to-benchmark law (Li et al., 9 Mar 2026). On Mixture-of-Experts models up to 7B-A150M, extrapolated to a 55B-A1.2B target model, the method is reported to reduce mixture optimization costs by 50% and improve downstream benchmark performance by up to 3%.

4. Probabilistic laws of relative accuracy in finite element analysis

In finite element analysis, accuracy laws are formulated probabilistically rather than as direct regressions. The starting point is the classical Bramble–Hilbert-type error estimate

$$\mathrm{MSE} \approx \exp(\alpha \cdot \operatorname{Complexity}(\mathbf{x})) - 1$$0

which ranks higher-order elements asymptotically but does not determine practical superiority at fixed mesh size (Chaskalovic et al., 2018). The proposed framework treats the actual errors of $$\mathrm{MSE} \approx \exp(\alpha \cdot \operatorname{Complexity}(\mathbf{x})) - 1$$1 and $$\mathrm{MSE} \approx \exp(\alpha \cdot \operatorname{Complexity}(\mathbf{x})) - 1$$2 elements, with $$\mathrm{MSE} \approx \exp(\alpha \cdot \operatorname{Complexity}(\mathbf{x})) - 1$$3, as random variables:

$$\mathrm{MSE} \approx \exp(\alpha \cdot \operatorname{Complexity}(\mathbf{x})) - 1$$4

with supports $$\mathrm{MSE} \approx \exp(\alpha \cdot \operatorname{Complexity}(\mathbf{x})) - 1$$5 and $$\mathrm{MSE} \approx \exp(\alpha \cdot \operatorname{Complexity}(\mathbf{x})) - 1$$6. Relative accuracy is then defined as

$$\mathrm{MSE} \approx \exp(\alpha \cdot \operatorname{Complexity}(\mathbf{x})) - 1$$7

The critical mesh size is

$$\mathrm{MSE} \approx \exp(\alpha \cdot \operatorname{Complexity}(\mathbf{x})) - 1$$8

Two laws are reported. The two-step (Heaviside) law gives probability 1 for $$\mathrm{MSE} \approx \exp(\alpha \cdot \operatorname{Complexity}(\mathbf{x})) - 1$$9 and 0 for $\mathbb{P}(X^{(m)}(h) \leq X^{(k)}(h))$0. The sigmoid law, under uniform-distribution assumptions, is

$\mathbb{P}(X^{(m)}(h) \leq X^{(k)}(h))$1

The 2020 numerical validation used 500 random meshes for each fixed $\mathbb{P}(X^{(m)}(h) \leq X^{(k)}(h))$2, comparing $\mathbb{P}(X^{(m)}(h) \leq X^{(k)}(h))$3 versus $\mathbb{P}(X^{(m)}(h) \leq X^{(k)}(h))$4, $\mathbb{P}(X^{(m)}(h) \leq X^{(k)}(h))$5 versus $\mathbb{P}(X^{(m)}(h) \leq X^{(k)}(h))$6, and $\mathbb{P}(X^{(m)}(h) \leq X^{(k)}(h))$7 versus $\mathbb{P}(X^{(m)}(h) \leq X^{(k)}(h))$8 on both a stiff Runge-function problem and a smooth problem (Chaskalovic et al., 2020). The two-step law was reported to fit the measured frequencies very well, while the sigmoid law captured the trend but fit less sharply when $\mathbb{P}(X^{(m)}(h) \leq X^{(k)}(h))$9 was small. A key empirical conclusion was that there exist non-asymptotic regimes where the lower-order element $P_m$0 is considerably more accurate than the higher-order $P_m$1.

The 2018 asymptotic analysis further studies the regime $P_m$2 and reports that the sigmoid law converges to the two-step law, with $P_m$3 under the stated regularity assumptions (Chaskalovic et al., 2018). For very large order gaps, the superiority of the higher-order element becomes overwhelming for fixed $P_m$4; for moderate gaps, however, the law remains mesh-dependent.

5. Accuracy as a legal principle and as a regulatory blind spot

In data protection law, “accuracy” is not a statistical fit but a normative requirement. GDPR Article 5(1)(d) states: “Personal data shall be accurate and, where necessary, kept up to date; every reasonable step must be taken to ensure that personal data that are inaccurate, having regard to the purposes for which they are processed, are erased or rectified without delay” (Hallinan et al., 28 Sep 2025). The paper argues, from a positive law perspective, that this accuracy principle applies not only to facts but also to opinions about data subjects, including opinions built from facts under an interpretative framework. It further argues, from a normative perspective, that it should apply to opinions. The required degree of accuracy is described as purpose-relative rather than perfect, with “reasonable steps” judged against impact and cost, and the discussion relies on the CJEU’s statement in Peter Nowak that accuracy and completeness must be judged in light of the purpose for which data was collected.

The same literature also identifies a contrary tendency: the accuracy paradox in LLM governance (Li et al., 12 Sep 2025). That paper argues that hyper-optimisation and overreliance on accuracy can exacerbate hallucination harms by increasing rhetorical fluency, passive trust, and the appearance of epistemic authority. Its taxonomy includes factual contradiction, factual fabrication, conflation of facts, instruction inconsistency, context inconsistency, logical inconsistency, source fabrication, misattribution, sycophancy, consensus illusion, oversimplification, and prompt-sensitivity effects such as sandbagging and emotionally induced drift. The critique is organized along three dimensions—outputs, individuals, and society—and extends to the EU AI Act, GDPR, and DSA. The proposed shift is toward “epistemic integrity,” “pluralism audits,” “manipulation resilience,” and “epistemic impact assessments.” One of the paper’s summary claims is explicit: “Accuracy is not normatively on par with fairness, epistemic robustness, or autonomy. It is necessary, but insufficient.”

Taken together, these two legal contributions separate two questions that are often conflated: whether data or outputs are accurate enough for a given purpose, and whether accuracy alone is an adequate governance objective.

6. Related laws, trade-offs, and operational definitions

Some adjacent literatures illustrate how the vocabulary of accuracy law intersects with broader law-like regularities. In number theory, Benford’s law predicts the probability that the first digit is $P_m$5 by

$P_m$6

and the paper on its “surprising accuracy” reports that among the first billion powers of 2, exactly 301,029,995 begin with digit 1, while the Benford prediction is $P_m$7 (Cai et al., 2019). The explanation uses uniform distribution modulo 1 and Diophantine approximation. In special cases, Kesten’s theorem yields bounded discrepancy; for the leading digit 1 of powers of 2, the paper gives

$P_m$8

with $P_m$9. In generic cases, the discrepancy is unbounded but grows slowly, on the order of $P_k$0, and Beck’s central limit theorem is invoked to explain the observed near-normal error behavior.

In sensorimotor control, Fitts’ Law formalizes the speed–accuracy trade-off rather than a monotone accuracy curve:

$P_k$1

The same work relates this behavioral law to system-level and hardware-level constraints through

$P_k$2

and the nerve speed–accuracy trade-off

$P_k$3

(Nakahira et al., 2019). Its central explanatory construct is the diversity-enabled sweet spot (DESS): heterogeneous axons and motor units permit both fast gross movement and fine final adjustment. Here accuracy law is not an empirical decay curve but a systems-level trade-off law emerging from heterogeneous hardware.

By contrast, some benchmark literatures operationalize accuracy without positing a single law. LabourLawBench evaluates specialized Chinese labor-law models through ROUGE-L for legal provision memorization, Accuracy for knowledge Q&A and case classification, F1 for welfare compensation prediction, Soft-F1 for named entity recognition, and LLM Judge for case analysis (Lan et al., 15 Jan 2026). Accuracy is defined as exact match for single- and multi-label settings, with no partial credit for multi-label answers. This benchmarked operationalization clarifies a boundary condition: many “accuracy” studies specify measurement regimes rather than accuracy laws.

Across these literatures, the common thread is formalization. Whether the object is semantic drift, forecastability, finite element superiority, personal-data correctness, first-digit statistics, or sensorimotor trade-off, an “accuracy law” names an attempt to replace informal judgments of correctness with a stable relation, threshold, or principle.