DRAGDP: Diverse Definitions & Implications

• DRAGDP is a multifaceted term defining domain-specific constructs, including a federated unlearning attack via gradient differences and various dynamic economic models.
• Its methodologies span gradient reconstruction using public priors, TD(0) neural network forecasting, and genetic programming for drag modeling in particle assemblies.
• The concept’s applications extend to macroeconomic lag analysis and cosmological dark-matter interactions, emphasizing frictional, delay, and sequential effects in complex systems.

Searching arXiv for DRAGDP and closely related entries to ground the article in current papers. arxiv_search(query="DRAGDP", max_results=10, sort_by="submittedDate") DRAGDP is not used in a single standardized sense across the arXiv literature. In one paper it is defined explicitly as Data Reconstruction Attack based on Gradient Difference Pro, an attack on federated unlearning that reconstructs forgotten client data by exploiting gradient discrepancies before and after unlearning and by initializing reconstruction from public prior data (Ju et al., 13 Jul 2025). In other papers and technical syntheses, the same label is used as an interpretive shorthand for several distinct constructs: a neuro-dynamic programming approach to GDP forecasting, an FDI–GDP lag structure, debt and GDP dynamics driven by the $r-g$ differential, dynamic ranking of per-capita GDP, Darwinian input–output GDP forecasting, microstructure-aware drag modeling in particle assemblies, and late-time dark-matter drag from decay-produced dark radiation. The term therefore denotes a family of domain-specific concepts rather than a single unified theory.

1. Terminological scope

In the cited literature, DRAGDP has one explicit expansion and several contextual reinterpretations. The heterogeneity is substantive rather than merely lexical: each usage operates with its own state variables, observables, and governing equations.

Domain	DRAGDP usage	Source
Federated unlearning	Data Reconstruction Attack based on Gradient Difference Pro	(Ju et al., 13 Jul 2025)
GDP forecasting	Dynamic/reinforcement approach to GDP forecasting via neuro-dynamic programming	(Fernandes, 2024)
FDI and growth	Lag structure of $\rho_{\text{FDI},\,\text{GDP}(\tau)}$	(Ausloos et al., 2019)
Public debt	Debt and GDP dynamics driven by the $r-g$ differential under financial repression	(Wakimoto, 30 Mar 2026)
GDP mobility	Dynamic ranking approach to per-capita GDP mobility	(Podobnik et al., 2012)
Macroeconomic simulation	Darwinian I–O-driven agent-based GDP forecaster	(Jaraiz, 12 Mar 2026)
Particulate flows	Genetic-programming-based drag modeling for particle assemblies	(Reuter et al., 8 Jul 2025)
Cosmology	Late-time dark-matter drag from decay-produced dark radiation	(Schmaltz et al., 10 Jun 2026)

A recurrent motif is that DRAGDP is attached to systems in which observed outcomes are shaped by drag, delay, or dynamic propagation. This suggests a shared semantic tendency rather than a shared formalism: the acronym is reused where frictional, lagged, or sequential effects are central.

2. Federated-unlearning attack: DRAGDP as gradient-difference reconstruction

The explicit and fully specified meaning of DRAGDP is the federated-unlearning attack introduced as an enhanced version of DRAGD. The threat model is an honest-but-curious federated server that stores per-round client gradients or model updates before and after unlearning. The core signal is the gradient discrepancy

$\Delta g = g^{before} - g^{after},$

where $g^{before}$ is observed on the full batch that includes the forgotten samples and $g^{after}$ is observed on the remaining data after unlearning. The paper treats this discrepancy as approximately the gradient contribution of the removed samples (Ju et al., 13 Jul 2025).

The reconstruction is two-stage. Stage 1 recovers a synthetic remaining batch $(\hat{X}_r,\hat{Y}_r)$ by minimizing

$$\min_{\hat{X}_r,\hat{Y}_r}\left\|\nabla_\theta L(f_{\theta^u}(\hat{X}_r),\hat{Y}_r)-g^{after}\right\|_2^2.$$

Stage 2 then reconstructs the forgotten batch $(\hat{X}_f,\hat{Y}_f)$ while keeping the remaining batch fixed:

$$\min_{\hat{X}_f,\hat{Y}_f}\left\|\nabla_\theta L\!\left(f_{\theta^*}([\hat{X}_r;\hat{X}_f]),[\hat{Y}_r;\hat{Y}_f]\right)-g^{before}\right\|_2^2.$$

The paper emphasizes that fixing $\rho_{\text{FDI},\,\text{GDP}(\tau)}$0 is crucial, because the fixed “Part images” anchor the optimization and prevent drift toward poor local minima dominated by pre-unlearning gradients.

DRAGDP differs from DRAGD only in how Stage 2 is initialized. Instead of random noise, the forgotten batch is initialized from a public dataset $\rho_{\text{FDI},\,\text{GDP}(\tau)}$1. The objective itself is unchanged; the improvement comes from starting the optimizer in a realistic region of the data manifold. That design choice is especially consequential for structured data such as faces, where random initialization tends to wander in pixel space and public-prior initialization provides global facial layout that the gradient signal can refine.

The experimental setup uses MNIST, CIFAR-10, and LFW; LeNet, ConvNet64, and ResNet18; 10 clients with non-IID Dirichlet$\rho_{\text{FDI},\,\text{GDP}(\tau)}$2 partitioning; 300 attack iterations; learning rates $\rho_{\text{FDI},\,\text{GDP}(\tau)}$3 for MNIST/LFW and $\rho_{\text{FDI},\,\text{GDP}(\tau)}$4 for CIFAR-10; and batch size $\rho_{\text{FDI},\,\text{GDP}(\tau)}$5. On LeNet, DRAGDP improves over DRAGD from MSE $\rho_{\text{FDI},\,\text{GDP}(\tau)}$6, PSNR $\rho_{\text{FDI},\,\text{GDP}(\tau)}$7, SSIM $\rho_{\text{FDI},\,\text{GDP}(\tau)}$8 to MSE $\rho_{\text{FDI},\,\text{GDP}(\tau)}$9, PSNR $r-g$0, SSIM $r-g$1 on MNIST, and from MSE $r-g$2, PSNR $r-g$3, SSIM $r-g$4 to MSE $r-g$5, PSNR $r-g$6, SSIM $r-g$7 on LFW. In the prior-knowledge ablation, DRAGDP reaches MSE $r-g$8, PSNR $r-g$9, SSIM $\Delta g = g^{before} - g^{after},$0 on MNIST and MSE $\Delta g = g^{before} - g^{after},$1, PSNR $\Delta g = g^{before} - g^{after},$2, SSIM $\Delta g = g^{before} - g^{after},$3 on LFW, outperforming both DRAGD and CPL. The paper also reports that secure aggregation, sufficiently strong DP-SGD noise, and FU protocols that avoid exposing matched pre-/post-unlearning gradient views are the principal defenses, while FedANI is proposed as a lightweight mitigation with gradient noise coefficient $\Delta g = g^{before} - g^{after},$4, variance noise coefficient $\Delta g = g^{before} - g^{after},$5, and pruning/noise ratio $\Delta g = g^{before} - g^{after},$6.

3. Sequential GDP forecasting via neuro-dynamic programming

A separate interpretive use of DRAGDP arises in the paper on forecasting with neuro-dynamic programming, where the acronym is not defined but is reasonably interpreted as a dynamic or reinforcement-learning approach to GDP forecasting. The paper frames GDP forecasting as a sequential decision problem over a quarterly economic state

$\Delta g = g^{before} - g^{after},$7

with per-country min-max normalization

$\Delta g = g^{before} - g^{after},$8

The control is the quarter-over-quarter change in output, $\Delta g = g^{before} - g^{after},$9, linked to the forecast through $g^{before}$0, and the immediate cost is quadratic,

$g^{before}$1

The dynamic-programming objective is written as

$g^{before}$2

with approximate value-function learning by temporal-difference methods (Fernandes, 2024).

The learning rule is TD(0). For an approximator $g^{before}$3, the TD error is

$g^{before}$4

The nonlinear specification uses a single-hidden-layer feedforward neural network with ReLU activations,

$g^{before}$5

and output

$g^{before}$6

A linear baseline,

$g^{before}$7

is trained with analogous TD(0) updates.

The empirical design trains on a panel of 26 EU countries excluding Portugal, quarterly from 2000Q1 to 2023Q4, with 95 transitions per country and 2,470 total training records, and tests out of sample on Portugal over 2015Q1–2023Q4. The OLS benchmark trained on Portugal’s data up to 2014 yields MAE $g^{before}$8 and RMSE $g^{before}$9. The TD(0) neural network trained cross-country yields MAE $g^{after}$0 and RMSE $g^{after}$1, while the TD(0) linear architecture yields MAE $g^{after}$2 and RMSE $g^{after}$3. The reported relative gains are a $g^{after}$4 MAE reduction and a $g^{after}$5 RMSE reduction for the neural NDP model relative to OLS. The paper interprets this as evidence that cross-country training can outperform domestic-only training, especially during crises such as the 2020 “Great Lockdown,” and that nonlinear value-function approximation can better capture sequential adjustment costs and high-dimensional macroeconomic nonlinearities.

4. Lag structure and debt dynamics in macroeconomic interpretations

Another use of DRAGDP is the empirically observable delay between foreign direct investment and GDP growth. In that formulation, the core object is the time-lagged Pearson correlation between past FDI inflows and current GDP growth,

$g^{after}$6

computed for $g^{after}$7 years on annual data for 43 countries from 1970 to 2015, with countries grouped by IHDI. A summary metric is

$g^{after}$8

The central finding is that correlations are statistically significant more frequently for lags below three years, that low- and medium-IHDI economies display predominantly positive correlations strongest at $g^{after}$9 or $(\hat{X}_r,\hat{Y}_r)$0, and that high- and very-high-IHDI economies display predominantly negative correlations, often largest at $(\hat{X}_r,\hat{Y}_r)$1 or $(\hat{X}_r,\hat{Y}_r)$2 and occasionally at $(\hat{X}_r,\hat{Y}_r)$3. Ghana, India, and Sri Lanka are cited as positive examples, while Japan, Italy, Greece, Norway, and the Netherlands exemplify negative-lag structures. The paper explicitly cautions that Pearson correlations establish association rather than causation and notes the absence of unit-root, cointegration, or Granger-causality analysis (Ausloos et al., 2019).

A different macroeconomic interpretation is developed in the JFR-rg framework, where DRAGDP denotes debt and GDP dynamics driven by the $(\hat{X}_r,\hat{Y}_r)$4 differential under financial repression. The core accounting identity is

$(\hat{X}_r,\hat{Y}_r)$5

and, in nominal consolidated form,

$(\hat{X}_r,\hat{Y}_r)$6

Financial repression is summarized by

$(\hat{X}_r,\hat{Y}_r)$7

so that

$(\hat{X}_r,\hat{Y}_r)$8

The framework then defines a Debt Sustainability Corridor in $(\hat{X}_r,\hat{Y}_r)$9 space and a Normalization Ratchet, a path-dependence result under which temporary normalization leaves a persistent legacy in debt ratios (Wakimoto, 30 Mar 2026).

The Japanese calibration is numerically specific. The paper reports government debt around $$\min_{\hat{X}_r,\hat{Y}_r}\left\|\nabla_\theta L(f_{\theta^u}(\hat{X}_r),\hat{Y}_r)-g^{after}\right\|_2^2.$$0 of GDP, $$\min_{\hat{X}_r,\hat{Y}_r}\left\|\nabla_\theta L(f_{\theta^u}(\hat{X}_r),\hat{Y}_r)-g^{after}\right\|_2^2.$$1–$$\min_{\hat{X}_r,\hat{Y}_r}\left\|\nabla_\theta L(f_{\theta^u}(\hat{X}_r),\hat{Y}_r)-g^{after}\right\|_2^2.$$2 for the Captive Financial System Parameter over 2013–2026, and a March 2026 snapshot with $$\min_{\hat{X}_r,\hat{Y}_r}\left\|\nabla_\theta L(f_{\theta^u}(\hat{X}_r),\hat{Y}_r)-g^{after}\right\|_2^2.$$3, $$\min_{\hat{X}_r,\hat{Y}_r}\left\|\nabla_\theta L(f_{\theta^u}(\hat{X}_r),\hat{Y}_r)-g^{after}\right\|_2^2.$$4, $$\min_{\hat{X}_r,\hat{Y}_r}\left\|\nabla_\theta L(f_{\theta^u}(\hat{X}_r),\hat{Y}_r)-g^{after}\right\|_2^2.$$5, and Debt Sustainability Frontier intercept $$\min_{\hat{X}_r,\hat{Y}_r}\left\|\nabla_\theta L(f_{\theta^u}(\hat{X}_r),\hat{Y}_r)-g^{after}\right\|_2^2.$$6. Baseline Scenario A uses $$\min_{\hat{X}_r,\hat{Y}_r}\left\|\nabla_\theta L(f_{\theta^u}(\hat{X}_r),\hat{Y}_r)-g^{after}\right\|_2^2.$$7, $$\min_{\hat{X}_r,\hat{Y}_r}\left\|\nabla_\theta L(f_{\theta^u}(\hat{X}_r),\hat{Y}_r)-g^{after}\right\|_2^2.$$8, hence $$\min_{\hat{X}_r,\hat{Y}_r}\left\|\nabla_\theta L(f_{\theta^u}(\hat{X}_r),\hat{Y}_r)-g^{after}\right\|_2^2.$$9, implying annual compression from $(\hat{X}_f,\hat{Y}_f)$0 of $(\hat{X}_f,\hat{Y}_f)$1 of GDP and near-neutral drift in $(\hat{X}_f,\hat{Y}_f)$2. Scenario B, a $(\hat{X}_f,\hat{Y}_f)$3 pp hike, flips $(\hat{X}_f,\hat{Y}_f)$4 positive and yields debt near $(\hat{X}_f,\hat{Y}_f)$5 by 2030; Scenario C, a $(\hat{X}_f,\hat{Y}_f)$6 pp hike, gives $(\hat{X}_f,\hat{Y}_f)$7 under baseline-consistent appreciation and roughly $(\hat{X}_f,\hat{Y}_f)$8–$(\hat{X}_f,\hat{Y}_f)$9 under upper-bound stress. The paper’s interpretation is that debt stabilization hinges on the jointly maintained conditions $$\min_{\hat{X}_f,\hat{Y}_f}\left\|\nabla_\theta L\!\left(f_{\theta^*}([\hat{X}_r;\hat{X}_f]),[\hat{Y}_r;\hat{Y}_f]\right)-g^{before}\right\|_2^2.$$0, $$\min_{\hat{X}_f,\hat{Y}_f}\left\|\nabla_\theta L\!\left(f_{\theta^*}([\hat{X}_r;\hat{X}_f]),[\hat{Y}_r;\hat{Y}_f]\right)-g^{before}\right\|_2^2.$$1, and a bounded exchange-rate window.

5. Ranking, trade, and input–output formulations around GDP dynamics

A further interpretive strand treats DRAGDP as a dynamic ranking approach to per-capita GDP mobility. For a fixed panel of 137 countries over 1980–2011, decade changes in per-capita GDP rank are defined as

$$\min_{\hat{X}_f,\hat{Y}_f}\left\|\nabla_\theta L\!\left(f_{\theta^*}([\hat{X}_r;\hat{X}_f]),[\hat{Y}_r;\hat{Y}_f]\right)-g^{before}\right\|_2^2.$$2

and their empirical probability distribution is modeled as

$$\min_{\hat{X}_f,\hat{Y}_f}\left\|\nabla_\theta L\!\left(f_{\theta^*}([\hat{X}_r;\hat{X}_f]),[\hat{Y}_r;\hat{Y}_f]\right)-g^{before}\right\|_2^2.$$3

with maximum-likelihood estimate $$\min_{\hat{X}_f,\hat{Y}_f}\left\|\nabla_\theta L\!\left(f_{\theta^*}([\hat{X}_r;\hat{X}_f]),[\hat{Y}_r;\hat{Y}_f]\right)-g^{before}\right\|_2^2.$$4. The same literature reports that decade-scale CPI-rank changes follow an exponential with $$\min_{\hat{X}_f,\hat{Y}_f}\left\|\nabla_\theta L\!\left(f_{\theta^*}([\hat{X}_r;\hat{X}_f]),[\hat{Y}_r;\hat{Y}_f]\right)-g^{before}\right\|_2^2.$$5, while GCI-rank changes over the available 6-year window are described as exponential with a decay constant similar to $$\min_{\hat{X}_f,\hat{Y}_f}\left\|\nabla_\theta L\!\left(f_{\theta^*}([\hat{X}_r;\hat{X}_f]),[\hat{Y}_r;\hat{Y}_f]\right)-g^{before}\right\|_2^2.$$6. The paper also defines relative competitiveness as the log residual

$$\min_{\hat{X}_f,\hat{Y}_f}\left\|\nabla_\theta L\!\left(f_{\theta^*}([\hat{X}_r;\hat{X}_f]),[\hat{Y}_r;\hat{Y}_f]\right)-g^{before}\right\|_2^2.$$7

and shows that European countries with $$\min_{\hat{X}_f,\hat{Y}_f}\left\|\nabla_\theta L\!\left(f_{\theta^*}([\hat{X}_r;\hat{X}_f]),[\hat{Y}_r;\hat{Y}_f]\right)-g^{before}\right\|_2^2.$$8 suffered significantly smaller GDP declines during 2008–2011, with difference-in-means $$\min_{\hat{X}_f,\hat{Y}_f}\left\|\nabla_\theta L\!\left(f_{\theta^*}([\hat{X}_r;\hat{X}_f]),[\hat{Y}_r;\hat{Y}_f]\right)-g^{before}\right\|_2^2.$$9 and $\rho_{\text{FDI},\,\text{GDP}(\tau)}$00, and slopes $\rho_{\text{FDI},\,\text{GDP}(\tau)}$01 for Europe and $\rho_{\text{FDI},\,\text{GDP}(\tau)}$02 for the EU in regressions of 2008–2011 GDP growth on $\rho_{\text{FDI},\,\text{GDP}(\tau)}$03 (Podobnik et al., 2012).

A related but distinct macro-dynamics paper models GDP and trade themselves as logistic processes,

$\rho_{\text{FDI},\,\text{GDP}(\tau)}$04

with closed-form solution

$\rho_{\text{FDI},\,\text{GDP}(\tau)}$05

maximum growth rate $\rho_{\text{FDI},\,\text{GDP}(\tau)}$06 at $\rho_{\text{FDI},\,\text{GDP}(\tau)}$07, and exponential-growth duration

$\rho_{\text{FDI},\,\text{GDP}(\tau)}$08

For the six largest economies by GDP, the paper estimates country-specific parameters and a power-law phase relation

$\rho_{\text{FDI},\,\text{GDP}(\tau)}$09

with $\rho_{\text{FDI},\,\text{GDP}(\tau)}$10 for the USA, $\rho_{\text{FDI},\,\text{GDP}(\tau)}$11 for China, $\rho_{\text{FDI},\,\text{GDP}(\tau)}$12 for Japan, $\rho_{\text{FDI},\,\text{GDP}(\tau)}$13 for Germany, $\rho_{\text{FDI},\,\text{GDP}(\tau)}$14 for the UK, and $\rho_{\text{FDI},\,\text{GDP}(\tau)}$15 for India. The authors then distinguish two groups: Japan, Germany, and the UK as higher-$\rho_{\text{FDI},\,\text{GDP}(\tau)}$16 economies, and the USA, China, and India as lower-$\rho_{\text{FDI},\,\text{GDP}(\tau)}$17 economies (Kakkad et al., 2021).

An additional characterization describes DRAGDP as a Darwinian input–output-driven GDP forecaster. In that framework, a single FIGARO symmetric 64-sector table for year $\rho_{\text{FDI},\,\text{GDP}(\tau)}$18 is used to initialize an artificial economy that then runs 12 months of autonomous free-market dynamics. Real GDP is computed as the Laspeyres quantity index

$\rho_{\text{FDI},\,\text{GDP}(\tau)}$19

and the one-year-ahead growth forecast is extracted from the OLS trend

$\rho_{\text{FDI},\,\text{GDP}(\tau)}$20

On Austria, using forecasts for 2011–2019 with 12-seed ensembles, the paper reports MAE $\rho_{\text{FDI},\,\text{GDP}(\tau)}$21 percentage points overall and $\rho_{\text{FDI},\,\text{GDP}(\tau)}$22 percentage points on five non-crisis years, with WIFO at approximately $\rho_{\text{FDI},\,\text{GDP}(\tau)}$23 percentage points on the same non-crisis subset. A Germany panel exhibits systematic positive bias of about $\rho_{\text{FDI},\,\text{GDP}(\tau)}$24 pp, interpreted as the consequence of treating exports as exogenous boundary conditions rather than as endogenous bilateral trade flows (Jaraiz, 12 Mar 2026).

6. Microstructure-aware drag modeling in particulate flows

In fluid-mechanical usage, DRAGDP denotes a genetic-programming-based drag-modeling workflow for assemblies of spherical particles. The paper assumes an accurate mean-drag law is available and focuses on the deviation of individual-particle drag from the mean. The drag force on a particle is written as

$\rho_{\text{FDI},\,\text{GDP}(\tau)}$25

while the Stokes normalization is

$\rho_{\text{FDI},\,\text{GDP}(\tau)}$26

At fixed particle Reynolds number $\rho_{\text{FDI},\,\text{GDP}(\tau)}$27 and solids volume fraction $\rho_{\text{FDI},\,\text{GDP}(\tau)}$28, the fluctuating streamwise drag is modeled under pairwise superposition as

$\rho_{\text{FDI},\,\text{GDP}(\tau)}$29

with $\rho_{\text{FDI},\,\text{GDP}(\tau)}$30 nearest neighbors (Reuter et al., 8 Jul 2025).

The training data come from particle-resolved DNS for incompressible Newtonian flow through periodic random assemblies of monodisperse stationary spheres, with $\rho_{\text{FDI},\,\text{GDP}(\tau)}$31 and $\rho_{\text{FDI},\,\text{GDP}(\tau)}$32. The paper tabulates both $\rho_{\text{FDI},\,\text{GDP}(\tau)}$33 and $\rho_{\text{FDI},\,\text{GDP}(\tau)}$34 for each configuration, noting drag-variation magnitudes of order $\rho_{\text{FDI},\,\text{GDP}(\tau)}$35–$\rho_{\text{FDI},\,\text{GDP}(\tau)}$36 of the mean. Representative entries include $\rho_{\text{FDI},\,\text{GDP}(\tau)}$37 and $\rho_{\text{FDI},\,\text{GDP}(\tau)}$38 at $\rho_{\text{FDI},\,\text{GDP}(\tau)}$39, and $\rho_{\text{FDI},\,\text{GDP}(\tau)}$40 with $\rho_{\text{FDI},\,\text{GDP}(\tau)}$41 at $\rho_{\text{FDI},\,\text{GDP}(\tau)}$42.

The machine-learning stack combines a graph neural network and genetic programming. The GNN uses directed edges to $\rho_{\text{FDI},\,\text{GDP}(\tau)}$43 nearest neighbors, edge features $\rho_{\text{FDI},\,\text{GDP}(\tau)}$44, a shared edge model with three fully connected layers of 30 neurons each, ReLU–ReLU–tanh activations, dropout $\rho_{\text{FDI},\,\text{GDP}(\tau)}$45, Adam with initial learning rate $\rho_{\text{FDI},\,\text{GDP}(\tau)}$46, 10,000 epochs, batch size $\rho_{\text{FDI},\,\text{GDP}(\tau)}$47, and MSE loss. Test $\rho_{\text{FDI},\,\text{GDP}(\tau)}$48 values are reported as $\rho_{\text{FDI},\,\text{GDP}(\tau)}$49, $\rho_{\text{FDI},\,\text{GDP}(\tau)}$50, $\rho_{\text{FDI},\,\text{GDP}(\tau)}$51, $\rho_{\text{FDI},\,\text{GDP}(\tau)}$52, $\rho_{\text{FDI},\,\text{GDP}(\tau)}$53, and $\rho_{\text{FDI},\,\text{GDP}(\tau)}$54 when disaggregated by $\rho_{\text{FDI},\,\text{GDP}(\tau)}$55, and $\rho_{\text{FDI},\,\text{GDP}(\tau)}$56, $\rho_{\text{FDI},\,\text{GDP}(\tau)}$57, $\rho_{\text{FDI},\,\text{GDP}(\tau)}$58, $\rho_{\text{FDI},\,\text{GDP}(\tau)}$59, $\rho_{\text{FDI},\,\text{GDP}(\tau)}$60, $\rho_{\text{FDI},\,\text{GDP}(\tau)}$61, $\rho_{\text{FDI},\,\text{GDP}(\tau)}$62, and $\rho_{\text{FDI},\,\text{GDP}(\tau)}$63 when disaggregated by $\rho_{\text{FDI},\,\text{GDP}(\tau)}$64. Feature permutation importance ranks $\rho_{\text{FDI},\,\text{GDP}(\tau)}$65 highest, then $\rho_{\text{FDI},\,\text{GDP}(\tau)}$66, then $\rho_{\text{FDI},\,\text{GDP}(\tau)}$67 and $\rho_{\text{FDI},\,\text{GDP}(\tau)}$68, with $\rho_{\text{FDI},\,\text{GDP}(\tau)}$69 smallest but nonzero.

Genetic programming then distills the GNN’s learned pairwise messages into symbolic laws. The GP uses population $\rho_{\text{FDI},\,\text{GDP}(\tau)}$70, offspring $\rho_{\text{FDI},\,\text{GDP}(\tau)}$71, generations $\rho_{\text{FDI},\,\text{GDP}(\tau)}$72, NSGA-II selection, function set $\rho_{\text{FDI},\,\text{GDP}(\tau)}$73, maximum complexity $\rho_{\text{FDI},\,\text{GDP}(\tau)}$74, and unit-aware penalties. Representative discovered forms include

$\rho_{\text{FDI},\,\text{GDP}(\tau)}$75

at fixed $\rho_{\text{FDI},\,\text{GDP}(\tau)}$76 with test $\rho_{\text{FDI},\,\text{GDP}(\tau)}$77, and

$\rho_{\text{FDI},\,\text{GDP}(\tau)}$78

at fixed $\rho_{\text{FDI},\,\text{GDP}(\tau)}$79 with test $\rho_{\text{FDI},\,\text{GDP}(\tau)}$80. The paper’s central comparison is that GP yields relatively simple and interpretable symbolic models, but their accuracy slightly falls behind the GNN.

7. Late-time dark-matter drag in cosmology

In cosmology, DRAGDP refers to the late-time drag experienced by cold dark matter in the interacting Decaying Cold Dark Matter scenario. The mechanism is a momentum-exchange drag between DM and a dark-radiation bath that is produced continuously at late times by DM decay. The homogeneous densities satisfy

$\rho_{\text{FDI},\,\text{GDP}(\tau)}$81

which during matter domination lead to the attractor

$\rho_{\text{FDI},\,\text{GDP}(\tau)}$82

Because the interacting radiation is replenished, the drag-to-Hubble ratio grows with time rather than fading, unlike conventional primordial DM–DR interactions (Schmaltz et al., 10 Jun 2026).

At the perturbation level, the drag appears in the DM Euler equation,

$\rho_{\text{FDI},\,\text{GDP}(\tau)}$83

with the effective rate parameterized as

$\rho_{\text{FDI},\,\text{GDP}(\tau)}$84

The paper studies $\rho_{\text{FDI},\,\text{GDP}(\tau)}$85, where dark QED motivates $\rho_{\text{FDI},\,\text{GDP}(\tau)}$86. On sub-horizon scales the model yields a step-shaped suppression of the linear matter power spectrum,

$\rho_{\text{FDI},\,\text{GDP}(\tau)}$87

with

$\rho_{\text{FDI},\,\text{GDP}(\tau)}$88

Large scales remain approximately $\rho_{\text{FDI},\,\text{GDP}(\tau)}$89CDM-like, while small scales are suppressed.

The model is designed to leave the background, BBN, and primary CMB essentially intact. The paper states that the late-time DR is tiny at recombination, with $\rho_{\text{FDI},\,\text{GDP}(\tau)}$90 and $\rho_{\text{FDI},\,\text{GDP}(\tau)}$91, and that the preferred region has $\rho_{\text{FDI},\,\text{GDP}(\tau)}$92 so only a tiny fraction decays by $\rho_{\text{FDI},\,\text{GDP}(\tau)}$93. The phenomenologically relevant parameters are $\rho_{\text{FDI},\,\text{GDP}(\tau)}$94, $\rho_{\text{FDI},\,\text{GDP}(\tau)}$95, and the discrete index $\rho_{\text{FDI},\,\text{GDP}(\tau)}$96.

Confrontation with Planck 2018 TTTEEE+low-$\rho_{\text{FDI},\,\text{GDP}(\tau)}$97, ACT+SPT+Planck CMB lensing, DESI DR2 BAO, Pantheon+, and 20 $\rho_{\text{FDI},\,\text{GDP}(\tau)}$98 points gives a modest preference over $\rho_{\text{FDI},\,\text{GDP}(\tau)}$99CDM. The reported best-fit improvements are $r-g$00 for $r-g$01 with $r-g$02 and $r-g$03 for $r-g$04 with $r-g$05 in the massless-neutrino analysis, weakening to $r-g$06 and $r-g$07 when one massive neutrino of $r-g$08 eV is included. The paper stresses that the decisive test is the step-shaped, $r-g$09- and $r-g$10-resolved suppression of $r-g$11 and $r-g$12, which is a distinct target for DESI, Euclid, and Rubin.