Fabrication-Risk Digit Randomness Screening (FDRS)

• FDRS is a forensic screening framework that assesses raw numerical data for digit randomness irregularities indicative of potential fabrication.
• It employs a six-step pipeline including digit extraction, chi-square tests, entropy measures, and progressive subsampling to quantify irregularities.
• The model integrates statistical metrics with machine learning to produce an ensemble risk score mapped to graded levels for prioritization.

Searching arXiv for the specified FDRS paper and closely related randomness-screening work. The Fabrication-risk Digit Randomness Screening model (FDRS) is a digit-forensic screening framework for raw numerical research data. Its purpose is not to decide that data were fabricated, but to detect whether the fine-scale decimal-digit structure of a dataset looks unusually non-random relative to what one would expect from instrument-generated continuous measurements with adequate precision. It targets non-random digit-pattern irregularity, especially in later decimal places where human preferences may leak into numbers that are supposed to arise from a measurement process, and it converts a single-column text file of raw numbers into an ensemble risk score and an ordinal grade (Cao, 5 Jun 2026).

1. Conceptual basis and scope

FDRS was developed in response to a gap in research-integrity practice: images can be screened for manipulation, manuscripts for plagiarism, and reported statistics for arithmetic inconsistency, but raw numeric columns are much harder to review systematically. The framework is therefore directed at absorbance readings, Ct values, fluorescence intensities, cell counts, densitometry, enzyme activity outputs, and related raw measurements, rather than at already-transformed summaries (Cao, 5 Jun 2026).

The model’s central assumption is domain-specific and modest. If one has a column of independent continuous measurements recorded with sufficient decimal precision and without strong rounding constraints, then the later decimal digits should be approximately random or at least comparable to a plausible empirical reference distribution. Humans manually inventing or altering values, by contrast, often show subtle preferences for certain digits or combinations of digits. FDRS operationalizes that distinction as a multi-stage workflow that evaluates single digits, joint decimal-digit combinations, information-theoretic dispersion, divergence from a reference null, and persistence of deviation as sample size increases (Cao, 5 Jun 2026).

The framework is explicitly bounded. Its output is an auxiliary screening signal, not proof of fabrication or misconduct, and should trigger verification of source records rather than adjudication. A high score means stronger, multi-metric evidence of persistent non-random digit structure; it does not identify the cause. The cause could be fabrication, but also instrument precision, bounded ranges, fixed formatting, batch workflows, or rounding conventions. This boundary condition is central to the interpretation of FDRS and distinguishes it from evidentiary or adjudicative tools (Cao, 5 Jun 2026).

2. Six-step workflow

FDRS is described as a six-step pipeline. First, the user provides raw numerical data in a single column, one value per row. Values are analyzed in their original numeric form whenever possible; scientific notation is converted to fixed-decimal form; if a value has fewer decimals than needed for the target positions, it is right-padded with zeros for digit extraction; and no extra normalization, transformation, or post-export rounding is applied (Cao, 5 Jun 2026).

Second, the framework performs decimal digit extraction. If the dataset is $X=\{X_1,\dots,X_n\}$, then for each value $X_i$, the digit at decimal position $j$ is denoted $D_{ij}\in\{0,\dots,9\}$. The model’s primary target in development was the third and fourth decimal places, especially the joint third-fourth decimal combination $D_{i,3:4}\in\{00,\dots,99\}$. For heterogeneous benchmark datasets, the analyzed positions were adapted to the original reporting precision: some datasets were screened at the third-fourth decimal combination, others at the first-second combination (Cao, 5 Jun 2026).

Third, FDRS computes full-sample statistical tests on the extracted digits. There is a single-decimal-digit module, usually on the third decimal digit, and a multi-decimal joint distribution module, primarily on third-fourth decimal pairs. These produce chi-square statistics, $P$-values, effect sizes, residuals, entropy measures, KL divergence, and digit-preference features (Cao, 5 Jun 2026).

Fourth, it performs progressive digit-randomness screening via progressive subsampling. Rather than relying on one full-sample test, the framework repeatedly subsamples the dataset at increasing sample sizes and recalculates the digit-randomness metrics each time. The point is not only whether a deviation is present, but whether that deviation attenuates toward randomness, or remains structurally present, as more observations are accumulated (Cao, 5 Jun 2026).

Fifth, the workflow assembles a dataset-level feature vector. The paper groups features into six classes: single-digit frequencies; joint-digit frequencies; distributional-deviation metrics; information-theoretic metrics; digit-preference indices; and progressive subsampling descriptors such as means, final values, maxima/minima, and slopes of $V$, entropy, and KL divergence over increasing sample size (Cao, 5 Jun 2026).

Sixth, these engineered features feed a semi-supervised machine-learning layer. Because confirmed fabricated raw datasets are rarely available, the models are trained on a mixture of simulated normal datasets, simulated digit-preference irregular datasets, and mixed-contamination datasets. The output is a model-based high-risk probability plus an anomaly score, integrated into a final ensemble risk score, which is mapped to a Grade 0–4 scale (Cao, 5 Jun 2026).

3. Statistical and information-theoretic structure

For a chosen decimal position $j$, the observed count of digit $k\in\{0,\dots,9\}$ is $O_k$, with $X_i$0. Under the null, the digit is assumed uniformly distributed:

$X_i$1

The expected count is

$X_i$2

FDRS uses a Pearson chi-square goodness-of-fit statistic:

$X_i$3

The framework also computes Cramér’s $X_i$4 as an effect size and standardized residuals

$X_i$5

to identify which digits contribute most to the departure (Cao, 5 Jun 2026).

The main joint test in development examines the third and fourth decimal digits together, producing categories $X_i$6. Let $X_i$7 denote the observed count of two-digit combination $X_i$8, with $X_i$9. Under the null:

$j$0

The expected count is

$j$1

The chi-square statistic is

$j$2

The motivation is that a person may not overuse isolated digits strongly enough to stand out one digit at a time, yet may prefer neat combinations like 00, 25, 50, 75, repeated pairs such as 55 or 88, or psychologically salient sequences. The joint 00–99 structure can therefore capture fabrication-risk signals invisible to a single-digit histogram (Cao, 5 Jun 2026).

FDRS uses Shannon entropy to quantify how dispersed the observed digit distribution is. If $j$3, then

$j$4

where terms with $j$5 are omitted. Because entropy depends on the number of categories, the model uses normalized entropy

$j$6

where $j$7 for single digits and $j$8 for joint pairs. Values near 1 indicate near-uniformity and lower values indicate concentration in fewer categories. The framework also computes Kullback–Leibler divergence from the expected uniform distribution:

$j$9

with $D_{ij}\in\{0,\dots,9\}$0 under the uniform null (Cao, 5 Jun 2026).

To capture interpretable human-choice tendencies, FDRS defines digit-preference indices. For single digits, these include $D_{ij}\in\{0,\dots,9\}$1, $D_{ij}\in\{0,\dots,9\}$2, odd-even ratio, high-low digit ratio, adjacent-digit dependence, repeated-tail ratios, and the explicitly listed indices for odd, extreme, and middle digits. For two-digit combinations, the methods emphasize repeated-pair and neat-combination measures, including

$D_{ij}\in\{0,\dots,9\}$3

$D_{ij}\in\{0,\dots,9\}$4

and

$D_{ij}\in\{0,\dots,9\}$5

These features make the system relatively interpretable, because a high-risk result can be explained in terms of concrete digit-structure behavior rather than only a classifier output (Cao, 5 Jun 2026).

4. Progressive subsampling and semi-supervised risk scoring

Progressive subsampling is one of the model’s distinctive innovations. If the complete dataset has size $D_{ij}\in\{0,\dots,9\}$6, FDRS defines increasing subsample sizes

$D_{ij}\in\{0,\dots,9\}$7

At each size $D_{ij}\in\{0,\dots,9\}$8, it repeatedly samples without replacement $D_{ij}\in\{0,\dots,9\}$9 times and recalculates

$D_{i,3:4}\in\{00,\dots,99\}$0

For each sample-size level, it summarizes the distribution of these metrics using mean, median, standard deviation, and empirical quantiles. In the RawData/ErrData analysis, except at the full-sample endpoint, each level was repeated 1,000 times (Cao, 5 Jun 2026).

To summarize trend direction, the authors regress Cramér’s $D_{i,3:4}\in\{00,\dots,99\}$1 on log sample size:

$D_{i,3:4}\in\{00,\dots,99\}$2

The same slope logic is applied descriptively to other metrics as well. If $D_{i,3:4}\in\{00,\dots,99\}$3, KL divergence, and $D_{i,3:4}\in\{00,\dots,99\}$4 decrease while $D_{i,3:4}\in\{00,\dots,99\}$5 and median $D_{i,3:4}\in\{00,\dots,99\}$6 increase with sample size, the apparent irregularity is attenuating, consistent with random fluctuation. If deviation metrics remain elevated or trend in the opposite direction, the signal is considered more persistent. This means that FDRS screens not only for static non-uniformity but for non-uniformity that survives progressive accumulation of data (Cao, 5 Jun 2026).

After feature engineering, each dataset becomes a feature vector $D_{i,3:4}\in\{00,\dots,99\}$7. The predictive models evaluated were Random Forest, Elastic-net Logistic Regression, radial-basis-function Support Vector Machine, Isolation Forest, and an integrated ensemble model. Random Forest was used to capture nonlinear feature interactions. Elastic-net Logistic Regression was used as a sparse, interpretable linear model with combined $D_{i,3:4}\in\{00,\dots,99\}$8 and $D_{i,3:4}\in\{00,\dots,99\}$9 penalty, and SVM used an RBF kernel. Isolation Forest was trained mainly on normal-reference patterns as an anomaly detector (Cao, 5 Jun 2026).

The training labels were synthetic. The paper generated three training families: simulated normal datasets from uniform, normal, log-normal, and gamma distributions; simulated digit-preference datasets with inflated frequencies of digits such as 2, 5, 8 or combinations such as 00, 25, 50, 75, 55, 88; and mixed-contamination datasets,

$P$0

with random contamination proportion $P$1. The final integrated risk is given generically as

$P$2

where $P$3 is the supervised-model risk and $P$4 is the Isolation Forest anomaly score. The article does not specify the exact fitted $P$5 used in the final implementation (Cao, 5 Jun 2026).

The ensemble risk score is mapped to five grades: Grade 0, $P$6 — no apparent digit-pattern irregularity; Grade 1, $P$7 to $P$8 — mild irregularity; Grade 2, $P$9 to $V$0 — moderate irregularity; further review recommended; Grade 3, $V$1 to $V$2 — high irregularity; raw records should be checked; and Grade 4, $V$3 — very high irregularity; detailed verification recommended (Cao, 5 Jun 2026).

5. Proof-of-concept validation on RawData and ErrData

The main proof-of-concept comparison uses RawData, an instrument-derived enzymatic absorbance dataset exported from a SpectraMax 190 plate reader, $V$4, and ErrData, a blinded manually simulated absorbance-like dataset, $V$5 (Cao, 5 Jun 2026).

For the single third-decimal-digit analysis, RawData showed $V$6, $V$7, $V$8, $V$9, $j$0, with no significant deviation from uniformity. ErrData showed $j$1, $j$2, $j$3, $j$4, with significant deviation. The paper also notes that ErrData had higher KL divergence than RawData in this single-digit analysis; from the figure, these values are approximately 0.0343 for ErrData versus 0.0157 for RawData (Cao, 5 Jun 2026).

For the joint third-fourth decimal digit 00–99 analysis, RawData showed $j$5, $j$6, $j$7, $j$8, $j$9, $k\in\{0,\dots,9\}$0, whereas ErrData showed $k\in\{0,\dots,9\}$1, $k\in\{0,\dots,9\}$2, $k\in\{0,\dots,9\}$3, $k\in\{0,\dots,9\}$4, $k\in\{0,\dots,9\}$5, $k\in\{0,\dots,9\}$6. Neither joint chi-square test was significant, but ErrData was consistently less uniform by effect size and information metrics: higher $k\in\{0,\dots,9\}$7, lower normalized entropy, higher KL divergence, and more uneven residual heatmap structure. This is exactly why FDRS uses multiple metrics rather than a single $k\in\{0,\dots,9\}$8-value (Cao, 5 Jun 2026).

The progressive subsampling results are central. At small subsample size $k\in\{0,\dots,9\}$9, both datasets behaved similarly because 100 categories are sparse. As sample size grew, the trajectories diverged. For RawData, deviation attenuated toward uniformity: $O_k$0 dropped from 97.6800 to 84.9447, median $O_k$1 rose from 0.5476 to 0.8419, $O_k$2 fell from 0.1809 to 0.0582, $O_k$3 rose from 0.6993 to 0.9628, and $O_k$4 fell from 1.3846 to 0.1715. For ErrData, some sparsity-induced noise still decreased, but the deviation signal persisted relatively more strongly: $O_k$5 rose from 99.3200 to 103.8235, median $O_k$6 fell from 0.5476 to 0.3503, $O_k$7 fell from 0.1825 to 0.0641, $O_k$8 rose from 0.6966 to 0.9485, and $O_k$9 fell from 1.3972 to 0.2371 (Cao, 5 Jun 2026).

This contrast is the paper’s central empirical argument: RawData looks like a dataset whose apparent irregularities wash out with more data; ErrData looks like a dataset with residual structure that remains directionally suspicious. The proof of concept is therefore not only that a manually simulated irregular dataset can deviate from a digit-randomness null, but that the persistence of deviation under progressive accumulation provides additional discriminatory value beyond full-sample significance testing (Cao, 5 Jun 2026).

6. Model performance, benchmark datasets, and graded interpretation

In internal validation, Elastic-net Logistic Regression achieved the highest AUC of 0.98395 and the lowest Brier score of 0.048439, with accuracy 0.923333, sensitivity 0.915, specificity 0.94, and balanced accuracy 0.9275. Random Forest achieved the highest accuracy of 0.926667 and the highest balanced accuracy of 0.935, with specificity 0.960, precision 0.978495, F1 0.943005, AUC 0.9809, and Brier score 0.055658. SVM radial showed accuracy 0.916667, balanced accuracy 0.925, AUC 0.97540, and Brier score 0.056181. Isolation Forest showed accuracy 0.863333, balanced accuracy 0.87, AUC 0.89220, and Brier score 0.206379. The ensemble achieved accuracy 0.913333, sensitivity 0.91, specificity 0.92, balanced accuracy 0.915, AUC 0.97895, and Brier score 0.065783 (Cao, 5 Jun 2026).

For the two primary demonstration datasets, model-specific predictions were as follows. RawData received Random Forest 0.170373, Elastic-net Logistic Regression 0.149674, SVM radial 0.054441, Isolation Forest 0.124020, and ensemble risk score 0.124627, corresponding to Grade 0: no apparent irregularity. ErrData received Random Forest 0.787406, Elastic-net Logistic Regression 0.950056, SVM radial 0.923895, Isolation Forest 0.301684, and ensemble risk score 0.740760, corresponding to Grade 3: high irregularity (Cao, 5 Jun 2026).

External benchmark evidence was then obtained from five independent real-world benchmark datasets divided into RealRawData1–3, datasets without identified public post-publication concerns for the analyzed records, and RealErrData1–2, datasets drawn from articles or figure panels with public data-integrity concerns, corrections, or institutional/journal handling records. Because these datasets differed in reported precision, different decimal positions were used: third-fourth decimal digit combinations for RealRawData1 and RealRawData2, and first-second decimal digit combinations for RealRawData3, RealErrData1, and RealErrData2 (Cao, 5 Jun 2026).

Dataset	Ensemble score	Grade
RealRawData1 ($X_i$00)	0.112644	0
RealRawData2 ($X_i$01)	0.257080	1
RealRawData3 ($X_i$02)	0.270803	1
RealErrData1 ($X_i$03)	0.447823	2
RealErrData2 ($X_i$04)	0.739743	3

The important result is graded concordance rather than perfect separation. All three RealRawData benchmarks were Grade 0 or 1, whereas the two datasets from questioned or institutionally handled publication contexts were Grade 2 or 3. The paper emphasizes that these are not ground-truth labels of misconduct, only external pragmatic categories. Mild irregularity in apparently unchallenged datasets and moderate rather than extreme irregularity in questioned datasets both underline that FDRS is a prioritization aid, not a binary adjudicator (Cao, 5 Jun 2026).

A common misconception is that a high FDRS score is equivalent to fabrication. The framework explicitly rejects that interpretation. Another misconception is that terminal digits should always be uniform. FDRS is domain-sensitive: later decimal digits are only meaningful if the measurement process truly leaves them free to vary. In datasets with fixed rounding, limited instrument resolution, bounded scales, reporting conventions, digital quantization, or natural digit constraints, apparent non-randomness may be entirely benign (Cao, 5 Jun 2026).

7. Relation to broader randomness-screening research, limitations, and future directions

Although FDRS is introduced as a digit-forensic framework for raw numerical research data, the broader literature shows analogous screening logic in physical random-number generation and hardware-variability characterization. Work on nano-intrinsic true random number generation using random telegraphic noise in amorphous SrTiO$X_i$05-based resistive memories links digit quality to measurable nanoscale noise properties, device variability, and readout architecture, and reports concrete validation metrics including entropy, autocorrelation, NIST results, and machine-learning predictability (Kim et al., 2017). A plausible implication is that FDRS belongs to a larger family of screening approaches in which low-bias, low-correlation, high-entropy output must be separated from architecture-induced or process-induced distortion.

A related hardware-health perspective appears in on-line anomaly detection and qualification of random bit streams, which uses NIST Adaptive Proportion and Repetition Count tests, complemented by Monobit and RUNS, to detect gross failures, subtle bias shifts, symbol-level entropy degradation, and time-dependent anomalies without interrupting generation (Caratozzolo et al., 2024). This suggests a methodological parallel: both FDRS and on-line TRNG qualification are multi-metric, persistence-aware screens that stop at the level of irregularity risk rather than root-cause proof.

Further related work on DRAM activation-failure TRNG enrollment, Rowhammer-based device fingerprinting, and DRAM startup/remanence behavior likewise frames screening in terms of process variation, stability, stress sensitivity, and statistical qualification (Kim et al., 2018); (Venugopalan et al., 2023); (Tehranipoor, 2018). These studies are not FDRS papers by name, but they reinforce the general principle that a useful screening model distinguishes intrinsic stochasticity from deterministic structure, and stable fabrication signatures from environmental or architectural confounds.

The limitations of FDRS are explicit. The biggest is training-label realism: the machine-learning models were trained on simulated normal and simulated irregular datasets, not on a large corpus of adjudicated fabricated raw data. Development data were limited to one instrument-derived absorbance dataset and one manually simulated irregular dataset. The method is domain-sensitive, depends on decimal precision, gives dataset-level risk rather than value-level localization, and was externally validated only preliminarily and heterogeneously. Statistical irregularity is not equivalent to fabrication. At most, it is suspiciousness of a specific kind: inconsistency between the observed digit structure and the assumed or referenced measurement process (Cao, 5 Jun 2026).

Future directions proposed for FDRS include building larger authentic raw-data reference banks across multiple experimental platforms; expanding simulations to include weaker and stronger preference patterns, local contamination, repeated-value insertion, copy-paste blocks, excessive rounding, variance compression, over-smoothing, and over-randomized artificial data; and adding external validation cohorts, SHAP-style interpretability, Bayesian risk modeling, likelihood-ratio approaches, and better calibration tailored to specific data-generating mechanisms (Cao, 5 Jun 2026). In that form, FDRS remains best understood as an interpretable, statistical-plus-machine-learning framework for detecting digit-pattern irregularities in raw numerical research data, with its strongest role in prioritization, triage, and follow-up review rather than in adjudication.