1. Introduction

Signature verification is one of the fundamental pillars of handwriting expertise and forensic document analysis. The growing need for objective, reproducible methods based on quantifiable metrics has driven the development of technological tools capable of complementing the expert’s judgment. In this context, graphic proportionality—understood as the mathematical relationship between the dimensions and external topology of a signature—has become a parameter of particular interest for evaluating similarities and differences between handwritten strokes.

Fibonacci Digital Modeller (FDM), integrated into the NEGA software, is presented as an innovative tool based on a mathematical model derived from the Fibonacci sequence. Its main function is to analyze the proportionality between two signatures, quantifying the degree of similarity using a percentage index. This approach reduces the subjectivity inherent in traditional visual comparison, providing an additional objective criterion.

The purpose of this study is to evaluate the effectiveness of FDM in different signature comparison scenarios, using a controlled scientific test to determine its reliability, internal consistency, and expert usefulness.

2. Goals

2.1. General Goal

Validate the performance of FDM as a proportional analysis tool for signature verification in the forensic field.

2.2. Specific goals

1. Evaluate FDM’s ability to identify signatures belonging to the same authorship.
2. Analyze their behavior when faced with identical signatures.
3. Determine its effectiveness in discriminating between signatures from different authors.
4. Examine your response to indeterminate cases resulting from evolutionary variations in the signature.

3. Methodology

3.1. Tool used

The study was conducted using the Fibonacci Digital Modeller (FDM) with new improved EVO 2 calculation module, available from version NEGA 6 v3.x.x onwards. This new evolution of the tool has significant improvements over previous versions. The FDM analyzes the external topology of the signature using a mathematical model based on proportions derived from the Fibonacci sequence, generating a percentage index that expresses the degree of proportional difference between two signatures.

3.2. Mathematical basis

The FDM method is based on the idea that certain graphic elements—such as lines, proportions, curvatures, or intensity profiles—maintain stable geometric relationships when they come from the same origin. To evaluate this stability, FDM:

• Extracts normalized numerical series from characteristic points in each image.
• Compare these series between a KNOWN image (reference) and a QUESTIONED image (to be analyzed).
• Calculate differences, gradients, proportions, and measures of dispersion between both series.
• Integrate these values into three main indicators:
  • Geometric percentage (IdentityScore)
  • Confidence percentage (ConfidenceAdjusted)
  • Final weighted score (FinalScore)

The system uses descriptive statistics, variability analysis, maximum normalization, and progressive penalty techniques to avoid false positives.

3.3. % Accuracy  (Geometric Identity Percentage)

The accuracy percentage is the geometric identity, represents the direct & fisical coincidence between the shapes of both numerical series.

3.3.1 Scientific principles used

• Normalization by absolute maximum to eliminate scale differences.
• Calculation of the point-to-point difference between both charts.
• Obtaining:
  • mean difference (meanDiff),
  • maximum difference,
  • relative rank,
  • median,
  • MAD (Median Absolute Deviation).

3.3.2 Scientific interpretation

The geometric percentage is obtained by applying a linear transformation to the mean difference:

“IdentityScore = (1 – meanDiff) × 100”

This means that the smaller the geometric difference between the two curves, the greater the coincidence.

This indicator is purely geometric, without external weightings.

3.4. % Trust (Identity Score)

The trust percentage assesses the statistical stability of the geometric match. It does not measure the similarity itself, but rather the reliability that this similarity is consistent and not the result of chance or noise.

3.4.1 Scientific variables used

• Average gradient between consecutive differences (local stability).
• MAD (robustness against outliers).
• Maximum difference (peak sensitivity).
• Relationship between gradient and mean difference (structural coherence).
• Relative standard deviation of percentages (proportional variability).

3.4.2 General method

1. Three normalized subscales between 0 and 1 are calculated.:
   • gradient stability,
   • statistical robustness,
   • absence of extreme peaks.
2. They are combined using a weighted average.
3. Progressive penalties apply when:
   • there are anomalous peaks,
   • the average difference is high,
   • internal variability is inconsistent.

3.4.3 Scientific interpretation

The confidence percentage is an indicator of the quality of the match, not of similarity. A match may be geometrically high but statistically unreliable.

3.5. % Fibonacci Similarity (Final Score)

The final result includes:

• geometric similarity (Identity Adjusted),
• statistical confidence (Confidence Adjusted),
• a system of non-linear penalties,
• a dynamic weight that regulates how much trust influences identity.

3.5.1 Applied scientific principles

• Adaptive weighting according to:
  • gradient,
  • average difference,
  • dominance among indicators,
  • overall consistency.
• Exponential penalties to avoid false positives.
• Attenuation in areas of low reliability.
• Global scaling to keep the result within an interpretable range.

3.5.2 Scientific interpretation

The final result is not a simple average, but an smart fusion of geometry and statistics, designed to:

• reward consistent matches,
• penalize doubtful matches,
• prevent a single parameter from dominating the result.

3.6. Automatic detection of perfect matches (TRACING mode)

The system incorporates a mechanism to detect near-identical matches:

“The system activates TRACING mode when all discrepancy indicators (mean difference, maximum difference, mean gradient, and MAD) are simultaneously below strict thresholds. This condition indicates that the two curves exhibit an almost perfect agreement in both geometric and statistical terms. TRACING mode allows exceptional matches to be recognized without compromising the integrity of the system or yielding trivial results.”

3.6.1 Average geometric difference:

• Meaning: represents the average discrepancy between both normalized series. A low value implies a high structural coincidence.
• Scientific definition: it is the arithmetic mean of the point-to-point differences between the two curves compared, Questioned and Known, after they have been normalized.
• Interpretation: quantifies the degree of overall similarity between the two shapes. Low values indicate that the two curves follow very similar trajectories.

3.6.2 Maximum point difference:

• Meaning: identifies the greatest local mismatch between the two curves, allowing anomalies or peaks that are inconsistent with a real match to be detected.
• Scientific definition: It is the highest value of discrepancy between both curves at any point of comparison.
• Interpretation: Detects peaks or sudden deviations that could indicate noise, distortion, or local mismatch.

3.6.3 Average difference gradient:

• Meaning: evaluates the local stability of the comparison. Low values indicate that the discrepancy between the two curves evolves smoothly and consistently, reinforcing the reliability of the pairing.
• Scientific definition: It is the average gradient (change between consecutive points) of the series of differences between both curves.
• Interpretation: Measures the local stability of the match. If the gradient is low, it means that the difference between the two curves changes smoothly, which is characteristic of genuinely matching patterns.

3.6.4 Median absolute deviation (MAD):

• Meaning: Quantifies the dispersion of differences in a robust manner against outliers. A low MAD indicates that the match is consistent and does not depend on isolated points.
• Scientific definition: MAD (Median Absolute Deviation) is a robust measure of dispersion, less sensitive to outliers than standard deviation.
• Interpretation: Evaluates the internal variability of the differences. A low MAD indicates that most discrepancies are concentrated around a small value, suggesting a stable match unaffected by noise.

In this mode:

• it is recognized that the match is practically perfect,
• but avoids giving a direct 100%,
• applying a special scaling to maintain the statistical integrity of the system.

This protects the method from manipulation or trivial coincidences.

3.7. Scientific reliability of the method used

The method used in the FDM tool is reliable because:

• uses robust statistics (MAD, medians, normalization),
• applies non-linear penalties that reduce false positives,
• combines multiple independent indicators,
• incorporates automatic detection of exact matches,
• maintains complete traceability of all intermediate values,
• generates reproducible reports.

Furthermore, the system does not rely on a single parameter, but rather on an ecosystem of complementary metrics, making it resistant to noise, lighting variations, scaling, and minor deformations.

3.8. Samples analyzed

Three test groups were selected:

1. Signatures from the same author: Six pairs KNOWN/QUEST (KNOWN-1 to KNOWN-6 / QUEST -1 to QUEST -6).
2. Forgery by tracing: A pair of matching signatures(KNOWN‑7 / QUEST ‑7).
3. Signatures from different authors: A questionable signature (QUEST-9) compared with eight unquestionable signatures (KNOWN-1 to KNOWN-8).
4. Undetermined case an additional pair: (KNOWN-9 / QUEST-10) with evolutionary changes in the signature.

3.9. Interpretation criteria

The FDM classifies the results into three ranges:

• ■ 0% – 60% → Clear divergence (different authorship or tracing).
• ■ 60% – 80% → Undetermined area (possible evolutionary changes).
• ■ 80% – 100% → Proportional match (same author).

4. How to properly prepare signatures for FDM

5. Tips for taking photos that we need to compare

 5.1. Document upload

1. Upload the document with the questioned signature using Button 1 (red LED).
2. Upload the document with the known signature using Button 2 (green LED).

5.2. Document calibration (optional but recommended)

Although the FDM tool does not require calibration to operate, it is recommended to perform calibration for the following reasons:

• Ensure accurate measurements in NEGA.
• Ensure that the images obtained are valid for subsequent analysis in the SPP tool.

Calibration procedure:

1. Start calibration with the known document.
2. Repeat the process with the questioned document.
3. Use forensic measurement patterns placed prior to digitization (camera or scanner).
4. In the absence of forensic patterns:
   • Select a measurable element from the document or signature.
   • Get your actual measurement.
   • Calibrate using the NEGA Dimension tool.
5. Do not save the generated dimensions, as they are used only for calibration.

5.3. Activation of collation mode

1. Activate collation mode to view both documents transparently, superimposed on each other.
2. Adjust each image using the individual zoom until you obtain a suitable working size.
3. Match the approximate size of both images.
   • An exact match is not necessary, as FDM works by proportions.

1.  5.4. Obtaining clippings

1. Select the Crop tool.
2. Perform a simultaneous cut of both signatures in comparison mode.
3. NEGA will generate two separate, perfectly aligned and centered cuts, automatically assigning them to:
   • Questioned signature button.
   • Known signature button.

5.5. Adjustment on the Negatoscope

1. Activate both images on the main screen of the Negatoscope.
2. SIf necessary, adjust the size using the DUAL zoom, which modifies both images simultaneously.
3. The DUAL zoom is only available in comparison mode and is activated by pressing the cyan blue LED next to the zoom control.

5.6. Verification and adjustment of inclination

1. First analyze the KNOWN signature.:
   • Verify that the baseline of the writing does not show any abnormal slant.
   • Consider that excessive slanting may be due to the signer turning the paper while signing.
2. Analyze the questioned signature:
   • Assess the slope of your baseline.
   • Use the Rotate tool to align it with the undoubted one..
3. This adjustment is critical to avoid:
   • That FDM interprets strokes in incorrect quadrants due to improper rotation.
   • Variations in the accuracy of proportional analysis.
4. Then, with comparison mode disabled, we will go to the cropped image Questioned image and take the photo by pressing directly on the red button in photo mode.
5. We will do the same with the cropped image Known using the red button in photo mode.

6. Audit test of signatures from the same hand

6.1 Signature Comparison KNOWN-1 / QUEST-1

6.2 Signature Comparison KNOWN-2 / QUEST-2

6.3 Signature Comparison KNOWN-3 / QUEST-3

6.4 Signature Comparison KNOWN-4 / QUEST-4

6.5 Signature Comparison KNOWN-5 / QUEST-5

6.6 Signature Comparison KNOWN-6 / QUEST-6

7. Audit test of matching signatures by tracing

7.1 Signature Comparison KNOWN-7 / QUEST-7

8. Audit test of signatures from different hands

8.1 Signature Comparison KNOWN-1 / QUEST-9

8.2 Signature Comparison KNOWN-2 / QUEST-9

8.3 Signature Comparison KNOWN-3 / QUEST-9

8.4 Signature Comparison KNOWN-4 / QUEST-9

8.5 Signature Comparison KNOWN-5 / QUEST-9

8.6 Signature Comparison KNOWN-6 / QUEST-9

8.7 Signature Comparison KNOWN-7 / QUEST-9

8.8 Signature Comparison KNOWN-8 / QUEST-9

8.9 Signature Comparison KNOWN-9 / QUEST-10

9. Results

10. Discussion

11. Final Conclusions