The study area corresponds to a mountainous valley system near Visoko, Bosnia and Herzegovina, represented in this analysis as a discrete spatial point dataset (Figure 4, Figure 5). Within a GIS environment, the dataset was defined as a closed universe comprising ten named, cartographically documented summit locations. Only features identifiable in official cartographic sources and satellite imagery were included. No inferred, unnamed, or visually selected terrain features were added.

Summit coordinates were derived from high-resolution LiDAR data and verified against independent cartographic references [3][4] (Figure 3). All summit locations were represented as fixed point features in a GIS point layer and treated as immutable inputs throughout the analysis. (Figure 5) The candidate universe was defined prior to hypothesis testing and was not modified during subsequent analyses to avoid post hoc selection effects.

The study area contains numerous minor topographic highs and local maxima beyond the summits included in the candidate set. Such features vary continuously with elevation threshold, prominence criteria, and scale of analysis. Rather than attempting exhaustive peak detection, which would require arbitrary parameter choices and introduce additional degrees of freedom, the candidate universe was restricted to named and cartographically documented summits that are consistently identifiable across official maps and satellite imagery.

This definition establishes a closed, reproducible spatial universe independent of the geometric hypothesis under evaluation. Inclusion was based solely on external cartographic documentation and not on spatial arrangement, alignment, or hypothesized significance. No unnamed, inferred, or visually selected terrain features were added during analysis.

All inferences drawn in this study are therefore conditional on the defined summit universe. The analysis does not assert uniqueness with respect to all possible local maxima in the landscape, but rather evaluates whether comparable geometric configurations arise within a fixed, independently defined set of spatial features.

Geographic coordinates were converted from latitude-longitude to a local planar coordinate system using the Gauss-Krüger projection (Figure 5). Given the spatial extent of the study area, projection-induced distortion is negligible for distance-based analysis. All spatial computations were performed using planar coordinates to ensure consistency across geometric operations.

Distances were measured in kilometers. For spiral fitting, spatial tolerance was defined as the maximum absolute radial deviation between an observed point and the corresponding location on the tested spiral curve. For clarity, tolerances are reported in meters, while all internal computations are performed using projected planar coordinates.

The geometric hypothesis evaluated in this study is a logarithmic spiral characterized by exponential radial growth with respect to angular displacement. The growth parameter corresponding to the golden ratio was used as the reference value [5][6]. The functional form of the spiral was fixed throughout the analysis.

Spiral placement was governed by explicitly defined constraints rather than by continuous optimization. The spiral was anchored exhaustively at candidate summit locations, with each summit in the evaluated subset permitted, in turn, to serve as the spiral origin. For each anchoring choice, orientation was evaluated systematically by rotating the spiral through the full angular range in discrete increments. No least-squares fitting, gradient minimization, or manual adjustment was performed.

For each admissible anchor-orientation combination, spatial correspondence was evaluated using a tolerance-based criterion. A configuration was classified as a valid fit only if all evaluated points lay within the predefined spatial tolerance of the spiral curve. The configuration yielding the smallest maximum point-to-curve deviation under these constraints was retained for descriptive reporting, but acceptance or rejection depended solely on whether the tolerance criterion was satisfied.

This procedure limits geometric degrees of freedom by replacing parameter optimization with exhaustive evaluation under fixed rules. As a result, correspondence between points and the spiral reflects constrained geometric structure rather than accommodation through flexible curve fitting.

Uniform random Monte Carlo simulation was used as an initial null model to assess whether a spiral-like configuration could arise from unconstrained spatial randomness [9][10]. In each simulation, five points were randomly distributed within the spatial extent of the study area. A logarithmic spiral was anchored at one of the simulated points, and the maximum deviation from the spiral was computed.

Simulations were performed under progressively stricter tolerance thresholds of 0.5 km, 100 m, and 50 m. Between 10,000 and 100,000 realizations were generated depending on the tolerance level (Figure 6, Figure 7).

To account for real-world spatial structure, a constraint-preserving null model was implemented. (Figure 8). Five points were randomly selected from the fixed ten-point universe while preserving their observed spatial locations. Each selected subset was tested against the spiral model using a 50 m tolerance. This procedure maintains terrain-driven spatial structure while removing intentional feature selection [7][8]. A total of 10,000 randomized subsets were evaluated.

Given the small size of the candidate universe, exhaustive enumeration was feasible. All possible five-point subsets were generated, yielding 252 unique combinations. Each subset was tested under identical geometric constraints, with spiral anchoring permitted at any of the five points. Exhaustive evaluation eliminates sampling uncertainty and avoids reliance on probabilistic approximation (Figure 9).

Multiple robustness tests were conducted to evaluate dependence on modeling assumptions:

Anchor-exclusion tests: Each point was excluded in turn from acting as a spiral anchor while all other conditions were held constant.Spiral growth sensitivity: The logarithmic growth parameter was varied ±10% around the reference value, producing 41 tested parameters.Rotation randomization: The spiral was rotated through 360˚ in 1˚ increments.Subset-removal tests: One point was removed at a time from the full dataset, and remaining subsets were evaluated.Positional noise tests: Gaussian noise with standard deviations of 5 m, 10 m, 20 m, and 30 m was added independently to point coordinates (Figures 10-16).

All robustness tests used a fixed spatial tolerance of 50 m unless otherwise noted.

To evaluate method selectivity, several non-equivalent spiral geometries were tested using the same data and tolerance criteria. These included clockwise and counter-clockwise logarithmic spirals, linear spirals (zero growth), expanded spirals (doubled growth), and compressed spirals (halved growth). These tests assess the likelihood of false-positive detections under incompatible geometric models (Figure 17).

Sensitivity to universe definition was assessed using a leave-one-out procedure. Each point was removed from the candidate universe in turn, and all possible five-point subsets of the remaining points were generated and evaluated under identical constraints. This procedure tests whether the detected structure depends on the precise definition of the candidate set (Figure 18).

An information-based comparison was conducted to assess descriptive efficiency. Two representations of the point configuration were evaluated: 1) independent encoding of point coordinates, and 2) representation relative to a logarithmic spiral with residual deviations measured orthogonally. Residual variance was used as a proxy for description compactness, with lower variance indicating greater information compression (Figure 19) [11]-[13].

All analyses address spatial geometry within a GIS framework only. No assumptions are made regarding chronology, construction processes, or cultural attribution. The methods are intended solely to evaluate whether a specified spiral configuration can be reproduced under defined null hypotheses (Figure 20).

Uniform random Monte Carlo simulations were used to evaluate whether a five-point logarithmic spiral configuration could arise under unconstrained spatial randomness. Under a spatial tolerance of 0.5 km, no simulated configuration satisfied the fitting criterion across 10,000 realizations. When the tolerance was reduced to 100 m, no valid configurations were observed across 100,000 simulations. Under the strictest tolerance of 50 m, no valid configurations were observed across 10,000 simulations. In all cases, maximum deviations exceeded the imposed tolerance thresholds.

Before comparison with alternative spatial and geometric configurations, quantitative deviation metrics were computed for the evaluated summit configuration relative to the logarithmic spiral. These metrics serve as descriptive baselines for subsequent null-model and robustness analyses and do not constitute independent statistical tests.

Spatial correspondence between summit locations and the spiral was quantified using point-to-curve deviation, defined as the shortest orthogonal distance from each summit to the spiral curve in projected planar coordinates. For the evaluated configuration, the maximum point-to-curve deviation across all summits was 34 m, with a mean deviation of 21 m.

To summarize overall dispersion, the residual variance of deviations was calculated as 0.0006 km² under the spiral-based representation. For comparison, representing the same summit locations as independent point coordinates yielded a residual variance of 0.018 km², reflecting substantially greater dispersion in the absence of a geometric constraint.

These values characterize the geometric compactness of the evaluated configuration under the specified spiral model and tolerance criteria. They are reported to provide a reference scale for interpreting the deviation distributions obtained under uniform random placement, summit-preserving randomization, exhaustive subset enumeration, and robustness tests presented below.

In the constraint-preserving null model, five points were randomly selected from the fixed ten-point universe while preserving their observed spatial locations. Each subset was evaluated using a 50 m tolerance. Across 10,000 randomized subsets, no configuration satisfied the spiral fitting criterion.

The complete candidate universe consists of ten spatial points, yielding 252 unique five-point subsets. Each subset was evaluated exhaustively under identical geometric constraints, with spiral anchoring permitted at any of the five points. None of the 252 subsets satisfied the fitting criterion within a 50 m tolerance.

Anchor-exclusion tests were conducted to assess sensitivity to the choice of spiral origin. Each point was excluded in turn as a potential anchor, while all other parameters were held constant. In every exclusion scenario, the remaining configuration failed to satisfy the spiral criterion.

The logarithmic spiral growth parameter was varied ±10% around the reference value, producing 41 tested parameter values. No alternative growth parameter yielded a valid configuration under otherwise identical conditions.

Orientation dependence was evaluated by rotating the spiral through a full 360˚ in 1˚ increments. For each orientation, anchoring was allowed at any point in the tested subset. No orientation satisfied the 50 m tolerance criterion.

One point was removed at a time from the full dataset, and the remaining nine points were evaluated for spiral coherence under identical constraints. No reduced dataset produced a valid spiral configuration.

Sensitivity to positional uncertainty was examined by perturbing point coordinates using Gaussian noise with standard deviations of 5 m, 10 m, 20 m, and 30 m. Multiple realizations were generated for each noise level. At all tested noise scales, no perturbed configuration satisfied the spiral fitting criterion.

Non-equivalent spiral geometries were evaluated using the same spatial data and tolerance criteria. These included clockwise and counter-clockwise logarithmic spirals, linear spirals with zero growth, expanded spirals with doubled growth, and compressed spirals with halved growth. None of the tested geometries produced a valid configuration.

Sensitivity to the definition of the candidate universe was evaluated using a leave-one-out procedure. Each point was removed from the universe in turn, and all five-point subsets of the remaining points were generated and tested. No valid spiral configuration was detected in any reduced universe.

An information-based comparison was conducted to assess the descriptive efficiency of representing summit locations relative to a single logarithmic spiral versus encoding each point independently. Two representations of the same spatial configuration were evaluated: 1) independent encoding of summit coordinates in planar space, and 2) encoding relative to the tested logarithmic spiral with residual deviations measured orthogonally.

Under independent point representation, the residual variance of summit locations was 0.018 km². When the same locations were represented relative to the spiral model, residual variance was reduced to 0.0006 km², reflecting a substantial reduction in dispersion under the geometric constraint.

These values indicate greater descriptive compactness of the spiral-based representation within the defined analytical framework. This comparison is descriptive and does not implement a formal minimum description length optimization or penalize model complexity. Instead, residual variance is used as a proxy for relative information compression under a fixed geometric hypothesis.

Across uniform and constraint-preserving null models, exhaustive enumeration, robustness tests, negative-control geometries, and information-based analysis, no alternative configuration reproduced the tested spiral geometry under identical constraints. This convergence of independent evaluation procedures indicates that the detected correspondence is not attributable to random placement, generic terrain structure, parameter flexibility, or alternative spiral formulations within the defined summit universe.

Identifying geometric structure in sparse spatial point datasets presents well-recognized methodological challenges in GIS and spatial analysis [11]-[13]. Flexible geometric forms, such as logarithmic spirals, permit wide variation in anchoring, orientation, and scale, increasing the risk that apparent alignments arise from parameter freedom rather than underlying spatial processes. Consequently, visual correspondence or isolated curve-fitting exercises provide insufficient evidence for constrained spatial organization.

The analysis presented here is explicitly framed as an evaluation of spatial geometry under defined assumptions. The initial identification of a candidate spiral configuration was exploratory and not statistically independent of the data. The subsequent analyses do not transform this exploratory observation into a confirmatory claim. Instead, they assess whether the configuration persists under systematic falsification using explicitly defined null models and robustness tests. This distinction is critical for avoiding overinterpretation of geometric patterns in GIS datasets.

A key outcome of the analysis is the convergence of results across multiple, non-redundant null models. Uniform random simulations demonstrate that the tested configuration does not arise under unconstrained spatial randomness. Constraint-preserving randomization further indicates that the configuration is not a generic consequence of terrain-driven spatial structure, as represented by the fixed-point dataset. Exhaustive enumeration eliminates ambiguity associated with stochastic sampling and confirms that no alternative subset within the candidate universe satisfies the same geometric constraints.

Robustness tests addressing anchor selection, parameter variation, rotation, subset removal, and positional uncertainty consistently eliminate the configuration when modeling assumptions are relaxed. Negative-control geometries further demonstrate that the detection procedure does not produce false-positive results under incompatible spiral forms. Together, these tests indicate that the observed configuration is highly constrained under the specific geometric and tolerance criteria applied.

The evaluated configuration exhibits pronounced sensitivity to perturbations in model assumptions, including positional noise, parameter variation, rotation, and subset modification. In spatial analysis, such sensitivity is often associated with overfitting when geometric correspondence arises from excessive parameter freedom or post hoc optimization. It is therefore important to distinguish explicitly between these cases.

Overfitting typically occurs when a flexible geometric form is adjusted to accommodate data through unconstrained optimization. In such cases, correspondence persists across a range of parameter values, anchor choices, or orientations because the model can absorb variation without structural loss. Apparent robustness under noise or parameter change is, in this context, a hallmark of geometric accommodation rather than meaningful structure.

The configuration evaluated here differs in a fundamental way. Correspondence arises only under a narrow and explicitly defined set of geometric constraints, and it collapses when those constraints are relaxed. The analysis deliberately restricts degrees of freedom by fixing the functional form, limiting anchoring options to candidate summits, and evaluating orientation systematically rather than optimizing it. Under these conditions, sensitivity to perturbation reflects the absence of compensatory flexibility rather than the presence of overfitting.

Importantly, describing the configuration as constrained does not imply intentional design, generative mechanisms, or causal processes. It indicates only that, within the defined analytical framework, correspondence is not a generic outcome of curve fitting and does not persist under relaxed assumptions. The observed fragility therefore delineates the boundary conditions under which the configuration exists, rather than serving as evidence for robustness or origin.

This distinction is central to evaluating a posteriori geometric hypotheses. The objective is not to identify patterns that are stable under broad parameter freedom, but to determine whether an observed correspondence survives explicit falsification under constrained alternatives. In this context, sensitivity to perturbation is an expected and informative outcome rather than a methodological failure.

The information-compression analysis provides a complementary, non-probabilistic perspective on the spatial structure of the dataset. Representing point locations relative to a single logarithmic spiral yields lower residual variance than independent point encoding, indicating greater descriptive compactness under the tested geometric model. This result suggests that the spiral captures a portion of the spatial regularity present in the data.

However, this finding should be interpreted cautiously. The analysis does not implement a formal minimum description length optimization, nor does it evaluate alternative low-parameter geometric models that might achieve comparable compression. Instead, the result supports the conclusion that the tested spiral provides a constrained summary of the observed configuration within the defined modeling framework.

Beyond the specific case study, the analytical framework demonstrated here has broader relevance for GIS research. The combination of constraint-preserving null models, exhaustive enumeration, robustness testing, and information-based evaluation provides a transparent approach to assessing geometric hypotheses in small spatial point sets. These methods are transferable to other applications where visual pattern recognition risks outpacing quantitative validation, including archaeological site alignment studies, ecological patch configuration analysis, urban landmark pattern assessment, and geological fracture or fault-line geometry evaluation.

Importantly, the framework emphasizes falsification over confirmation. By explicitly defining the conditions under which a hypothesized pattern fails, the approach reduces the likelihood of overfitting and post hoc interpretation. This perspective aligns with best practices in GIS-based spatial analysis and supports reproducible hypothesis testing in complex geographic datasets. While exhaustive enumeration is feasible only for limited candidate universes, the same logic can be extended to larger datasets through Monte Carlo subset sampling or constraint-based pruning strategies.

The author conceived the study, defined the research questions, designed and performed the spatial and statistical analyses, interpreted the results, and wrote the manuscript.

This research received no external funding.

Not applicable. This study does not involve human participants or animals.

Not applicable.

All data used in this study are derived from publicly available LiDAR datasets and official cartographic sources. Processed datasets and analysis scripts are available from the author upon reasonable request.

The study was conducted in accordance with applicable ethical standards. No protected cultural heritage sites were disturbed, and no excavation or physical modification of the landscape was undertaken as part of this research.

Artificial intelligence-based tools were used to assist with language refinement, structural editing, and formatting during manuscript preparation. All scientific content, data analysis, methodological design, interpretation of results, and conclusions are the sole responsibility of the author. The use of AI tools did not influence the analytical outcomes or introduce new scientific claims.