Zero poses a unique challenge for numerical cognition because it denotes absence in cardinal contexts, yet functions as a formal entity in ordinal or interval systems. We examined how adults compare zero to positive single-digit numbers across symbolic, nonsymbolic, and mixed formats. Results replicated classic distance and end effects for positive numbers. However, the cognitive status of zero was found to be strictly format-dependent. In the symbolic format, the distance effect was driven by boundary values (0 and 1) and vanished when they were removed, suggesting symbolic zero functions as a structural anchor that defines the semantic transition from absence to quantity. In contrast, nonsymbolic zero was integrated into a continuous, nonlinear magnitude gradient, following the same psychophysical power-function pattern as other quantities. Furthermore, the [0, 1] pair elicited a unique facilitation in nonsymbolic and mixed formats, but not in the symbolic format. Together, these findings suggest that zero is not merely a point on a scale but a multi-faceted construct: while nonsymbolic zero is processed as a perceptual category of absence, symbolic zero acts as a semantic boundary that redefines the internal structure of the mental number line. However, the cognitive status of zero was found to be strictly format-dependent. In the symbolic format, the distance effect was driven by boundary values (0 and 1) and vanished when they were removed, suggesting symbolic zero functions as a structural anchor that defines the semantic transition from absence to quantity. In contrast, nonsymbolic zero was integrated into a continuous, nonlinear magnitude gradient, following the same psychophysical power-function pattern as other quantities. Furthermore, the [0, 1] pair elicited a unique facilitation in nonsymbolic and mixed formats, but not in the symbolic format. Expanding on these developmental findings, Nieder ( 2016 ) proposed a four-stage framework for understanding how the empty set is integrated as a numerical quantity. Initially, the absence of a stimulus corresponds to inactive neural states without a specific representation. In the second stage, behavioral relevance emerges as ‘absence’ becomes a meaningful category through reinforcement learning, allowing individuals to distinguish ‘something’ from ‘nothing’. The critical third stage involves integrating this categorical representation into the quantitative system, where empty sets acquire numerical meaning and are positioned at the lowest end of the MNL. C . A homogeneous stimulus of two nonsymbolic quantities, that includes nonsymbolic zero. The bunny is the task-relevant stimuli, and the carrot is the irrelevant stimuli Method The task, the raw data and the analysis can be found in the following link: https://osf.io/gkpxa/overview?view_only=0366249443ba48d58fd24661e738042e . Participants Using G*Power 3.1.9.2 (Kang, 2021 ), we calculated the sample size to be 50, for a three-way within-subjects ANOVA with an alpha of 0.05, power of 0.95, and an effect size of 0.15. Accordingly, the experiment included 50 neurotypical adults (31 females, 46 right-handed), aged 18–35 (Mean age = 28 years, SD = 5.3 years). Second, the use of bootstrap methods and robust post-hoc tests ensured that findings were not unduly influenced by data sparsity or unequal variances, and that conclusions about model fit and group differences reflected genuine cognitive effects rather than analytical artifacts. The MATLAB code and the raw data can be found in the data repository (see link at the beginning of the method section). To formally test the nature of the distance effect, one-sample t-tests were conducted on the bootstrapped exponents (b) to determine if they significantly differed from 1 (the value expected under a strictly linear relationship on the original scale). For all six conditions, bootstrap-derived 95% confidence intervals for the linear slope excluded zero, demonstrating robust and significant distance effects in each condition (Table 1 ; Fig. 4 ). Table 1 Summary of linear and power-law fits for each condition: Means and bootstrap 95% confidence intervals for the linear RT-distance slopes and power exponents Condition Mean Linear Slope 95% CI Slope Mean Power Exponent (b) 95% CI (b) t (b vs. 1) p (b vs. As shown in Table 1 , the power function consistently yielded higher R 2 values compared to the linear model. Crucially, the improvement in fit was most pronounced in the symbolic condition with zero, where R 2 increased from.48 (linear) to.72 (power), representing a substantial 50% increase in explained variance. In contrast, improvements in the nonsymbolic conditions were more modest (e.g., from 0.93 to 0.99 for nonsymbolic with zero), suggesting that the power function captures a specific non-linear psychophysical behavior rather than merely benefiting from additional degrees of freedom. Individual slopes were calculated using bootstrap analysis (5,000 iterations) to determine the stability of the distance effect when anchored by these specific values. A summary of these comparisons revealed that the presence of an endpoint significantly facilitated processing speed, particularly when zero was involved (see Table 2 ; Fig. 5 ). Bootstrap analysis revealed shallower distance-effect slopes for zero-anchored pairs in these nonsymbolic and mixed formats, indicating a unique categorical facilitation (Fig. 5 ). Specifically, the substantially higher R 2 values for the power function, most notably in the symbolic condition with zero, where explained variance increased by 50% indicate that zero-related comparisons follow specific non-linear psychophysical patterns. These statistical results suggest that while zero serves as a semantic anchor, its integration into the magnitude system reflects a complex, non-monotonic association rather than a simple linear progression.

een found. This example demonstrates that different laboratories using slightly different stimuli and methods sometimes find very different exponents. Exponents of the power function seem to depend substantially on the experimental setup. Second, some studies found that the power function fits adequately only to averaged data but not to individual data ( averaging effect ). For example, Freides and Phillips ( 1966 ) and Steingrimsson and Luce ( 2006 ) revealed a lack of fit of the power function when applied to individual data. Generally, differences between psychophysical functions seem to emerge when these are fitted on the individual level rather than on the group level (Bernasconi & Seri, 2016 ). Other studies, however, also reported a good fit of the power function when fitted to individual as well as to aggregated data (Algom & Marks, 1984 , 1990 ; Marks & Stevens, 1966 ). Besides the averaging effect , the estimated exponent of the power function strongly depends on the range of stimuli used in the experiments ( range effect ). Engen ( 1956 ) was the first to report that larger ranges of stimulus intensity go along with smaller exponents of the power function. Poulton ( 1968 ) reviewed the previous literature and revealed that 30% of the variance of exponents can be explained by the range of stimuli applied in the different experiments. Fourth, the location of the reference stimulus within the stimulus range influences the resulting power functions ( location effect ). The exponent tends to be larger when the reference stimulus is placed in the center of the range and smaller when it is located closer to one of the extremes of the stimulus set. For example, Engen and Levy ( 1955 ) reported such a location effect for both brightness and weight judgments (for replications, see Ahlström & Baird, 1989 ; Fagot & Pokorny, 1989 ; Pradham & Hoffman, 1963 ). Thus, in the past, severe points of criticism concerning the magnitude estimation method have been raised. In t