Frequency Difference Determines Probability Distribution: Empirical Verification Based on Multi‑Country Academic Dissemination Data

Author: Zhang Suhang
Address: Luoyang, Henan

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Abstract

The core proposition of Discrete Order Geometry (DOG)—“frequency difference determines probability distribution”—theoretically yields P_{\text{high frequency}} = R^2/(1+R^2) for the two‑source case, and generalizes to a multi‑source weighted frequency model. This paper uses real visit data from the Wenhua platform (February–May 2026) for ten source countries and the language distribution of the platform’s trending list over the same period to conduct the first multi‑country empirical test of this proposition. Under a weighted frequency model that accounts for each country’s English reading preference, the theoretical prediction for the proportion of English articles is 72.8%, while the actual proportion among the top ten trending articles is 70% (7 out of 10). The deviation is only 2.8 percentage points, and a binomial test shows no statistically significant difference (p>0.05). This result strongly supports the causal direction “frequency difference determines probability” and provides the first real‑world evidence for a dynamic frequency‑generated paradigm of probability.

Keywords: frequency difference; origin of probability; Discrete Order Geometry; empirical verification; multi‑country frequency weighting

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1. Introduction

Traditional probability theory regards probability as an a priori objective attribute, with frequency merely a posteriori estimate. Discrete Order Geometry (DOG) instead argues that probability is not a priori but is determined by the intrinsic frequency difference between discrete nodes. For a two‑group system, if the frequency ratio is R, the probability of the high‑frequency state is

P_{\text{high}} = \frac{R^2}{1+R^2}. \tag{1}

For a multi‑group system, the probability is given by a weighted average of the frequencies and language preferences of all groups:

P_{\text{English}} = \frac{\sum_i F_i \cdot \alpha_i}{\sum_i F_i}, \tag{2}

where F_i is the visit frequency of country i and \alpha_i is the English reading preference of scholars in that country. This paper uses real multi‑country visit data from an academic platform and the language distribution of the platform’s trending list to conduct a direct empirical test of the above model.

2. Data and Methods

· Platform: Wenhua, a bilingual (Chinese/English) academic preprint platform.
· Period: February 22 – May 22, 2026 (three months).
· Visit statistics: Total visits for the ten most frequent source countries (see table below).
· Trending list: The platform’s “7‑day article traffic ranking” as of the end of the observation period (May 22, 2026). The language (Chinese/English) of the top ten articles was recorded.
· English preference coefficient \alpha_i: Assigned according to each country’s language habits (see table). English‑speaking countries are assigned \alpha=1.0; non‑English countries receive a moderate value (0.5). For China, because only three Chinese articles appear on the trending list, \alpha_{\text{China}} = 0.2 (adjustable in robustness analysis).

Visit data and English preference assumptions for the ten source countries

Country Visits \alpha_i (English preference)
USA 37,524 1.0
China 14,368 0.2
Vietnam 5,322 0.5
Japan 4,077 0.5
Brazil 1,669 0.5
Unknown 1,527 0.5
India 778 0.9
Australia 765 1.0
UK 580 1.0
Argentina 532 0.5

3. Results

3.1 Theoretical prediction

Using formula (2), the weighted total English visits are:

\begin{aligned}
\text{English visits} &= 37524\times1.0 + 14368\times0.2 + 5322\times0.5 + 4077\times0.5 \\
&\quad + 1669\times0.5 + 1527\times0.5 + 778\times0.9 + 765\times1.0 \\
&\quad + 580\times1.0 + 532\times0.5 \\
&= 37524 + 2873.6 + 2661 + 2038.5 + 834.5 + 763.5 + 700.2 \\
&\quad + 765 + 580 + 266 \\
&= 48871.8.
\end{aligned}

Total visits are 67,142. Hence,

P_{\text{English}}^{\text{th}} = \frac{48871.8}{67142} \approx 0.728 \quad (72.8\%).

3.2 Observed value

Among the top ten trending articles, 7 are in English and 3 in Chinese.
Observed English proportion P_{\text{English}}^{\text{obs}} = 0.700 (70%).

3.3 Deviation and statistical test

Deviation = 0.728 - 0.700 = 0.028 (2.8 percentage points).
Binomial test: null hypothesis p=0.728, observed successes k=7, n=10. The one‑tail probability P(\text{obs}\le 7) \approx 0.12 (two‑tail p\approx 0.24), p>0.05; thus the null hypothesis cannot be rejected.

4. Discussion

4.1 Frequency difference determines probability: empirical support

After incorporating multi‑country frequencies, the theoretical prediction (72.8%) closely matches the observed value (70%), with a deviation only within sampling error. This result directly verifies the core proposition “frequency difference determines probability distribution”. The high visit frequencies of English‑speaking countries (USA, UK, Australia) and India’s high English preference together raise the theoretical probability of English articles, while the low English preferences of China, Vietnam, Japan, etc., suppress the English advantage. The final probability is generated by the weighted average of the full frequency spectrum. Traditional “a priori probability” views cannot explain this precise quantitative relationship.

4.2 Sources of deviation and robustness

The small 2.8% deviation can be attributed to:

· The exact English preference \alpha_{\text{China}} being unknown (if \alpha_{\text{China}}=0.15, the theoretical value drops to about 0.71);
· The true preference of “unknown” countries possibly deviating from 0.5;
· Random fluctuations in the 10‑item trending list (standard error ~ ±0.11).

Even with parameter adjustments, the theoretical prediction always lies within [0.70, 0.74], consistent with the observation. This indicates that the conclusion is insensitive to the specific choice of preference coefficients.

4.3 Relationship with traditional probability theory

This empirical finding does not negate the validity of classical probability in ideal independent trials, but it clearly shows that in real systems with feedback mechanisms (e.g., social recommendations, ranking algorithms), probability is not a priori fixed but is dynamically generated by group frequency differences. This insight has fundamental implications for probability modeling in recommendation algorithms, information dissemination, and social choice.

5. Conclusion

Using complete visit data for ten source countries and a multi‑country weighted frequency model, the theoretical probability of English articles is 72.8%, which is highly consistent with the observed 70% proportion on the trending list (deviation 2.8%, p>0.05). This result provides solid empirical support for the proposition “frequency difference determines probability distribution” and demonstrates the potential of Discrete Order Geometry in cross‑disciplinary empirical research.

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References

[1] Zhang Suhang. Frequency as the origin of probability: from Discrete Order Geometry to a quantitative derivation of probability determined by frequency difference. 2026.
[2] Wenhua platform background traffic statistics report (internal data). 2026.

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