Computational Visualization of Symmetric Prime Pair Distributions Around Even Integers
This paper develops a substantially expanded computational and visual study of symmetric prime-pair distributions around even integers. For an even integer $2n$, a symmetric prime pair $(p_1,p_2)$ satisfies $p_1+p_2=2n$, and the nearest symmetric distance is $d_n=\min{k\geq0:n-k\ \text{and}\ n+k\ \text{are prime}}$. The normalized quantity $R_n=d_n/n$, introduced by Daniel and Ogunsola, measures the relative displacement of the nearest Goldbach pair from the midpoint. An exhaustive deterministic computation is carried out for every midpoint $2\leq n\leq500{,}000$, corresponding to all $499{,}999$ even integers $4\leq2n\leq10^6$. The analysis combines exact enumeration, record-value analysis, logarithmic binning, empirical quantiles, frequency distributions and asymptotic heuristics. The largest observed ratio is $R_{22}=9/22\approx0.4090909$, attained at $2n=44$, and no later value in the tested range exceeds it. Scale-dependent means, medians and upper quantiles decline sharply, while the running maximum stabilizes at $9/22$. Mathematical propositions establish elementary structural properties of $d_n$ and $R_n$, and a Hardy--Littlewood/Cramér-type heuristic predicts a typical scale $d_n$ of polylogarithmic order and hence $R_n\to0$ heuristically. The findings provide stronger finite-range evidence for boundedness and relative decay, but they neither prove Goldbach's conjecture nor establish a universal upper bound.