Golden Ratio Scaling in the Arithmetic Density of Non-trivial Tate–Shafarevich Groups of Elliptic Curves over ℚ Author: Elias Oulad Brahim (@Cloudhabil) Date: January 25, 2026 Abstract We investigate the incidence of non-trivial Tate–Shafarevich groups (E/Q) in families of elliptic curves ordered by conductor. Using an empirically validated dataset spanning conductors N 1, 499, 999, we study the conditional probability that | (E/Q) | > 1 as a function of N. We find that the observed density is well-approximated by a power law of the form P\! (| (E/Q) |>1 N) C N^. A log-log regression yields an estimated exponent 0. 2584 with R² = 0. 91. This estimate is closer to the golden-ratio based hypothesis = () 2 0. 2406, = 1+52, than to alternative benchmarks such as (2) /2 or Euler–Mascheroni derived constants, under the deviation metrics considered. In rank-stratified analyses, the empirical prevalence of ||>1 differs sharply by analytic rank: rank 0 curves exhibit substantially higher incidence than rank 1 curves, consistent with the distinct analytic BSD terms present when L (E, 1) 0. Finally, summary statistics of sampled L-function zero data (skewness 1. 91, kurtosis 5. 43) show pronounced non-Gaussian behavior and do not align with the specific Tracy–Widom skewness baseline (0. 29) under the reported comparison. Interpreted cautiously, these findings support an arithmetic density model with a phi-linked scaling exponent and provide a reproducible statistical target for further study across databases and curve orderings. Technical Summary Dataset and Verification A total of 3. 06M curves were processed. Analytic BSD computations were performed for 1. 17M rank-0 curves, with numerical agreement reported to 8 decimal places for the BSD quantity used in the inference. Model Comparison A previously explored mapping from elliptic curve invariants to a fluid-dynamics Reynolds-number proxy is not supported by the data (R² = 0. 05). In contrast, the arithmetic density model based on conductor scaling yields R² = 0. 91 under the stated regression procedure. Rank Stratification The incidence of ||>1 shows a pronounced disparity by rank: Rank 0: 19. 04% incidence (reported sample size N = 1. 82M curves). Rank 1: 1. 34% incidence (reported sample size N = 1. 11M curves). This corresponds to an approximate 14-fold difference in incidence, consistent with the distinct regulator and L (E, 1) structure in the rank-0 analytic BSD setting. Random Matrix Comparison For the reported zero-statistics sample, skewness 1. 91 and kurtosis 5. 43 indicate strongly non-normal behavior and diverge from the cited Tracy–Widom skewness reference value (0. 29) in this comparison. Popular Summary Elliptic curves underpin modern cryptography, yet some of their deepest arithmetic invariants, especially the Tate–Shafarevich group, are difficult to predict. This study reports evidence that the frequency of non-trivial values in large curve families grows with conductor according to a simple scaling law. The estimated scaling exponent is empirically close to () /2, where is the golden ratio. While any interpretive connection to broader “phi-based” frameworks remains speculative, the statistical regularity itself is concrete and testable. Conclusions and Key Findings Scaling law (Brahim’s Theorem, empirical form). P (||>1 N) C N^, 0. 2584, \ R² = 0. 91. Phi-linked exponent hypothesis. The data are more consistent with = () /2 0. 2406 than with (2) /2 or Euler–Mascheroni derived alternatives under the stated deviation comparisons. Rank dependence. Rank 0 curves show much higher incidence of non-trivial than rank 1 curves, approximately 14× in the reported aggregates. Rejected auxiliary hypothesis. A Reynolds-number style proxy model is not supported (R² = 0. 05) relative to the arithmetic density scaling model. Zero-statistics signature. The reported skewness and kurtosis values suggest arithmetic structure not captured by the particular Tracy–Widom baseline used in the comparison. Reproducibility Statement These results can be replicated using the Cremona database under the following workflow: Filter elliptic curves with conductor N 1 as a function of N, using a specified binning or smoothing scheme. Fit a log-log linear regression to estimate in P (||>1 N) C N^, and report goodness-of-fit metrics, including R², confidence intervals, and sensitivity to binning choices.
Elias Oulad Brahim (Fri,) studied this question.