A Geometric Scaling Law for Cosmic Distances

This preprint presents a semi-refined numerical exploration that builds upon and synthesizes findings from two of our previous working papers. It is crucial to emphasize from the outset that this work does not claim to establish definitive physical laws, nor does it propose a new cosmological mechanism. > Instead, this document represents a modest, independent effort to report an intriguing empirical pattern. By taking two fixed constants directly from M-theory literature—the dimension of the exceptional Lie group $G_2$ (14) and the moduli dimension of a specific Calabi-Yau orbifold (41)—we observe that a simple Fibonacci-like recurrence generates a fixed 15-circle system. When normalized, these constants align with real astronomical distances across thousands, millions, and billions of light-years with a mean error of 9.87%, statistically outperforming 99.93% of randomly generated fixed sets ($p = 0.0007$). Additionally, the geometric spiral constructed from these seeds exhibits an unexpected convergence to the universal percolation threshold $\ln(3)$. We explicitly present these findings not as a completed theory, but as a series of structured numerical observations and open questions. It is our hope that this humble contribution invites scrutiny, discussion, and potential theoretical interpretation from the wider academic community to determine whether these alignments are merely profound geometric coincidences or hints of an underlying physical framework.