The Arithmetic Gauge: Cross-Ratios and Projective Invariants Across Fields

The fine-structure constant admits an exact dimensionless expression as a ratio of two electron length scales: $\alpha = r_e / \bar{\lambda}_C$. This ratio, expressible as a degenerate four-point cross-ratio, exemplifies a broader principle: **projective invariants — with the cross-ratio as their canonical representative — survive the choice of field between Archimedean ($\mathbb{C}$) and non-Archimedean ($\mathbb{Q}_p$) geometries.** We demonstrate that the Minkowski question mark function $?(x)$ is a groupoid isomorphism between $\text{PGL}(2,\mathbb{Z})$ and the Thompson group $F$, that tree-to-line projection produces apparent disorder through metric mismatch (quantified by a kurtosis spectrum ranging from $0$ to $\infty$), and that the Bruhat-Tits building for $\text{PGL}(2,\mathbb{Q}_p)$ — a $(p+1)$-regular tree — serves as the unified geometric object supporting invariants across arithmetic gauges. Farey neighbor distances on the Stern-Brocot tree boundary form a power-law distribution with universal exponent $\alpha \approx 1.484$ and infinite kurtosis. The primarity constraint on the primes reduces this infinite kurtosis to approximately $4.9$, providing a geometric mechanism for the apparent randomness of prime gaps: they are the projection shadow cast by the divisibility tree through a primarity filter onto the Euclidean line.