Arithmetic Gauge

Identity: A unified framework exploring the cross-ratio as the fundamental projective invariant across physics, mathematics, and computation. The investigation demonstrates that projective invariants — with the cross-ratio as their canonical example — survive the choice between Archimedean ($\mathbb{C}$) and non-Archimedean ($\mathbb{Q}_p$) fields, that tree-to-line projection produces apparent disorder through metric mismatch (quantified by a kurtosis spectrum), and that the Bruhat-Tits building serves as the unified geometric object for invariants across arithmetic gauges.

Core Thesis: The choice of field is a "gauge choice" for projective invariants. Cross-ratios are the canonical PGL(2)-invariants. The Minkowski $?(x)$ is a groupoid isomorphism between $\text{PGL}(2,\mathbb{Z})$ and the Thompson group $F$. Tree geometry, when projected onto the Euclidean line, produces a power-law distance distribution with infinite kurtosis — a precise mechanism for the apparent randomness of number-theoretic sequences.

Key Documents:

• 0.39.md — Final synthesis: "The Arithmetic Gauge"
• 0.28.md — Systematic investigation of genuine convergence signals
• 0.31.md — Formalization of the Projection Principle
• 0.34.md — Bruhat-Tits building as unified geometric object
• 0.38.md — Complete PGL(2,Z) → Thompson group conjugation table

Status: Active investigation. Mathematical framework established. Testable predictions proposed and computationally investigated.