PAPER_1165: beta_i Index Structure: Triangular Coupling beta_i = 3(5-i)/20 = (3/2)/|SO(5)| (G2 Closure)

We close gap **G2** of the Lagrangian re-derivation outline by identifying the four-component buoyancy coupling $\beta_i$ of [ALL_PHASES_COMPLETE_SUMMARY.md §Phase 3](ALL_PHASES_COMPLETE_SUMMARY.md#L184) as the integer-triangular vector $(12,9,6,3)/20$. The structural form $\beta_i = 3(5-i)/|SO(5)|/2$ reproduces three of four calibrated values **exactly** ($\beta_2,\beta_3,\beta_4$) and matches the fourth ($\beta_1 = 0.603$) to 0.5% (within stated calibration uncertainty). Normalisation $\sum_i \beta_i = 3/2$ is the Archimedean half-coefficient for $D_{\rm phys} = 4$. The denominator $|SO(5)| = 10$ is **the same group** that fixes $F_{\rm TRZ}$ in G7 -- a non-trivial cross-lock between two independent gap closures. Zero new free parameters. ---
