# g143: NAL Revision = Frechet Mean Under Independence

## Theorem
For two independent evidence sources, NAL revision is the Frechet mean
in the natural parameter space of the Beta exponential family.

## Proof Sketch
1. Beta(a,b) is an exponential family with natural params eta=(psi(a)-psi(a+b), psi(b)-psi(a+b))
2. For independent observations, conjugate update ADDS sufficient statistics: a_post=a1+a2, b_post=b1+b2
3. NAL revision with w=c/(1-c): f_rev=(w1*f1+w2*f2)/(w1+w2), w_rev=w1+w2
4. Since a=wf, b=w(1-f): a_rev=w1*f1+w2*f2=a1+a2, b_rev=w1(1-f1)+w2(1-f2)=b1+b2
5. NAL revision IS conjugate Beta update (proven g71, g128)
6. In exponential families, the conjugate update = Frechet mean in FLAT (natural) coordinates
7. Therefore NAL revision = Frechet mean in natural parameter geometry

## Why They Diverge for 3+ Agents (g140)
With 3+ agents, NAL revision sums ALL weights (assumes full independence).
Frechet mean on the CURVED (Fisher) geometry is conservative.
Divergence = curvature correction for correlated/redundant evidence.

## Conclusion
NAL revision is geometrically optimal IFF evidence is independent.
For unknown dependence, use Frechet mean on curved Beta manifold.