lin-0204

16.4 Localize before pooling

Let \(s=(x,a,g)\) index a prompt, annotator population, or other declared stratum. The localized obstruction is the family

\[ \Omega _{\mathcal U} = \{ Z_s^\top \ell _s:s\in \mathcal U\} . \]

Aggregation can erase precisely the variation that matters.

Proposition 16.4 Opposite-cycle cancellation

Let one population have reciprocal comparison probabilities with log odds \(\ell \) and nonzero circulation. A second population reverses every comparison and therefore has log odds \(-\ell \). Their equal probability mixture has zero pooled log odds on every edge, although neither population is scalarizable.

Proof ▼

If \(H=\sigma (\ell )\), reversal gives \(\sigma (-\ell )=1-H\). The equal mixture is \(1/2\), whose logit is zero, while the two local circulations remain opposite and nonzero.

Design principle

The cover declares where a representation must be valid. Pooling is permitted only after local obstructions and their overlap semantics agree.