lin-0203

16.3 Tangent transport and reward gauge

Suppose \(\ell _{x,\theta }\) is differentiable and a declared perturbation \(\dot\theta \) induces \(\dot\ell _x=J_{\ell _x}(\theta )\dot\theta \). The tangent obstruction is

\[ T\Omega _x(\dot\theta ) = Z_x^\top J_{\ell _x}(\theta )\dot\theta . \]

It distinguishes directions that preserve, create, or repair circulation, even when their base fields agree.

Proposition 16.3 Naturality and gauge

The base and tangent obstructions are invariant under admissible response relabeling. They are also unchanged by \(r_x\mapsto r_x+c_x\mathbf1\).

Proof ▼

Signed edge permutation transports \(B_x\), \(Z_x\), and \(\ell _x\) equivariantly. The resulting permutation matrices cancel in \(Z_x^\top \ell _x\) and its derivative. Gauge invariance follows from \(B_x^\top \mathbf1=0\).

Prompt-common reward offsets are thus quotiented as representational gauge. They cannot change a pairwise decision and should receive no structural credit.