1.3 Transfer from observable characteristics

Definition 11 Randomized bi-Lipschitz pushforward

The exposure law is generated from an observable characteristic law by one common stochastic map with bounded slack:

\[ B_i=T(X_i,U_i)=t(X_i)+\delta _i(X_i,U_i),\qquad X_i\sim C_i,\qquad \lVert \delta _i(X_i,U_i)\rVert \le \tau _i, \]

with the common carrier \(t\) bi-Lipschitz, \(\ell \, d(x,y)\le \lVert t(x)-t(y)\rVert \le L\, d(x,y)\) for \(0{\lt}\ell \le L\). The constants \(L\), \(\ell \), and the slack radii \(\tau _i\) are maintained modelling restrictions, not quantities the observed distances identify or the empirical design estimates.

Theorem 12 Upper transport transfer under slack

✓

Under the upper Lipschitz bound and bounded per-endpoint slack, the optimized squared exposure transport cost of two finitely supported clouds is at most \(L^2\) times the optimized squared characteristic cost, plus a slack term carrying the maintained support diameter.

Proof ▼

Push the certificate coupling of the characteristic clouds through \(T\), bound the resulting cost pointwise, and absorb the slack endpoints.

Theorem 13 Lower transport transfer under slack

✓

Under a positive lower Lipschitz constant and bounded per-endpoint slack, the optimized squared exposure transport cost is at least \(\ell ^{-2}\) times the optimized squared characteristic cost, less a slack term carrying the maintained support diameter.

Proof ▼

Symmetric to the upper direction, using injectivity of \(T\) supplied by the lower bound.

Corollary 14 Metric-level bracket

At the level of the metric rather than the squared optimized cost, the slack enters additively and no support diameter appears:

\[ \ell ^{-1}W_2(C_i,C_j)-(\tau _i+\tau _j) \le W_{2,\Gamma }(P_i,P_j) \le L\, W_2(C_i,C_j)+(\tau _i+\tau _j). \]

This sharper form is what the empirical diagnostic uses.

The upper direction is machine-checked directly at this level: two synchronous-coupling legs bound the slack, one contraction leg carries the transmitted laws, and the \(\mathcal P_2(\mathcal H)\) triangle inequality chains them. The lower direction still needs the reverse Lipschitz half, which supplies injectivity of \(T_\Gamma \) on its image, and remains a prose result. The node keeps notready because it asserts both directions.

Corollary 15 Covariance bracket the diagnostic tests

Substituting the metric bracket into the envelope brackets \(\kappa ^{\max }_{ij}\) in observable characteristic distances and the maintained \((L,\ell ,\tau )\). Coverage of this bracket, and the share of unresolved dyads, is the primary reported diagnostic.