80. A margin that has probability meaning
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Let \(\mathcal L_d=\{(m,v):m\geq\|v\|_2\}\), with mass row \(\lambda=(1,0,\ldots,0)\). A mass preserving invertible matrix \(T\) transports a proposed predictive domain to this cone. Its normalized base in the original coordinates is
Write \(F_a(u)=TB_a(u)T^{-1}\) for the transported operator density. Observations lie in \([-1,1]\) with reference measure \(du/2\). All operator densities in this block are polynomial and all equalities involving integrals use that reference measure. For each action define
The subscript \(\rm sp\) denotes spatial coordinates, not a separate measured signal. This margin is expressed before normalization. It simultaneously controls mass orientation and distance from the curved boundary. It is coordinate dependent, so its uncertainty budget must use the same \(T\).
Proposition 71: Robust oriented margin
Suppose \(\sigma_a(u)\geq\kappa_a(u)\) and an uncertain family has transported errors \(\Delta_a(u)\) satisfying, for every normalized input,
If \(\kappa_a(u)\geq e_{a,0}(u)+e_{a,1}(u)\), every uncertain branch preserves \(\mathcal L_d\). If the uncertain family has exact integrated mass and its initial state belongs to \(K_T\), it defines a valid probability model at every finite horizon. With a strict inequality, the residual slack is at least \(\kappa_a-e_{a,0}-e_{a,1}\).
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Proof
For \(w=F_a(u)(1,v)\) and \(d=\Delta_a(u)(1,v)\),
The right side is nonnegative under the hypothesis. Homogeneity extends the assertion from normalized inputs to the entire cone. The only zero-mass cone element is zero. Integrated mass conservation and successive cone preservation give nonnegative normalized observation laws by the same induction as in Proposition 52. A positive-density branch may be normalized. A zero-density branch is not assigned a posterior. \(\square\)
This is a uniform deterministic guarantee. The inequalities must cover the whole observation interval and the whole normalized domain. A noise scale estimated from a few residuals does not automatically provide those inequalities. For continuous observations the displayed margin is a density margin relative to the declared measure, not a probability assigned to a single observation value.