82. An explicit repair and the price of its slack
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In Lorentz coordinates let
This reference instrument reports a state-independent density and resets to the center. It is a mathematical model modification, not a new observation. Given an estimate \(\widehat F\), define
Proposition 73: Interior repair with an explicit error budget
Suppose a reference family \(F^*\) is valid on \(\mathcal L_d\), and the combined mass and spatial error of \(\widehat F-F^*\) is at most \(d_a(u)\) on normalized inputs. Then the margin of \(\widetilde F_a(u)\) is at least
An additional uncertainty family with combined output error at most \(\nu_a(u)\) is certified whenever
Assume exact integrated mass for the estimate. If its integrated base-norm discrepancy from \(F^*\) is at most \(\varepsilon\), the corrected discrepancy is at most
When the corrected model is valid on the same cone, with the same initial state, its length-\(T\) law under any common policy differs from the reference law by at most
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Proof
The margin of \(\widehat F\) is at least \(-d_a\) by Proposition 71's inequality applied to \(F^*\). Adding the reference changes only output mass, by \(\eta g_a\), while scaling the remaining output by \(1-\eta\). This proves the first bound and the robust implication. Integrated mass is preserved by mixing.
For the error, write
On a normalized input, both \(N\) and \(F^*\) distribute total cone mass one. Their integrated difference has base norm at most two. Integrate the triangle inequality to obtain the stated budget. The final implication is the continuous observation-law bound of Proposition 59. It requires validity of both comparison models on the same cone. \(\square\)
For constant budgets, a sufficient repair weight is \((d+\nu)/(g_{\min}+d)\) when this number is at most one. This is not an optimality claim. The reference example instead fixes a repair weight and verifies the result independently. If an exact model already has a robust structured certificate, adding a reset may be unnecessary and would introduce avoidable bias.
The term \(2\eta\) is deliberately retained. A larger repair creates more room for uncertainty but can change future predictions more. The worst-case law bound can become uninformative at long horizons. Neither positivity nor mixing alone proves long-horizon predictive accuracy.