76. What a residual can prove about misspecification
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A certified operator family can still be the wrong model for the observed process. Predictable residuals offer an explicit way to find that mismatch without pretending that internal validity proves empirical adequacy.
Let \(P_{t,j}\) and \(Q_{t,j}\) denote the true and proposed next-observation laws on the same actual history and action. The proposed law must be defined there. Since \(0\leq\phi_{t,j}\leq1\),
Proposition 68: Sequential evidence gives a lower error certificate
On the simultaneous event of Proposition 66,
Thus crossing the boundary rejects zero conditional drift for that rule, with the registered false-rejection budget. Failing to cross supplies no general upper bound on total variation, no rank certificate and no guarantee for untested actions.
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Proof
The triangle inequality gives \(|D_{t,j}|\geq(|R_{t,j}|-b_{t,j})_+\). Also \(|D_{t,j}|\leq\sum_s|m_{s,j}-p_{s,j}|\leq\sum_s\|P_{s,j}-Q_{s,j}\|_{\mathrm{TV}}\). Combine these inequalities. \(\square\)
76.1 Why one aggregate residual can miss conditional errors
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The reference path has binary observations. Its true conditional probability of one is \(4/5\) after a one and \(1/5\) after a zero. A deliberately inadequate model predicts \(1/2\) at every step. Its long-run aggregate frequency can be near one half even though its history-conditioned predictions are wrong.
Three tests are registered before the path is generated. One tests the raw observation. A second tests it only after the preceding observation was one. The third applies the same gate but uses the correct conditional predictor. Gates are functions of past data, not selected after seeing the current measurement.
For the supplied 4,096-step seeded path, the aggregate mean residual is approximately \(0.00732\). The gated mean residual is approximately \(0.14917\), while its time-uniform mean boundary is approximately \(0.05114\). The lower certificate for average conditional total variation is approximately \(0.09803\). The correctly predicted gated residual is approximately \(-0.00303\).
These are illustrative path calculations, not estimated coverage rates. The theorem supplies the probability guarantee. The example supplies an explicit reason to test conditional structure rather than relying only on cancellation-prone aggregate residuals.
The exact reference suite also checks a dependent likelihood-ratio construction on every binary path through depth eight. Its expectation under the declared null is exactly one. This is an additional finite martingale calculation, not a replacement for the general argument or a claim that every possible test has been run.