19. A single propagation law for typed defects
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A reconstruction error, an operator-fit error and a refinement error have different origins. Once each has been translated into an error in the state used by the next operation, their propagation can be analyzed by the same identity.
Allow the representation to change with time. Let \(\gamma_t:X\to Z_t\) be the declared representation at step \(t\), with a norm on each \(Z_t\). Let \(x_{t+1}=F_t(x_t)\) be the true transition and let \(\widehat f_t:Z_t\to Z_{t+1}\) be the implemented step. Its defect is
This definition already includes a representation switch. It is not restricted to one fixed feature vector.
Proposition 18: Compositional defect propagation
Suppose \(\widehat f_t\) is \(\lambda_t\)-Lipschitz on a domain containing both \(\gamma_t(x_t)\) and the predicted state \(\widehat z_t\). Suppose \(\|r_t(x_t)\|\leq\delta_t\) and \(\widehat z_{t+1}=\widehat f_t(\widehat z_t)\). Then, with \(e_t=\|\gamma_t(x_t)-\widehat z_t\|\),
Empty products are one. If the output \(h_t(x)\) has a decoder \(H_t\) with Lipschitz constant \(\kappa_t\) and \(\|h_t(x_t)-H_t(\gamma_t(x_t))\|\leq\eta_t\), then
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Proof
Add and subtract \(\widehat f_t(\gamma_t(x_t))\) to obtain \(e_{t+1}\leq\delta_t+\lambda_te_t\). Repeated substitution yields the product-sum formula. Add and subtract \(H_t(\gamma_t(x_t))\) for the output inequality. \(\square\)
For a fixed linear representation \(R\) and an estimated transition \(\widehat B_u\), the defect is \((RL_u-\widehat B_uR)x\). For a reference reduced operator \(B_u\), it splits exactly as
The first term is representational nonclosure. The second is operator-estimation error. A fit statistic for the second cannot certify that the first vanishes. Proposition 16 supplies a conditional bound on the second in its declared fixed-space setting.
If \(\lambda_t\leq\rho<1\) and \(\delta_t\leq\delta\), then
This is a genuinely recursive prediction bound. It is not merely a one-step comparison made using the true history at every step. If the Lipschitz constants exceed one, the same theorem remains valid but can permit error growth.
19.1 Stability under changing operations
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For linear candidate operators, one sufficient way to obtain a common contraction norm is to exhibit a positive-definite matrix \(M\) and \(0\leq\rho<1\) with
Then \(\|z\|_M=(z^\top Mz)^{1/2}\) gives \(\|\widehat B_uz\|_M\leq\rho\|z\|_M\) for all actions, directly from the displayed inequality. Proposition 18 therefore applies under arbitrary switching among them. This is the common quadratic Lyapunov construction from control theory [R8], used here as an explicit certificate rather than inferred from a list of eigenvalues.
Individual spectral stability does not suffice. The matrices
are both nilpotent, but \(BA=\operatorname{diag}(0,4)\). Alternating them amplifies an appropriate state by four every two steps. Even a single stable nonnormal matrix can produce large transient amplification in a specified norm.
If a learned policy chooses actions from predicted states while the reference process uses actions chosen from true states, the two action sequences may differ. Proposition 18 then needs a bound for that action mismatch, or a Lipschitz bound for the full closed-loop map. The result is not an automatic control-policy guarantee.