RST: Routes for Certifying and Constructing Prescribed Information in Variational Autoencoders

Posterior-collapse diagnostics can establish that a VAE uses its latent representation, but they do not determine whether a specified information view crosses a representation/readout interface fixed before training. Reconstruction-Student-Teacher (RST) makes that stronger requirement explicit: R denotes Reconstruction, S the fixed Student witness, and T the frozen Teacher specification. Let $T_X=T(\cdot \mid X)$ be a full-support categorical Teacher, $C\mid X\sim T_X$, $\bar{T}={\mathbb{E}}_X{T}_X$, and let an audited representation $Z$ satisfy $C-X-Z$. With $I_T:=I(C;X)$, $L_{\mathrm{TS}}:=\mathbb{E}\mathrm{KL}(T_X\Vert S(\cdot \mid Z))$, and $G_T:=I_T-L_{\mathrm{TS}}$, we prove $I(C;Z)\ge G_T$ and $L_{\mathrm{TS}}\ge I_T$ for every input-independent $Z\perp X$, with equality attained by the centered RST witness. Thus $G_T>0$ certifies transmission of the declared Teacher information through the specified interface. In the main experiments $Z=u_{\varphi }(X)$ is the deterministic encoder mean, while reconstruction uses $z\sim q_{\varphi }(z\mid x)$. A centered regular-simplex witness turns the certificate into an explicit construction: a minimum-dimensional closed-form Teacher code, its complete affine solution fiber, exact witness-visible/witness-null routing, a margin-energy path, strict function-space departure from the centered input-independent point, certified preservation cylinders, and orthogonal multi-view composition. Five-dataset experiments test boundary crossing, stronger KL pressure, Teacher counterfactuals, witness-null routing, escape, and preservation. On CIFAR-100, learned-head auxiliary (Aux) codes are probe-readable while their predeclared margins are negative. Cross-seed Aux-head reuse falls from $36.57/21.88$ native accuracy to $5.20/0.94$, whereas one fixed RST witness directly reads all five Conv-RST encoders at $43.84/33.58$. The same witness is also certificate-positive on five independently optimized ResVAE RST encoders, with coarse/fine margins $G_c=0.869\pm 0.057$ and $G_f=1.083\pm 0.105$; Conv$\leftrightarrow$Res learned-head reuse is near chance, while a ridge-affine adapter makes all $100/100$ transferred margins positive. These tests validate a functional consequence of prescribed alignment; they do not claim fixed simplex classifiers or compatible representations as new.