Certified selective predictors attain whatever coverage they attain; operators impose an automation floor: answer at least a $β$-fraction of shifted target traffic with at most an $α$-fraction of answers wrong. Under bounded-ratio covariate shift we prove the Floor Certification Map: once that floor must be certified alongside the selection-conditioned risk $α$, certification acquires a feasibility frontier and a two-resource complexity map, additive up to constants: risk in labeled source, the floor in unlabeled target samples. The rates are local, needing a regular frontier margin, slack below the local-regime threshold, and lattice conditions: pre-registered with a lattice margin for the upper bounds, compatible per-slack for the lower. The displayed split is the operational route; oracle weights also allow a labeled-source floor estimate. Three model-tagged results: a lower bound (Model-B), a matching oracle-weight upper bound (Model-A), and an implementable upper bound (Model-B') valid under a pre-registered exact stratified-shift model with nuisance cost priced explicitly. The match is across these models rather than a single-model minimax theorem, and necessarily so: over the full bounded-ratio class no unknown-weight procedure matches at any sample size (Model-B is inconsistent, witnessed at $α=β=1/2$). The nuisance's necessity is only partially settled. Complexity tracks a localized accepted-region functional, not global effective sample size (ESS), on both sides, though a fixed-ESS separation theorem is left open; both lower-bound axes vanish as $β\to0$, so the floor creates the map. Empirically, the registered bite family diverges with log-log slope $-2.002$ within its pre-registered band; a 1,024-cell audit records 0 violations where the formal certificates fire; and a single-corpus SQuAD-to-NewsQA feasibility audit returns honest refusal.
Certified world models estimate how long their predictions remain valid. We turn this validity horizon into an operational sensing clock: a rule for when an agent should stop coasting and re-sense. Starting from an audited equivariant world model, we derive a deadline for no-sensing intervals and show that deployable deadlines in learned world models must be drift-aware: on-manifold Lyapunov rates alone overestimate coasting validity, while calibrated native rollout-drift envelopes carry the deployed guarantee. On a frozen 3D VN-JEPA model, the resulting clock controls held-out interval-simultaneous certificate violation across seeds and data shards. In a cue-conditioned theorem-bed (a synthetic bench where all schedulers share the exact model, isolating the scheduling rule), the clock remains valid on the deployment distribution and substantially reduces eventful-tail violations relative to exact-mixture expected-belief scheduling at matched sensing budget. We also report limits: in the short-horizon frozen VN-JEPA regime, empirical conformal horizons match the deployed clock on validity and budget, and a partial-reset exploration finds no clean budget-matched advantage for the spectral term. Thus the contribution is a certified sensing-clock primitive and drift-aware deployment method, not a claim that spectral clocks empirically dominate all non-spectral schedulers.
We introduce PLACE (Persistence-Landmark Analytic Classification Engine), a closed-form pipeline for classifying point clouds and graphs through their persistent-homology signatures. Three quantitative guarantees -- a margin-based excess-risk rate, a closed-form descriptor-selection rule, and a per-prediction certificate -- are derived from training labels alone, with no learned weights or held-out calibration. The embedding sums Mitra-Virk single-point coordinate functions over a sparse landmark grid; closed-form weights maximize a structural distortion constant $λ(ν)$ (a Lipschitz lower bound on $\mathcal{D}_n$ under non-interference). (i) An $O(kR/(Δ\sqrt{m_{\min}}))$ margin bound, driven by class-mean separation $Δ$ and embedding radius $R$, matched by a sample-starved minimax lower bound. (ii) The Mahalanobis margin under Ledoit-Wolf-shrunk covariance is the strongest closed-form descriptor selector on a heterogeneous 64-descriptor chemical-graph pool (mean Spearman $ρ\approx +0.54$ across 10 benchmarks, positive on 9 of 10); the isotropic surrogate $Δ/\sqrt\ell$ admits a closed-form selection-consistency rate on homogeneous (14-15 descriptor) protein/social pools. (iii) A training-time-decided certificate with no per-prediction overhead, in non-asymptotic Pinelis and asymptotic Gaussian plug-in forms. Empirically, PLACE is the strongest diagram-based method on Orbit5k and matches the strongest topology-based baseline within statistical noise on MUTAG and COX2. The remaining gaps fall into two diagnosable regimes: descriptor blindness on NCI1/NCI109, and pool-coverage limits elsewhere. Both radii exceed the firing threshold $\hatΔ/2$ on every benchmark at our training-set sizes, dominated by the $\sqrt\ell$ scaling of the multivariate-norm bound; the per-prediction certificate is constructive but not yet operational at these sizes.