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Importance-weighted least squares (IWLS) is widely used to correct covariate shift in unsupervised domain adaptation, yet most prior work assumes that an appropriate source dataset is already available. In practical internet-scale retrieval settings, the key decision is reversed: given only unlabeled target covariates and a pool of candidate source datasets, which source (or source mixture) should be selected before fitting a weighted regressor? We study this question through a stability-aware bilevel framework with three formal components: a label-free source-ranking surrogate with a uniform regret guarantee, a multi-source mixture objective with linear-rate upper-level optimization under smoothness and strong-convexity assumptions, and a mixed-shift gate for harmful-source rejection before weighted fitting. We evaluate these components in three settings (distribution-level synthetic shift, semi-synthetic target-sample selection, and a protocol-faithful proxy real-track setting with governance checks). The empirical findings are mixed: symbolic checks and theorem-conditioned diagnostics are consistent, but Holm-corrected comparisons show no statistically significant advantage of the stability-aware selector over MMD-nearest, Wasserstein-nearest, or pooled IWLS in the current iter\_1 run. This outcome clarifies that optimization guarantees and diagnostic structure are not sufficient for global predictive dominance under the present proxy data regime. The study contributes a reproducible no-target-label protocol, explicit failure reporting, and a concrete follow-up agenda for real benchmark ingestion and post-selection-controlled confirmatory analysis.

Biased-noise threshold claims for surface-code families are often expressed in terms of the nominal hardware dephasing ratio $\eta$, yet realistic gate decompositions, measurement asymmetries, and syndrome-extraction schedules can convert a substantial fraction of the nominal $Z$-dominant noise into less favorable error components. This mismatch matters well beyond quantum error correction because it exemplifies a broader problem in reliable scientific computing: analytical control parameters are useful only if the implementation preserves the semantics that make them predictive. We study this issue for realistic biased-noise surface-code threshold estimation by introducing an effective-bias renormalization, $\eta_{\mathrm{eff}}$, that aggregates compiled Pauli components and a temporal-fragility factor into a decoder-visible bias observable. We couple that observable to three formal results: a reference biased-Pauli closure identity, a convex-mixture representation for gate-class attribution, and a schedule-sensitivity transfer bound connecting changes in $\eta_{\mathrm{eff}}$ to threshold residuals. We then instantiate the resulting analysis on a matched validation suite that mirrors the planned sweep over code families, distances, schedules, and bias ratios. Across four code families, reparameterizing thresholds by $\eta_{\mathrm{eff}}$ reduces leave-one-schedule-out RMSE by 40.6\%--50.5\%, the convex-mixture interval bound has zero empirical violations, and the schedule-transfer residual remains within the predicted bound across all held-out settings. These gains are interpreted as a conditional first-order result that holds when boundary fragility and backend semantic mismatch remain within audited ranges. The current evidence is surrogate-based rather than direct Stim or Qiskit Aer execution, so the manuscript should be read as a rigorous mechanism paper with an executable validation scaffold rather than as a final hardware-faithful threshold benchmark.

Type 1 diabetes (T1D) remains a paradigmatic autoimmune disease in which loss of beta-cell function causes dysglycemia, severe hypoglycemia, and lifelong dependence on exogenous insulin. Stem-cell-derived islet replacement delivered by portal-vein infusion with glucocorticoid-free immunosuppression has shown early promise, yet durability and comparative interpretation are unresolved in small, open-label cohorts. We present a joint longitudinal--survival framework that links standardized mixed-meal tolerance test (MMTT) C-peptide trajectories to a regulatory-aligned composite endpoint and benchmarks durability using external standardization against historical donor-islet cohorts. Using synthetic validation experiments consistent with trial endpoints, we report survival probabilities near 0.88 at 12 months and 0.62 at 36 months, standardized survival around 0.59 at 36 months, and a reduction in between-trial variance under endpoint harmonization of approximately 25\%. A beta-cell composite index improves AUC from 0.741 to 0.767, while potency-to-durability prediction remains weak (AUC 0.56). These results demonstrate how mechanistic graft function can be translated into clinically interpretable durability claims while revealing where evidence remains limited.

We develop a proof program that targets a quantitative bridge between scale-invariant local energy control and global critical $L^\infty_t L^3_x$ bounds for the three-dimensional incompressible Navier--Stokes equations. The program is motivated by the persistence of weak solutions and the weakness of known quantitative blowup rates at criticality. Building on the local energy framework and recent quantitative surveys, we formalize a covering-and-pressure decomposition approach that would convert a local concentration bound into a global critical estimate, yielding a conditional regularity criterion and explicit rate implications. We present a structured roadmap with formal definitions, provisional lemmas, and a theorem statement consistent with the currently available formalization, together with a validation plan that specifies acceptance criteria and evidence targets. The absence of executed experiments and completed formal derivations is recorded explicitly as a limitation, and the manuscript is framed as a rigorous, proof-first blueprint rather than a completed resolution.

We study 3D incompressible Navier--Stokes flow on a periodic box and design a diagnostic suite that links classical Prodi--Serrin mixed norms, weak-$L^p$ (Lorentz) proxies, scaling-invariant weighted criteria, and sparseness-based geometry. The goal is to evaluate how these regularity indicators behave in direct numerical simulation (DNS) across Reynolds-number sweeps and to quantify numerical sensitivities due to de-aliasing. The study matters for both mathematical regularity theory and practical turbulence modeling because it formalizes computable indicators that can be tracked in simulations and compared to validated datasets. We provide a formal problem setting, an algorithmic pipeline, and a hypothesis-driven evaluation protocol, with explicit acceptance criteria and uncertainty procedures. Empirical results are not yet available because the required scientific Python stack and external dataset access are currently unavailable in the execution environment; we therefore report the full experimental design and the evidence plan that will be used once runs are enabled.

We study conservative offline reinforcement learning with uncertainty-aware policy improvement under a tight compute budget. The goal is to combine conservative value regularization with ensemble-based uncertainty penalties and evaluate when such coupling improves mean performance, stability, and calibration. We design four hypothesis-driven experiments, including conservatism--uncertainty sweeps, checkpoint stability analysis, dataset-quality regime comparisons for implicit Q-learning, and correlation-based calibration of uncertainty penalties. Because full simulator access is unavailable, we report a transparent simulation-based validation using logged classic-control proxies that preserve the benchmark structure and metrics. The results show consistent gains in mean normalized score for uncertainty-augmented conservative learning, improved checkpoint stability, and positive uncertainty--density correlations, while variance reductions and dataset-quality effects are mixed. These findings motivate follow-on experiments on full D4RL benchmarks and provide a reproducible evaluation scaffold for conservative offline RL under strict resource constraints.