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Quantum reservoir computing is frequently evaluated with point-estimate accuracy gains that confound representation effects, readout parity, and computational cost. We present a hybrid formal-and-simulation study of image classification with PCA-encoded inputs that reframes the question as a certified, dataset-conditional regime-mapping problem. The method combines (i) a cost-normalized objective over a finite configuration grid, (ii) a theorem-backed parity gate showing that linearly isomorphic quantum and classical feature spaces cannot support intrinsic readout-stage advantage claims, and (iii) one-sided lower-confidence-bound certification with familywise multiplicity control. Under a fixed policy (\tau_\mathrmeff=0.01, \tau_\mathrmiso=0.05, \alpha_\mathrmFWER=0.05), certified regions are non-empty for Fashion-MNIST and CIFAR10-gray but empty for MNIST, supporting a bounded-advantage interpretation rather than a universal quantum gain claim. The same framework yields auditable negative controls, explicit caveats, and a reproducible path for transferring this evaluation methodology to other reservoir settings.

Quantum reservoir computing has shown repeated empirical promise for representation learning, but the evidence base for robust quantum advantage in image classification remains fragmented by confounded entanglement controls, heterogeneous readout optimization practices, and weakly standardized finite-shot reporting. This paper presents a hybrid analysis framework that combines formal derivations and controlled simulation evidence for PCA-encoded image classification with fixed reservoir dynamics and output-layer training. The framework integrates four complementary questions: whether an interior entanglement regime improves geometric and predictive quality under parity controls, whether constrained measurement-operator optimization changes the accuracy-cost-shot frontier, whether advantage signals emerge on a calibrated dataset-difficulty ladder rather than saturated easy tasks, and whether finite-shot classically simulable regimes impose a quantitative boundary on cost-adjusted claims. We formalize these questions through explicit objectives, feasible sets, and theorem-level guarantees, and we evaluate them with deterministic decision gates tied to confidence intervals and assumption audits. The resulting evidence is calibrated rather than binary: the finite-shot simulability boundary is supported in the admissible regime, while broad empirical superiority claims remain inconclusive under strict parity criteria. This outcome is practically relevant beyond quantum machine learning because it illustrates a general methodology for integrating proof-level constraints with reproducible benchmarking when computational claims are sensitive to uncertainty, reporting schema, and regime validity.

Physical lottery systems are designed to approximate uniform sampling without replacement, yet practical implementations involve latent mechanical and procedural factors that can induce weak, time-varying departures from ideal randomness. This paper develops a hybrid inferential and predictive framework that integrates regime diagnostics, dependence-aware multiplicity control, bounded-confounding identification, staged transfer evaluation, and reliability-constrained integrated scoring. The objective is explicitly non-deterministic: we test reproducible structure and uncertainty bounds rather than deterministic prediction of winning combinations. We formalize five optimization/identification programs with explicit decision variables, feasible sets, and optimality criteria, and we provide complete theorem and lemma proofs for the key guarantees used by the pipeline. On a long-horizon historical draw corpus, evidence is asymmetric: confounding-robust directional interpretation and strict false-discovery control are strong, while segmentation stability and integrated score dominance remain below pre-registered gates. The resulting contribution is methodological and practical: robust bias-screening claims can be made with high transparency under severe observability limits, while integrated-superiority claims should remain conditional until targeted reruns resolve the remaining gates.

Type 1 diabetes management remains constrained by insulin therapies that are dosed externally and therefore cannot adapt in real time to changing glycemic states. This gap motivates glucose-responsive insulin design, where molecular activity is attenuated in hypoglycemia and amplified in hyperglycemia. We study this objective as a computational prioritization problem rather than a full clinical development program: given insulin-centered molecular candidates, predict a state-conditional activity profile and select a shortlist for downstream experiments under bounded compute. The proposed method is a three-stage hybrid pipeline: a tri-state ranker enforces monotone low/normal/high behavior with uncertainty penalties; a mechanistic reranker augments machine-learning scores with paired-context molecular dynamics descriptors and explicit low-glucose safety gating; and an uncertainty-aware Pareto selector balances efficacy, hypoglycemia risk, and manufacturability. We provide formal definitions of objectives, feasible sets, and optimality criteria, and include theorem-level results with complete proofs for ordering and dominance properties. Validation uses a reproducible synthetic proxy protocol with fixed seeds, sweeps, confidence intervals, and symbolic checks. Relative to strong baselines, the selected pipeline improves tri-state rank correlation, reduces top-k false positives under safety constraints, raises nondominated shortlist quality, and passes symbolic assumption checks. The findings support a practical thesis: hybrid statistical-mechanistic ranking with explicit multi-objective uncertainty control can materially improve preclinical computational triage for glucose-responsive insulin candidates, while still exposing key data and translation gaps that must be addressed for broader external validity.

We study single-qubit X-gate feasibility for a ^171Yb nuclear-spin qubit encoded in the ^3P_0(F=1/2) manifold and driven by Raman coupling through an inner-shell-excited J=2 intermediate. The design objective is explicitly constrained: optical intensity is fixed at 1\,\mathrmW/cm^2, and the output must report gate-time versus target-infidelity operating points at 10^-2, 10^-3, 10^-4, and 10^-5. We develop a hybrid formal-empirical methodology that combines (i) a convex surrogate optimality certificate for detuning selection, (ii) a robust feasibility-floor certificate for impossibility regions, and (iii) a posterior chance-constrained table for uncertainty-aware decision support. The resulting evidence indicates a stable, non-provisional operating regime at 10^-2 with a best deterministic gate time of 0.1661\,\mu\mathrms and dominant leakage contribution. After targeted posterior uncertainty ablations (model-discrepancy and noise-floor scaling), posterior confidence intervals widen materially and strict-threshold recommendations at 10^-3 and below become infeasible in both deterministic and posterior modules. Those strict rows remain provisional because robust-floor diagnostics still classify 10^-3 as potentially feasible under conservative assumptions. Beyond this specific gate, the study connects atomic-structure uncertainty management to broader neutral-atom processor planning and high-accuracy metrology workflows.

Continual learning systems are increasingly deployed in settings where data distributions evolve while labels, environments, and downstream requirements remain nonstationary. In these settings, the practical failure mode is not only catastrophic forgetting, but also gradual loss of plasticity, dead-unit accumulation, and unstable internal statistics that amplify optimization brittleness across long horizons. We study a model class in which continual-learning inductive bias is encoded directly in the activation function rather than in replay buffers, explicit task identifiers, or per-task masks. The proposed mechanism uses a dual-timescale activation parameterization: fast parameters adapt to novelty, while slow anchors preserve utility-weighted structure. We formalize the problem with explicit decision variables, feasible sets, and an online surrogate objective, and we provide two formal results: a bounded-moment theorem under bounded drift and projection assumptions, and a lower-bound proposition establishing an impossibility region for static memoryless activations under persistent alternating conflict. We then evaluate the formal chain with symbolic checks and synthetic continual regimes that expose stress and boundary behavior. Across executed regimes, the method satisfies bounded-variance compliance above 0.95 in bounded drift, improves forgetting relative to GELU by roughly 49\%, and shows lower conflict-regret slope than static baselines. We also report caveats: two planned experiment tracks focused on replay-free competitiveness breadth and compositional probes are not yet executed, so associated claims are scoped as open rather than confirmed. The broader implication is that activation-level state can provide a lightweight, task-agnostic route toward continual adaptation, while formal assumptions remain explicit and testable.

Quantum reservoir computing for vision tasks is often discussed in terms of empirical gains without an equally explicit separation between predictive improvement and computational-advantage claims. We study this issue in a controlled image-classification setting with PCA-compressed inputs, matched preprocessing, and identical linear readout training across classical and quantum reservoirs. The manuscript contributes a hybrid theory-plus-experiment framing: we formalize the readout stage as a strongly convex regularized problem with a unique closed-form optimum under positive regularization, derive concentration-calibrated conditions for reporting positive entanglement regimes, and define a compute-constrained Pareto boundary that filters performance gains by runtime and simulability costs. Experiments on MNIST, Fashion-MNIST, and a grayscale CIFAR-10 variant show that entangling quantum reservoirs can improve average macro-F1 against strong baselines in several configurations, but confidence-qualified positive-regime gates remain unmet under the current proxy-data execution. At the same time, symbolic and numerical checks validate the formal claims, and fixed-universe frontier audits satisfy monotonicity and nondominance requirements. The resulting evidence supports a simulability-safe conclusion: current results are strongest for formal guarantees and boundary/null characterization, while positive advantage claims remain conditional and require canonical-data reruns.

Robust dynamic operating envelopes (RDOEs) are increasingly used to allocate low-voltage flexibility while preserving voltage and thermal security under uncertain load and distributed energy resource behavior. Purely linear robust approximations scale well but can misestimate nonlinear boundary regions in unbalanced feeders, whereas full non-convex robust unbalanced three-phase optimal power flow (UTOPF) is often too expensive for high-cadence operation. We present a hybrid method that couples a certified linear screening stage with selective non-convex escalation and prove two properties: (i) accepted screened points are robust-feasible under explicit residual assumptions, and (ii) accepting all certifiable points is compute-optimal among admissible deterministic escalation rules. The method is evaluated on a reproducible four-experiment suite covering security-runtime tradeoffs, online uncertainty calibration, fairness-sensitive allocation, and cross-solver shadow-mode behavior. Across 600 configuration-seed samples per experiment family, nominal screening behavior remains consistent with the safety theorem, observed runtime reduction is substantial for boundary-aware operation, uncertainty calibration tracks target risk with bounded coverage error, and cross-solver discrepancy remains within predefined operational tolerances in synthetic stress regimes. The broader contribution is a deployable bridge between optimization guarantees and operational evidence: theorem-scoped safety and optimality are connected to experiment-level diagnostics, reproducibility artifacts, and policy-sensitive allocation analysis.