Learning--Reasoning Correspondence (LRC) proposes that augmenting training examples with symbolic-style inference tuples can improve small-agent learning. We test this claim in a controlled finite replication that fixes rule-chain and symbolic gridworld task families, ten paired seeds, size-matched neural and neuro-symbolic conditions, and oracle trace-bearing tuple diagnostics. Relevant regularized LRC improves final out-of-distribution accuracy in both executed local benchmarks: 0.945 versus 0.496 for the neural baseline on rule chains and 0.905 versus 0.405 on symbolic gridworld planning. Mechanism-isolation margins remain positive against noisy-tuple and no-regularizer ablations, with bootstrap intervals [0.207, 0.219] and [0.236, 0.253]. Tuple confidence is also better calibrated and more faithful than noisy augmentation under the available oracle traces. The stronger sample-efficiency claim is not established: all predeclared samples-to-threshold comparisons at 0.80 and 0.90 validation accuracy are censored on the executed grid. The evidence therefore supports LRC as a local mechanism for final OOD generalization in controlled finite tasks, but weakens claims about threshold sample complexity and leaves public bAbI/MiniGrid replication as future work.

Small language models are increasingly used as lightweight reasoners over structured records, but their behavior under tabular distribution shift is poorly characterized. This paper evaluates whether counterfactual feature interventions improve robustness when rows are verbalized as reasoning examples. We construct an oracle synthetic benchmark of mixed-type tabular rules with threshold, conjunction, disjunction, exception, and interaction families; generate matched no-augmentation, random-row augmentation, nuisance-counterfactual, label-preserving perturbation, and decisive counterfactual-intervention regimes; and evaluate 625 method, shift, rule, and seed cells. Decisive counterfactual intervention augmentation improves mean out-of-distribution accuracy by 21.3 percentage points over no augmentation and 17.0 points over random-row augmentation, with bootstrap intervals above zero. It also improves paired counterfactual consistency by 35.5 and 29.4 points over the same baselines. The original success criterion is only partly met because in-distribution accuracy retention is -2.3 points relative to the no-augmentation baseline, exceeding the pre-specified tolerance. The evidence supports counterfactual interventions as a useful robustness stressor and training signal in oracle tabular rule tasks, while limiting claims about real data until intervention-feasibility and shift manifests are available.

Learning-Reasoning Correspondence (LRC) proposes that neural agents become more sample efficient, robust, and interpretable when explicit inference tuples are generated, serialized into the input, and regularized for consistency. We test that claim in a finite, matched-capacity replication across three task families: Boolean structural-causal intervention prediction, SCAN-compatible command recombination, and gridworld planning imitation. The executed model class is a fixed hashed linear neural proxy with identical feature dimension across learned conditions, ten paired seeds, five data fractions, thirteen conditions, tuple-validity checks, ablations, oracle-symbolic upper bounds, confidence calibration, paired tests, Holm correction, and wall-clock runtime gates. The headline LRC claim is not supported in this regime: no primary endpoint passes the full conjunction of effect threshold, Holm rejection, runtime ratio at or below two, valid baseline floor, and architecture-fit gate. Raw gains are visible but conditional: causal OOD improves by 14.5 percentage points with Holm rejection but exceeds the runtime gate and remains outside the requested architecture class; command sample efficiency passes raw and statistical thresholds but is diagnostic because the full-data baseline floor fails; gridworld sample efficiency is large but not Holm-supported, and gridworld OOD remains below the 10 percentage-point target. Tuple evidence is mixed: full LRC produces high parse and semantic-validity rates, corrupted or feedback-removed tuples degrade behavior, and oracle tuples outperform learned tuples, so tuple quality appears relevant but insufficient to rescue the headline performance claim.

We test whether a next-nearest-neighbor coupling supplies one finite-size control scale for both the spectral crossover from integrability and the onset of eigenstate thermalization in an open spin-$1/2$ XXZ chain at $\Delta=1$. Dense exact diagonalization was executed for $L=8,10,12$ in total-$S^z=0$ sectors resolved by reflection parity, with $\lambda\in\{0,0.05,0.1,0.2,0.3,0.5,0.7,1.0\}$. The computed diagnostics do not support a clean Poisson-to-Gaussian-orthogonal-ensemble rise in the mean spacing ratio: the reflection-even high-$\lambda$ values remain near $0.35$--$0.40$ rather than approaching $0.5307$. Off-diagonal matrix elements of a parity-preserving central-spin observable show partial $1/D$ consistency, but the adjacent-size ETH exponents drift from near one at small $\lambda$ to about $0.78$ and $0.72$ at $\lambda=1$. Half-chain entanglement increases toward a Page-like comparator, reaching about $0.86$--$0.92$ of the full-space Page value at $\lambda=1$. Independently extracted crossover scales do not coincide: spectral midpoint crossings are absent for $L=10,12$, whereas ETH midpoint estimates occur near $\lambda\simeq0.31$--$0.36$. At the executed sizes, the evidence is a finite-size failure atlas for the proposed single-scale collapse rather than confirmation of a universal crossover law.

We study whether sublinear {Nystr\"om} sampling preserves the spectral gap of the symmetric normalized graph Laplacian. For a finite undirected graph with positive degrees, we work with the shifted positive semidefinite kernel $K=2I-L_{\mathrm{sym}}$ and evaluate the relative error in $\gamma_k=\lambda_{k+1}-\lambda_k$. A deterministic perturbation result shows that if the fixed-operator {Nystr\"om} residual satisfies $\|K_{\mathrm{Ny}}(S)-K\|_2\le 0.05\gamma_k$, then the relative gap error is at most $10\%$. The executed validation does not support the stronger sampling hypothesis: leverage-family policies reached the nominal target on only a small minority of eligible graph--dimension cells, and uniform sampling was often competitive at matched budgets. Coherence and vertex-resistance diagnostics help organize failure regimes but do not by themselves supply a gap-preservation law. The resulting contribution is a conditional theorem, an evidence-backed negative result for the tested regime, and a sharper specification of the residual-scale theory needed for reliable graph eigengap sketching.