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The third law of thermodynamics has been verified experimentally, but how to perfectly express such a law in theory has become a cross-century problematic issue. It is found that by introducing an innovative method, the Nernst equation can be obtained directly from the experimental data of chemical reactions at low temperatures without the need for artificial additional assumptions appearing in textbooks, so that the Nernst theorem should be replaced by the Nernst statement. It is also found that the heat capacity statement can be obtained from the experimental data of the heat capacity at low temperatures. The heat capacity statement and the Nernst statement are proved to be mutually derivable and the two are equivalent. The unattainability principle of absolute zero temperature is only a corollary of the Nernst statement or the heat capacity statement. Simultaneously, the defects and deficiencies related to the contents of the third law of thermodynamics appearing in textbooks are pointed out and corrected. The results obtained show clearly that the Nernst theorem and the unattainability principle of absolute zero temperature should be withdrawn from the statements of the third law of thermodynamics. It is important to find that the Nernst statement and the heat capacity statement are two equivalent statements of the third law of thermodynamics, which can solve the centennial debate problems of the third law of thermodynamics and supply the perfect statements for the third law of thermodynamics.

Fault-tolerant quantum computing requires understanding how error-correcting codes perform on diverse physical hardware. This is typically assessed via noisy stabilizer simulation of logical circuits at HPC scale, combined with a noise model that yields a logical error rate for the relevant code distances and depths. The uniform depolarizing model is the standard baseline, but its homogeneous assumptions fail to capture the heterogeneity, asymmetries, and correlations of real devices, where Pauli, measurement, and spatio-temporal errors are not weakly coupled. Yet these same structured features create opportunities for joint code-hardware co-design, motivating noise models that more faithfully reflect target hardware while remaining tractable to simulate. We introduce FTPrimitiveBench, a systematic benchmarking approach for studying how logical primitives interact with hardware-motivated noise. It supports both custom specifications and representative structured noise families: Pauli bias, measurement bias, and spatial or spatio-temporal non-uniformity -- together with generators for core surface-code Clifford primitives: logical memory, lattice surgery, transversal logical Hadamard, and the logical phase gate via lattice surgery. We find that structured noise affects these primitives in qualitatively distinct ways, with outcomes shaped by the interplay between noise model, primitive, and decoder choice. These results extend memory benchmarks to active logical computation, where the interaction between noise structure and primitive implementation matters. By standardizing the link between noise-model specification and primitive construction, FTPrimitiveBench enables reproducible comparative studies of QEC protocols and decoders, supporting hardware-aware co-design of fault-tolerant architectures. Code: https://github.com/ShuwenKan/FTPrimitiveBench.

We study a class of branching processes in which the offspring distribution is not specified directly but is induced by a cycle of internal colony growth, catastrophic reduction and structured dispersal. The parameters governing growth, survival and dispersal are allowed to vary deterministically or randomly from one generation to the next, giving rise to branching processes in varying and random environments with implicitly defined offspring laws. We show that survival and extinction are governed entirely by the associated log-mean process, exactly as in the classical theory. The paper treats four qualitatively different dispersal mechanisms and establishes a universal ordering of the induced offspring means. For Poissonian growth with binomial survival, explicit thresholds are obtained that determine extinction or survival uniformly over all four mechanisms. A series of ecologically motivated examples with Yule-Simon growth illustrates the versatility of the framework.

We introduce PALACE (Persistence Adaptive-Landmark Analytic Classification Engine), the data-adaptive companion to PLACE, paying a small cross-validation tier on three knobs (budget, radii, bandwidth; $\leq 5$ choices each). A cover-theoretic core (Lebesgue-number criterion on the landmark cover) yields four closed-form guarantees. (i) A structural lower distortion bound $λ(τ;ν)$ on $\mathcal{D}_n$ under cross-diagram non-interference, with a $(D/L)^2$ budget reduction over the uniform grid when diagrams concentrate. (ii) Equal weights $w_k = K^{-1/2}$ maximizing $λ$, and farthest-point-sampling positions $2$-approximating the optimal $k$-center covering radius; both derived from training labels alone, no gradient training. (iii) A kernel-RKHS classification rate $O((k-1)\sqrt{K}/(γ\sqrt{m_{\min}}))$ with binary necessity threshold $m = Ω(\sqrt K/γ)$ from a matching Le Cam lower bound, and a closed-form filtration-selection rule. The kernel-Mahalanobis margin $\hatρ_{\mathrm{Mah}}$ is the strongest closed-form ranker across the chemical-graph pool (mean Spearman $ρ\approx +0.60$); the isotropic surrogate $\hatγ/\sqrt{K}$ admits a selection-consistency rate, and $\widehatλ$ from (i) provides an independent data-level signal (positive on COX2 and PTC). (iv) A per-prediction certificate, in non-asymptotic Pinelis and asymptotic Gaussian forms, with no calibration split. Empirically, PALACE is the strongest closed-form diagram-based method on Orbit5k ($91.3 \pm 1.0\%$, matching Persformer), leads every diagram-based competitor on COX2 and MUTAG, and is competitive on DHFR (within 1 pp of ECP). At $8\times$ domain inflation, adaptive placement maintains $94\%$ while the uniform grid collapses to chance ($25\%$ on 4-class data).

Audio-Visual Intelligence (AVI) has emerged as a central frontier in artificial intelligence, bridging auditory and visual modalities to enable machines that can perceive, generate, and interact in the multimodal real world. In the era of large foundation models, joint modeling of audio and vision has become increasingly crucial, i.e., not only for understanding but also for controllable generation and reasoning across dynamic, temporally grounded signals. Recent advances, such as Meta MovieGen and Google Veo-3, highlight the growing industrial and academic focus on unified audio-vision architectures that learn from massive multimodal data. However, despite rapid progress, the literature remains fragmented, spanning diverse tasks, inconsistent taxonomies, and heterogeneous evaluation practices that impede systematic comparison and knowledge integration. This survey provides the first comprehensive review of AVI through the lens of large foundation models. We establish a unified taxonomy covering the broad landscape of AVI tasks, ranging from understanding (e.g., speech recognition, sound localization) to generation (e.g., audio-driven video synthesis, video-to-audio) and interaction (e.g., dialogue, embodied, or agentic interfaces). We synthesize methodological foundations, including modality tokenization, cross-modal fusion, autoregressive and diffusion-based generation, large-scale pretraining, instruction alignment, and preference optimization. Furthermore, we curate representative datasets, benchmarks, and evaluation metrics, offering a structured comparison across task families and identifying open challenges in synchronization, spatial reasoning, controllability, and safety. By consolidating this rapidly expanding field into a coherent framework, this survey aims to serve as a foundational reference for future research on large-scale AVI.