arXiv:2606.04277v1 Announce Type: cross
Abstract: In quantum technologies, continuous-variable systems offer advantages over their discrete counterparts. However, continuous-variable tomography suffers from exponentially growing sample complexity. We propose protocols using quantum mirrors to transfer the complete information of incident photonic states onto a control atomic system. This enables full photonic state characterization through measurements on the control atom alone, realized via kernel functions, direct wavefunction reconstruction, and pointwise Wigner function measurements. Our approach overcomes the limitations of conventional photon counting, statistical inference, and inverse transformation, providing a robust framework for benchmarking and verifying non-Gaussian states in continuous-variable quantum optics.
Science Journals
arXiv:2606.04174v1 Announce Type: cross
Abstract: Computed Axial Lithography is a volumetric additive manufacturing method that selectively cures photosensitive resin through the 3D superposition of patterns of light, offering advantages over layer-based processes including rapid print times, reduced layer artifacts, and compatibility with high-viscosity materials. However, diffusive effects, primarily those of free-radical quenchers such as oxygen, blur the boundary between cured and uncured regions, limiting resolution and preventing the reproduction of sharp, high-spatial-frequency features. By comparing micro-CT data to computational dose models convolved with kernels across a range of diffusivities, we establish a framework for extracting a single diffusion kernel from any standard uncorrected print to account for all observed deviations from the target. In this work, we correct diffusion-induced blurring by co-optimizing for its effects alongside the inherent blur of the computed tomography reconstruction, demonstrating improved fidelity over previous approaches of pre-compensating the target geometry via deconvolution.
arXiv:2606.04173v1 Announce Type: cross
Abstract: It is well-known that any class of simple graphs, that is characterized by finitely many forbidden minors, also admits a characterization by finitely many forbidden topological minors; furthermore, the list of forbidden topological minors may be derived from the list of forbidden minors.
We prove a similar result in Matching Theory. Our Main Theorem states that any class of matching covered graphs, that is characterized by finitely many forbidden $S$-minors that are cubic, also admits a characterization by finitely many forbidden conformal minors that are cubic as well; once again, the list of forbidden conformal minors may be derived from the list of forbidden $S$-minors.
In order to establish the above, we first prove that every matching covered graph has one of two graphs as a conformal minor -- either $K_4$, or the $\Theta$ graph (that is, two vertices joined by three edges). (In fact, we need and prove a much stronger statement.) This is reminiscent of a theorem due to Lov\'asz: every nonbipartite matching covered graph has one of two graphs as a conformal minor -- either $K_4$, or the triangular prism $\overline{C_6}$.
As applications of our Main Theorem, we deduce known 'forbidden conformal minor characterizations' of pfaffian near-bipartite graphs, and of pfaffian solid graphs, using their respective known 'forbidden $S$-minor characterizations'.
arXiv:2606.04138v1 Announce Type: cross
Abstract: The coordinated interactions between organisms enhance collective functionality, a feature that artificial systems such as enzymatic nanomotors seek to replicate. A key objective, yet still a major challenge, is to achieve chemical communication among nanomotors. Progress has been limited by the difficulties in verifying effective signaling processes, including chemical signal propagation and the response of receiving nanomotors. Here, we address this challenge using an enzymatic nanomotor system that demonstrates communication between two populations through generically non-reciprocal phoretic response. A primary swarm of glucose-responsive nanomotors migrates toward a glucose gradient while producing H2O2 as a diffusible communication signal. This self-generated chemical gradient then acts as a chemoattractant for a secondary swarm of catalase-powered nanomotors. Through carefully designed experiments, we visualize the propagating H2O2 gradient and quantify the spatiotemporal response of the receiver nanomotors to the chemical front. Combined experimental and theoretical analysis has revealed that the synergy between different combinations of chemo-attractive and chemo-repulsive mobilities and catalytic rates of consumption and production of substrates and products gives rise to a wealth of different collective responses in the system. This work represents a step toward programmable synthetic systems at the collective level, broadening the functionality of chemical nanomotors and opening opportunities for future hybrid living-synthetic systems.
arXiv:2606.04123v1 Announce Type: cross
Abstract: Trajectory optimization is a critical component for enabling safe and reliable autonomous operations in space exploration. As space missions increase in frequency, complexity, and scope, there is a growing need to rapidly formulate mathematically sound trajectory optimization problems that accurately reflect mission objectives and operational constraints. However, translating mission intent into tractable analytical formulations for trajectory optimization requires substantial domain expertise. This paper presents a framework that leverages large language models (LLMs) to translate natural language descriptions of mission requirements and constraints into executable trajectory optimization code and corresponding mathematical formulations. Experiments in spacecraft rendezvous scenarios demonstrate a high success rate in reconditioning a convex trajectory optimization problem from semantic mission requirements. Ultimately, this work highlights the potential of LLMs to bridge high-level intent and formal optimization models, enabling more flexible and efficient trajectory design of spacecraft.
arXiv:2606.04093v1 Announce Type: cross
Abstract: In numerical experiments simulating Turbulent Radiative Mixing Layers (TRMLs) it is observed that as the cooling time in the mixed gas, $t_{\rm cool}$, becomes very short compared to the dynamical time of the turbulence, $t_{\rm eddy}/t_{\rm cool} \gg 1$, there is a change in the scaling behavior of the total energy radiated in the TRML as a function of this ratio, also known as the Damk\"{o}hler number, ${\rm Da} \equiv t_{\rm eddy}/t_{\rm cool}$, from $\dot{E}_{\rm cool} \propto {\rm Da}^{1/2}$ to $\dot{E}_{\rm cool} \propto {\rm Da}^{1/4}$. The latter, so-called "fast-cooling," regime is of particular interest as many astrophysical mixing layers lie in this regime. We demonstrate that the origin of this change is the suppression of turbulent folding of the surface by the ram-pressure of the inflowing gas, which becomes much greater than the turbulent pressure in this regime. We present an argument that reproduces the $\dot{E}_{\rm cool} \propto {\rm Da}^{1/4}$ behavior by appealing to the suppression of the fractal structure of the interface by the ram-pressure of the inflowing gas.
arXiv:2606.04087v1 Announce Type: cross
Abstract: Turbulent Radiative Mixing Layers (TRMLs) are of fundamental importance to the transport of energy and momentum in multi-phase, astrophysical fluids. We use measurements of the "micro" and "macro" properties of these layers in high-resolution \texttt{AthenaK} simulations to investigate when their properties can be considered \textit{well}-resolved. In particular, we demonstrate that the previously noticed resolution independence of total cooling, $\dot{E}_{\rm cool}$, in these simulations is due to a remarkable, and perhaps fortuitous, cancellation of the countervailing effects of numerical dissipation and numerical viscosity. This calls into question the degree to which we can trust the results of these experiments, as there is no physical picture that explains this cancellation. We also demonstrate that in order to correctly resolve the phase structure in these layers, important for accurate predictions of their observable properties, one must resolve the scale on which turbulent diffusion acts on time-scales comparable to the cooling time. This "turbulent Field length", $\lambda_{\rm F,turb}$, is where the eddy turnover time is equal to the cooling time ($t_{\rm eddy}(\lambda_{\rm F,turb}) = t_{\rm cool}$). We demonstrate that resolving this scale results in converged phase-structure and spatially resolved transitions in the gas phases.
arXiv:2606.04080v1 Announce Type: cross
Abstract: We develop a semantic framework for higher-order quantum computation based on a boundary-centric presentation of compact closed categories, building on Kelly--Laplaza and Abramsky.Morphisms are polarized boundary linkings composed by execution, with a unit-free monoidal sum providing reversible control and branching.
We identify a notion of \emph{essential unitarity} generalizing unitarity from first-order processes to higher-order interfaces;at first order it coincides with standard unitarity, and at higher order it characterizes when information is preserved relative tothe boundary. Essential unitarity is the unique predicate compatible with dagger-monoidal structure, coherence reindexing, and currying, and reducing to ordinary unitarity at first order. Every morphism of the quantum core is essentially unitary.
The framework realizes the coherent quantum switch and other one-slot, equal-ratio, purity-preserving supermaps as coherent pure-comb dilations.
Extended Abstract appears in QPL 2026
arXiv:2606.04079v1 Announce Type: cross
Abstract: Quantum computing platforms based on arrays of tweezer-confined neutral atoms have recently emerged as a competitive modality thanks to a direct path toward high qubit count, rapidly advancing operation fidelities, and their ability to execute circuits with arbitrary qubit connectivity. These features will enable the use of efficient error correction schemes with high encoding-rates, time-efficient decoding, and resource-efficient architectures based on transversal gates. With these goals in mind, recent state of the art neutral atom demonstrations focus on the transition from the use of physical qubits to error-corrected logical qubits, but to date there has been no demonstration of repeated error correction scalable to arbitrary depth. Here, we demonstrate many cycles of syndrome extraction in a toric quantum error correcting code, using mid-circuit measurement and replacement of lost qubits, including reloading of a qubit reservoir for indefinite coherent operation. We characterize the logical error rate after up to 90 cycles, showing that logical information can be preserved through multiple rounds of qubit reloading. Comparing two distances of the code up to 8 rounds of syndrome extraction shows a lower absolute logical error rate for the larger distance code.
Finite-Iteration Local Dynamics and Warm Starts for Alternating Power Iteration in Spiked Tensor PCA
arXiv:2606.04065v1 Announce Type: cross
Abstract: We study simultaneous alternating power iteration for fixed-order asymmetric rank-one spiked tensor models. Our main contribution is a finite-iteration local theory that is independent of any particular initialization. Once the iterates enter a sufficiently small neighborhood of the planted rank-one direction, their error decomposes into a geometrically decaying transient and an intrinsic noise floor caused by fixed orthogonal noise contractions at the planted point. The deterministic finite-sample conditions are stated explicitly, but under a coarse fixed-order multilinear noise event they reduce to a conservative high-signal regime for fixed or slowly expanding local radii.
We then separate the warm-start mechanism from any specific spectral construction. A generic one-sweep principle shows that, if a sign-compatible initializer has correlation \(\gamma_N\), first-sweep noise level \(a_N\), and \(a_N/(\gamma_N^{d-1}\omega_{N,d})\to0\), then one can choose an expanding radius \(r_N=o(\omega_{N,d})\) for which the first sweep enters the local basin. After entry, the local affine contraction yields convergence to the unique informative local fixed point in that basin. For centered-Gram initialization, we verify the required correlation and same-sample first-sweep noise bound under i.i.d. finite-fourth-moment noise by a signal-preserving noise-only leave-one comparison and an averaged leave-one slice-contraction estimate, which we call a pressed-back estimate. The leave-one comparison keeps the spike fixed and averages over the deleted coordinate, so planted coordinates enter through \(\ell_2\)-weighted sums rather than worst-case incoherence bounds.
arXiv:2606.04784v1 Announce Type: new
Abstract: Let R denote the class of all computable, causal functionals that are rate-independent in the classical sense (invariant under monotone time reparametrizations), and let Pi_n be the Preisach extremum stack of an input sequence u_{0:n}. We prove a characterization theorem establishing that every F in R satisfies Fu = f(Pi_n) for a computable f, and derive two information-theoretic results. First, under any probability measure on u_{0:n}, the equality I(u_{0:n}; Fu) = I(Pi_n; Fu) holds for every F in R and is an immediate corollary of the characterization theorem. Second, the main result: Pi_n is a Shannon-minimal sufficient statistic in the sense that I(u_{0:n}; Pi_n) <= I(u_{0:n}; S) for every random variable S from which all R-queries are computable. The proof uses the finite indicator family of [Frydrych, 2026] to reconstruct Pi_n from any sufficient S. As a corollary, online maintenance of Pi_n suffices for rate-independent estimation: the NNLS estimator of the Preisach measure mu can be assembled from the incremental stack process (Pi_t)_{t=0}^n in O(k * L^2) memory per step, where k = |Pi_t| and L is the grid resolution.
arXiv:2606.05130v1 Announce Type: new
Abstract: Individual-level mobility prediction is central to urban simulation, transportation planning, and policy analysis. Supervised sequence models achieve strong accuracy but require task-specific training and offer limited decision-level transparency. Recent LLM-based methods improve interpretability, yet mostly rely on static prompts and single-pass inference, limiting their ability to seek additional evidence when mobility signals are weak or conflicting. We propose \method{}, a training-free LLM-driven agent framework that formulates next-location prediction as adaptive evidence-controlled decision making. \method{} resolves routine cases through a fast path based on historical regularity, while ambiguous cases trigger iterative tool use over recent trajectories, historical behavior, stay-move likelihood, and geographical evidence. Across three mobility datasets, AgentMob achieves the strongest overall performance among training-free LLM-based methods, with GPT-5.4 reaching 71.42\% Acc@1 on BW, 33.14\% on YJMob100K, and 33.50\% on Shanghai ISP. On BW non-fast-path cases, the LLM controller improves Acc@1 from 30.65\% to 48.62\% over a same-tool statistical baseline, showing that its main benefit lies in resolving ambiguous predictions through adaptive evidence gathering. Our code is available at https://github.com/Unknown-zoo/AgentMob.
arXiv:2606.04533v1 Announce Type: cross
Abstract: A vibrational paradigm of atomic dynamic in dense fluids is known to provide useful insight on the transport and thermodynamic properties of fluids in three dimensions. In this paper, a vibrational model is generalized to describe the excess entropy of two-dimensional (2D) fluids. A simple practical implementation of this model is demonstrated to deliver accurate results for various systems, such as one-component plasmas with Coulomb and logarithmic interactions, a 2D fluid of dipole particles, and a 2D Yukawa fluid. The applicability limits, relevance to three-dimensional fluids, relations to other 2D phenomena, and potential practical applications are briefly discussed.
arXiv:2606.05129v1 Announce Type: new
Abstract: Preserving data privacy is an important topic in structural data management and data mining. However, the issue of privacy leakage in distributed causal structure learning is a persistent challenge, especially in cases where data transmission and computation are required. In this paper, we propose a method based on fully homomorphic encryption (FHE) that performs calculations on ciphertexts, keeping data encrypted in transition and computation. Nevertheless, adopting FHE to causal structure learning is challenging due to the high computation cost and limited support on division as well as logarithm operations in FHE. To tackle this challenge, we propose a series of novel techniques including (i) circuit simplification for better efficiency, (ii) approximation of division and logarithm through Newton-Raphson Reciprocal and Taylor expansion, and (iii) a batching technique with SIMD-acceleration to enhance the whole learning process. Additionally, our method can be easily extended beyond FHE by demonstration of its portability to support differential privacy. Empirical results show that our method achieves high consistency and comparable causal structure with the plaintext version in the datasets tested. Last, our method is efficient and practical to complete learning causal structures in tens of minutes even under the privacy protection of FHE.
arXiv:2606.05127v1 Announce Type: new
Abstract: Foundation models in atomistic machine learning encode interaction physics across diverse atomic environments, but whether that structure can be transferred when building specialist potentials at quantum-chemical accuracy remains open. Here we show that knowledge distillation from a pretrained universal machine-learning interatomic potential (MLIP), followed by coupled-cluster fine-tuning with single and double excitations and perturbative triples [CCSD(T)], transfers not only low-cost labels but a physically meaningful prior on interaction length scales, anisotropy, and the repulsive-dispersive balance, which CCSD(T) data then sharpens to quantum-chemical accuracy. For He--benzene, fine-tuning with 30% of the CCSD(T) data outperforms direct training using the full 80%; a 60% reduction in the high-fidelity compute budget. A symmetry-adapted perturbation theory (SAPT)-informed adaptive short-range/long-range architecture further lowers the validation MAE from 0.75 1/cm to 0.49 1/cm. Across a circumarene series of polycyclic aromatic hydrocarbons (PAHs), swapping the MLIP teacher under an otherwise identical pipeline changes the coronene error by an order of magnitude while leaving the larger PAHs stable, direct evidence that distillation transfers physical structure, not labels alone. Together, these results identify the choice of pretrained teacher as a primary design axis for data-efficient quantum-chemical-accuracy potentials, alongside architecture and training protocol.
arXiv:2606.05126v1 Announce Type: new
Abstract: Liveness detection has evolved from a safeguard against presentation and replay attacks in biometric authentication to a broader requirement for distinguishing human users from non-human agents in modern digital systems. The emergence of generative and agentic AI further amplifies this need, positioning liveness as a fundamental security primitive. Existing approaches face key limitations, including reliance on explicit user interaction, specialized hardware, vulnerability to increasingly realistic spoofing, and limited scalability in real-world deployments. We present A-Live, a passive liveness detection framework that operates solely on inertial measurement unit (IMU) signals available in commodity devices. A-Live is based on the observation that neuromuscular micro-motions inherent to human motor control produce subtle but measurable signatures in inertial data, which are often treated as noise in prior work. We design a lightweight feature extraction pipeline and a compact classifier suitable for real-time on-device deployment, and introduce a controllable physical micro-motion platform to evaluate robustness against engineered non-human motion. Extensive evaluation across Android and iOS devices, including both automated and real-user settings, shows that A-Live achieves over 99.5\% accuracy with low false acceptance and rejection rates. Our results demonstrate that neuromuscular micro-motion signatures provide a scalable and passive foundation for liveness detection under emerging AI-driven threat models.
arXiv:2606.05124v1 Announce Type: new
Abstract: After the success of 3D Gaussian Splatting (3DGS) for novel view synthesis, many works have explored how to also use it for geometric surface representation. However, extracting accurate geometric information directly from 3DGS remains challenging and can often reduce the appearance rendering quality. In this work, we show that 3DGS in its default form is inheritedly unsuited to represent texture and geometry at the same time, by training with complete ground-truth texture and geometry information. We also propose a simple solution by applying a single additional geometry opacity parameter to each splat, together with an optional transparency-curated optimization pipeline. Our experiments, both with ground-truth and vision foundation model geometric input, show that this change leads to improved rendering and geometry performance on a wide variety of dataset, and especially complex scenes with transparent objects benefit significantly from our method.
arXiv:2606.05122v1 Announce Type: new
Abstract: Large language models are increasingly evaluated by other models, raising a natural question: can a model predict how a judge will score its own output? We find that the ability is largely present before any targeted training: prompted few-shot, a base model already predicts an external judge's multi-attribute quality scores on open-ended responses well above chance across three benchmarks. We introduce Self-Evaluation Elicitation (SEE), a method that surfaces this latent ability through a short cycle comprising a calibration-coupled reinforcement learning phase that improves the answer and predicts the judge, followed by a masked distillation phase that sharpens the prediction while leaving the answer untouched. From 160 unique examples, roughly 31x fewer than a reinforcement learning baseline, SEE improves held-out calibration across three benchmarks while preserving answer quality. The elicited self-evaluation is sharply localized within the model's own token distribution and stable across judges it was never trained against, indicating a transferable notion of quality rather than a single judge's preference. These results reframe judge-aligned self-evaluation as a problem of elicitation rather than acquisition.
arXiv:2606.05121v1 Announce Type: new
Abstract: Audio is an inherently interactive modality, yet today's Large Audio Language Models (LALMs) are offline, and streaming audio models each handle only a single task such as streaming ASR or voice chatting. It is time to unify them into one online LALM: a model that, through an always-on perceive-decide-respond loop, listens to sound, environment, and instructions in real time and reacts on the fly. We formalize this regime as the Audio Interaction Model, and realize it with Audio-Interaction, a unified streaming model that retains offline task execution while adding online general audio instruction following, from dialogue to full voice chatting, deciding when to respond from the semantics of the stream. To enable this, we propose SoundFlow, a framework that instantiates the perceive-decide-respond loop end to end, from data to training to deployment, through streaming-native data construction, comprehension-aware training, and asynchronous low-latency inference for stable real-time interaction. We further construct StreamAudio-2M, a 2.6M-item streaming corpus spanning 7 fundamental abilities and 28 sub-tasks, and Proactive-Sound-Bench for evaluating proactive audio intervention. Across 8 benchmarks, Audio-Interaction preserves competitive performance on mainstream audio tasks while unlocking capabilities inaccessible to offline LALMs, including real-time ASR, streaming audio instruction following, and proactive help.
arXiv:2511.12298v4 Announce Type: replace
Abstract: We introduce a new direct multilevel method for solving arbitrary complex square linear systems that uses a regular smoother and an arbitrary but equal number of pre- and post-smoothings. Through careful analysis of the error propagation operator, we cluster the spectrum of this operator. This allows us to write a direct K-cycle version of the method.
arXiv:2404.11309v2 Announce Type: replace
Abstract: Achieving rotational invariance in deep neural networks without data augmentation is a research hotspot. Intrinsic invariance enables features to capture targets' inherent properties, enhancing deep learning performance in visual tasks. Based on various types of non-learnable operators, this paper proposes a comprehensive set of convolution operations that are natually invariant to arbitrary rotations. Unlike most prior methods, these rotation-invariant convolutions (RIConvs) have the same number of learnable parameters and a similar computational process as standard convolutions, making them interchangeable. Using the MNIST-Rot dataset, we validate their invariance across rotation angles and compare them with previous rotation-invariant CNNs, where two gradient-based RIConvs achieve state-of-the-art results. Then, we integrate RIConvs with classic CNN backbones and evaluate them on texture recognition, aircraft type recognition, and remote sensing image classification tasks. Results show that RIConvs significantly improve accuracy, particularly with limited training data, and enhance performance even with data augmentation.
arXiv:2606.05118v1 Announce Type: new
Abstract: This paper examines whether and how artificial intelligence (AI) advances scientific creativity. Drawing on scientific publications, the primary output of researchers, we analyze over one million publications from OpenAlex to investigate the relationship between AI adoption and multiple dimensions of scientific creativity, including novelty (recombinant novelty and object novelty) and impact (3-year short-run citation impact and 10-year long-run citation impact). We find that AI publications are significantly more likely to achieve top-decile creativity relative to non-AI publications, with 5.5 to 10.2 percentage point higher likelihood to rank in the top creativity decile. Critically, we uncover substantial heterogeneity across AI research modes. Tool-oriented AI research, which applies existing AI models to domain tasks, is associated with the largest gains in recombinant-based creativity, while Adaptation-oriented AI research, modifying AI models for domain-specific problems, is associated with relatively higher object-based creativity. These findings reveal that AI does not advance science through a single mechanism but through structurally distinct creative pathways that depend on how AI is incorporated into the research process. Our results contribute to ongoing debates about AI's role in science and carry direct implications for research evaluation and science policy, highlighting the need for assessment frameworks that can distinguish between recombinant and conceptual forms of creativity and that recognize how different modes of AI adoption produce fundamentally different types of scientific contribution.
arXiv:2606.05108v1 Announce Type: new
Abstract: We show that in discretised first-price auctions with complete information, if the buyers learn to bid with online gradient ascent, in time-average the outcome is (almost) the efficient outcome of the second-price auction. Our proof rests on two novel innovations in the analysis of online gradient ascent in normal-form games, which may be useful in a wider range of applications. First, we develop a potential-function-based argument for the analysis of gradient ascent in normal-form games, allowing us to deduce that certain strategies will not be played in time-average. We provide sufficient conditions which ensure this argument can be applied iteratively, resulting in a procedure reminiscent of iterative elimination of dominated strategies. Second, we develop a novel class of cubic "candidate potential functions", classifying a family of quadratic strategy modifications on the probability simplex against which online gradient ascent incurs no regret.
arXiv:2606.05107v1 Announce Type: new
Abstract: We propose a label-free approach to adapt powerful but generic vision foundation models to specialized scientific domains. Standard supervised fine-tuning is often ill-suited to these settings: labels are scarce, and task-specific training can collapse the model's generality and hurt robustness. We instead leverage metadata to adapt representations to new domains in a self-supervised manner. Our method, FINO, combines a standard self-supervised objective with flexible metadata guidance that handles both highly granular discrete metadata and continuous metadata. It encourages the representation to preserve informative factors while suppressing spurious ones. Across subcellular fluorescence microscopy, Earth observation, wildlife monitoring, and medical imaging, FINO consistently outperforms standard unsupervised domain adaptation and fully supervised adaptation. It also exceeds highly-specialized domain-specific state of the art, while using no task labels for backbone adaptation and only lightweight probes for supervision.
arXiv:2606.05106v1 Announce Type: new
Abstract: We investigate whether methods of human mathematics pedagogy can guide the training of language models toward arithmetic reasoning. Building on the GASING method -- an Indonesian pedagogy that solves basic arithmetic through a left-to-right procedure aligned with the causal order of token generation -- we operationalize each operation as a computational procedure whose execution trace is serialized into natural-language Chain-of-Thought (CoT) supervision. A small GPT-2 decoder (86M parameters) with a syllabic-agglutinative TOBA tokenizer for Indonesian is trained from scratch on this data using only a next-token prediction objective, without reinforcement learning or reward-based optimization. Monitoring training reveals three distinct learning phases, and mechanistic analyses -- attention-masking interventions on the CoT information graph, residual-stream probing, and logit-lens inspection -- show that the model first internalizes a procedural pathway and subsequently develops an associative, ``mental-arithmetic'' capacity that retrieves intermediate results without explicit step-by-step computation. The trained model reaches over 80% accuracy on held-out problems and attains competitive performance against substantially larger language models, indicating that targeted, pedagogically grounded training can yield strong and economical arithmetic capability at small scale.