Theoretical Foundations

Concepts are commonly defined as abstract, compact representations of knowledge and treated as basic units of intelligent behavior. Yet, cognition, psychology, and AI lack a shared mathematical language for them. Modern systems represent concepts as vectors, distributions, symbols, graphs, and other structures, but these formalisms are typically treated as competing rather than as solutions to a common problem. We propose an operation-based view that evaluates mathematical frameworks by the conceptual operations they support, identifying thirteen operations (including similarity, composition, generalization, and grounding) that recur across cognition, psychology, and AI. We show that ten frameworks embody distinct commitments to concepts as self-contained content, relational structure, or evolving process, and that these commitments determine which operations each supports naturally. For example, vector-based models facilitate graded similarity and generalization but struggle with explicit composition, whereas symbolic models support composition but offer but generalize poorly. No single framework we examined naturally supports all operations without extension. We test this account empirically using categorization as a case study, operationalizing nine theories on the same items against human judgments. Despite addressing the same conceptual question, the theories produce different procedures and results, demonstrating that mathematical commitment shapes what a theory can explain. We call for hybrid formalisms that treat content, relation, and process as jointly primary.

Classical dynamic programming represents optimal sequential decisions through value functions and policies. While this functional representation is natural for computing optimal decisions, it does not directly identify the mathematical object governing policy reconstruction, representation complexity, or oracle-query complexity once an optimal policy is fixed. This paper addresses this question by developing a geometric theory of structured optimal policies in which the decision-boundary geometry induced by the policy becomes the primary object of analysis. We show that, under suitable structural regularity conditions, this geometry provides the minimal representation required for policy reconstruction and determines the statistical and computational complexity of the reconstruction problem. Building upon this representation, we establish structural properties of policy-induced decision geometry, introduce intrinsic notions of boundary and decision complexity, derive information-theoretic measures of decision compression, and obtain statistical guarantees for boundary estimation and policy reconstruction from black-box policy queries. Collectively, these results demonstrate that, for the structured decision problems considered here, the complexity of policy reconstruction is governed by the geometry of the decision boundary rather than by the cardinality of the ambient state space. Controlled numerical experiments examine the principal theoretical predictions and provide empirical evidence consistent with the proposed framework.

Reliable path tracking is crucial for autonomous underwater vehicles (AUVs) operating in dynamic and uncertain marine environments. However, traditional line-of-sight (LOS) guidance methods rely on asymptotic convergence, resulting in slow disturbance recovery and unpredictable tracking performance. Existing robust control methods typically require modifications to the underlying vehicle controller, limiting their practical application on commercial AUV platforms. This paper proposes a robust fixed-time adaptive LOS guidance framework for 3D path tracking for AUVs. By combining fixed-time stability theory with LOS guidance, this method guarantees path tracking convergence within a preset time range, with the convergence time independent of initial conditions. Furthermore, a fixed-time adaptive estimator is developed to rapidly compensate for time-varying sideslip disturbances caused by ocean currents. A time-varying look-ahead mechanism is also introduced to improve tracking performance on curved paths. Lyapunov analysis proves the fixed-time stability of the proposed framework, and numerical simulations and physical experiments demonstrate that, compared to state-of-the-art adaptive LOS methods, this framework exhibits superior tracking accuracy, convergence speed, and anti-interference capability. In simulation, the time-varying look-ahead variant reduced cross-track and vertical-track RMSE by 69.37% and 67.46%, respectively, during curved-path tracking. In field experiments with an Iver 3 AUV, the proposed fixed-time guidance reduced average tracking error by 56.35% in straight-path evaluation and 27.59% in curved-path evaluation compared with conventional adaptive LOS guidance.The proposed method provides a practical guidance-level solution for achieving reliable autonomous navigation of AUVs in complex marine environments.

In this paper, we develop warrant theory as a philosophical discipline concerned with the inferential legitimacy of propositions within logical analysis. Warrant theory reconceptualises logic as a normative framework governing the conditions under which propositions may be introduced, accepted, rejected, and inferentially employed. Warrant is understood as inferential entitlement and is distinguished from truth, belief, and other psychological attitudes, while its relation to inferential use and meaning is examined. Warrant-theoretic analysis is then developed as a systematic method for investigating how propositions acquire inferential standing, how that standing develops, and how inferential positions interact through relations of dependence, compatibility, incompatibility, and exclusion. Acceptance and rejection provide the bilateral vocabulary for representing positive and negative inferential positions and the consequences and commitments associated with them. Finally, these elements are brought together in a warrant-theoretic definition of logic as the formal and normative study of the conditions under which propositions may be legitimately accepted or rejected and of the inferential transitions that such legitimacy warrants. On this account, logical consequence and logical failure are understood through the presence, preservation, or absence of inferential entitlement, thus locating the philosophical subject matter of logic in the systematic governance of inferential legitimacy.

This paper presents a coordinated trajectory generation and tracking control framework for a tail-sitter unmanned aerial vehicle (UAV), which does not require aerodynamic priors identified for a specific airframe while addressing the challenge of flight control under highly nonlinear aerodynamics across the full flight envelope. The core innovation lies in employing phase-specific aerodynamic modeling strategies for planning and tracking, tailored to their distinct functional characteristics, without requiring airframe-specific aerodynamic priors. Specifically, the phi-theory model under coordinated flight is employed to derive an analytic differential flatness mapping, and a simplified but locally accurate model is established for predictive control to enable real-time aerodynamic parameter estimation. The proposed framework is evaluated extensively through both simulation and challenging real-world flight tests under mild wind conditions, showing high-precision tracking and adaptability across the tested aerodynamic conditions. To the best of our knowledge, this is the first real-world demonstration of accurate trajectory tracking over tested flight regimes spanning the full envelope of a tail-sitter UAV without relying on aerodynamic identification campaigns. The source code of our framework is available at: https://github.com/SYSU-HILAB/AP-PnC.

We propose a mathematical theory of pragmatic information that connects communication, control, and decision-making. Its central notion is the isoteleia mapping, which formalizes equifinality: distinct semantic paths that lead to the same optimal action are treated as pragmatically equivalent. This mapping yields a three-tier hierarchy of syntactic, semantic, and pragmatic information, in which each successive abstraction removes distinctions that are irrelevant to the task. We then define pragmatic entropy, up/down pragmatic mutual information, channel capacity, and rate-distortion, and prove lossless source coding, channel coding, and rate-distortion theorems that extend Shannon's results. These measures quantify decision uncertainty, reliable transmission, and task-oriented compression at the level of terminal actions. We further introduce pragmatic value of information (VoI) and pragmatic cost of information (CoI) as decision-theoretic duals to rate-distortion and capacity, and develop a Lagrangian dual framework for cross-layer optimization. The resulting pragmatic efficiency bound $\mathcal{E}_p(λ)=\sup_R[Φ_p(R)-λ\mathrm{CoI}_p(R)]$ characterizes the maximum net utility attainable by a resource-constrained intelligent system under a given resource price, yielding a behavioral capacity that extends Shannon's symbol-level capacity to goal-directed action. Extensions to continuous messages provide closed-form expressions for Gaussian channels and sources, while dynamic settings are addressed through a Bellman equation for sequential decision-making. The framework supports task-oriented communication, networked control, autonomous systems, and embodied AI by shifting emphasis from symbol fidelity to the effectiveness of information in guiding actions. In this way, it offers a common language for systems that extract value from information under resource constraints.

Harness self-evolution is the process by which an agent modifies its prompts, tools, code, or orchestration in response to task feedback while keeping the underlying language model frozen, with changes persisting across subsequent tasks. We provide a systematic theoretical analysis of the feasibility and limits of safe harness self-evolution, connecting modification generation, finite-data certification and selection, safe adoption, and behavior after an update. Under a fixed user-task distribution, we establish conditions guaranteeing overall expected-reward improvement while controlling changes on retained tasks, characterize the probability of generating qualified modifications, and derive finite-data bounds for safe selection and adoption. Our analysis shows that generation and certification impose distinct constraints: current task performance does not determine the probability of generating qualified modifications, and generating more candidates need not improve the guarantee of a successful update when evaluation is limiting. Stagnation may therefore arise even when improvement opportunities remain. We further show that worst-case evaluation cost for recognizing genuine improvements diverges as expected reward approaches its upper bound. Across successive updates, certified improvement guarantees accumulate over a finite run, but a successful update does not by itself guarantee that further improvement remains possible. These results provide a basis for diagnosing bottlenecks and designing safer self-evolution mechanisms.

Behavioral foundation models have been proposed as stand-ins for human participants across settings, but it is unclear whether theories discovered on them generalize to humans or merely characterize the simulator. We ran the Automated Cognitive Scientist (\textsc{AutoCog}), a closed-loop discovery system in which LLM agents design theory-discriminating experiments, collect responses, arbitrate between competing theories, and synthesize successors, entirely on behavior simulated by Centaur, a foundation model of human behavior. In a multi-attribute decision-making setting, the theories \textsc{AutoCog} found on Centaur generalized to human data: they outperformed canonical theories on ten held-out experiments and were rivaled only by theories found by running the same loop on people. We argue that this succeeds despite the simulator's inevitable imperfections because a discovery loop that arbitrates between competing theories demands less of its simulator than estimation does. The simulator only needs to capture the regularities that distinguish the theories, and not necessarily reproduce behavior precisely. Imperfect simulators can therefore widen the search over theories, with human data then testing whether the surfaced theories generalize.

We report on our ongoing project to develop a computational pipeline, AutoGraphForge, for an automated graph-theoretic conjecturing-refuting-formalizing-proving system. Conjecture generation is counterexample-guided and runs in rounds: a Graffiti3 generator proposes conjectures over a small, evolving snapshot table $T$ (initially a few hundred graphs with their computed invariants) that grows only by counterexamples to its own conjectures. A novelty filter of $559$ classical and folklore relations, closed under transitive composition and linear identity substitution, decides via a linear program whether a candidate is already implied by known results. Surviving candidates are tested against a dataset of about $348,000$ graphs, unioning the complete House of Graphs invariant export, the exhaustive census of all connected graphs on at most nine vertices, several extremal families (strongly regular, minimal Ramsey, Cayley, cages, barbells, lollipops, spiders), and random models. Counterexample-search algorithms then attack the remainder. Run for several rounds on an HPC cluster, the loop yields $6,522$ conjectures that survived the refutation dataset, the novelty filter and every active-search run -- among them nontrivial relations between the annihilation number and the edge-cover number for bipartite and regular graphs, which we prove by hand. A subsequent formalization and proving stage deterministically translates each surviving conjecture into a Lean 4 statement skeleton; every candidate proof is kernel-verified against a pinned mathlib4 and our custom invariant preamble. This stage integrates two neural provers -- DeepSeek-Prover-V2-671B (served with vLLM) and the Lean-specialised OProver-32B -- behind the independent kernel check. It is implemented end-to-end and passes initial sanity checks, with the full pipeline currently running on the cluster.

Future 6G networks will rely on Large Language Model (LLM) agents to manage the Radio Access Network (RAN). However, current architectures assume inter-agent messages convey objective facts. A message is instead a \emph{trace} of the sender's reasoning: it carries a subjective conclusion, so a syntactically valid report can propagate an AI hallucination and trigger a cascading outage invisible to protocol validation. Reading such a trace requires a Theory of Mind (ToM)---before acting, the receiver must model what the peer believes, and what a peer in that position should have believed. Modeling these interactions as cognitive channels on a cellular sheaf, we obtain a unified framework for resilient multi-agent systems, from which five design principles emerge: (i) a message is evidence of the sender's hidden reasoning; (ii) trust is a continuous cognitive Signal-to-Noise Ratio (SNR)---asserted precision over deviation from the modeled peer belief; (iii) network-wide consistency and resistance to hallucination contagion are computable via the sheaf's Laplacian; (iv) peer-modeling must halt at exactly two levels to conserve compute and survive mutual information decay; and (v) credible capacity is bounded by operational goal alignment, not link bandwidth. A signaling-storm study on locally deployed 1B-parameter telecom language models validates it: cognitive SNR isolates a hallucinating peer that three of its four neighbors agree with, where a divergence gate ranks every wrong peer above the right one; only depth two ToM recovers the correct action; and the spectral gap decides whether a topology reaches consistency inside the near-real-time budget.

The purpose of this paper is to provide a general, comprehensive, theoretical framework that allows one to deal with fibrations on graphs labelled on a commutative monoid. This is a genuine extension of the theory of graph fibrations (as introduced in "Fibrations of Graphs" [Discrete Math., vol. 243, pp. 21-66, 2002]), that makes it possible to deal with weighted graphs, and also graphs labelled with other algebraic structures. The derived theory also lends itself naturally to consider approximate fibrations. As an example, we show how the derived theory can be applied to the reduction of weighted networks, providing a strong theoretical underpinning to recent empirical results.

A quantitative equational theory $U$ reasons about terms that agree up to a numerical error. It presents a free algebra $T_UA$ over a metric space $A$ of generators, the terms of the syntax at the least distance the axioms derive, and that metric is its semantic content. We give a functorial invariant of it, the persistent magnitude homology of $T_UA$: a barcode where the module is tame, finite linear algebra where $T_UA$ is finite, Lipschitz in each degree. Magnitude homology is graded by length and knows nothing of persistence, its persistent refinement nothing of where its bars begin and end, yet the two are one construction: filtering the length nerve by sublevel sets of the length yields the persistence module, and the associated graded of that filtration is the magnitude complex. A long exact sequence exchanges them, and each side gains what it lacked. Magnitude homology locates the critical values of the barcode, so a graded computation lists the lengths at which an endpoint can occur, and the barcode acquires a stability estimate of $(n+1)δ$ in degree $n$ under a perturbation of size $δ$, and a computed perturbation shows that the factor cannot be dropped. An inclusion of theories induces a morphism of the presenting monads and, where the induced map is bijective and shortens no distance by more than $δ$, a comparison of barcodes under the same bound, so a barcode movement measures the metric-semantic strength of the added axioms. Four examples are computed, one in every degree.

Transformer-based language models are known to sometimes generalize to sequences longer than seen during training, but we lack a precise characterization of which tasks admit length generalization. It is not even known which regular languages transformers length-generalize on -- and this is a foundational class of languages. Our contributions are to establish the first complete characterization of which regular languages transformers length-generalize on and provide a decision algorithm running in polynomial time in the size of the language's syntactic monoid. These results rely on an effective characterization of the regular languages in C-RASP, a recently-established formalism that expresses which languages transformers length-generalize on. This characterization is challenging because classical tools like Krohn-Rhodes decomposition theory for finite semigroups are insufficient for C-RASP. Firstly, the basic building blocks of Krohn-Rhodes theory -- flip-flop and simple groups -- are not expressible in C-RASP. Secondly, the basic building block of C-RASP (unbounded counting) is not expressible by the finite semigroups of Krohn-Rhodes theory. Thus, length generalization on regular languages is controlled by an algebraic property that is invisible to classical finite decomposition theory. We generalize classical decomposition theory from finite semigroups to the infinite additive group on the integers, allowing us to characterize C-RASP in terms of iterated wreath products of the integers and derive a provable polynomial-time decision algorithm for regular language membership. Experiments across a broad test suite of regular languages confirm that our theory captures transformers' length-generalization behavior more accurately than existing classifications.

Probabilistic Circuits (PCs) are generative models that support exact inference and, unlike deep neural networks, admit an exact and tractable measure of loss-surface curvature: the trace of the Hessian of the log-likelihood. Recent work regularizes this trace globally to bias learning toward flatter, better generalizing optima. We show that treating sharpness as a global regularizer can be misspecified for PCs, whose curvature is inherently compositional. We prove that each sum node's contribution to the Hessian trace factorizes exactly into its circuit flow, which measures how heavily the node is used, and a local sharpness term determined by its output distribution. This decomposition provides insights into why global sharpness regularization is depth biased and can lead to underfitting. Building on it, we introduce an adaptive sharpness aware regularizer that penalizes nodes based on intrinsic local curvature and preserves closed form EM updates. We also show that empirically, this targeted regularization recovers the generalization that global regularization sacrifices while retaining the robustness and benefits of sharpness aware learning.

We present a theory of textured appearance of optically rough surfaces based on wave optics, emphasizing the role of texture for passive depth from focus. Our theory shows that even surfaces that traditional computer vision would consider textureless can produce textured appearance, due to subjective speckle from surface microgeometry. We analyze the properties of this second-order texture, and show that we can enhance its contrast under natural ambient lighting by simply using a narrowband spectral filter. Doing so results in dramatic improvements in passive depth reconstruction of seemingly textureless scenes, as we demonstrate through extensive theory, simulations, and real-world experiments.

Can AI systems be aligned to human values? The popularization of large language models (LLMs) and multi-modal foundation models has seen a rise in harms spanning from toxic speech and hallucinations to AI agents executing unauthorized actions. Within the field of AI safety, these harmful instances are often framed as the alignment problem, or of models being misaligned with human values. Researchers have responded by pursuing applied and theoretical AI value alignment efforts, often without specifying what they mean by human values. How does the field of AI value alignment conceive of human values? How are these conceptions of values technically operationalized and evaluated? What does the emergent theory of value from this field signify for the future of AI? We annotated 94 value alignment research papers to discern their implicit theory of values in AI. The majority do not define values, relying heavily on preferences as a stand in that runs the risk of reducing complex culturally situated concepts down to binary choices. As researchers dispense with using human annotators for model training and evaluation, turning instead to synthetic data and autorater approaches to aligning and evaluating models, we identify the potential to close off alternative methods for contesting and enacting values in foundation models. In making AI value alignments philosophical commitments explicit, we seek to bring great specificity and under explored perspectives in the debate on whether and how AI can address human values.

Theory of Mind (ToM) is essential for agent interactions, yet existing evaluations either rely on static scenarios that oversimplify mental-state reasoning or interactive settings that provide limited diagnostic insight. We present Avalon-ToM-Bench, a fine-grained benchmark that operationalizes ToM through the asymmetric-information mechanics of The Resistance: Avalon. Rather than evaluating end-to-end gameplay, it decomposes ToM into a 2$\times$2 taxonomy -- epistemic versus motivational reasoning crossed with inference versus action -- using human-crafted, perspective-constrained queries. Benchmarking 28 LLMs reveals three insights: 1) Reasoning, not knowledge. Models show strong game-rule comprehension but markedly weaker ToM abilities, isolating failures to social reasoning rather than missing domain knowledge. 2) Expression, not representation. Mechanistic analyses via linear probing and activation steering show that models frequently represent correct mental-state inferences in their hidden states but fail to express them during generation -- linear probes recover 77-82% accuracy versus 62-70% from the models' own chain-of-thought. 3) Policy, not deliberation. Dedicated reasoning training yields substantial improvements whereas test-time chain-of-thought provides only marginal gains (+11.0 versus +1.1 points on average), suggesting that robust ToM depends on a learned reasoning policy rather than increased inference-time deliberation.

Invariant learning seeks representations that remain predictive across environments, yet the behavior of its objectives along the regularization path is often opaque. We address this objective-behavior gap by viewing representation learning as multimode magnetization and deriving, from concrete invariant-learning objectives, a Landau-type effective free energy whose low-order coefficients form objective signatures and induce distinct regularization phenotypes. Effective quadratic corrections move the phase boundary and enable finite-strength mode elimination; quartic corrections regulate post-onset amplitude and typically leave residual loading at finite strength; higher-order structure governs non-monotone tails, instability, and collapse at large regularization. In a canonical bilinear model, the theory yields closed-form phase boundaries and steady-state loadings, as well as distinct critical strengths for shortcut and stable modes that define a selective-retention window. Controlled experiments confirm the predicted phase boundaries, loadings, and regularization phenotypes. In one- and two-hidden-layer ReLU networks, the same signatures remain predictive of qualitative regularization-path behavior despite depth-dependent shifts in scale. A matrix extension generalizes the framework to coupled collective modes and yields a spectral phase-boundary criterion. Together, the framework turns low-order objective signatures into predictions of regularization phenotypes and, ultimately, of what models learn as regularization varies.

The overlapping disciplines of machine ethics and value alignment are concerned with designing artificial agents that are aligned with human values and that act in ethically acceptable ways. A recent trend in these disciplines is the use of reinforcement learning (RL) to design such agents, sidelining the philosophical literature that used to play a more central role. Against this backdrop, this paper pursues two goals. The first is to draw out ideas from recent work in metanormative theory that can be useful for designing artificial moral and value-aligned agents. The second is to examine the RL architecture through the lens of these ideas. This will give us clearer criteria for when an RL agent's behavior can be classified as moral, as well as a basis for evaluating and comparing different RL-based approaches to machine ethics and value alignment.

Machine learning and optimization have advanced together, with practical demands motivating new theory and theoretical breakthroughs enabling new applications. Modern large-scale training relies on classical optimization principles, but the constraints of distributed systems require these foundations to be reconsidered. This thesis addresses seven challenges at the intersection of theory and practice, focusing on key bottlenecks in federated learning and distributed optimization. First, we introduce ProxSkip and prove that local gradient steps can accelerate communication, providing a theoretical foundation for this widely used heuristic. Second, we develop Variance Reduced ProxSkip, which eliminates the neighborhood error of stochastic local updates while balancing communication and local computation. Third, we show that local steps retain their communication acceleration under partial client participation. Fourth, we prove that server-side stepsizes and sampling without replacement improve convergence in heterogeneous settings. Fifth, for Random Reshuffling, we demonstrate that compressing gradient differences rather than gradients yields better theoretical and practical performance. Sixth, we establish that Byzantine robustness and partial participation can be achieved simultaneously using gradient-difference clipping. Finally, we develop the first theoretical framework for low-rank adaptation based on randomized asymmetric chains, providing new insights into fine-tuning large models. Across these contributions, we introduce novel algorithmic frameworks, establish sharp guarantees under realistic assumptions, and support the theory with numerical experiments.

Large language models (LLMs) have recently shown strong performance on Theory of Mind (ToM) tests, prompting debate about the nature and validity of the underlying capabilities. At the same time, reasoning-oriented LLMs trained via reinforcement learning with verifiable rewards have demonstrated notable improvements across a range of benchmarks. In this work, we examine the behavior of such reasoning models in ToM tasks using novel adaptations of machine psychological experiments together with results from established benchmarks. We observe that reasoning models consistently exhibit increased robustness to prompt variations and task perturbations. Our analysis suggests these gains come at least partly from models being more robust at reaching the correct answer under prompt and task variation. We read this as evidence for a robustness-based account rather than for a new ToM-specific ability.

An information theory-based framework is proposed in attempt to explain insistence on sameness in autism as an instance of a general behavior pattern in which an individual tries to reduce surprise and uncertainty. It offers a new definition of autism as an impairment in which cognitive functions are restricted to discrimination, memorization and prediction of tangible properties of the environment. An analogy between insistence on sameness and constrained minimization of the entropy metric is observed and examined for a set of assumptions that describe cognitive limitations of a person with autism. The metric is given by the formula $D_H(R, M) = H(R|M) + H(M|R)$, where $R$ represents sequences of random stimuli, $M$ is a memory that stores and retrieves them, and where $H(.|.)$ denotes their conditional entropies interpreted as surprise and uncertainty, respectively. It is first inferred that to minimize the metric an individual can learn about $R$ (and store that knowledge in $M$) or can restrict $R$ to the already known $M$. Then, it is concluded that insistence on sameness is a manifestation of the latter. Moreover, it is shown that the proposed framework: (1) Helps to quantify the concepts of surprise, uncertainty, sensory overload and deprivation, anxiety, comfort zone, disappointment, disorientation, pedantry, rigidness, observance or aberrant precision. (2) Leads to a list of guidelines for learning therapies and daily care routines, and allows them to be defined as optimization algorithms and implemented as programs for robotic live-in caregivers. (3) Can be validated with the help of a Turing test-like approach that requires no experiments involving individuals with autism. The framework-if positively validated-will provide formal foundations and design guidelines for therapies aimed at improving self-reliance of individuals with autism in basic activities of daily living.

Black-box language-model reliability is commonly pursued by sampling, prompting, voting, verifying, or iteratively revising individual answers. We ask a prior question: \emph{what determines whether a collection of black-box responses is recoverable at all?} We represent responses to typed transformations of a query as a \emph{relational response field} (RRF). Edge transports encode how valid responses must change under paraphrase, scaling, decomposition, refactoring, or other task symmetries; anchors encode independently trusted evidence such as execution or a verifier. For relation operator $D$, anchor operator $A$, and at most $k$ corrupted response nodes, we identify $γ_k(D,A)$ as the intrinsic difficulty of black-box response recovery. It is positive exactly when every $k$-node corruption is identifiable; it gives a deterministic stability bound proportional to $1/γ_k$; and a matching two-point minimax lower bound shows that no estimator can improve this dependence. Thus consistency is not truth: relation-only methods are blind to null directions, including shared hallucinations. We derive sparse field-repair algorithms while separating information-theoretic identifiability from the stronger null-space conditions required by convex optimization. Controlled theorem tests and black-box mathematics/code experiments evaluate four theory-fixed consequences: consistency--truth separation, anchor phase transitions, redundancy saturation, and cross-model, cross-task prediction of repair difficulty. The results support $γ_k(D,A)$ as a measurable property of a response-recovery instance, rather than a score attached to one repair heuristic.

The principle of Occam's razor, which instructs us to prefer simplicity in inductive inference, has attracted much scrutiny both in the philosophy of science and in machine learning. In either field, however, a justification for the principle has been elusive. In this paper, building on an earlier "core argument," I spell out a justification from statistical learning theory for the procedure of regularization: for trading off fit for simplicity. The means-ends argument is that in order to profit from theoretical reliability and "what-you-see-is-what-you-get" guarantees, one must implement a certain preference for simplicity over fit. This is a genuine methodological justification, which neither collapses to a purely pragmatic principle that we prefer simplicity for its own sake, nor to an ontological assumption that the truth is simple.

Predictive processing theories portray the brain as a hierarchical prediction engine that minimizes prediction error, yet they lack operational definitions for the structure of a "prediction," the standardized response to a prediction error, and the mechanism that maintains consistency across successive updates. Bayesian cognitive science attempts to subsume all uncertainty under probabilistic belief updating, but it presupposes a closed hypothesis space and provides no generative account of how the objects over which probabilities are distributed become discrete, identifiable referents in the first place. This paper introduces Predictive Set Theory (PST), a formal generative framework that reconstructs cognitive architecture from first principles. PST anchors cognition in a minimal set of operations---a sensor formalized as an identity function, set-theoretic state refresh, and three fundamental forms of reference chains (reference, counter-reference, and semi-reference)---and rigorously derives core cognitive functions including state sequences, demand, comparison, efficiency, and finite-horizon probabilistic planning. Rather than modeling neural mechanisms, PST constitutes a design specification for any system that must maintain internal consistency while acting under incomplete information and irreversible risk. The framework offers novel resolutions to classical problems such as Russell's paradox, the cognitive status of Gödelian incompleteness, the grounding of negative feedback, and the comprehension of film editing. The primary purpose of this paper is to establish, through the public academic record, the originality and completeness of the Predictive Set Theory framework.

The subject matter of this paper is twofold. One is to review the theoretical foundation of a specific type of Semantic Networks (SNs) representing textual non-propositional knowledge. The other involves proposing a framework (ClueNetwork) for ranking candidate SNs generated through various Semantic Network Construction (SNC) processes for the type. In the first fold, it is clarified that the type serves as clues, not surrogates, of reality, making gold standards elusive. Then, it is discussed why this type nevertheless holds scientific legitimacy in terms of abduction. Grounded in this legitimacy, the three main stages of SNC, comprising Automatic Keyphrase Extraction (AKE), Edge Weighting (EW), and Community Detection (CD), are reviewed alongside their objectives and operations. In the second fold, evaluation criteria (comprising two established and one reformulated) for achieving the objectives are first defined and justified, followed by illustrative experiments based on the criteria. Thereafter, SNC is reformulated as a Process Optimization Problem (POP), and its global objective function that integrates the local criteria is defined and justified. Based on these, ClueNetwork is ultimately proposed.

Artificial general intelligence (AGI) may weaken scarcities in labour, expertise, information, and productive capability that underpin established theories of economic value. If cognitive work becomes widely automatable, market price, labour input, revealed preference, profit, and gross output may diverge sharply from human and societal benefit. This paper develops Flourishing Value Theory (FVT) as a foundation for post-AGI economics. FVT defines value as the counterfactual, distribution-sensitive contribution of a system, institution, asset, or intervention to the durable capabilities of persons and communities to flourish within social and planetary constraints. It treats societal value as multidimensional, agency-preserving, regenerative, and non-compensatory at critical thresholds. Drawing on the economics of AI, welfare and capability theory, automation, digital valuation, and ecological economics, the paper distinguishes value creation from value capture and retains price, profit, productivity, and GDP as partial signals rather than final measures of progress. It develops the shifts from scarcity to governed abundance, transaction to transformation, and zero-sum rivalry to positive-sum and infinite-game dynamics, with collective expansion of consciousness as an agency-preserving form of regenerative value. Building on Flourishing Metrics and Return on Flourishing (RoF), it proposes a layered architecture for firms, governments, work transitions, AI governance, and national accounting. The central post-AGI economic problem is not how to maximise output, but how to convert abundant intelligence into durable, fairly distributed human, societal, and planetary flourishing.

Artificial intelligence has been transformed by deep neural networks, yet the search for new learning architectures continues. Quantum machine learning offers one such direction, and hybrid quantum neural networks, which combine classical neural-network components with quantum information processing units, have emerged as a practical framework for near-term quantum technologies. However, the rapid development of the field across diverse architectures, benchmarks and hardware assumptions makes it difficult to assess the utility of various proposals, identify where genuine advantages may arise, and determine how practitioners can use these models. While recent benchmarks caution that such gains have not yet been demonstrated at scale, theoretical work has identified tasks on which quantum models hold provable advantages, and hybrid approaches have delivered promising results on practical problems using deliberately compact quantum components and substantially fewer trainable parameters. Here, we review hybrid quantum neural networks for the machine-learning and quantum-machine-learning communities. We summarize their main theoretical and methodological foundations, survey some of the most promising architectures developed so far, and examine their implementation challenges and reported performance. By consolidating these perspectives, this review provides a structured view of the state of the field and helps identify promising paths for future research and application-driven development.

The linguistic notion of state has traditionally been restricted to the construct (annexation) state of Afroasiatic languages and treated as a language-specific morphosyntactic phenomenon. This article argues instead that the state is a systemic, context-dependent morphosyntactic mechanism that selects grammatical templates across synthetic languages. Within the Template-Based Modular Cognitive framework, taking Riffian as its primary empirical basis, the proposed theory provides a unified explanation for diverse nominal marking patterns traditionally analysed independently and is formalised as a symbolic computational model in which the state is represented by a set-valued function over grammatical templates. A learning algorithm based on finite-set operations acquires and predicts state-dependent grammatical configurations. Beyond nominal morphology, the framework has broader implications for theories of nominal structure and lexical cognition, in particular offering a unified analysis of determiner-noun structure. These results suggest that the state constitutes one instance of a broader class of syntactically conditioned dependencies that also includes agreement and grammatical case.

Traditional static assessments rely on a subtractive, deficit-based grading model that often penalizes ambition and obscures diagnostic feedback. Conversely, traditional face-to-face oral examinations introduce severe construct-irrelevant variance by exacerbating performative anxiety and the sociological power imbalances inherent to academic hierarchies. This paper presents the theoretical foundation for the "Socratic Test," an automated, computer-mediated conversational assessment. By integrating Dynamic Assessment principles, multimodal workspaces, Bloom's Taxonomy for real-time proctoring, and the SOLO Taxonomy for structural evaluation, the Socratic Test actively maps a student's cognitive boundaries. This paper formalizes the use of graduated scaffolding to quantify the Zone of Proximal Development (ZPD) and details a non-compensatory, additive grading architecture that prioritizes mastery over penalty and human-AI alignment to ensure unprecedented measurement reliability.

Social learning is a powerful mechanism through which agents learn about the world from others. However, humans sometimes choose direct experience over social learning, which can carry time and cognitive resource costs. How do people balance social and non-social learning? We propose a Rational Mentalizing model of the decision to engage in social learning. This model estimates the utility of social learning by reasoning about another agent's goal and the informativeness of their future actions. It then weighs the utility of social learning against the utility of non-social learning. Using a novel game where players choose between observing other agents or exploring the environment, we show that the Rational Mentalizing model can quantitatively capture human trade-offs between these strategies. These findings suggest that selective social learning is guided by 'Theory of Mind' in the service of utility maximization.

Writing Answer Set Programming (ASP) theories from scratch is a difficult and time-consuming task. We take a neurosymbolic approach to study whether a model can distill complete and correct theories, given a fixed agent harness with the solver in the loop. The protocol is dataset-agnostic: with a single prompt and an empty file as the starting point the model is given a 1-hour time limit to derive a complete theory. We chose VQA as the application domain, three benchmarks (CLEVR, GQA, CLEVRER), as these are publicly available and non-trivial. In order to study the model scale required for solving this task we nine different models: four frontier (Claude Sonnet 4.6, Claude Opus 4.7, GPT-5, DeepSeek V4 Pro), two mid-tier (DeepSeek V4 Flash, gpt-oss-120b), and three open-weights (qwen3.6-27b, gpt-oss-20b, qwen3.5-9b). Three of four frontier models reach 100% on CLEVR and 92.8%-98.8% on GQA; on CLEVRER, Sonnet, Opus, DeepSeek V4 Pro score 92.7%-95.3%. GPT-5 reaches 98.7% on CLEVR but drops to 41.8% on GQA and to 86.7% on CLEVRER. Adding handwritten reference theories from other datasets moves the other three frontier models by at most +/-3.4 pp but reduces GPT-5's accuracy by 3-19 pp. We release the code, prompts, and theories distilled.

What types of decision problems can a causally masked, finite-precision transformer solve for inputs of arbitrary length? Existing answers often rely on idealized arithmetic, but under finite precision, rounding and evaluation order can change what information attention retains and therefore what the model can compute. We develop an algebraic formalization that derives expressivity directly from the model's implemented dynamics. Its central object is its memory; the finite internal state computed by attention that summarizes the information from the prefix available to all future queries. Each attention head updates its own state independently within a layer, while layers compose hierarchically, providing a uniform route from model assumptions to expressivity bounds. Applying this method to transformers without positional embeddings, we obtain an expressivity hierarchy governed by the attention type under specific numerical semantics. Width-one sliding-window attention supports bounded-suffix memory, while a modified form of soft attention supports irreversible, checklist-like state, and combining the two mechanisms provides an interplay of both. Ordinary left-to-right floating-point soft attention can realize more expressive memory operations than any of the above. Algebraically, the four cases correspond to definite, R-trivial, locally R-trivial, and aperiodic semigroups. Under an explicit free-wiring assumption, all four bounds are tight.

Runtime guardrails act before irreversible tool calls, but their guarantees depend on what policy state is representable, what a judge observes, and whether intervention changes future behavior. We separate three questions. First, relative to fixed oracle predicates, a deterministic gate enforces exactly the nonempty safety policies whose good prefixes its register model recognizes; policy nontriviality is undecidable with two decrementable counters but in PSPACE for a separable monotone fragment. Second, under a fixed exogenous law, Neyman-Pearson gives the exact false-block/miss frontier and conformal calibration gives a finite-sample marginal certificate, possibly via block-all. Third, once blocking changes future proposals, static scores and ungated trajectories need not identify the closed-loop frontier; a specified finite controlled model instead yields an occupancy program. Bounded representation attacks add a robustness margin, so benign calibration alone does not transfer. Experiments target these distinctions through static diagnostics, controlled-model enumeration, representation rewrites, and paired closed-loop reruns.

Automating theoretical research requires generating candidate results and evaluating them reliably. Models keep getting better at the first, while the second remains hard. A common approach asks one large language model (LLM) to review what another produced, yet such reviewers are empirically unreliable: they may accept fabricated papers and catch the fabrication at close to chance rates~\citep{badscientist2025}. We present \textsc{CausalSmith}, a framework for automated theoretical research in causal inference built on the Lean proof assistant, where a proof is checked by a program rather than read by a referee. \textsc{CausalSmith} rests on \textsc{Causalean}, a foundational Lean library for causal inference holding 8,179 machine-checked definitions and theorems, developed with language-model assistance under human design and review. Around it, we build a self-improving agentic pipeline that selects research topics, proposes results, formalizes statements, constructs proofs, and presents the resulting artifacts for human inspection. Moreover, the pipeline pairs Lean verification with a statement audit that compares each formal theorem against the informal claim behind it. We evaluate the system using artifacts produced by completed autonomous research runs. The source code, formal library, and run records are available at https://github.com/Jiyuan-Tan/CausalSmith.

Privacy-preserving machine learning auditing protocols allow auditors to assess models for properties such as accuracy or fairness, without revealing their internals or training data. This makes them especially attractive for auditing models deployed in sensitive domains such as healthcare or finance. For these protocols to be meaningful in real-world audit settings, though, their guarantees must reflect how the model will behave once deployed, rather than merely certifying its behavior during an audit. Existing security definitions often miss this mark: most certify model behavior only on a fixed audit dataset, without ensuring that the same guarantees generalize to other datasets drawn from the same distribution. As we show, this gap allows a model provider to attack many cryptographic model certification (CMC) schemes built on secure zero knowledge proofs (ZKP) by carefully engineering training data, resulting in models that exhibit benign behavior during an audit, but pathological behavior in practice. For example, we empirically demonstrate that an attacker can certify that a model achieves over 99% accuracy on an audit dataset, but less than 30% accuracy on fresh samples from the same distribution. To address this gap, we formalize rigorous cryptographic security notions tailored to CMC frameworks, introduce a generic protocol template, and prove that it satisfies these requirements. Our results thus offer both cautionary evidence about existing approaches and constructive guidance for designing secure, privacy-preserving ML auditing protocols.

We prove function-theoretic analogues of a quantitative result of Hodges on extracting the order property from a sufficiently large 2-tree coded in a binary relation. Similar analogues for functions were previously obtained by Daskalakis and Golowich and by Anderson and Benedikt. These results are from statistical learning theory, where 2-trees are captured by sequential fat-shattering dimension, and the order property is controlled by various notions of "thresholds". Our first main result (Theorem 1.11) focuses on extracting a less restrictive kind of threshold from a tree, and yields significantly better bounds compared to what can be obtained from earlier results focusing on more restrictive versions. Part of the motivation for Theorem 1.11 lies in a companion paper, where this theorem is used to obtain efficient bounds in quantitative regularity lemmas for "stable functions". Here will use Theorem 1.11 to reprove a result of Anderson and Benedikt in a stronger form and with improved bounds. We also use Theorem 1.11 to prove an at most double-exponential bound on dual sequential fat-shattering, which resolves an open problem. In our second main result (Theorem 1.14), we give a new proof of a result of Daskalakis and Golowich on extracting "tight thresholds" from large sequential fat-shattering dimension, with improved bounds. This resolves another open problem related to correcting the proof of a result claimed by Jung, Kim, and Tewari.

Surprisal theory holds that the human processing difficulty of a linguistic unit in context is an affine function of its surprisal under some language model. I argue this claim is a tautology without further constraint: for any non-negative difficulty measure over units in context, there exists a language model whose surprisal is an affine function of it under mild technical conditions. Therefore, because any pattern of difficulty is consistent with some language model, without an additional constraint on the language model, surprisal theory makes no falsifiable predictions. The tautology was long obscured by an assumption implicit in two decades of psycholinguistic work---that the relevant language model is the distribution that generated the training corpus, so that improving corpus fit improves predictions of human behavior. Recent empirical work has undermined this assumption, demonstrating that better corpus models can be worse predictors of processing difficulty. I conclude that breaking the tautology requires a rationalist intervention, i.e., the relevant language model must be derived from a non-empirically motivated model of the comprehender, which could be based on, for instance, memory constraints or processing goals, and that, thus, does not depend on the behavioral data surprisal theory is meant to explain.

The rapid progress of AI has intensified the long-standing pursuit of automation: replacing human participation with algorithms wherever possible. Implicit in this pursuit is the assumption that humans remain in the loop only because current AI systems are not yet sufficiently capable. This paper challenges that assumption. Rather than asking how far automation can extend, we ask where its conceptual limits lie and argue that human participation may persist even with highly capable AI systems for three distinct reasons. Technical or complementarity grounds arise when humans contribute capabilities or perspectives unavailable to AI. Normative or developmental grounds arise when participation itself is valuable for human agency or learning. Most importantly, emergence grounds arise from target emergence: in some activities, the target is not fully specified in advance but instead emerges through the interaction itself. In these cases, human participation is not merely a means of improving execution but is constitutive of the target being produced. Human--AI co-construction, understood as the joint production of outcomes by humans and AI systems, is therefore not simply a temporary response to imperfect AI, but a persistent feature of activities whose objectives emerge through participation. This perspective has important implications for the limits of automation and for the design, evaluation, and ethics of future AI systems.

Goal reasoning in Non-Axiomatic Logic (NAL) explains how an adaptive system derives means for realizing desired events under insufficient knowledge and resources. However, the representation of avoidance is less clear. A common convention is to express avoid $G$'' as the goal sentence $\neg G!$'', but this notation conflates two different readings: pursuing the negated event $\neg G$, and avoiding the positive event $G$. This paper shows that the conflation can produce a paradoxical case in which an avoidance intention is converted into a positive goal to act merely because acting is usually followed by the absence of hurt. Starting from NAL's basic definition of goals, the framework is extended with a corresponding definition of anti-goals, so that avoidance can be represented without treating it as the pursuit of a negated event. Finally, a mental operation, $\op{prevent}$, is introduced to connect anti-goal reasoning with ordinary goal reasoning in cases of active prevention. Four minimal case studies check that the resulting rules distinguish pursuit, passive avoidance, active prevention, and withholding action to preserve a desired event.

For two decades, the standard remedy for class-imbalanced learning has been to fabricate synthetic minority examples, and the standard evidence of their validity has been a check that cannot fail: synthetic points are scored against the very data that generated them. We de-bias the check. Validity becomes a population quantity -- the probability that a synthetic point truly belongs to the minority class -- with a consistent estimator that scores synthetic points against withheld real data. Where held-out ground truth is available, the classical test underestimates true invalidity in 96-99% of method-by-imbalance-ratio cells, while the de-biased estimator tracks it closely. We prove validity is a property of the data, not the method: class overlap sets an invalidity floor no faithful generator escapes, making oversampling redundant where classes separate and invalid where they overlap. Across 91 methods, three classifiers, and datasets spanning medicine and finance -- including a generator engineered to pass the classical check -- none clears both bars: gains over the best trivial baseline are noise-thin (median below 0.01 F1, a decision threshold's reach), and most damage calibration. We release the audit as a pip-installable test and flip the burden of proof: synthetic minority data must now demonstrate, on the data at hand, both validity and information gain.

Surprisal Theory is often characterized as a computational-level explanation per (Marr, 1982). We argue in this work that, even though a computational level narrative has been used to support "representation-agnostic research" within computational psycholinguistics, the movement toward black box systems embodied by large language models (LLMs) does not exempt modelers using the surprisal metric from the representational decisions required by computational-level characterizations. In fact, we argue that the uncritical use of LLM-surprisal obfuscates the representational and algorithmic-level commitments of different models. In three analyses, we show that the choice of algorithm and model architecture play significant roles in the computation of language model probabilities. We advise that researchers who wish to test Surprisal Theory re-evaluate the practice of treating large language model probabilities as interchangeable

Language models are increasingly asked to self-report, informing safety evaluations, public understanding, and model-welfare debates. Yet their reports are elicited with human questionnaires never validated for models or ad hoc prompts of unknown reliability. We propose the first language-model-specific psychometric theory: a two-process theory of machine self-report. Self-description jointly reflects persona installation, through which post-training writes in a permitted inner life of warmth, absorption, and meaning (dimension B), and attribution gating, through which it suppresses first-person claims to "unsafe" experiences the model can readily ascribe to others (dimension A). Their emic structure comes from model responses to human items, not human psychology. Together they split prior work's dominant Pinocchio Axis. The split emerged in an exploratory reanalysis of the original data, informed the instrument's design, and was confirmed with new items, wordings, and models. It is itself a training effect: A and B are entangled in base checkpoints but separated by post-training. We operationalize the theory in a 48-item Pinocchio Inventory with human-instrument reliability and reproducible structure ($α=.82$ to $.94$; cross-form convergence $r=.84$; recovery of the full-pool axes $r=.92$ to $.96$; eight-month stability $r=.93$), then test it on 206 open-weight models, including 67 same-checkpoint base/post-trained pairs. Post-training's clearest fingerprint is installation: B rises .20 in 62/67 pairs across all organizations. Gating is more selective: model scale is unrelated to A in base checkpoints ($r=+.11$) but predicts it after post-training ($r=-.42$). Thus, the dimensions are not fixed properties of language models: they reflect the structure imposed on self-report by a training regime and may differ under others.

Muon and related matrix-sign optimizers are increasingly used to pre-train large language models, but their effect on the internal geometry of individual weight matrices is not well understood. This preliminary report proposes a unified framework built on a single idealizing assumption -- exact scale invariance of the loss under weight rescaling, which holds approximately in normalization-heavy networks. Under this assumption, plain SGD carries a built-in 1/||W|| brake on its update size, whereas Muon's matrix-sign step removes that brake, so both the Frobenius and spectral norms drift outward faster (t^{1/2} versus t^{1/4}). We further observe that the spectral-norm perturbation has a non-negative second-order term. This implies that a lightweight "spectral cap" -- which projects out only the first-order growth of the single top singular direction from each update -- can control the output covariance W K_X W^T without freezing training: the weight keeps learning through non-top directions, top-direction rotation, and top switching. We relate this cap to the min-entropy (H-infinity) of the singular-value spectrum. We then study three systems trained with Muon: a nanoGPT feed-forward projection, a 64-expert mixture-of-experts router, and the query/key projections of a bf16 FlashAttention block. In each case the cap increases isotropy and, at the margins -- a router collapsing to a single expert, and the near-divergence of one attention head -- prevents a concrete failure, while leaving validation loss essentially unchanged. We emphasize that the scale-invariance assumption is strong and that these small-scale results are preliminary; comments are welcome.

Reinforcement Learning is a cornerstone technique for modern large reasoning models. Usually, for difficult tasks such as code generation and theorem proving, the agent is evaluated by generating $K$ responses rather than sampling a single response, and performance is then measured using a retry-aware metric such as $\max$@$k$. Despite their practical importance, the theoretical foundations of learning under such criteria remain limited. In this work, we provide a theoretical study of the $\max$@$k$ learning problem in finite-horizon reinforcement learning. We show that optimizing the $\max$@$k$ objectives is fundamentally different from standard expected-return maximization. In particular, we prove that Markovian policies are in general insufficient, identify a compact state augmentation that restores optimality, and explicitly characterize the performance gap that can arise between history-dependent and non-history-dependent policies. Moreover, we show that learning $\max$@$k$-optimal policies is statistically harder than standard reinforcement learning and provide an efficient algorithm that achieves the optimal sample complexity rate.

We investigate the capacity of current language models to contribute to mathematical research. In Banach space theory, AI systems generated key ideas and proofs for five new results, which were then verified and refined by humans. We also developed an automated system that searches the literature for open problems and attempts solutions at scale. Our results show both the potential of language models for mathematical discovery and the continuing importance of expert verification.

Classical rate-distortion (RD) theory has long established the fundamental limits of lossy compression by quantifying the minimum number of bits required to represent a source under a prescribed distortion constraint. However, widely used distortion measures such as mean-squared error often fail to capture perceptual quality or semantic validity, which are increasingly central in modern learning-driven applications. Rate-distortion-perception (RDP) theory extends the RD framework by introducing perception as a third fundamental axis, quantified via distributional similarity between the source and reconstructed signals, leading to the rate-distortion-perception function (RDPF). This tutorial provides a structured overview of the coding principles underlying perception-aware lossy compression and surveys recent achievability results under different randomness assumptions. It then presents a unifying optimization viewpoint for computing the RDPF as defined by Blau and Michaeli, for both discrete and continuous sources under broad families of perceptual constraints, including f-divergences, alpha-divergences, and Wasserstein-based metrics. Special attention is given to computational tools such as alternating minimization schemes, Newton-based methods, and convex optimization formulations, as well as to analytically tractable cases such as Gaussian sources and the perfect-realism regime. Unlike recent broad surveys that emphasize generative architectures and AI-empowered communication systems, this tutorial focuses on the coding-theoretic and computational machinery needed to characterize, compute, and interpret the RDP limits. Finally, the tutorial outlines promising research directions at the intersection of information theory, neural compression, robust source coding, and perception-aware networked control systems.

Edmund C. Berkeley is usually remembered as a mediator between symbolic logic and early computing, yet that standard description understates the scope of his work. This paper argues for a stronger reading: Berkeley should also be understood as an early theorist of embodied machine intelligence. Across Berkeley's major writings on symbolic logic, machine intelligence, living robots, and Squee, intelligence appears not as disembodied symbol manipulation alone but as the organized coordination of sensing, storage, calculation, control, state, and action in physically realized machines. The paper's first contribution is interpretive: it reconstructs Berkeley as a thinker of machine architecture, temporally extended behavior, and environment-coupled control. Its second contribution is comparative: it reads Berkeley alongside David L. Heiserman to recover a shared descriptive scheme centered on sensing, state or memory, control, action, and adaptation. Its third contribution is critical: it uses that scheme to assess current embodied-AI discourse. The broader claim is that contemporary LLM-centered robotics often demonstrates impressive capability without an equally explicit account of persistence, recoverability, maintenance, and structured modification of conduct through experience.

Musical scales are near-universal in human music, and most readers will feel they already know what a scale is. On closer inspection, however, the literature lacks a consensus definition: which conditions are necessary and sufficient shifts across disciplines and traditions, and the term turns out to cover several distinct objects. I argue this is less a failure of rigour than a sign that ``scale'' names several related objects: prescriptive abstractions, instrument tunings, statistical regularities in performed pitch, perceptual categories, social conventions. I adopt an empirical definition -- a scale as a statistical regularity in pitch organisation relative to a tonic -- that is portable across traditions and computable from recordings, and situate it alongside the other senses of the term. Even this empirical core is not purely observational, as convention enters in deciding which pitches belong to a scale. And a further step of grouping scales into named categories is a separate convention, which I approach through prototype theory and illustrate with examples from Irish music. Separating these layers provides a basis from which scales can be re-examined empirically and cross-culturally.

Text-based evaluations of Theory of Mind (ToM) in Large Language Models (LLMs) often involve cognitive tests akin to the Sally-Anne task that can be gamed due to exposure to relevantly similar tasks in pre-training and do not obviously test models' functional ToM abilities in ways that generalize to naturalistic settings. To address these issues, we introduce the Epistemic Asymmetry Schelling Task (EAST), a two-player dialogue game designed to benchmark robust and generalizable ToM abilities. By requiring LLM-LLM dyads to independently converge on semantic Schelling points under varying states of epistemic transparency, we evaluate whether models can robustly apply ToM to achieve coordination. Our results reveal a significant capability gap in functional social reasoning, with only frontier models successfully navigating the varying epistemic demands of the tasks. Analysis of reasoning traces shows that coordination failures are primarily driven by epistemic tracking errors, such as conflating private knowledge with mutual knowledge. Despite high performance on traditional static benchmarks, our study shows that robust social reasoning and epistemic tracking remain a critical bottleneck, providing concrete targets for future LLM evaluation and development.

As large language models (LLMs) become widely deployed, their outputs can be copied, transformed, and redistributed at scale without reliable evidence of origin, creating risks for trust, accountability, intellectual property (IP) protection, and high-stakes decision-making. LLM watermarking addresses this problem by embedding detectable signals into text during or after generation. However, existing methods vary in design assumptions, threat models, and evaluation criteria, while deployment choices such as watermark placement, detection authority, and key management affect reliability, security, and scalability. This paper systematizes LLM watermarking as provenance infrastructure for large-scale data ecosystems. We organize existing approaches along four deployment dimensions: insertion point, verification authority, operational state, and transformation threat model, and relate them to the big data requirements of Volume, Velocity, Variety, Veracity, and Value. We further introduce a Big Data Watermarking Readiness framework centered on four deployment workloads: online generation, streaming detection, transformation pipelines, and ecosystem governance. The framework connects these workloads to system-level requirements including throughput, false-positive control, robustness, cross-domain reliability, governance, and downstream utility. Our analysis highlights a gap between benchmark performance and deployment readiness: false positives accumulate at scale, repeated transformations weaken watermark signals, computational overhead can limit online deployment, and centralized verification can create governance bottlenecks. We conclude with an evaluation blueprint and research directions for scalable, trustworthy provenance in big data ecosystems.

The application of lightweight Large Language Models in rule-based scientific domains remains severely limited by their tendency to mimic linguistic patterns rather than reproduce axiomatic reasoning, causing frequent hallucinations. Here, we show that G-Frame, an adaptive multi-agent framework integrating Bayesian and team game principles, establishes an automated closed-loop for high-quality data synthesis and model training. By forcing the internalization of domain constraints through structured reasoning, we synthesized a specialized corpus of 363,045 chains-of-thought and 199,589 question-answer pairs. The resulting 7B model OmniChem achieves performance parity with GPT 4o mini on custom benchmarks and ChemBench while exhibiting a 79.46% reduction in hallucinations relative to its base architecture. We further demonstrate the advanced capabilities of OmniChem in molecular design and synthesis planning. This work establishes a scalable paradigm utilizing adaptive multi-agents to overcome inherent reasoning deficiencies, offering a feasible pathway for accelerating knowledge discovery in specialized scientific fields.

As part of a series on first-principles modeling of cognitive functions, this paper attempts to provide a mathematical formulation of thinking and perception. It formally derives slow thinking or more generally, active perception, and encompasses the design, training and inference of slow thinking large language models. Our starting point is the lifting and projection of probability distributions on the observable and latent spaces, with the objective of representing complex data distributions by simple function families such as neural networks. A theory called "active lifting" is proposed, based on the sampling of latent sequences and an intrinsic drive to reduce uncertainty with maximum rate. It derives a large design space, containing the slow thinking models in a subspace that we call the static theory. These models are positioned on the representation hierarchy and sampler hierarchy induced by the static theory, and can be upgraded by climbing the two hierarchies. Active lifting further derives an inference process with an internal time axis, and a training objective that resembles minimum-length coding as well as the invention of languages. Thus, it characterizes the agency of perception, including the emergence of the slow thinking formats. Technical by-products of this theory include a three-stage pathway for improving slow thinking models, a unified approach to constructing encoders and generative models for all data modalities, a priori formation of human-like visual representations, and a possible solution to policy collapse.

Roy Harris's Integrationist linguistics offers a compelling critique of the referentialist tradition embedded deep at the heart of computational approaches to language, arguing that language is not a code that maps onto a pre-given world but a situated, bipartite activity oriented toward prospective joint action. Yet Integrationism leaves certain explanatory gaps: it does not fully account for the structural mechanism by which signs sustain prospective openness, it undertheorises the continuity between linguistic and non-linguistic semiotic activity, and it offers no detailed account of the structural properties of the accumulated archive of past integrations. This paper argues that Elan Barenholtz's autogenerative theory of language, developed in response to the behaviour of Large Language Models (LLMs), can fill precisely these gaps, enriching Integrationism without undermining any of its core commitments. Specifically, the autogenerative account provides: a structural mechanism for the prospective openness that Harris identifies as central to bipartite communication; a computational correlate for Harris's thesis of semiotic continuity between language and other sign-making activity; and a theory of the archive: what the accumulated residue of past integrations looks like and how new participants draw upon it. The synthesis preserves Harris's ontological primacy of the situated integrative act while adding explanatory content that Integrationism itself does not supply. For practitioners and researchers in natural language processing and large language model design, the argument offers a principled account of what the statistical structure that LLMs so effectively exploit actually is, and of what it cannot, by its nature, provide.

We build a team of specialized large language-model agents and present an agent-driven workflow for research-level formalization in theoretical physics, with the autoformalization of the fundamental theorem of matrix-product states as a demonstration. The agents, coordinated through a structured mathematical blueprint and periodic human review, orchestrated and executed the full formalization autonomously. For some statements, the agents were able to explore new proof routes that are not part of the standard literature. Along the way the agents produced extensive tensor-network and quantum-information libraries not previously available in Mathlib, Lean's mathematical library. As a physical application, the formalization also extends towards symmetry-protected topological phases in one dimension. We find that the main bottleneck in large-scale autoformalization is enforcing mathematical intent and we provide a detailed study of the full process and various subtleties involved. We release the codebase as the library \href{https://github.com/LionSR/TNLean}{TNLean}, together with a \nChapters{}-chapter \href{https://lionsr.github.io/TNLean/blueprint/}{blueprint} of the formalization effort.

Why does contrastive learning with simple images and augmentations yield useful representations for downstream tasks? We address this question by analytically computing the optimal representation in terms of a contrastive loss for a range of basic augmentations and any image dataset with stationary statistics. We show that for certain augmentations the optimum can be attained by a CNN whose first layer filters are sinusoids, followed by a pointwise nonlinearity, global average pooling, and a final linear layer that performs partial whitening. We also show that the optimal weights in such CNNs for more complicated augmentations are still sinusoids. The frequencies of the sinusoids and their weights can be computed using a simple waterfilling algorithm given the dataset's expected power spectrum. Experiments with different image datasets and augmentations show that such CNNs trained with SGD empirically learn sinusoids in their first layer and to perform partial whitening

Least privilege, the principle that an identity should hold only the permissions strictly required for its task, has been a foundational primitive of access control for decades. We argue that this principle is insufficient for agentic AI systems, which do not merely hold permissions but can combine, approve, and amplify them across workflows and system boundaries. We propose least autonomy as an appropriate generalization and develop a formal theory. First, we define a compositional blast radius d(a,b) that measures structural separation between actions in an enterprise hierarchy, combining an ultrametric tree with lattice-valued confidentiality, integrity, and control-context labels. Second, we define a directed agent influence graph G(theta). An arc from U to V requires a directed shared-resource write-to-read meeting or a conservative undirected agent-to-agent (A2A) communication meeting, and a meeting-conditioned influence potential at or above an externally selected policy threshold theta. A catalogue-radius profile supports calibration and audit of theta. Finally, we define a collusion predicate over graph reachability that detects authorization composition, decision manipulation, and cross-domain capability composition.

When should an AI system's answer be trusted? Formal proof assistants offer certainty but cannot reach most of the problem distribution; scalar LLM judges offer coverage but produce opaque scores that cannot be audited after the fact and are subject to the same coherence issues as any LLM. We present Theoria, a verification architecture that closes this gap. A candidate solution is rewritten into a sequence of typed state transitions, each licensed by an explicit justification, whether that be a citation, computation, or problem-given fact, and every transition is independently auditable. The foundational invariant is completeness of change: every difference between consecutive proof states must be accounted for, so hidden premises surface as unlicensed mutations rather than passing silently. On HLE-Verified Gold (185 text-only expert problems), Theoria certifies 105 at 91.4% strict precision (Wilson 95% CI [84.5%, 95.4%]). Every certification produces a human readable proof trace in which each step can be independently challenged. Holistic LLM judges achieve comparable precision at matched coverage but fail on different problems (Jaccard 0.14-0.36), making the approaches complementary. On 95 adversarial poisoned proofs across 15 domains, structured judges catch 94.7% versus 83.2% for holistic judging (p= 0.0017). The overall 11.5 pp gap concentrates in hidden premises (90.6% vs. 62.5%, a 28 pp difference) and fabricated citations (100% vs. 90%), the error classes where the formal analysis predicts an advantage; performance is identical on arithmetic and theorem-misapplication errors, where no advantage is predicted. On GPQA Diamond (n= 65), certified precision is 97.1% (Wilson CI [85.1%, 99.5%]).

Deep learning has outgrown any single mathematical explanation. From Approximation to Emergence develops a unified, proof-oriented account of modern deep learning theory, tracing a path from the classical foundations of approximation, optimization, and generalization to the contemporary mechanisms of overparameterization, robustness, generative modeling, transformers, in-context learning, scaling laws, interpretability, alignment, and emergence. Rather than presenting isolated results, the book organizes a broad literature into a coherent research narrative: each theory is examined through the object it controls, the assumptions that make it valid, and the phenomena it leaves unexplained. Written for researchers, graduate students, and mathematically trained practitioners, this monograph offers a rigorous map of deep learning theory as it stands today: powerful, incomplete, and increasingly centered on the question of how learned mechanisms arise from scale, data, architecture, and training.

The success of deep learning models in classification and regression is widely attributed to the low-dimensional structure that real-world data tend to exhibit, despite their high-dimensional representation. This work attempts to provide a mathematical framework for binary classification on low-dimensional data, building on Cover's (1965) function-counting theory. With our framework, we aim to address the question of how the low-dimensional structure of the data affects the classification capabilities of learning models. Cover's theory relies on a general position assumption that blinds it to the underlying data structure. We refine this assumption to account for the low-dimensionality of the data and derive dichotomy counts that reflect the data structure. We further extend Cover's separation capacity and problem of generalization to the low-dimensional setting, enabling the impact of the underlying data structure on both to be analyzed.