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Search indexed arXiv papers on artificial intelligence, large language models, computer vision and robotics. Read source abstracts and follow links to arXiv.

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arXiv2026

A Formalizable Proof of the No-Supervenience Theorem: A Diagonal Limitation on the Viability of Physicalist Theories of Consciousness

The no-supervenience theorem limits the capacity of physicalist theories to provide a comprehensive account of human consciousness. The proof of the theorem is difficult to formalize because it relies on both alethic and epistemic notions of possibility. This article outlines a formalizable proof using predicate modal logic in which the epistemic inferences are expressed in terms of an existing mathematical formalism, the inference device (Wolpert, 2008). The resulting proof shows definitely that any physicalist theory which describes a self-aware, intelligent system must be internally inconsistent.

q-bio.NC↗

arXiv2026

Extracting total Amb programs from proofs

We present a logical system CFP (Concurrent Fixed Point Logic) supporting the extraction of nondeterministic and concurrent programs that are provably total and correct. CFP is an intuitionistic first-order logic with inductive and coinductive definitions extended by two propositional operators: Restriction (binary), a strengthening of implication, and a unary operator for total concurrency. The source of the extraction is formal CFP proofs, the target is a lambda calculus with constructors and recursion extended by a constructor Amb (for McCarthy's amb) which is interpreted operationally as globally angelic choice and is used to implement nondeterminism and concurrency. The correctness of extracted programs is proven via an intermediate domain-theoretic denotational semantics. We demonstrate the usefulness of our system by extracting a nondeterministic program that translates infinite Gray code into the signed digit representation. A noteworthy feature of CFP is the fact that the proof rules for restriction and concurrency involve variants of the classical law of excluded middle that would not be interpretable computationally without Amb. This is a revised and extended version of the conference paper presented at ESOP 2022 with the same title that contains full proofs of all major results.

cs.LO↗

arXiv2026

Locked Polyomino Tilings

A locked $t$-omino tiling is a grid tiling by $t$-ominoes such that, if you remove any pair of tiles, the only way to fill in the remaining $2t$ grid cells with $t$-ominoes is to use the same two tiles in the exact same configuration as before. We exclude degenerate cases where there is only one tiling overall due to small dimensions. It is a classic (and straightforward) result that finite grids do not admit locked 2-omino tilings. In this paper, we construct explicit locked $t$-omino tilings for $t\geq3$ on square and rectangular grids of various dimensions. We show that locked 3- and 4-omino tilings exist on finite square grids of arbitrarily large size. The result for 4-omino tilings in particular is remarkable because they are so rare and difficult to construct: Only a single tiling is known to exist on any grid up to size $40\times40$. We also find locked 5- and 6-omino tilings, and provide a computer-assisted proof that locked 7-omino tilings do not exist on any rectangular grid. In a weighted version of the problem where vertices of the grid may have weights from the set $\{1,2\}$ that count toward the total tile size, we demonstrate the existence of locked tilings on arbitrarily large square weighted grids with only 6 tiles. Finally, we construct locked $t$-omino tilings of the infinite grid for arbitrarily large $t$. Locked $t$-omino tilings arise as obstructions to widely used political redistricting algorithms in a model of redistricting where the underlying census geography is a grid graph. Most prominent is the ReCom Markov chain, which takes a random walk on the space of redistricting plans by iteratively merging and splitting pairs of districts (tiles) at a time. Locked $t$-omino tilings are isolated states in the state space of ReCom. The constructions in this paper are counterexamples to the meta-conjecture that ReCom is irreducible on graphs of practical interest.

math.CO↗

arXiv2026

QubiC 2.0: An Extensible Open-Source Qubit Control System Capable of Mid-Circuit Measurement and Feed-Forward

Researchers manipulate and measure quantum processing units via the classical electronics control system. We developed an open-source FPGA-based quantum bit control system called QubiC for superconducting qubits. After years of qubit calibration and testing experience on QubiC 1.0, we recognized the need for mid-circuit measurement and feed-forward capabilities to implement advanced quantum algorithms, and upgraded the system to QubiC 2.0 on an AMD ZCU216 RFSoC evaluation board with all these enriched features. The system uses portable FPGA gateware with a simplified processor to handle commands on-the-fly. For design simplicity and straightforward scaling, we adopted a multi-core distributed architecture, assigning one processor core per qubit. The pulses combine the unique pulse envelope and carrier information specified in a command. Each pulse envelope is pre-stored in the FPGA's block RAMs, ensuring speed and reusability throughout the quantum circuit, while pulse parameters including amplitude, phase, and frequency can be updated from pulse to pulse. The software stack is developed in Python, running on both the FPGA's ARM core and a host computer via XML-RPC. Quantum circuits can be described in a high-level language supporting both pulse-level and native-gate-level programming with high-level control flow constructs, and are compiled into binary commands loaded into the FPGA. With QubiC 2.0, we achieved multi-FPGA synchronization and low-latency data communication, and demonstrated feed-forward experiments on conditional circuits. The system also serves as a platform for artificial intelligence and machine learning driven quantum control, such as FPGA-based real-time quantum state discrimination. The enhanced QubiC system provides researchers with powerful tools to explore and implement advanced quantum algorithms and applications.

quant-ph↗

arXiv2026

Computational reach of machine learning models is independent of untrained quantum resources

This paper proves that if an untrained quantum circuit is used as a resource in a machine learning model, then on average no resource quantum circuit is better than any other that can achieve the same set of computational effects. The paper also proves a supporting theorem that even if certain mild idealisations are omitted, the average quantum advantage remains negligible at best. These results cast serious doubt on several proposals to use untrained quantum circuits in machine learning models: at best such claims should be substantiated empirically, as this paper proves there is no a priori theoretical reason to suppose that introducing an untrained quantum circuit will increase performance.

quant-ph↗

arXiv2026

The Recursive Behavior of the Number of Irreducible Factors of Certain Rational Transformations over Finite Fields

Constructing irreducible polynomials over finite fields via rational transformations is an active field of study. In this paper, we show that the method can be used for more than that. By considering normalized rational functions $Q$ that generate an invariant subfield $\mathbb{F}_q(x)^G$ of $\mathbb{F}_q(x)$, for a subgroup $G\le Aut(\mathbb{F}_q(x)/\mathbb{F}_q)$, and exploiting the factorization pattern of their rational transformations, we are able to prove that the sets arising from considering whether the rational transformation $f^Q$ has exactly $deg(Q)$ irreducible factors of the same degree as $f$, or an irreducible factor of higher degree, admit a recursive behavior, i.e. their numbers satisfy a recursive equation. These sets occasionally can be described in terms of arithmetical properties the roots of the irreducible polynomials satisfy, e.g. their roots satisfy certain trace or power/non-power conditions. These can often be translated to properties of their minimal polynomials. With that, we prove a new theorem about the number of certain irreducible factors of a class of linearized polynomials, which gives a new context to a recent paper by Granger.

math.NT↗

arXiv2026

Marital Sorting, Household Inequality and Selection

Using CPS data for 1976 to 2025, we explore how wage inequality has evolved for married couples with both spouses working full time full year, and its impact on household income inequality. We also investigate how marriage sorting patterns have changed over this period. To determine the factors driving income inequality we estimate a model explaining the joint distribution of couples' wages that accounts for spouses' employment decisions. This requires a multivariate distribution regression model that incorporates a double selection rule. We find that income inequality has increased for these households and increased wage sorting has exacerbated the inequality resulting from individual wage growth. We find that positive sorting partially reflects the correlation across unobservables influencing the wages of both members of the marriage. We decompose the changes in sorting patterns over the 50-year sample into structural, composition and selection effects and find that the increase in positive sorting primarily reflects the increased skill premia for both observed and unobserved characteristics.

econ.EM↗

arXiv2026

Multiple SLEs and Dyson Brownian motion: transition density and Green's function

We consider multiple chordal Schramm-Loewner evolution (SLE) with $κ\in (0,4]$. Under common-time parameterization, we show that the transition density of the driving function of multiple chordal SLEs can be given by the transition density of Dyson Brownian motion and Green's function. This is a companion paper of [FWY24].

math.PR↗

arXiv2026

Crowdsearch

A common economic process is crowdsearch, wherein a group of agents is invited to search for a valuable object, such as a software vulnerability or a proof-of-work nonce. We study a binary model of crowdsearch in which agents with heterogeneous search costs decide whether or not to search. Inviting more agents directly expands the pool of potential searchers, but it also increases competition for the prize and lowers individual search incentives. This crowding-out effect may dominate, so that inviting more agents can lower the probability of finding the object. We show that inviting an unlimited crowd need not guarantee success. We also provide sufficient conditions, based on the lower-tail behavior of the cost distribution, under which the probability of success is eventually increasing or eventually decreasing in the number of agents.

econ.TH↗

arXiv2026

A General Framework for Importance Sampling with Markov Random Walks

Although stochastic models driven by latent Markov processes are widely used, the classical importance sampling methods based on the exponential tilting for these models suffer from difficulties in computing associated eigenvalues and eigenfunctions and the plausibility of the indirect asymptotic large deviation regime for the variance of the estimator. We propose an importance sampling framework that twists the observable and latent processes separately using a link function that directly minimizes the estimator's variance. An optimal link function is chosen within the locally asymptotically normal family. We show the logarithmic efficiency of the estimator. As applications, we estimate an overflow probability under a pandemic model and the CoVaR, a measurement of the co-dependent financial systemic risk; both are beyond the scope of traditional importance sampling methods due to nonlinear features.

stat.CO↗

arXiv2026

Optimal Clustering of Multi-layer Networks under Mixture Block Models

We study clustering for multi-layer networks, where each observation is an undirected network measured on the same set of vertices. Under the mixture multi-layer stochastic block model (MMSBM), each network is generated from one of two stochastic block models, and the goal is to recover the latent network labels. We establish the minimax optimal network clustering error rate, which takes an exponential form and is characterized by the Chernoff information between the edge probability distributions of the component networks. We propose a novel two-stage estimator that combines tensor-based initialization, node and sample splitting, and a likelihood-based refinement. The refinement step exploits estimated node communities and achieves the minimax rate with high probability under extreme network sparsity. We further extend the likelihood-refinement framework to count-weighted networks through a mixture multi-layer Poisson block model. Numerical studies and two real-data analyses illustrate the finite-sample behavior of the proposed method and its use for both external-label benchmarking and unlabeled exploratory network clustering.

math.ST↗

arXiv2026

Asymptotics of the free path length in the periodic Lorentz gas in the Boltzmann-Grad limit

We study the distribution of the free path length in the periodic Lorentz gas in the Boltzmann-Grad limit. For any unimodular Euclidean lattice and any smooth initial velocity distribution, we establish an asymptotic formula for the distribution function of the free path length with a polynomial rate of convergence. The proof combines the dynamical interpretation of the free path length in terms of expanding translates of horospherical orbits on the space of affine lattices, due to Marklof--Strömbergsson, with the effective equidistribution results of Kleinbock--Margulis and Kim. This extends the asymptotic formula of Boca--Zaharescu for the planar case to arbitrary dimension.

math.PR↗

arXiv2026

Operator quantum singular value transformation: Spectral mapping, design and topological invariants

Quantum singular value transformation (QSVT) provides a unifying framework for quantum algorithms by applying the polynomials of quantum signal processing (QSP) to block-encoded matrices. We connect this framework with spectral analysis, design and topological protection in condensed matter physics. Extending QSVT beyond matrices to possibly infinite-dimensional operators (operator QSVT), we establish the following. First, it transforms every block of the operator by the QSP polynomials; no singular value decomposition is needed. Second, a spectral mapping theorem determines the full spectrum of the QSVT operator, including continuous spectra, from the embedded operator through the QSP polynomial, allowing spectral analysis without translation invariance. Third, the spectral mapping turns QSP into spectral design: the QSP phases can stretch, fold and separate the bands and place the eigenvalues. Finally, the kernel sectors that conventional QSVT usually sets aside carry a topological index, invariant under the choice of QSP phases, local perturbations and gap-preserving deformations, which guarantees topologically protected states. These results unify two quantum walks: Szegedy's walk from quantum algorithms and the split-step walk from condensed matter physics. Their kernel sectors encode the graph's Euler characteristic and the difference of bulk winding numbers, respectively. For the split-step walk, QSP breaks chiral symmetry but not the topological protection, and can move the protected state into a continuous band to form a bound state in the continuum (BIC).

quant-ph↗

arXiv2026

Unambiguous Efficiency of Random Allocations

When allocating indivisible objects via lottery, planners often use ordinal mechanisms, which elicit agents' rankings of objects rather than their full preferences over lotteries. In such an ordinal informational environment, planners cannot differentiate between utility profiles that induce the same ranking of objects. We propose the criterion of unambiguous efficiency: regardless of how each agent extends their preferences over objects to lotteries, the allocation is Pareto efficient with respect to the extended preferences. We compare this with the predominant efficiency criterion used for ordinal mechanisms. As an application to mechanism design, we characterize all efficient and strategy-proof mechanisms satisfying certain regularity conditions.

econ.TH↗

arXiv2026

Envelopes of Horospheres and Weingarten Surfaces in Hyperbolic 3-Space

We derive basic differential geometric formulae for surfaces in hyperbolic space represented as envelopes of horospheres. The dual notion of parallel hypersurfaces is also studied. The representation is applied to prove existence and regularity theorems for Weingarten surfaces in H^3, which satisfy (1-a)K = a(2-H), for an a < 0, and have a specified boundary curve at infinity. These surfaces are shown to be closely connected to conformal mappings of domains in S^2 into the unit disk and provide Riemannian interpretations for some conformal invariants associated to such mappings. This paper was originally written in 1984, before I learned to use TeX, and was typed by one of the secretaries in the Princeton Math Department. It was more or less, my first original work after my dissertation. For some reason, I was not able to get this paper published in a timely manner. The results and perspective in this paper have proved to be useful to a variety of people, some of whom asked me to render the article into TeX and post it to the arXiv. I had been seriously thinking about doing this, when Martin Bridgeman sent me a transcription of my original article into TeX. I am extremely grateful to him for the effort he has put into this project. The paper is now formatted in a more or less modern AMS-article style, but for lots of additional punctuation, a few corrections and some minor stylistic changes, the content has been largely reproduced as it originally was. Remarks about the 'state-of-the-art' in hyperbolic geometry are obviously way out of date, as there has been enormous progress in many aspects of this still rich subject.

math.DG↗

arXiv2026

Opinion Dynamics in Social Multiplex Networks with Mono and Bi-directional Interactions in the Presence of Leaders

We delve into the dynamics of opinions within a multiplex network using coordination games, where agents communicate either through one-way or two-way interactions, and where a designated leader may be present. By employing graph theory and Markov chains, we illustrate that despite zero diagonal elements in transition probability matrices or decomposable layers, opinions converge under specific conditions on the layers, leading to a consensus, and we show by a counterexample that a spanning tree in the union of the layers does not suffice. We further scrutinize the convergence rates of opinion dynamics in networks with one-way versus two-way interactions. We find that in networks with a designated leader, opinions converge towards the initial opinion of the leader, whereas in networks without a designated leader, opinions converge to a convex combination of the opinions of agents. Moreover, we emphasize the crucial role of designated leaders in steering opinion convergence within the network. Paired experiments on 1,200 random multiplex networks show that a leader slows down convergence, and that two-way interactions slow it down in the presence of a leader but speed it up in its absence.

eess.SY↗

arXiv2026

Quantum theory of molecular orientation

We formulate a quantum phase space for rotational and nuclear-spin states of rigid molecules. For each nuclear spin isomer, we re-derive the isomer's admissible angular momentum states from molecular geometry and nuclear-spin data, introduce its angular position states using quantization theory, and develop a generalized Fourier transform converting between the two. We classify molecules into three types -- asymmetric, rotationally symmetric, and perrotationally symmetric -- with the last type having no macroscopic analogue due to nuclear-spin statistics constraints. We discuss two general features in perrotationally symmetric state spaces that are Hamiltonian-independent and induced solely by symmetry and spin statistics. First, we quantify when and how an isomer's state space is completely rotation-spin entangled, meaning that it does not admit any separable states. Second, we identify isomers whose position states house an internal pseudo-spin or "fiber" degree of freedom, and the fiber's Berry phase or matrix after adiabatic changes in position yields naturally robust operations, akin to braiding anyonic quasiparticles or realizing fault-tolerant quantum gates. We outline how the fiber can be used as a quantum error-correcting code and discuss scenarios where these features can be experimentally probed

quant-ph↗

arXiv2026

String attractors and bi-infinite words

String attractors are a combinatorial tool coming from the field of data compression. It is a set of positions within a word which captures an occurrence of every factor. While one-sided infinite words admitting a finite string attractor are eventually periodic, the situation is different for two-sided infinite words. In this article, we characterise the bi-infinite words admitting a finite string attractor as the eventually periodic words, the characteristic Sturmian words and their morphic images. For words that do not admit finite string attractors, we study the structure and properties of their infinite string attractors.

math.CO↗