Papers

Total quadratic ball discrepancy measures the deviation of codeword counts from uniformity over all Hamming balls. We prove that, whenever a Hamming space and cardinality admit a perfect code, the discrepancy minimizers among all subsets of that cardinality are precisely the perfect codes. This extends Barg's binary minimizing result and characterizes equality over arbitrary finite alphabets. At the necessary parameter sets for correcting one or two errors, we give explicit lower bounds whose attainment is equivalent to perfect tiling, even when existence is unresolved. For alphabets of size at least four, the numerical values follow from known universal energy bounds; we establish their discrepancy normalization and the attainment criterion. The ternary discrepancy potential falls outside the completely monotonic regime of those bounds, and separate exact certificates settle the ternary Hamming and Golay cases.

Prefix caching reuses the key–value (KV) states of shared prompt prefixes and can substantially reduce the time to first token (TTFT) of large language model (LLM) inference. In a wireless edge network, however, prefix states are distributed across geographically separated GPU nodes. A nearby node offers a short radio path but may provide little reuse, whereas a more distant node may cache a longer matching prefix but incur additional communication and queueing delay. Moreover, persistent prefixes and active-request KV states compete for the same GPU memory, so aggressive caching can reduce inference concurrency and create queueing hotspots. This paper develops a stochastic-geometry and queueing framework for this spatial communication–caching–computation tradeoff. We represent the prompt workload by a prefix forest and define an ancestor-closed cache profile that may contain multiple reusable prefixes. Edge GPU nodes form a Poisson point process and are independently marked by cache profile, yielding analytically tractable spatial tiers. We derive the profile-association probability, conditional serving-distance distribution, token-level computation-offloading ratio, and TTFT coverage probability under a load-aware association policy. A fixed-point formulation captures the coupling between spatial association and multi-server GPU queues, while an outer optimization selects the cache-profile distribution subject to static-memory and stability constraints. Analytical and Monte Carlo results agree closely. The results show that the latency-optimal node need not be the nearest node, that TTFT can be non-monotonic in cached-prefix depth because of GPU-memory coupling.

Training error is what we can observe on a training set; test error is the quantity we care about. We study squared-error linear regression in a deterministic $(d+1)$-dimensional single-spike model. Each training vector has spike coordinate $\sqrt{\gamma}$, $\gamma>1$, and equal-norm, mutually orthogonal nuisance components. Training labels are $1$. Fresh test points are centered Gaussian with covariance $\operatorname{diag}(\gamma,1,\ldots,1)$ and noise-free label $x[1]/\sqrt{\gamma}$. We compute the exact train–test tradeoff for linear predictors in the training span. When the spike direction is unknown, allowing all linear predictors does not improve this tradeoff, on average or in the worst case over directions consistent with the data. There is a range of training-set sizes where every span predictor that generalizes well must fit the training data worse than the zero predictor. We call this benign misfitting, or the fourth quadrant. At its sharpest, test error tends to zero while training error diverges, with both zero-predictor errors normalized to one. The best span predictor begins to generalize when $n\gg d/\gamma^2$, while interpolation needs $n\gg d/\gamma$. In the intervening window $d/\gamma^2\ll n\ll d/\gamma$, useful span prediction lies beyond interpolation: training predictions overshoot the labels. One-pass stochastic gradient descent with a large constant learning rate attains small test error throughout this window, matching the best span predictor up to a logarithmic factor, while leaving large empirical training error. Finally, the nuisance component responsible for the training misfit also controls adversarial sensitivity.

We introduce and characterize age dispersion as a measure of temporal consistency in status update systems. Age dispersion is defined as the difference between the ages of the two most recently received updates, and its higher-order extension, the {$k$-th} order age dispersion, captures the difference between the ages of the most recent update and the $(k+1)$-th most recent one. We analyze age dispersion in an M/G/1/1 queueing system. Furthermore, we establish connections between the {$k$-th} order age dispersion and the {$k$-th} order age of information (AoI), where the latter quantifies the age of the {$k$-th} most recently received update.

Solomonoff Induction, or SolInd, provides an ideal unbounded model of a priori sequence prediction but cannot naturally describe extrapolation from a given training dataset, as performed by Large Language Models. We apply de Finetti's theorem on exchangeable distributions to SolInd to produce what we call Hierarchical Solomonoff Induction, or HSI, which maintains a hyperprior over all Solomonoff priors that can be conditioned on previously observed sequences. We extend Wood et al.'s proof that universal mixtures of semimeasures are equivalent to SolInd to show that universal mixtures of these mixtures are also equivalent, proving that HSI=SolInd. We also prove that HSI's excess error on any distribution, compared to its true generator, is bounded by that generator's complexity in the hyperprior. This result is directly comparable to SolInd's prediction error being bounded by the Kolmogorov complexity of the sequence being predicted, and forces HSI's average excess error to converge to 0 as a dataset grows, leading to optimal prediction in the limit. We claim that HSI is an ideal unbounded model of sequence prediction given a dataset in the same way that SolInd is ideal over individual sequences.

The Koopman operator provides a linear framework for analyzing nonlinear dynamical systems through spectral properties. Extended Dynamic Mode Decomposition (EDMD) approximates this operator from data, but non-invariant dictionaries can introduce spurious eigenvalues. We introduce the Projected Koopman Operator Approximation framework for constructing Filtered EDMD operators. The framework projects the Koopman action onto admissible dictionary subspaces that need not be invariant, while exactly preserving every represented Koopman eigenpair with nonzero eigenvalue. A forward–intersection chain provides a canonical hierarchy of compatible subspaces, connecting the full dictionary to its maximal invariant core while retaining useful intermediate models. We analyze two projection geometries: a coordinate-orthogonal projector, which requires no function-space Gram-matrix estimate but is basis-dependent, and a function-space orthogonal projector, which recovers population EDMD at the unfiltered level. We characterize their relationship to EDMD and existing subspace-selection methods. We also develop SVD-based algorithms for constructing sampled forward–intersection spaces and implementing the coordinate projector. Under independent noiseless sampling and exact-rank identifiability, the resulting empirical operators converge almost surely to their population counterparts. Experiments on a Kronecker flow, a polynomial system, and the Van~der~Pol oscillator demonstrate reduced spectral pollution. For Van~der~Pol, the coordinate projector recovers the local equilibrium spectrum independently of the sampling measure, whereas the $L^2(\mu)$ projector approximates the limit-cycle spectrum on the same certified subspace.

We consider First-Order Logic (FOL)-based semantic communication for neuro-symbolic decision-making in collaborative environments such as autonomous driving networks. Each connected autonomous vehicle (CAV) converts its partial sensor observations into a natural-language scene description and corresponding grounded FOL evidence. Under an uplink budget, a semantic encoder at each car selects the observations most informative for evaluating traffic rules and transmit to a Road Side Unit (RSU). The RSU fuses all received evidence, evaluates collaborative rules, performs logical deduction for vehicle-specific safety and right-of-way information for constrained downlink transmission. Each CAV combines the received deductions with its local description, enabling a local LLM agent to select a high-level driving action. We develop a principled, verifiable semantic communication method using a random-support Dirichlet–Categorical model of inductive logical probability, providing a modern statistical reinterpretation of Carnap's and Hintikka's systems. From this model, we derive a goal-oriented semantic information-bottleneck formulation that prioritizes evidence transmission by its reduction of uncertainty over task goals. Using 152 traffic rules extracted from the California Driver Handbook, we evaluate the framework on MDrive simulator in CARLA. Under identical communication budgets, semantic evidence selection completes every scenario without safety hazards, whereas uniform evidence selection produces collisions, showcasing semantic communication's superiority.

Uncrewed aerial vehicle (UAV)-enabled federated learning (FL) can provide flexible, on-demand edge intelligence for large-scale IoT deployments, but operating in shared unlicensed bands makes uplink update delivery interference-coupled and unreliable. In this paper, we develop a packet-level transmission framework that captures buffer overflow, delay violations, and transmission errors, and uses the resulting packet delivery ratio (PDR) to represent partial-update reception through a packetized, Bernoulli-masked FL aggregation process. We then formulate a fairness-consensus bilevel (FCB) optimization that jointly controls (i) transmission thresholds to maximize the average PDR while reaching consensus under partial observability and (ii) transmission powers to improve the worst PDR and enforce fairness across IoT learners. To solve this problem, we propose an alternating FCB optimizer composed of a consensus-based threshold controller (CTC), which drives the IoT learners toward a PDR-efficient consensus on transmission thresholds, and a fairness-based power controller (FPC), which updates transmission powers to improve the worst PDR and ensure fairness under the resulting consensus thresholds. Numerical results on CNN-based FL tasks show that the FCB optimizer improves FL aggregation and training performance by enhancing packet-level update delivery, consistently outperforming baseline transmission policies.

The Service Rate Region (SRR) serves as a critical metric for evaluating the concurrent service capacity of distributed storage systems. While several works have characterized the SRR for MDS codes and first order Reed-Muller codes, for high-order Reed-Muller codes the problem becomes way more complicated and only partial results were given by Ly, Soljanin, and Lalitha [IEEE ISIT 2025]. In this paper, we refine the SRR analysis by explicitly characterizing the intersection patterns of recovery sets for high-order Reed-Muller codes, deriving the exact region for the case m=r+1 and providing new types of strictly tighter constraints to bridge the gap between existing approximations and the exact SRR polytope.

We construct an explicit symplectic realization of the Barnes-Wall lattice that yields a family of multimode Gottesman-Kitaev-Preskill (GKP) codes with encoding rate $R = \frac{1}{2}\log_2 N$ and a deterministic $O(N\log^2 N)$ bounded-distance decoder. The recursive generator $G_{m+1} = \bigl(\begin{smallmatrix} G_m & 0 \\ G_m & R_m G_m \end{smallmatrix}\bigr)$ with $R_m = I + \Omega$ simultaneously guarantees symplectic integrality for valid quantum stabilizers and preserves the exact Barnes-Wall decoding structure through a chain of isometric isomorphisms. The code distance is constant at $\Delta^2 = 1$ (in units of $2\pi$), representing an explicit distance–rate tradeoff in which logarithmic encoding efficiency is achieved at the cost of non-scaling protection. This construction provides a deterministic, space-efficient paradigm for GKP error correction in platforms supporting non-local modular connectivity.

In the seminal paper (Yager 2015), Yager defined the negation of a probability distribution $\mathbf{p}=(p_1,\dots,p_n)$, as the distribution $\overline{\mathbf{p}} = (\overline{p}_1,\dots,\overline{p}_n)$, where $\overline{p}_i = ({1-p_i})/({n-1}),$ for $ i=1, \ldots , n.$ In this paper, we present a comprehensive information-theoretic analysis of Yager's negation and its generalizations. Using tools from information theory and majorization theory, we unify, extend, and strengthen a number of previously known properties of Yager's negation within a common framework. Overall, our results offer strong theoretical justification for Yager's negation as the most natural and principled definition of probability distribution negation under various information theoretic criteria.

DNA molecules are so small that it might be practical to use their frequency vectors to encode messages. More precisely, a sender can inject $M_X$ copies of the string $X =$ CATCATCAT into a pool and the receiver can recover $M_X$ by sequencing the pool. There are, however, two sources of uncertainty: (a) $M_X$ is usually too big to be counted exactly, but is estimated by sampling. (b) The DNA sequencer could be noisy; it may have difficulty distinguishing CATCATCAT from CATGATCAT. Recently, Tamir, Weinberger, and Guillén i Fàbregas clarified the amount of information the frequency vector can carry under (a). They showed that each string can carry about $\log_4 R$ bits, where $R$ is the average number of times each string is read. They also showed that $\log_4 R$ bits can be achieved by a low-complexity uncoded scheme under the condition that there are at least $\sqrt R$ distinct strings. In this paper, we show that a low-complexity coded scheme can achieve the same $\log_4 R$ bits unconditionally. We then generalize the scheme to handle sequencing noise, (b), and show that the noise penalizes the total number of bits by $\log_2 \det W$, together with a linear term due to the use of Fourier transforms in our proof. The former penalty $\log_2 \det W$ is asymptotically the same as that obtained by Gerzon, Shomorony, and Weinberger; our scheme trades a small amount of rate for practical complexity.

Bandwidth-limited visual communication systems require efficient transmission of high-dimensional data under dynamic wireless conditions. Existing approaches either rely on joint source-channel coding, which tightly couples representation learning with channel models and lacks flexibility across varying environments, or adopt generative reconstruction techniques that may introduce semantically inconsistent outputs. In this paper, we propose RQ-NAC, a channel-agnostic semantic compression framework for visual communication. The proposed method leverages residual quantization to produce scalable discrete semantic representations, enabling fine-grained and predictable control over the rate-distortion tradeoff. To further enhance compression efficiency, we integrate an n-gram-driven arithmetic coding module that exploits contextual dependencies among latent indices for lossless entropy coding. Extensive experiments demonstrate that RQ-NAC achieves over 600$\times$ compression relative to uncompressed visual data while preserving high perceptual quality. The results show that our approach enables efficient, flexible, and reliable semantic transmission under bandwidth-constrained conditions.

Integrating non-terrestrial networks (NTN) with terrestrial infrastructure is a key enabler of next-generation wireless systems, providing ubiquitous connectivity while meeting stringent rate and latency requirements. In particular, high altitude platform stations (HAPS) can complement terrestrial networks and jointly form vertical heterogeneous networks (vHetNets), extending coverage while delivering high-capacity, reliable, and low-latency connectivity for user equipments (UEs) including ground users and uncrewed aerial vehicles (UAVs). However, the high altitude deployment of HAPS establishes strong line-of-sight (LoS) links to UEs, creating highly correlated channels among UEs. Moreover, the wide coverage footprint of HAPS enables it to serve a large number of UEs, forcing limited radio resources to be shared among many UEs and resulting in significant intra-resource block (RB) interference. To address this challenge, we propose an interference management scheme based on UE clustering and rate-splitting multiple access (RSMA). Specifically, the network is modeled as a heterogeneous graph, and a graph neural network (GNN) is developed to efficiently allocate the common and private RSMA powers, maximizing the minimum spectral efficiency (SE) in a fast and scalable manner. Simulation results demonstrate that the proposed GNN-RSMA interference management algorithm outperforms conventional multiple access schemes while achieving fairness and worst-user performance comparable to successive convex approximation (SCA)-based optimization at only a fraction of its computational cost.

We give new lower bounds for the Shannon capacities of small odd cycles: $\Theta(C_7)\geq3.258805369885\ldots$, $\Theta(C_{11})\geq5.294502522149\ldots$, $\Theta(C_{13})\geq6.302455083464\ldots$, $\Theta(C_{15})\geq7.301600534487\ldots$, $\Theta(C_{19})\geq9.357192705918\ldots$, $\Theta(C_{21})\geq10.342455853338\ldots$, and $\Theta(C_{23})\geq11.328224257774\ldots$. The bounds are obtained by an iterative procedure due to Gao (2026) which is based on a method by Itty, Rosin, Carstensen and Reichman (2026). The bounds are fully formalised in Lean.

We investigate the forward-backward-splitting version of the Plug and Play (PnP) method for linear ill-posed problems with MMSE estimators as denoisers. In contrast to existing literature, we consider estimators which are specialized for (degenerate) Gaussian noise with possibly non-diagonal covariance matrices. We further deviate from the classical iteration by replacing parts of the descent step with a linear operator that relates the observation noise to that of the MMSE estimator. Under mild assumptions, we derive several properties of the denoiser and prove recovery guarantees of the iteration both pointwise and in the Wasserstein distance of the underlying probability distributions. Crucially, our analysis shows that the denoiser cannot be chosen in a physics-agnostic way, that is, independently of the forward model. We extend our results to the case where the MMSE denoiser is parametrized by a neural network and derive the corresponding recovery bounds.

Batteryless overlays couple passive throughput to Wi-Fi sounding overhead and channel state information (CSI) aging. This paper investigates channel sounding for ultra-high reliability (UHR) operation in a Wi-Fi 8/IEEE 802.11bn-inspired downlink multi-user multiple-input multiple-output (MU-MIMO) system with a batteryless passive overlay. We optimize the post-sounding transmission interval to maximize the aggregate throughput of the active Wi-Fi and passive links, while jointly accounting for sounding overhead, CSI aging, modulation and coding scheme (MCS), passive attenuation, and passive data rate. A packet-level cross-layer model evaluates the cycle-average throughput, and a data-driven search identifies the optimal interval under different operating conditions. Simulations demonstrate that passive overlay reshapes the conventional sounding tradeoff: depending on the MCS and passive-link configuration, the additional passive throughput may or may not compensate for the associated Wi-Fi reliability loss, causing the optimal interval to shift. The results provide design guidance for reliable and low-power MU-MIMO WLANs.

Let $X,X'$ be independent and identically distributed discrete real-valued random variables of finite Shannon entropy, and write $H(X)$ for the Shannon entropy of $X$. We prove that \[ \max\{H(X+X'),\,H(XX')\} \ge \frac87 H(X)-O(\log H(X)). \] This is the entropic analog of the celebrated sum-product phenomenon, and answers a question of Goh, which simply asked for a coefficient strictly larger than 1. An example by the author, Gavalakis, and Kontoyiannis showed the coefficient cannot exceed $\frac43$. Previous work by Gavalakis, Goh, and Kontoyiannis was able to prove a result of a weaker form, which could not translate to a coefficient strictly larger than 1 because of examples where the min-entropy is significantly smaller than the Shannon entropy. By splitting the distribution of $X$ into uniform pieces, which costs $O(\log H(X))$ entropy, we obviate this issue, establishing a coefficient of $\frac{10}{9}$. We augment this to $\frac87$ by adapting the work of Solymosi, which established the combinatorial sum-product phenomenon with coefficient $\frac43$ by bounding the multiplicative energy, to the entropy setting, again via a uniformization technique.

An asymmetric random variable $X$ in the reals is said to be variance symmetrization resistant if every independent random variable $Y$ in the reals that produces a symmetric sum $X+Y$ has a greater variance than that of $X$. Asymmetric Bernoulli random variables were shown to be variance symmetrization resistant by Kagan, Mallows, Shepp, Vanderbei, and Vardi (1999); Pal (2008) gave a proof using stochastic calculus. We introduce the notion of entropic symmetrization resistance on locally compact groups– this means that the entropy of any independent symmetrizer $Y$ must exceed that of $X$. We show that asymmetric Bernoulli random variables exhibit entropic symmetrization resistance, and show a multidimensional generalization. We also explore basic aspects of the entropic symmetrization resistance problem in compact groups. In particular, we show that any distribution on a finite group that is entropic symmetrization resistant must lie on the boundary of the probability simplex, and describe precisely the class of all entropic symmetrization resistant distributions on $\mathbb{Z}_3$ and $\mathbb{Z}_4$.

We realize the GML estimator by an expectation-maximization algorithm initialized near the true positions and verify the theory by simulation.

The upper-mid band, particularly the 7-8 GHz range within frequency range 3 (FR3), has emerged as a leading spectrum candidate for wide-area sixth-generation (6G) cellular networks. Its shorter wavelength enables hundreds of antenna elements to be integrated within the physical aperture of an existing 5G base-station panel. In principle, the resulting aperture gain can compensate for the increased path loss and enable extreme MIMO (E-MIMO) with 256 or more antenna ports while reusing current cell sites. In practice, however, simply scaling the 5G New Radio (NR) architecture from tens to hundreds of ports encounters fundamental system-level limitations. This paper identifies where 5G-style MIMO scaling breaks and develops a research roadmap for practical upper-mid-band E-MIMO. We first review the evolution of FR3 spectrum, its propagation and channel characteristics, and the emerging 6G system requirements. We then organize the principal challenges into four coupled areas: maintaining effective coverage across all physical channels and protocol states; implementing wideband, energy-efficient RF devices and radio units; developing new low-power array and beamforming architectures; and acquiring sufficiently refined channel state information with manageable sounding and feedback overhead. Representative system studies illustrate the coverage asymmetry between user-specific data transmission and common or channel-acquisition signals, as well as the spectral- and energy-efficiency tradeoffs among fully digital, hybrid, tri-hybrid, dynamic-metasurface, and fluid-antenna architectures. Finally, we discuss how distributed apertures, integrated sensing, AI-assisted channel acquisition, and environment-aware operation can transform fixed-aperture scaling into a deployable 6G E-MIMO architecture.

Inspired by a classical algorithm for online prediction, we propose a novel denoising scheme which is universal for families of probability distributions both in terms of the source that generates the noiseless signal and in terms of the channel that generates the noisy signal. The new denoising scheme does not rely on assumptions of exact channel knowledge or finite signal alphabets. Our analysis provides an upper bound for the performance gap compared with the Bayes envelope. For the special case in which the signal is an i.i.d. sequence and the family is countable, we characterize the consistency condition under which our scheme approaches the Bayes envelope as the length of the sequence tends to infinity. We also show that in general, approaching the Bayes envelope is not possible for a universal denoising scheme when this consistency condition is not satisfied. Furthermore, we show that, as in the online prediction case, the so-called plug-in approach (which relies on a maximum likelihood estimation of the underlying distribution parameter) does not approach the Bayes envelope in general. We also include numerical evaluations of our scheme for denoising of a Poisson signal.

Deep joint source–channel coding (Deep JSCC) enables visual semantic transmission by mapping inputs directly to channel symbols and task outputs, but its performance can deteriorate under distribution shifts between training and deployment domains. We study single-source domain adaptation for task-oriented Deep JSCC and formulate a classification-capacity-invariance (CCI) function to characterize how the available channel capacity and class-conditional cross-domain invariance affect target domain classification accuracy. A scalar linear analysis of source-domain-optimal solutions and a controlled shallow nonlinear validation show that target domain classification accuracy can vary non-monotonically with the invariance constraint and with available capacity along separate control paths obtained by varying the transmitted dimension or CSNR. We then propose a domain-adaptive Deep JSCC framework that combines pseudo-label-based class-level adversarial alignment with supervised contrastive learning on confidence-filtered target samples. Experiments on digit and PACS datasets over AWGN and Rayleigh fading channels demonstrate improved target domain generalization without introducing additional inference-time networks. On SVHN $\rightarrow$ MNIST, the proposed method achieves 98.15% target-domain accuracy at a CSNR of 10 dB.

Let $N=2^k-1$ and let $\operatorname{wt}(n)$ denote the binary Hamming weight. The Tu-Deng conjecture asserts that, for every $1\le t\le N-1$, at most $2^{k-1}$ pairs $(a,b)\in\{0,\ldots,N-1\}^2$ satisfy $a+b\equiv t\pmod N$ and $\operatorname{wt}(a)+\operatorname{wt}(b)<k$. Partial results are known. We give a complete proof of this conjecture. We first show that the Tu-Deng counts equals the number of cyclic carry solutions for which $\operatorname{wt}(B)-\operatorname{wt}(A)<0$ and $A+t\equiv B\pmod N$. The enumerator of the cyclic carry solutions factors as $$C_v = 1+(X+Y-1)J_v+X^{\operatorname{z}(v)+1}Y^{\operatorname{o}(v)+1},$$ where $t=10v$ is the binary expansion of $t$(least significant bits first) and $J_v$ enumerates the language $$\operatorname{Sub}(v)\mathbin{\dot\cup}\{u\in\partial_1\operatorname{Sub}(v):u<_{\rm lex}v\}.$$ Estimating the strict negative half-plane mass of $C_v$ gives the desired bound.

We make progress on the question of constructing explicit evaluation places for leakage-resilient Shamir's secret sharing, over composite order fields. Previously, Maji et al. (EUROCRYPT 2024) showed that random evaluation places yield Shamir's secret sharing over the composite order field $\mathbb{F}_{p^d}$ that is statistically secure against physical-bit leakage. Later, Nguyen (EUROCRYPT 2025) established a dichotomy that linear code-based secret-sharing scheme over the field $\mathbb{F}_{p^d}$ is either statistically secure or completely insecure against such leakage. Building upon Nguyen's dichotomy, we present a partial derandomization of evaluation places, improving upon the Maji et al. result for a restricted regime of parameters. We replace the random choice of $n$ independent evaluation places by the iterates $x_j = \Phi^j(x_0)$ of a simple fixed rational function $\Phi$, where the initial point $x_0 \in \mathbb{F}_{p^d}^*$ is randomly chosen. The randomness in the evaluation places thus drops from $nd \log p$ bits to $d\log p$ bits. Our construction is valid for the regime $n = O(d/\log_p d)$, and any reconstruction threshold $k \ge 2$; in fact, the scheme attains perfect security (statistical distance exactly zero) against single-block leakage. Our technique is a partial fraction nondegeneracy argument that exploits the distinct poles of the rational iterates.

We construct an infinite family of $q$-ary primitive narrow-sense BCH codes whose minimum distance strictly exceeds the Bose distance; in fact, the gap between the two can be arbitrarily large as the length of the code tends to infinity. The key idea is to embed these BCH codes in a suitably large punctured generalized Reed–Muller code, whose codeword weights obey divisibility conditions supplied by Ax's theorem. This divisibility forces the minimum distance of the BCH codes far above the Bose distance. In particular, our family disproves a longstanding conjecture of Charpin asserting that this difference is at most four.

We study the convergence properties of the random-sweep Gibbs sampler for Gaussian graphical models with a thin-membrane prior. We demonstrate that the convergence rate of the Gibbs sampler is significantly accelerated in the dual model, which is obtained by applying the Fourier transform to the local factors of the normal factor graph representing the original model. In both domains, we derive the exact convergence rates for homogeneous $k$-regular graphs. We prove that, for all homogeneous models whose graphical representations contain cycles, the convergence rate in the dual domain is universal and independent of the underlying graph topology. Moreover, we show that the effective convergence rate in the dual domain is governed by the algebraic connectivity of the graph, providing an additional acceleration without increasing the computational complexity per sweep. We further establish an explicit algebraic relation between the covariance structures of the primal and dual models, enabling marginal statistics of the primal model to be recovered directly from those of the dual model. Finally, numerical experiments on several graph families confirm our theoretical results and demonstrate substantial improvements in the convergence rates in various settings.

Robert Smithson's 1970 land artwork Spiral Jetty, located in the north arm of Utah's Great Salt Lake, provides a fixed remote-sensing target whose visual complexity reflects hydroclimatic conditions. A companion study analyzed 1,744 co-registered Landsat 4-9 and Sentinel-2 image chips spanning every year and month from 1984 to 2025. It found robust relationships between coarse-scale permutation entropy, mean intensity, and the third principal component of ResNet50 avg-pool embeddings, and lake elevation, regional temperature, and cumulative CO2. It also showed that image complexity leads lake stage by about three years and that the long-term trend is non-monotonic, with decline after 2015. Building on those findings, this paper develops a two-stage forecasting pipeline. Stage 1 applies IPCC AR6 SSP1-2.6, SSP2-4.5, and SSP5-8.5 temperature changes to forecast regional temperature, north- and south-arm lake elevation, salinity, and cumulative CO2 for 2030 and 2050. Stage 2a projects 14 image-complexity features using linear and Random Forest regressions trained on the 42-year record. All six scenario-year combinations move the inputs outside the 1984-2025 training distribution. The Random Forest model therefore saturates near the post-2015 dry-playa regime, while a threshold-based hydrological interpretation classifies Spiral Jetty as fully exposed in every scenario. Stage 2b specifies a climate-conditioned latent-diffusion framework using Stable Diffusion XL fine-tuned with LoRA on the 1,744 chips, ControlNet conditioning on the climate vector, and a hydrological physics mask for ensemble image synthesis. Code is provided, and validation results will be released after diffusion training. We conclude by discussing ethical implications of generative speculation on cultural heritage and outlining a roadmap for forecasting at Sun Tunnels, Double Negative, Lightning Field, and Roden Crater.

We propose Kohn-Sham Spectral Embedding (KSSE), an energy-based model replacing the top-layer classifier of convolutional networks with a sparse-graph spectral embedding at the Nishimori temperature of an associated Random-Bond Ising Model the spectral detectability threshold where class structure becomes marginally distinguishable from disorder. Mapping pre-trained features onto quasi-cyclic low-density parity-check graphs, we construct a regularized Laplacian (Bethe-Hessian) as an effective Kohn-Sham Hamiltonian, yielding D independent spectral problems-one per feature channel-solvable in $O(N log N + k_{mode}^{2} N)$ time by FFT on circulant blocks (Pontryagin self-duality), with low-mode Rayleigh-Ritz refinement ($k_{mode}=5$). Physically, this is a k.p effective-mass reduction on a one-dimensional ring crystal: the circulant support is the perfect crystal, the data weights a slowly varying impurity potential, and the Nishimori crossing a Fermi level at the band edge. Star-domain surgery optimizes the graph: instead of eliminating all frustrated cycles impossible without destroying the codewords-edge shifts create certified convexity around codewords with bounded residual frustration, with multi-scale fractal certification (basins $D_{2}<1$ vs rough landscapes $D_{2}>3$). The theory includes a generalized Ihara-Bass identity with a sharp spectral threshold, a non-backtracking growth trichotomy with frustration as a gauge-invariant $Z_{2}$ flux, a trapping-set spectral test, exact channel separability with a cup-product obstruction, plus loop-series, convexity, surgery, and quasi-stationarity bounds. On ImageNet-1000 with frozen EfficientNet-B4 features (D=1792) under a transductive protocol, KSSE achieves 88.93% Top-1 accuracy with ~21.24M parameters-beating Swin-L (197M, 86.4-87.3%) and matching the lower end of ViT-H/14 (632M, 88.0-89.5%) with 10x and 30x fewer parameters.

The finite-block maximum-likelihood (ML) guarantee of soft-input GRAND requires querying noise-effect patterns in nonincreasing conditional-likelihood order. Under correlated Gaussian noise, additive reliability metrics and independent-block approximations need not preserve this order because the matched metric contains cross-coordinate interactions; the first codebook hit need not induce an ML codeword. We develop Low-Pathwidth GRAND (LP-GRAND) for binary phase-shift keying (BPSK) with precision matrix $Q$. The candidate-dependent part of the Gaussian negative log-likelihood is an observation-dependent quadratic pseudo-Boolean energy whose interaction graph has edge $\{i,j\}$ exactly when $Q_{ij}\neq0$. If $Q$ has half-bandwidth at most $\nu$, this energy admits a trellis with at most $2^\nu$ states per layer; a path decomposition of width $w$ yields at most $2^{w+1}$ bag assignments per layer. In real arithmetic, suffix dynamic programming and best-first complete-path enumeration enumerate patterns in nondecreasing energy. With complete enumeration and no abandonment, the first codebook hit induces an ML codeword for any nonempty binary codebook with equiprobable codewords. LP-GRAND agreed with exhaustive codeword ML in all $10{,}000$ frames for two $[20,12]$ codes. At nominal $E_b/N_0=2$ dB, its empirical BLER was lower than that of each block-based approximation for six $[64,52]$ codes.