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arXiv2026

Blind Random Search with Noisy Loss Measurements: Averaging, Thresholding, and Almost Sure Convergence

Blind random search repeatedly draws a candidate point and replaces the current estimate whenever the candidate has a lower loss. In the absence of noise, the true loss is observed directly. It decreases strictly at every accepted update and is monotone nonincreasing over all iterations. Measurement noise can make a worse candidate appear better and thereby break this monotonicity. To recover almost sure convergence under noise, we incorporate averaging and thresholding into the original decision criterion. These two classical tools are coupled. As the sample sizes grow, the positive threshold shrinks at a matched rate. These modifications allow blind random search to recover eventual monotonicity of the true loss under noisy measurements and to converge almost surely.

math.OC↗

arXiv2026

Difference equations of average entropies

Exact cumulants of entanglement entropies of random state ensembles have traditionally been studied within the random matrix framework. In this work, we propose an alternative approach based on the intrinsic connection to integrable systems. The central idea is to embed entropic quantities into tau functions satisfying Toda-type lattice equations, which in turn yield linear difference equations for their averages. Directly solving the difference equations recovers exact entropy formulas in the literature. The integrable systems approach bypasses the case-by-case, ensemble-dependent derivations required by random matrix methods. The approach also suggests a possible route towards unified and more efficient higher-order cumulant calculations by exploring integrable hierarchies.

math-ph↗

arXiv2026

It Takes Three to Converse: Empirical Observations on How the Developer, the Convener and the Participant Shaped 119 Polis Conversations

Polis is a popular democratic innovation tool that allows asynchronous citizen engagement through atomic statements: short statements that together describe a complex question, inviting the citizen to vote Agree or Disagree on each. This paper uses 119 conversations with 100 or more participants and an extensive data export, drawn from a wider set of 271 collected processes. The paper asks what determines the output of such a process. Three parties shape the result. The developer of the platform has made important design choices that restrict the outcome: the number of groups the platform is able to report (restricted to 2--5) and which statements are prioritized. The convener defines the assignment: the initial statements that set the tone, the policy that accepts or rejects new statements and who can be invited. Finally, the participant works within these boundaries. With access to less than half of the generated statements, they end up responding to more statements when their conversation seems to have an achievable number of statements to complete, than when they are presented with more statements. Due to choices such as warm path clustering, the exported resulting clustering cannot be reproduced based on the voting data. Conveners may want to re-analyse their own conversations once the process is closed, to consider the data in its entirety, and make their own analysis priorities explicit.

cs.CY↗

arXiv2026

Commit-first LLM judging inherits the judge's own errors

LLM judges, models that score another system's output, can be gamed by the systems they score. Recent work identifies one defence that works: the judge solves the task itself first and commits to that answer, then accepts a candidate only if the two match. We call this commit-first judging, and ask whether shipped software implements it, and what it costs. We audit the default judge configurations of eight widely used evaluation frameworks. Of the 24 configurations in scope, none implement it. Nine implement a variant the literature measures as ineffective, and share one ancestor prompt, traceable through a copied typographical error. In a controlled experiment, an ordinary best-of-N search with no access to correct answers optimises code against one of these configurations, used exactly as documented. On an interval merging task the judge accepted 90 of 96 candidates in one seed and 93 of 96 in the other; every accepted candidate passed every test the search could see and failed a held-out suite it could not. The judge identified the defective line and cited it as grounds for a perfect score. Commit-first judging removed the effect: 0 of 96 in both seeds. On a second task it made matters worse in both seeds: the judge's committed answer was wrong, and in one seed the population converged on it. This is our main finding. Commit-first judging does not remove the anchor that gets gamed, it moves it from the candidate to the judge's own answer, so evaluation is only as good as the judge is at the task. That precondition is cheap to measure in advance, and is task local rather than scale dependent: a smaller judge solved a task the frontier judge failed and resisted gaming where it did not. We also validate our own instruments: five of fifteen claims in our criteria were wrong against verbatim sources, and two held-out checks were unjustified by their specifications.

cs.SE↗

arXiv2026

An Explicit Family of Log-Concave Counterexamples to the Gaussian Completely Monotone Conjecture

We construct smooth, strictly log-concave counterexamples to the Gaussian completely monotone conjecture in every dimension. In one dimension, they form an explicit family $f_m$ whose signed $m$th entropy derivative at time zero is negative for every sufficiently large $m$; the inequality persists for all sufficiently small positive times. Tensorization with a broad Gaussian factor gives the higher-dimensional examples. The argument is analytic and self-contained. It reduces the sign to a two-frequency entropy calculation on the circle and transfers the resulting asymptotic to the real line through an exact heat-flow formula for Gaussian-windowed Fourier modes. The proof was developed by GPT-5.6 Sol Pro under the authors' guidance.

cs.IT↗

arXiv2026

One note in three: a verified census of three deployed AI scribes, and the instrument that counted it

Ambient AI scribes draft clinical notes under the reassurance that a clinician signs every note. We audited three commercial AI scribes on the same 142 consultations: 565 notes from recorded UK primary-care and US ambulatory encounters plus authored scenarios. Twelve discovery passes proposed 13,678 candidate errors; the 5,898 clearing an importance filter went to an adversarial panel of two models from different families, each told to refute what it could, and 618 survived. One note in three (31.3% [27.0, 35.6]) carries a verified failure, concentrated in allergy and medication information, invented patient identity, and history written up as examination on telephone consultations that can contain none. No product was given a patient record; setting aside the two classes a record would have prefilled, invented identity and dates, the rate is 24.8% [20.8, 29.0]. One failure mode did not fit our scheme, drawn from published scribe-error taxonomies: a treatment the clinician retracts, recorded as delivered care. Two clinicians adjudicated blind, disjoint samples: a physician author upheld 20 of 21 findings (95.2% [77.3, 99.2]) and an independent clinician, not an author, 12 of 12 ([75.8, 100]); both judged every sampled refusal genuine. A failure rate depends on the instrument as much as the scribes. With model, evidence and settings fixed, the review instruction alone moves the share of candidates verified from 9.3% to 79.0%, and the reviewing family moves it too: alone at that instruction the gentler flags 54.8% of notes against 27.8%. Between 28% and 97% of sampled notes carry a failure depending on the standard. Published audits disagree among themselves by a margin instrument differences alone can produce: omission is 54-86% of their errors against our 23.1%. We release all 618 findings with transcript-side evidence, every prompt and model version, and the re-runnable pipeline.

cs.CL↗

arXiv2026

The marginal is pretty good

One-shot information theory measures often require an optimization over states, but the form of these optimizers can be complicated or depend on the initial problem in nonlinear ways. In this note, we show that in many instances using the marginal instead of the optimal state is sufficiently good and only changes the result by a small factor. We prove that for the Petz-Rényi divergence of order $α\in[1/2,1)$, replacing the optimizing state on $B$ by the marginal $ρ_B$ results in a multiplicative overhead of at most $1/α$. We also show a similar relation for the fidelity, and in the case of pure or quantum-classical states for the sandwiched Rényi divergence.

quant-ph↗

arXiv2026

Secrecy Outage Analysis over Correlated Composite Generalized-Gamma Fading Channels

This paper investigates physical-layer security (PLS) over correlated composite generalized-Gamma (GG)/GG fading channels, where both shadowing and small-scale fading follow GG distributions. Using Mellin transforms and Fox-H functions, closed-form expressions are derived for the single-link probability density function (PDF), joint distribution, survival function, and zero-rate secrecy outage probability (SOP)/probability of non-zero secrecy capacity (PNZSC). The general-rate SOP is expressed as an exact double series with one residual onedimensional integral per term. The model includes the Nakagamim/GG and Nakagami-m/Gamma channels as special cases. Numerical results validate the analysis and demonstrate the impact of the fading parameters on secrecy performance.

cs.IT↗

arXiv2026

Generalized Tan-Arlery-Rabaste-Lehmann-Ovarlez Lower Bound on Ambiguity Function of a Set of Sequences With Mismatched Filters

In this paper, a lower bound on the maximum ambiguity function (AF) sidelobes of a set of unimodular sequences is formulated for the desired low-ambiguity-zone (LAZ). Our main idea is to introduce a set of mismatched filters associated to a set of unimodular sequences and two weight vectors for the delay and Doppler shifts, respectively. The length of mismatched filter maybe different to the length of unimodular sequence. The proposed lower bound on the maximum AF sidelobes for the desired LAZ can be treated as an extension of Tan-Arlery-Rabaste-Lehmann-Ovarlez lower bound, published in 2020, which dealt with the conventional correlation of sequences.

cs.IT↗

arXiv2026

The Rate-Distortion-Deception Tradeoff

The problem of finding the optimal compression rate for a given random variable has been traditionally studied under two main constraints: distortion and perception. The distortion constraint enforces the fidelity of our reconstruction with respect to the observed realization of the random variable, while the perception constraint ensures that the reconstruction is close to a sample from the distribution of the random variable of interest. In this work, we explore the possibility of reconstruction, such that the reconstructed sample is still within a desired fidelity level with our original realization of the random variable, but at the same time, it resembles a sample from a different target distribution. We term this criterion as the deception constraint and find the fundamental tradeoffs of rate-distortion and deception.

cs.IT↗

arXiv2026

Sharp mean-field analysis of permutation mixtures and permutation-invariant decisions

We develop sharp bounds on the statistical distance between high-dimensional permutation mixtures and their i.i.d. counterparts. Our approach establishes a new geometric link between the spectrum of a complex channel overlap matrix and the information geometry of the channel, yielding tight dimension-independent bounds that close gaps left by previous work. Within this geometric framework, we also derive dimension-dependent bounds that uncover phase transitions in dimensionality for Gaussian and Poisson families. Applied to compound decision problems, this refined control of permutation mixtures enables sharper mean-field analyses of permutation-invariant decision rules, yielding strong non-asymptotic equivalence results between two notions of compound regret in Gaussian and Poisson models.

math.ST↗

arXiv2026

Look It Up: Analysing Internal Web Search Capabilities of Modern LLMs

Modern large language models increasingly integrate internal web-based retrieval to provide real-time answers, yet it remains unclear how effectively these systems identify information need, trigger retrieval, and use retrieved evidence. To understand these parameters better, we evaluate the necessity and effectiveness of internal web search through an external lens in closed-source LLMs, without access to model parameters or internal configuration. Our evaluation method comprises a static split of 783 temporally anchored factual queries answerable from pre-cutoff knowledge, designed to test whether retrieval is invoked based on factual uncertainty, and a dynamic split of 288 post-cutoff queries that require up-to-date information, designed to evaluate retrieval effectiveness under unavoidable information need. We experiment across four models across two model families and scales. Across models, enabling retrieval yields substantial accuracy gains on the static split, but systematically degrades confidence calibration. On the dynamic split, models frequently invoke retrieval yet remain below 70 percent accuracy, with failures dominated by query formulation and source selection errors rather than integration of retrieved content. While retrieval is inexpensive to invoke and can be selectively beneficial, repeated retrieval attempts rarely recover from early failures, and confidence becomes inflated once retrieval is available. Overall, internal web-based retrieval functions effectively as a low-latency verification mechanism, but falls short as a reliable IR pipeline, highlighting the need for improved retrieval triggering, query formulation, and evidence-aware confidence calibration in web-enabled LLMs.

cs.CL↗

arXiv2023

Differential uniformity properties of some classes of permutation polynomials

The notion of $c$-differential uniformity has recently received a lot of attention since its proposal~\cite{Ellingsen}, and recently a characterization of perfect $c$-nonlinear functions in terms of difference sets in some quasigroups was obtained in~\cite{AMS22}. Independent of their applications as a measure for certain statistical biases, the construction of functions, especially permutations, with low $c$-differential uniformity is an interesting mathematical problem in this area, and recent work has focused heavily in this direction. We provide a few classes of permutation polynomials with low $c$-differential uniformity. The used technique involves handling various Weil sums, as well as analyzing some equations in finite fields, and we believe these can be of independent interest.

cs.IT↗

arXiv2026

QUACK: Questioning, Understanding, and Auditing Communicated Knowledge in Multimodal Social Deduction Agents

Social deduction games have become a popular testbed for probing reasoning, deception, coordination, and belief modeling in Large Language Model (LLM) agents. However, most environments are scored only by game outcomes such as win rates and largely remain to text-only interaction, making it difficult to tell whether an agent's language is actually grounded in what it perceived and did, or to identify the failure modes underlying its behavior. To address this gap, we introduce QUACK, an open-source environment and evaluation framework for auditing the grounding of agent language in multimodal social reasoning. QUACK evaluates agents at three levels: game outcomes, behavioral trajectories, and utterance-level consistency. Its core Statement Verification Pipeline reconstructs each agent's ground-truth trajectory from engine logs and checks every discussion claim against it, automatically flagging spatial hallucination, unsupported accusation, deception collapse, and language-action inconsistency. Evaluating three frontier VLMs in both homogeneous and cross-model adversarial settings, we find that even the strongest agent hallucinates 15.1% of its verifiable spatial claims and 11.5% of accusations are strictly unsupported. We release the full engine, evaluation framework, toolkit, and logs in https://github.com/AAAAA-Academia-Attractions/QUACK.

cs.CL↗

arXiv2026

Bellman-sufficient Information Complexity

We introduce Bellman-sufficient information complexity for minimax analysis of sequential decision problems. A Bellman-sufficient state retains enough of the history to close the controlled recursion, while an index $Y=χ(Ω)$ specifies the decision-relevant information being charged. The upper bound is a log-penalized Bellman program; the lower bound is a Bellman--Fano comparison along an algorithm-dependent reference trajectory. If the two values match at a common localization scale and the stated admissibility, calibration, and growth conditions hold, they form an information-risk sandwich. UCB, E2D, and AMS/EBO control or relax the upper Bellman bracket in different ways. For the main application, we give a negative answer to a widely studied form of the GP--UCB minimax-optimality question. For every $0<α<1/4$, we construct one bounded continuous kernel whose minimax regret is $Θ(T^{1-α})$ along an infinite sequence of horizons, while two globally calibrated GP--UCB rules incur linear regret under one fixed truth. An epochwise finite-marginal action-index AIR Bellman policy, implemented through robust AIR/AMS/EBO control, attains the minimax order. The construction separates realized information from the cost of uniform optimism: many low-value directions inflate the exploration multiplier and change the trajectory. Through the canonical RKHS feature map, it also yields a finite-horizon polynomial minimax separation for the specified maximal-information-calibrated LinUCB rule. A reproducible experiment illustrates the mechanism.

cs.LG↗

arXiv2026

Honesty in Causal Forests: When It Helps and When It Hurts

Causal forests estimate how treatment effects vary across individuals, guiding personalized interventions in areas like marketing, operations, and public policy. A standard practice is honest estimation: dividing the data into two samples, one to define subgroups and another to estimate treatment effects within them. This is intended to reduce overfitting and is the default in many software packages. But is it the right choice? We show that honest estimation can reduce the accuracy of estimates of individual treatment effects, especially when effect heterogeneity is substantial and datasets are large enough to detect it. The reason is a bias-variance trade-off: honesty lowers the risk of overfitting but increases the risk of underfitting by limiting the data available to detect and model heterogeneity. Across more than 7,000 benchmark datasets, we find that the cost of using honesty by default can be as high as requiring 27% more data to match the performance of models trained without it. Honesty is best understood as a form of regularization. Whether to adopt it should depend on the goals of the application and its empirical performance, not on reflexive default use.

cs.LG↗

arXiv2026

A Complete Characterization of Tensorizable $f$-divergences

Csiszar's formulation of the $f$-divergence introduced a vast family of functionals for quantifying dissimilarity between probability distributions. However, many applications in statistics and information theory rely only on a few $f$-divergences, such as the Kullback-Leibler divergence, the $χ^2$-divergence, and the squared Hellinger distance. These divergences are especially useful because they admit simple compositional formulas under product measures, a property sometimes referred to as tensorization. In this work, we refine a formalism of tensorization previously introduced in the literature. Then, we show that any possible tensorization formula has a multi-affine form characterized by a single parameter, and identify all tensorizable $f$-divergences under our adopted notion of tensorization.

cs.IT↗

arXiv2026

Posterior Tempering Explains Variance Inflation in Linear and Generalized Linear Thompson Sampling

We study a variant of the Thompson Sampling (TS) algorithm, called $α$-TS, for solving stochastic generalized linear bandit problems. Existing analyses of TS require inflating the posterior variance to derive near-optimal regret guarantees. We formalize the idea of variance inflation by introducing $α$-TS that uses a fractional or $α$-posterior instead of the standard posterior. Our main contribution is to identify general regularity conditions on the prior and reward distributions that enable a regret analysis of $α$-TS without assuming any tractable approximation of the posterior distribution, unlike previous works. For a specific choice of $α\propto d^{-1}$, our general regret bound yields the best known regret bound of $O(d^{3/2}\sqrt{T}\log T)$ for both the exponential and sub-Gaussian families of reward distributions. We further provide an $α$-dependent lower bound showing that the regret constant depends on the product $αd$, and that when $α\propto d^{-1}$ the regret scales as $Ω(d^{3/2}\sqrt{T})$, explaining the origin of the $d^{3/2}$ factor in the upper bound. Our proof technique adapts and combines recent advancements in the analysis of linear bandit problems with first- and second-order posterior concentration theory from the Bayesian statistics literature.

stat.ML↗