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1,013 records · Page 6Linked to original sources
arXiv2026

What Will This Copper Look Like Later? Forecasting Surface Appearance and Rendering It as a PBR Material

Digital design requires predicting how a metal surface will look later in its oxidation; this paper presents such a pipeline for copper. Given a fixed-camera observation, the system forecasts appearance 10 accelerated units ahead and converts it into the albedo, normal, roughness and metallic maps a renderer consumes. Forecasting is evaluated as an authoring tool would use it, on a copper specimen the system has not observed: an entire recording is held out, so training and checkpoint selection use one specimen and the test set is the whole of a second, recorded on a different day and condition. Under this protocol a learned spatio-temporal model with a monotone oxidation state, the most accurate forecaster within a single recording, is less accurate than copying the last observed frame on an unseen specimen, in both directions, as are three further trained architectures. The only forecaster that transfers is a closed-form global color extrapolation with no trained parameters, improving on copy-last-frame by 13.4% and 50.6%, with a margin that increases with horizon to +16.7% and +55.5% at t+10. Two controls qualify this: correcting every frame for the photometric drift measured on a non-oxidizing reference region leaves both margins intact, ruling out uncontrolled exposure as their source, and a moving-block bootstrap over the 6 independent windows each recording contains separates the larger margin from zero but leaves the smaller one not individually significant. The mechanism is measured: a learned susceptibility map encodes where corrosion begins on the training specimen and misleads on a new one, whereas the global color trajectory is what specimens share. The pipeline therefore deploys the closed-form forecaster for unseen specimens and the learned model only for continuing one already observed. Code, splits, protocol and leakage audit are released.

cs.GR↗
arXiv2026

Observer-robust energy condition verification for warp drive spacetimes

Whether a warp drive metric requires exotic matter is decided by energy conditions quantified over all observers, not only the Eulerian. Each of the null, weak, strong and dominant conditions is equivalent, at a point, to feasibility of a $4\times4$ linear matrix inequality $A_{ab}+σg_{ab}\succeq0$, by the S-lemma, with $A_{ab}$ the stress-energy tensor or its trace reverse and the dominant condition a conjunction of two such tests. It forms no eigendecomposition of $T^a{}_b$, imposes no rapidity cap and assumes no Hawking-Ellis type, so it decides all four alike, Types I and IV not being exhaustive; its multiplier margin is exactly half the null-cone minimum, so the same test returns the severity. Composed with an interval enclosure of the curvature chain it decides a point from the metric itself, not from a floating-point copy of its stress-energy. At Type I each condition reduces instead to an eigenvalue inequality holding for all observers at once. The type label is numerical and tolerance-bound; the reported severities are rapidity-capped diagnostics, not certificates. Everything decided uses only boost-invariant data and stays well posed through $v_s=1$. On a flat slice the Eulerian momentum that opens the Type-IV wall vanishes only for a gradient shift, so among four matched drives the irrotational Rodal geometry is Type I identically, its shift curl-free by an exact profile identity, while Alcubierre and Natário are Type-IV dominated at every sampled speed and Van den Broeck above its transition. A single-frame reading of Rodal misses about 73% of its wall weak-energy violations. All four violate the pointwise null energy condition at every sampled speed, consistent with the Santiago-Schuster-Visser no-go, whose null step is conditional. Both are realized in warpax, a JAX toolkit building $T^a{}_b$ by automatic differentiation.

gr-qc↗
arXiv2026

Preference Shapes Relevance: Cross-component Hierarchical Semantic Alignment for Personalized Generative Retrieval

Generative Retrieval (GR) has emerged as a promising paradigm by mapping queries directly to Semantic IDs (SIDs) with powerful representation capabilities for candidate items. However, existing SIDs derived solely from item content create a semantic gap, failing to align dynamic query intents with static item representations. Furthermore, current generative paradigms rarely model user behavior sequences and are always bottlenecked by the high inference latency of beam-search autoregressive decoding. To address these challenges, we propose $\textbf{C}$ross-component $\textbf{H}$ierarchical semantic $\textbf{A}$lignment for $\textbf{P}$ersonalized generative retrieval ($\textbf{CHAP}$), a novel personalized GR framework from a hierarchical perspective. First, we design a Hierarchical Semantic Alignment module to align query's latent space with item's quantization path and synchronize multi-granular semantics. Second, we construct a personalized GR framework that models user behavior by synergizing discrete SIDs for structural guidance and continuous representations for fine-grained semantic refinement. Notably, we introduce a Residual Cascading Generation mechanism to restrict the costly multi-step Transformer Decoder to a single-pass inference, boosting inference throughput while mitigating information loss. Extensive experiments on three public datasets, one proprietary industrial dataset, and online A/B tests demonstrate CHAP's superiority, validating the effectiveness and practical value of our approach. The code is publicly available at https://github.com/zzzgm/CHAP.

cs.IR↗
arXiv2026

Efficient GPU Retrieval for Semantic Search

Semantic Search on LinkedIn must retrieve relevant profiles from a corpus of hundreds of millions in response to natural-language queries such as "a fintech founder in Berlin who worked in payments." The deployed relevance policy is bottleneck-oriented: every active non-negotiable facet must be satisfied, and a pre-existing LLM Graded Relevance (GR) judge operationalizes this through a fixed min/median aggregation over facet grades. Cosine similarity instead averages evidence, letting a strong match on one facet mask failure on another, capping the recall of the first-stage (L0) retriever. We present a policy-aligned retrieval framework: embeddings are partitioned into eight category-supervised segments whose scores follow the same min/median rule at serving time; for multi-vector retrieval, this segment score is computed independently per tagged document slot and maximized across slots. A lightweight single-slot Stage-1 scorer generates high-recall candidates, while scale-invariant relative-norm gating keeps category activation consistent across training, evaluation, and serving. On 21K held-out queries, this representation improves offline relevance over a matched-capacity baseline, with gains broadly distributed across facet combinations. We serve this framework with a two-stage GPU architecture: an FP8 coarse ranker scores the full corpus, increasing per-shard capacity by 71% and Stage-1 matmul throughput by 36%, then an FP16 stage exactly re-ranks an oversampled candidate set, recovering 99.6-99.8% of full-FP16 recall at over 500 QPS per shard replica. In a member-randomized A/B test, exploratory-query Precision@10 under the unchanged GR judge rises from 63.7% to 79.0% and navigational Precision@1 from 65.5% to 74.7%, with a blinded human evaluation independently confirming the Precision@10 gain.

cs.AI↗
arXiv2026

Beyond Semantic IDs: Encoding Business-Value Ranking into Document Identifiers for Generative Retrieval

Generative Retrieval (GR) formulates retrieval as a sequence-to-sequence generation task, assigning each document a document identifier (DocID) and retrieving it through autoregressive decoding, making DocID design a critical factor in retrieval quality. However, existing schemes based on discrete representation learning suffer from inherent collision issues and create a mismatch between the DocID's encoding objective and the system's business optimization target. To address these limitations, we propose \textbf{Cluster-Ranked Identifier (CRID)}, which decouples DocID into \textit{semantic clustering} and \textit{business-value ranking}, yielding collision-free identifiers that support incremental updates via intra-cluster reranking. We further introduce an analytical framework that decomposes retrieval gains into \textit{personalized preference} and \textit{statistical prior} generalization, revealing how semantic cluster size governs the balance between the two components. Experiments on a Taobao e-commerce corpus of over 300M items show that CRID surpasses the strongest embedding-based retrieval baseline on top-K Hitrate, and delivers +1.06\% GMV in full-traffic deployment.

cs.IR↗
arXiv2005

The maximum entropy state

We give an algorithm for calculating the maximum entropy state as the least fixed point of a Scott continuous mapping on the domain of classical states in their Bayesian order.

math.PR↗
arXiv2026

Semantic Trimming and Auxiliary Multi-step Prediction for Generative Recommendation

Generative Recommendation (GR) has recently transitioned from atomic item-indexing to Semantic ID (SID)-based frameworks to capture intrinsic item relationships and enhance generalization. However, the adoption of high-granularity SIDs leads to two critical challenges: prohibitive training overhead due to sequence expansion and unstable performance reliability characterized by non-monotonic accuracy fluctuations. We identify that these disparate issues are fundamentally rooted in the Semantic Dilution Effect, where redundant tokens waste massive computation and dilute the already sparse learning signals in recommendation. To counteract this, we propose STAMP (Semantic Trimming and Auxiliary Multi-step Prediction), a framework utilizing a dual-end optimization strategy. We argue that effective SID learning requires simultaneously addressing low input information density and sparse output supervision. On the input side, Semantic Adaptive Pruning (SAP) dynamically filters redundancy during the forward pass, converting noise-laden sequences into compact, information-rich representations. On the output side, Multi-step Auxiliary Prediction (MAP) employs a multi-token objective to densify feedback, strengthening long-range dependency capture and ensuring robust learning signals despite compressed inputs. Unifying input purification and signal amplification, STAMP enhances both training efficiency and representation capability. Experiments on public Amazon and large-scale industrial datasets show STAMP achieves 1.23--1.38$\times$ speedup and 17.2\%--54.7\% VRAM reduction while maintaining or improving performance across multiple architectures.

cs.IR↗
arXiv2025

Turing complete Navier-Stokes steady states via cosymplectic geometry

In this article, we construct stationary solutions to the Navier-Stokes equations on certain Riemannian $3$-manifolds that exhibit Turing completeness, in the sense that they are capable of performing universal computation. This universality arises on manifolds admitting nonvanishing harmonic 1-forms, thus showing that computational universality is not obstructed by viscosity, provided the underlying geometry satisfies a mild cohomological condition. The proof makes use of a correspondence between nonvanishing harmonic $1$-forms and cosymplectic geometry, which extends the classical correspondence between Beltrami fields and Reeb flows on contact manifolds.

math.DG↗
arXiv2026

Geometric mean and Lebesgue-type decomposition of completely positive maps

We introduce the geometric mean and the parallel sum of completely positive (CP) maps between von Neumann algebras, based on the Pusz--Woronowicz theory of positive sesquilinear forms. We provide a concrete characterization via a block matrix positivity condition and establish their fundamental properties, including the AM--GM--HM inequality with respect to the CP order. In finite-dimensional settings, our construction is compatible with the Choi--Jamiolkowski correspondence, under which the geometric mean of CP maps corresponds to the Kubo--Ando geometric mean of their Choi matrices. This yields a natural operator-theoretic framework for interpolating quantum channels. As an application, we obtain index-type inequalities for conditional expectations in subfactor theory. Finally, we establish a Lebesgue-type decomposition of CP maps via a parallel sum construction, thereby providing a unified framework that simultaneously generalizes Ando's decomposition of bounded positive operators and Kosaki's decomposition of normal positive functionals on von Neumann algebras.

math.OA↗
arXiv2026

Two Adjoint Perspectives on Fokker-Planck Optimization: A Microscopic-Macroscopic Correspondence

The Fokker-Planck equation admits both a macroscopic Eulerian description through probability densities and a microscopic Lagrangian description through stochastic trajectories. Consequently, optimization problems constrained by the Fokker-Planck equation can be formulated from either perspective. Surprisingly, the corresponding adjoint equations appear to be fundamentally different: the macroscopic adjoint is governed by the backward Kolmogorov equation, whereas the microscopic adjoint evolves pathwise along stochastic trajectories. In this note, we reconcile these two formulations by establishing their correspondence in the continuum setting. We further show that, although their discrete gradients no longer coincide after discretization, both provide consistent numerical approximations of the continuum gradient. Explicit convergence rates are established for both discretization strategies.

math.NA↗
arXiv2026

Multiplicative comparisons of Rényi entropies for weighted Bernoulli sums

We establish improved multiplicative bounds relating the Rényi entropies of different orders for weighted sums of independent Bernoulli random variables. In particular, we prove a logarithmic bound between the zeroth-order and infinity-order Rényi entropies, which yields a polynomial improvement over the square-root bound of Jain, Sah, and Sawhney. Additionally, we obtain explicit constant-factor bounds for comparisons among Rényi entropies of nonzero orders.

math.PR↗
arXiv2026

Set Theory in the Foundation of Math; Internal Classes and External Sets

Usual math sets have special types: countable, compact, open, occasionally Borel, rarely projective, etc. Each such set is described by a single set theory formula with parameters unrelated to formulas. Exotic expressions involving sets related to formulas of unbounded quantifier depth appear mostly in esoteric or foundational studies. Recognizing the internal to math (formula-specified) and external (parameter-based) aspects of math objects greatly simplifies foundations. I postulate that external sets (not internally specified, constituting the domain of quantifiable variables) are hereditarily countable and independent of purely formula-defined classes, i.e. with finite algorithmic information about them. Variables for classes are not explicitly quantified. This opens a way to eliminate all non-integer quantifiers in set theory sentences. The restrictions seem to require almost no changes in math papers, only reinterpreting some formalities.

cs.LO↗
arXiv2026

Harmonic higher weight distributions, Simonis' approach of MacWilliams identity and moments

We present a combinatorial proof of Simonis type MacWilliams identity for harmonic higher weight distributions of linear codes. Furthermore, we investigate the statistical moments of the harmonic higher weight enumerators for random linear codes. Defining the enumerators via rank functions of the generator matrices of linear codes, we prove that its expectation vanishes for all non-trivial harmonic functions due to the inherent symmetry of random matrices, and we also derive an explicit, non-trivial formula for the covariance.

math.CO↗
arXiv2026

Shannon's problem on the monotonicity of entropy and a Conjecture of Tao

Let $X_1,X_2,\ldots$ be i.i.d. finitely supported random variables in a torsion-free abelian group, and write $S_k=X_1+\cdots+X_k$, and $H(S_k)$ is the Shannon entropy $S_k$, for all $k \ge 1$. We prove that, for every fixed $n\geq1$, \[ H(S_{n+1})-H(S_n) \geq \frac12\log\frac{n+1}{n} -o_{H(X_1)\to\infty}(1), \] uniformly over the ambient group and the input law. This proves a conjecture of Tao [29] in 2010.

math.PR↗
arXiv2026

Geometric integrators for adiabatically closed simple thermodynamic systems

A variational formulation for non-equilibrium thermodynamics was developed by Gay-Balmaz and Yoshimura. In a recent article, the first two authors of the present paper introduced partially cosymplectic structures as a geometric framework for thermodynamic systems, recovering the evolution equations obtained variationally. In this paper, we develop a discrete variational principle for adiabatically closed simple thermodynamic systems, which can be utilised to construct numerical integrators for the dynamics of such systems. The effectiveness of our method is illustrated with several examples.

math-ph↗
arXiv2026

A counterexample to Kusner's conjecture on equilateral sets

We disprove Kusner's 1983 conjecture that every equilateral set in $\ell_p^n$ with $2 57$. This is the first equilateral set of more than $n+1$ points in $\ell_p^n$ for any finite $p\ge2$. The construction persists on an open interval of exponents around $5$; since Ge, Xu and Zhou recently proved the conjecture for $2\le p\le4$, the infimum of exponents at which it fails lies in $[4,5)$. The configuration is the unique solution of an explicit polynomial system with rational coefficients in a rational box, established in exact arithmetic.

math.MG↗
arXiv2026

Edge codes constructed from unicyclic graphs

Jaramillo-Velez recently introduced edge codes, a new class of toric evaluation codes constructed from the edges of a (hyper)graph $\mathcal{H}$. In the case that $\mathcal{H}$ is a tree, Jaramillo-Velez computed both the minimum distance and the weight distribution of the associated code. In this paper, we study edge codes associated to unicyclic graphs. Our most striking result is that computing the parameters of these codes is subtle in the case that the induced cycle has an even length because these values will depend on certain conditions regarding the length of the cycle and the size of the base field.

math.CO↗