Gauge

Comparing and selecting task-oriented LLM agents increasingly relies on a low-cost offline evaluation gate: persona-driven LLM user-simulators converse with each candidate, an LLM-as-a-judge scores the transcripts, and the higher-scoring agent is promoted. We introduce GAUGE, a reusable offline protocol that measures whether this gate's ranking matches a grounded verifiable reward across 25 agents from six providers on the $\tau^2$-bench and SimulatorArena benchmarks, separating two kinds of evaluation validity that release practices conflate: ranking validity and construct validity. First, a satisfaction-success gap: satisfaction carries essentially no information about task success, as conversations rated satisfied by our blind panel are decorrelated from actual success, with 57.5% of them failing the customer's task, a pattern consistent across five rater populations, both benchmarks, and every subjective dimension we rated. Second, while the gate's ranking is robust across the broad capability span, it loses resolution among the near-equal strong agents: this decision-disagreement rate jumps from $<$1% on wide-reward pairs to 31% on close pairs. The gate is thus human-validated yet mis-anchored. As a remedy, we propose a calibrate-then-trust cadence in which a judge-free completion bit is a zero-cost tripwire for truncation regressions.

Large language models (LLMs) are increasingly used in peer review at major AI conferences, yet novelty remains a persistent weak point. Existing benchmarks assess novelty as a single holistic score, making it difficult to diagnose which dimension a model misjudges or whether its evidence is faithful. We present NovGauge, a human-anchored benchmark for fine-grained novelty assessment diagnosis. The benchmark contains 619 paper pairs and 50 multi-paper sets, drawn from two expert sources: ICLR reviewer overlap claims and survey co-citations. Instances are independently labeled along three dimensions: task, problem, and method, capturing application goals, technical challenges, and solution approaches. We propose a cascading diagnostic pipeline that verifies per-dimension correctness, evidence grounding, and logical support. Evaluation of 18 LLMs shows hallucination rates ranging from 0% to 39% across dimensions, and among non-hallucinated correct-positive judgments, over 70% cite evidence fails to logically support the stated reason. The best-performing model, GPT-5.5, achieves 43-72% Verified F1 across dimensions, while most models retain less than half of their raw F1 after faithfulness verification. These results suggest that current LLMs remain far from reliable scientific novelty assessment, particularly when correctness is conditioned on faithful evidence grounding.

For trained operators, gauge reading requires little specialized knowledge, low cognitive effort, and high repeatability. Yet Multimodal Large Language Models (MLLMs) remain unreliable in continuous-valued measurement despite strong results on general multimodal benchmarks. Existing benchmarks expose this weakness but isolate measurement from realistic, knowledge-grounded settings, with limited situated context, specialized instruments, real-world noise, and matched diagnostic annotations, reducing realism and constraining root-cause analysis. We introduce InSituMeasure to evaluate situated measurement grounding. It contains 2,922 real industrial monitoring scenes across eight functional categories of professional engineering instruments, with dense gauge-attribute annotations and noise tags for failure diagnosis. We define metrics for numerical accuracy under predefined tolerances and unit consistency, rejection of fake or unanswerable tasks, and alignment between model failures and annotated error factors. Across 24 state-of-the-art MLLMs, the best model reaches only 25.7% joint value-unit accuracy and 51.8% confidence-diagnosis F1, revealing a substantial gap between general multimodal competence and reliable situated measurement. Further analysis identifies failures from text-induced shortcuts, overconfident responses, and authentic industrial noise, including mixed disturbances, viewpoint deviation, occlusion, and environmental interference.

Physics engines facilitate large-scale training and evaluation for embodied intelligence, while generative video world models are emerging as implicit simulators of future states and interactions. However, existing evaluations of physical fidelity are often conducted in isolation and rely heavily on perceptual similarity or human judgments, providing limited insight into which physical principles or parameters are violated. We introduce GAUGE, a real-world-grounded diagnostic benchmark for jointly evaluating how numerical simulators and generative video world models reproduce or deviate from real-world physics. It comprises 22 controlled task families covering rigid bodies, flexible cables, textiles, and volumetric deformable objects. Grounded in real-world trajectories and paired with calibrated physical metadata, uncertainty annotations, and task-specific observables, these tasks cover fundamental physical processes including collision, friction, momentum transfer, oscillation, self-contact, and deformation across diverse materials and conditions. We benchmark Isaac Sim, Genesis, and Newton on 14 task families using generalized trajectory errors, and evaluate 6 image-to-video models on 5 rigid-body tasks by testing physical-law consistency and the temporal stability of inferred parameters. Our results reveal no uniformly faithful physics engine, with the largest discrepancies arising in impulsive contact, rapid textile motion, and volumetric deformation. We further find that video world models can produce trajectories with the expected equation form while recovering incorrect accelerations, momentum transfer, and oscillation timing. GAUGE lays the groundwork for developing more physically faithful simulators and world models for embodied intelligence.

Multimodal classification typically assumes all modalities are available, yet real-world inputs are often incomplete. Imputation and dynamic fusion can mitigate such incompleteness, but existing methods operate at a coarse modality level and thus cannot retain reliable components while suppressing misleading ones within the same recovered modality, compromising prediction reliability. To address this issue, we propose GAUGE, a lightweight counterfactual gating framework for incomplete multimodal classification. GAUGE first imputes missing modalities with a frozen imputer and encodes observed and recovered inputs uniformly as fine-grained evidence units. Rather than intervening on each unit explicitly, GAUGE scores the counterfactual effect of replacing every unit with a reference representation through prediction-aware Taylor evidence scores, all obtained in a single forward-backward pass. These scores are mapped to continuous gates, which are converted into additive attention-logit biases for unit-wise evidence modulation without altering the backbone architecture. Experiments across six benchmarks demonstrate that GAUGE outperforms strong baselines across diverse incomplete-input settings. Furthermore, a Taylor remainder theoretical analysis characterizes the error of the first-order approximation relative to the exact counterfactual effect, establishing GAUGE as a principled and scalable framework for fine-grained evidence control under modality incompleteness.

Gradient descent on a factored model $W = UV^\top$ is implicitly biased toward low-rank solutions, while Adam, starting from the same small initialization, is not. We trace the difference to the gauge symmetry of the loss, its invariance under $(U, V) \mapsto (UQ, VQ)$. Gradient flow's low-rank mechanism is available to an optimizer only if that optimizer is gauge-equivariant, a condition necessary for the transfer but not sufficient for low-rank recovery. Gradient descent, momentum, "shared-scalar" Adam, Muon, and Shampoo satisfy it. Adam, RMSProp, and the other coordinate-wise methods do not. A structure theorem characterizes the memoryless equivariant rules as exactly the Gram-determined left preconditioners, and a transfer theorem carries gradient flow's pathwise properties to common-scalar flows. We then sort nine update rules on underdetermined matrix sensing by recovery error against the planted ground truth. A one-parameter family from coordinate-wise to shared-scalar preconditioning restores the bias monotonically, isolating anisotropy as the cause. A "spectral schedule" reconciles two opposing reports about Muon: equal-rate updates recover exactly low-rank targets but lose their edge as the spectral tail grows. In transformers, Adam separates two gauge-equivalent initializations at the first step, where the equivariant optimizers stay at float precision, and ends with the per-head invariants $W_Q^\top W_K$ 56% apart in relative Frobenius distance, a gap no per-head rotation can close. On two hyperspectral datasets at matched training loss, gradient descent cuts held-out error by 43-44% at the lowest sampling density, and at lower effective rank. Basis choice is therefore not a tuning detail but a decision about which interpolant the optimizer selects.

Financial models combine public disclosures with analyst assumptions to produce forecasts and valuations. While some components can be checked mechanically, forecasts, discount rates, and target prices often admit multiple reasonable answers. Existing benchmarks nevertheless tend to grade such outputs against a single expert reference. Using independently built analyst models for the same companies, we find that across 108 directed pairs covering 65 companies, the median single-reference score is 0.33, 92.6% score below 0.70, and no same-vintage pair agrees on implied price within 10%. Point-tolerance grading can therefore penalize disagreement already present among professionals. We introduce GAUGE, a benchmark for evaluating agent-built valuation models against observed analyst practice rather than a single point answer. GAUGE uses 1,001 vendor-classified analyst workbooks and a 196-task evaluation set, with a three-layer observed-practice envelope, 56 auditable facets, eight validity gates, and deterministic structural checks. We validate the benchmark with a 55-participant known-groups study, company-grouped cross-fitting, and judge-stability audits. On the failure-aware score $φ_0$, senior analysts average 88.3, juniors 66.0, and finance students 43.2. Across 24 agents and 1,011 scored generations, the best agent scores 53.4, above the student mean but below every senior and most juniors. It passes 93% of mechanical facets and 78% of judgment facets, with a fleet-median gap of 26 points. Current agents are substantially stronger at model construction than valuation judgment. We release the methodology, a gated de-identified data tier, a controlled training split, a versioned 48-task evaluation core, and a withheld refresh pool.

We present DualityCert, a symbolic verifier for candidate Seiberg-duality claims in four-dimensional N=1 quiver gauge theories. The verifier evaluates 't Hooft anomaly matching, superpotential R-charge consistency, central-charge matching, and a bounded chiral-ring proxy. A claim that passes receives a consistency certificate, which states that no tested inconsistency was found, not that the duality is proven. We use the verifier as a repair environment for language-model agents, which receive a deliberately broken claim and must edit it until it certifies. On a preregistered benchmark of 145 broken claims, with the analysis fixed before the first confirmatory model call, verifier-gated retry improves final repair success over a single attempt by +8.3 percentage points (pp) on deepseek-chat and +7.1 pp on qwen-plus (Holm-adjusted p<0.002). Under an equal budget of eleven attempts, the stop-first strategy portfolio underperforms independent verifier-filtered resampling by 10.3 percentage points on deepseek-chat but outperforms it by 14.7 points on qwen-plus, reversing the ordering of the two tested verifier-exploitation policies across the two confirmatory models. On qwen-plus, category-level verifier feedback is worth +8.7 pp over content-free retry, and interpretable obligation identities alone are worth +6.4 pp over structurally identical masked feedback. Neither effect is detected on deepseek-chat. Separately, a preregistered MiniMax-M2.5 extension again finds an iteration gain and independent verifier-filtered resampling outperforming the strategy portfolio. Which policy is better thus differs between the two models, while every winning policy uses the same cheap certificate. The verifier, benchmark, protocol, and all per-attempt records are released.

Transformers are known to have internal continuous symmetries that leave outputs invariant, while modifying quantization. GaugeQuant leverages this in-training by introducing a LogSumExp term to the loss that breaks the symmetries, thus selecting a basis that minimizes activation outliers. A stop-gradient operator ensures that only rotation matrices are updated, yielding the language modeling objective completely unaltered. Our requires no specific calibration data, no quantization simulation, and adds negligible training overhead. With the LLaMA-2 7B model under W4A4 quantization with group size 128, perplexity drops from 8.22 to 6.73, competing with post-training methods that require frozen models and calibration datasets. Under W4A16, perplexity drops from 11.16 to 5.45. Code is available at https://github.com/MPedraBento/gauge-quant.

The multiplicative repetition penalty shipped across the LLM inference ecosystem (HuggingFace, vLLM, llama$.$cpp, and a dozen further engines) branches on the sign of each raw logit (divide positives by theta, multiply negatives). But the softmax is unchanged by adding a constant to every logit, so a model's logit zero-point is arbitrary (a gauge choice), and the sign-branch reads it. Two measurable consequences follow. (1) The penalty is not well-defined: re-centering a model's logits by a constant is a provable no-op at theta=1, yet at a routine theta=1.3 it changes 58-96% of greedy tokens, while subtractive and normalized penalties change none; real checkpoints sit at widely different zero-points, so a fixed repetition_penalty is a different operation on every model. (2) It corrupts structured output: on 200 real-world JSON schemas, theta=1.3 drops the rate of valid, schema-conformant output from 97% to 23%. Applying the penalty to normalized log-probabilities instead of raw logits removes the gauge dependence by construction; HuggingFace's beam search has applied its processor chain, penalty included, to log-probabilities since at least v4.0.0, so repetition_penalty already names two different operators depending on decoding strategy. Because equal theta is not equal strength across the two operators, we also compare them at matched suppression, calibrated per model by search: there the normalized operator is statistically no worse on any quality metric measured, but on four of seven models it cannot match the raw operator's suppression at theta >= 1.15, and on six of seven at theta=1.3, the setting where the corruption was measured. This note gives the mechanism, the measurements (five models up to 7B; two code models; both effects replicated inside vLLM and llama$.$cpp through their own samplers), the per-model calibration map, and the normalized variant.

Spectral positional encodings (PEs) for \emph{directed} graphs face two obstacles: magnetic Laplacians require an $O(n^3)$ Hermitian eigendecomposition per potential, and their complex eigenvectors are defined only up to unitary gauge, which prior work handles with basis-invariant architectures. We propose learnable spectral PEs of the form $h_θ(A_q)\,R$, where $A_q$ is a normalized magnetic operator, $h_θ$ a learnable scalar spectral response, and $R$ a block of random probes. Because the PE is a \emph{matrix function} of the operator, it is gauge-invariant by construction. We compute it in a Hermitian block Krylov subspace from sparse matrix--vector products only, prove that $k = O(\log(1/\varepsilon))$ block steps suffice uniformly over heat--resolvent response families, and give a covering-number argument for why low-dimensional structured families generalize where free per-eigenvalue weights overfit. On a directed SBM whose symmetrization is uninformative by construction, direction-blind PEs stay at chance while magnetic Krylov PEs converge to the exact-eigendecomposition oracle as the depth grows. The same probes yield gauge-invariant pairwise features with $1/\sqrt{s}$ Monte-Carlo error, and the undirected $q{=}0$ case improves heterophilous benchmarks over no-PE and polynomial baselines.

Adaptive optimizers carry hidden states that change how visible gradients become parameter motion. We develop optimization geometrodynamics as a variational theory of this hidden geometry. Infimal pushforward eliminates all hidden states realizing the same action and composes across optimizer hierarchies. Under smooth nondegeneracy, it yields hidden susceptibility and the Schur-complement curvature seen after relaxation. For affine pre-reduction perturbations, the induced interaction curvature is the negative-semidefinite operator $-G^*H^{-1}G$, whose mixed entries integrate to finite mechanism contrasts. Our main realization is the determinant-one affine-invariant SPD action map $P\mapsto PA$. We prove a global analytic bundle with closed totally geodesic fibers and a unique analytic nearest-controller section. A strongly convex fiber theorem and an explicit logarithmic action residual give a globally linearly convergent solver from every feasible initializer, together with nonasymptotic value, controller-distance, and residual bounds and observable posterior stopping certificates. A conditional inexact result propagates supplied rigorous residual-error and radius majorants. The dense spectral kernel is confined to an active subspace of dimension $r\le 2m$, yielding an explicit spectral-arithmetic operation bound and a strict dimensional reduction when $r<d$. For nested shape-normalized quadratic actions, canonical multi-secant projections satisfy an exact CAT(0) Pythagorean decrease and recover the determinant-one inverse Hessian shape at the sharp rank threshold $d-1$, provided the scalar gauge $c_H=(\det H)^{1/d}$ is known. These results turn the action bundle into an exact iterative computation with posterior certificates and a finite-identification theory.

The rapid scaling of over-parameterized machine learning architectures, particularly LLMs, raises a profound crisis: do these systems exhibit genuine intelligence, or are they merely sophisticated statistical pattern matchers? Classical flat Euclidean statistics cannot differentiate continuous interpolation from the autonomous discovery of novel causal laws. To resolve this, we introduce Statistically Meaningful Geometry (SMG), a framework modeling over-parameterized learning systems as infinite-dimensional non-parametric Orlicz fiber bundles. We prove that under persistent out-of-distribution (OOD) stimuli governed by unmodeled causal mechanisms, continuous optimization fails. Unmodeled variance is rejected by the visible horizontal base manifold, leaking into the unobservable vertical fiber space and generating an accumulation of Active Acausal Tension. Driven by the statistical manifold's non-linear curvature, this tension inevitably strikes a conjugate focal boundary ($T_{\text{crit}} = π^2 / K_{\text{max}}$), triggering localized volumetric collapse and a catastrophic matrix singularity ($[G_f]^{-1} \to \infty$). We demonstrate this geometric breakdown acts as the strict non-equilibrium trigger for a Gauge Symmetry Break (GSB). The system purges hidden tension from unobservable gauge redundancies, spontaneously crystallizing a new, mathematically independent horizontal coordinate axis. This non-parametric phase transition registers as a discrete $+1.0$ integer step-jump in observable Structural G-Entropy. By decoupling parameter charts and subjecting emergent axes to a Minimal Energy Path Criterion and a Causal Invariance Filter, we distinguish genuine discovery from malignant hallucinations. Ultimately, SMG provides a parameter-free, falsifiable dashboard to mathematically certify true intelligence, transforming AI for Science into an engine of autonomous paradigm shifts.

A deep network's loss is invariant to continuous symmetries of its parameters: the logit shift, the ReLU rescaling, the LayerNorm scale, the per-head attention rotation. Adam's per-coordinate preconditioner drifts along each symmetry orbit, which pulls the trajectory off the symmetry quotient where the optimization lives and blurs the singular-learning rate the quotient makes readable. We build DDC, a Dead-Direction Conditioner that lifts a base optimizer into a $G$-equivariant one: it conditions the optimizer's state in the orbit decomposition of a $G$-invariant metric, so the trajectory stays a preconditioned gradient flow on the quotient $\barΘ= Θ/G$. The construction carries four architectural gauges (cross-entropy shift, ReLU and SwiGLU rescaling, LayerNorm and RMSNorm scale, and a per-head $O(d_{\rm head})$ attention rotation matched to RoPE), proves exactly equivariant on an Adam base, and composes with a Muon base through a gauge-equivariant orthogonaliser. Respecting the symmetry changes both the minimum the optimizer reaches and what it leaves measurable there. On a language model trained past the point of fit, DDCAdam resists the over-training collapse AdamW falls into, holding a validation-train loss gap of 0.67 against 5.88, and reads the dead-direction rate in 32 of 65 layer-by-observable cells where AdamW reads it in 7. A vision transformer trained from scratch reaches lower validation loss (1.71 against 2.12) while compressing spare feed-forward capacity a matched AdamW leaves intact. On a Muon base, where the rotation gauge composes exactly, DDCMuon groks ten of eleven seeds at depth 24 that a plain Muon never reaches. Built into the optimizer, a network's gauge symmetry sharpens the minimum it finds and turns that minimum's geometry into something the trajectory can measure.

Many important observables in physics and geometry are cup products of cochains. The adjusted cup product neural layer has been introduced in this paper. It is a neural primitive that hard wires the cup product with an adjustment term from higher gauge theory. This creates a readout that is gauge invariant by design. Their main theoretical result shows that on a closed cycle the output relies entirely on the adjustment coefficient. Setting this coefficient to zero removes the output completely regardless of other parameters. Thus the adjustment is the only source of gauge invariant signal. They prove this observable is a nonzero quadratic form and is exactly invariant under one and two gauge transformations.

Existing work on population dynamics inference often focuses on flows arising from vector fields that are the gradients of scalar potentials. Among all admissible flows that are compatible with the population dynamics, gradient flows are optimal in a specific sense: they minimize kinetic energy. The selection of fields based on different criteria corresponds to a gauge freedom when determining population dynamics, which we leverage in this work. We propose Non-Gradient Inference Flows (NGIF), an algorithm to infer non-gradient population dynamics using a weak formulation of the continuity equation. This allows us to parameterize general vector fields and choose other selection criteria beyond minimal kinetic energy. We demonstrate on a variety of low- and high-dimensional physics problems that this more general approach improves distributional accuracy over gradient-restricted baselines and better captures non-potential transport.

Muon's matrix-level update couples two distinct effects: spectral control via a polar map, and equivariance under orthogonal changes of multiplicity-space basis (Schur gauge-equivariance). We separate them with PolarAdamW, a controlled hybrid that preserves Muon's polar spectral-norm control but breaks the gauge-equivariance, since AdamW's coordinatewise preconditioner is basis-dependent. Algorithmically, PolarAdamW applies Muon's Newton-Schulz polar map to AdamW's preconditioned direction rather than to raw momentum, at per-iteration wall-time comparable to Muon. We prove that Muon's polar step is Schur gauge-equivariant on multiplicity matrices while AdamW's coordinatewise step is not. On DeiT-Tiny trained from scratch on four independently sampled 100-class subsets of ImageNet-1k, where multiplicity-basis freedom is trivial, PolarAdamW outperforms Muon by +1.93 pp in test accuracy on average and AdamW by +9.5 pp; under the 300-epoch DeiT-style recipe, it remains ahead of Muon by +1.37 pp and AdamW by +5.80 pp on average. On SO(3)-equivariant 3D point-cloud regression, where multiplicity-basis freedom is non-trivial, the ordering reverses: Muon outperforms PolarAdamW at every audited capacity, and the gap widens with capacity. Both matrix-polar optimisers continue to outperform AdamW. This double dissociation separates spectral control from Schur gauge-equivariance: the first composes well with AdamW preconditioning on standard transformers, while the second becomes consequential when multiplicity-basis freedom is structurally non-trivial.

Dirac-Frenkel instantaneous residual minimization evolves nonlinear parametrizations of PDE solutions in time, but ill-conditioning can render the parameter dynamics non-unique. We interpret this non-uniqueness as a gauge freedom: nullspace directions that leave the time derivative unchanged can be used to select better-conditioned parameter velocities. Building on Onsager's minimum-dissipation principle, we introduce a history variable -- interpretable as momentum -- and inject it only along the nullspace directions. The resulting Dirac-Frenkel-Onsager dynamics preserve instantaneous residual minimization, in contrast to standard regularization that can introduce bias, while promoting temporally smooth parameter evolutions. Examples demonstrate that the approach leads to increased robustness in singular and near-singular regimes.

Local gauge symmetry underlies fundamental interactions and strongly correlated quantum matter, yet existing machine-learning approaches lack a general, principled framework for learning under site-dependent symmetries, particularly for intrinsically nonlocal observables. Here we introduce a gauge-equivariant graph neural network that embeds non-Abelian symmetry directly into message passing via matrix-valued, gauge-covariant features and symmetry-compatible updates, extending equivariant learning from global to fully local symmetries. In this formulation, message passing implements gauge-covariant transport across the lattice, allowing nonlocal correlations and loop-like structures to emerge naturally from local operations. We validate the approach across pure gauge, gauge-matter, and dynamical regimes, establishing gauge-equivariant message passing as a general paradigm for learning in systems governed by local symmetry.

The digital transformation of industrial manufacturing increasingly relies on the ability of autonomous robots to interact with legacy infrastructure, particularly analog gauges. While Vision-Language Models (VLMs) have demonstrated potential in zero-shot instrument recognition, their deployment in measurement systems remains constrained by an inherent inability to accurately analyze high-frequency temporal events and needle vibrations. This paper evaluates state-of-the-art models, including GPT-5 and Gemini 3, against the strict requirements of metrology and uncertainty quantification. To facilitate this evaluation, we introduce a novel dataset comprising video sequences of various gauge types: circular, linear, and Vernier, under diverse motion speed profiles. Our findings indicate that current VLMs exhibit limited ability in interpreting needle trajectories and scale semantics, failing to provide the traceability and reliability needed for safety-critical monitoring. The results demonstrate that these models have not yet achieved the performance necessary to be classified as trustworthy synthetic instruments under existing IEEE and ISO standards.

Equivariant network architectures are a well-established tool for predicting invariant or equivariant quantities. However, almost all learning problems considered in this context feature a global symmetry, i.e. each point of the underlying space is transformed with the same group element, as opposed to a local ``gauge'' symmetry, where each point is transformed with a different group element, exponentially enlarging the size of the symmetry group. Gauge equivariant networks have so far mainly been applied to problems in quantum chromodynamics. Here, we introduce a novel application domain for gauge-equivariant networks in the theory of topological condensed matter physics. We use gauge equivariant networks to predict topological invariants (Chern numbers) of multiband topological insulators. The gauge symmetry of the network guarantees that the predicted quantity is a topological invariant. We introduce a novel gauge equivariant normalization layer to stabilize the training and prove a universal approximation theorem for our setup. We train on samples with trivial Chern number only but show that our models generalize to samples with non-trivial Chern number. We provide various ablations of our setup. Our code is available at https://github.com/sitronsea/GENet/tree/main.