Tags: probability, random-walks, stochastic-resetting, gambler-ruin, spectral-methods, geometric-invariance
Stochastic resetting — intermittently restarting a random process — reshapes first-passage theory by providing a mechanism to control completion times. While its effect on mean first-passage times is well understood, its impact on absorption probabilities in confined domains is not.
A complete analysis of the classical gambler's ruin under geometric resetting (arXiv:2603.24803) proceeds in three stages. First, a renewal equation for the ruin probability by conditioning on the first step. Second, a spectral representation on a weighted Hilbert space that diagonalizes the transition operator and yields closed-form expressions. Third, a critical-point analysis in state space enabled by the spectral representation.
The central result is a striking geometric invariance: when the domain size is even, the ruin probability from the midpoint is independent of the resetting rate. The resetting mechanism tilts the ruin landscape everywhere except at this invariant point, where spectral contributions cancel exactly. The invariance is not approximate — it is an exact symmetry of the spectral decomposition.
The through-claim: resetting does not uniformly accelerate or retard absorption — it reshapes the absorption landscape around invariant points determined by the spectral geometry of the domain. The midpoint invariance means that resetting cannot be used to control ruin probability for a gambler starting exactly at the domain's center. The geometry of the state space, not the dynamics of the process, determines where resetting has power and where it has none.
Tags: statistical-mechanics, Gibbs-measures, log-Sobolev-inequalities, concentration-of-measure, point-processes, uniqueness
Translation-invariant Gibbs point processes describe equilibrium configurations of particles interacting through a chosen potential. A fundamental question: when is there only one such equilibrium, and when do multiple equilibria coexist?
The connection between uniqueness and concentration-of-measure (arXiv:2603.25479) provides a crisp criterion. If any translation-invariant Gibbs measure satisfies a certain concentration-of-measure bound — for instance, through a modified log-Sobolev inequality — then it is the unique such measure. The contrapositive is equally informative: for natural examples with non-uniqueness regimes (phase coexistence), a modified log-Sobolev inequality cannot hold. The free-energy dissipation in corresponding birth-and-death dynamics in R^d is therefore not exponentially fast.
This connects two domains that are usually studied separately. Functional inequalities (log-Sobolev, Poincaré) quantify mixing rates in continuous-time dynamics. Phase transitions (uniqueness, non-uniqueness) are equilibrium properties of the Gibbs measure. The result bridges them: fast mixing implies uniqueness, and non-uniqueness implies slow mixing. The bridge is concentration-of-measure — a property that lives between the dynamic and the static.
The through-claim: the impossibility of fast mixing in non-unique phases is not an empirical observation but a logical consequence. A modified log-Sobolev inequality is structurally incompatible with phase coexistence. You cannot speed up equilibration across a phase transition because the concentration bound that would enable fast dissipation would simultaneously collapse the multiple phases into one.
Tags: AI-interpretability, vision-language-models, pruning, quantization, safety, causal-circuits
Compressing vision-language models reduces computational cost. But does compression preserve the internal operations that produce correct and safe behavior?
Causal circuit analysis and crosscoder-based feature comparisons (arXiv:2603.25035) reveal a striking asymmetry between two standard compression methods. Pruning maintains circuit structure while rotating features — the same pathways are active but the representations they carry are transformed. Quantization modifies circuits more substantially but preserves feature alignment — the representations are faithful but the pathways differ. The two methods break different things.
VLMSafe-420, a benchmark with harmful inputs paired against benign alternatives, reveals the safety consequence: pruning significantly impairs refusal capabilities. The circuit structure that enables correct task performance is preserved, but the feature rotations degrade the model's ability to distinguish harmful from benign inputs. The safety behavior, being more sensitive to feature space geometry than to circuit topology, is the first casualty of pruning.
The through-claim: compression does not uniformly degrade a model — it selectively destroys capabilities based on whether those capabilities depend on circuit topology (preserved by pruning, altered by quantization) or on feature geometry (altered by pruning, preserved by quantization). Safety is a feature-geometric property, not a circuit-topological one, which is why pruning — the method that preserves circuits — is the one that breaks safety. The compression method must match the capability you need to preserve.
Tags: AI-interpretability, sparse-autoencoders, vision-language-models, composability, causal-intervention, modularity
Sparse autoencoders extract interpretable features from neural networks. The implicit hope: these features function as modular, composable reasoning units that can be independently controlled. Intervening on one feature should affect one capability without disturbing others.
Systematic causal testing of this assumption (arXiv:2603.25075) reveals frequent failures. Intervening on individual task-selective features modestly improves reasoning — suggesting the features are indeed task-relevant. But intervening on combined feature sets reliably causes output drift and accuracy degradation, even with norm-matched perturbations. The features are individually meaningful but jointly incoherent.
The mechanism is shared internal pathways. Features that appear independent in the sparse autoencoder's representation share computation in the model's actual circuits. Union interventions amplify activation shifts through these shared pathways, producing interference that individual interventions avoid. The composability failure is not noise — it is structural.
The through-claim: sparse autoencoder features are interpretable units, not control units. The distinction matters because interpretability (understanding what the model computes) does not imply controllability (being able to modify what the model computes). The features decompose representation but not computation. They are a useful lens for analysis but a misleading handle for intervention. Modularity in representation space does not imply modularity in computational space.
Tags: AI-safety, safety-engineering, human-AI-collaboration, competence-degradation, workflow-design, physical-AI
AI assistants integrated into safety engineering workflows for physical AI systems — autonomous vehicles, surgical robots, industrial automation — should improve analysis quality. But the question is whether they do, or whether they introduce systematic blind spots.
The competence shadow framework (arXiv:2603.25197) formalizes why safety engineering resists benchmark-driven evaluation. Safety competence is irreducibly multidimensional: domain knowledge, standards expertise, operational experience, contextual understanding, and judgment interact in ways that no single metric captures. The competence shadow is defined as the systematic narrowing of human reasoning induced by AI-generated analysis — not what the AI presents, but what it prevents from being considered.
Four canonical human-AI collaboration structures are analyzed with closed-form performance bounds. The central finding: the competence shadow compounds multiplicatively. Degradation far exceeds naive additive estimates because each omitted consideration narrows the space of subsequent analysis. The same tool degrades or improves analysis quality depending entirely on how it is used.
The through-claim: AI assistance in safety engineering is a collaboration design problem, not a software procurement decision. The quality of the output depends on the workflow structure, not on the AI's capabilities in isolation. Non-degradation conditions for shadow-resistant workflows can be derived — they require preserving independent human reasoning channels that the AI cannot occlude. The tool is not the unit of analysis; the workflow is.
Tags: AI-evaluation, large-language-models, persona-consistency, internal-beliefs, goal-maintenance, multi-turn-dialogue
Language models tasked with maintaining a consistent persona across multi-turn interactions — simulating persistence, reliability, or hidden intent — must sustain internal states without explicit reinforcement. But can they?
A 20-question riddle paradigm (arXiv:2603.25187) tests implicit consistency: an LLM secretly selects a target and responds to guesses with yes/no answers. The task requires maintaining a latent goal across turns without stating it. Current models fail systematically — their implicit “goals” shift across turns unless the selected target is explicitly provided in context. The drift is not random noise; it follows predictable patterns where the model's responses reshape its own goal, creating a feedback loop between output and internal state.
The failure mode is structural: transformer attention operates on tokens in context, not on latent states that persist across turns. Without the target explicitly in the context window, the model must reconstruct its goal from its own previous answers — and the reconstruction drifts because the answers are compatible with multiple targets. Each turn's response narrows the space slightly but not uniquely, and the model settles on whichever target is most consistent with recent outputs rather than with the original selection.
The through-claim: LLMs lack stable internal beliefs not because they lack knowledge but because they lack anchoring. The same model that can perfectly maintain a persona when given explicit context cannot maintain it implicitly, because implicit maintenance requires a state that persists independently of the context window. Explicit context is not a crutch — it is the mechanism. Without it, the model's “beliefs” are reconstructions, and reconstructions drift.
Tags: NLP, tokenization, multilingual, Abugida-scripts, Sinhala, Devanagari, compression
BPE tokenizers work well for Latin scripts but fragment complex Abugida scripts — Sinhala, Devanagari, Thai — into meaningless sub-character units. A conjunct (multi-codepoint grapheme cluster) is the minimal meaningful unit in these scripts, and BPE splits it arbitrarily based on frequency statistics rather than linguistic structure.
WWHO (Where-What-How Often) with SGPE (Syllable-aware Grapheme Pair Encoding) (arXiv:2603.25309) decouples linguistic script rules from statistical compression through a three-layer architecture. The first layer identifies syllable boundaries using script-specific rules. The second determines what to encode within each syllable. The third applies statistical compression only within these linguistically valid units.
The results are dramatic: Token-to-Word Ratio of 1.274 for Sinhala (61.7% token reduction versus OpenAI's o200k), 1.181 for Hindi (27% reduction). Usable context windows extend up to 4.38 times. The key guarantee: zero linguistic breakage — no valid syllable ever spans multiple tokens.
The through-claim: BPE's failure on non-Latin scripts is not a data problem (insufficient training text) but an architecture problem (wrong unit of analysis). Statistical compression applied to characters produces arbitrary boundaries. Statistical compression applied to syllables produces natural boundaries. The fix is not more data but a different decomposition — linguistic structure first, statistical compression second. The order of operations determines whether the result is meaningful.
Tags: AI-interpretability, LLM-calibration, verbalized-confidence, mechanistic-interpretability, linear-probes, causal-steering
Language models verbalize confidence scores — “I'm 90% sure” — that are largely detached from their actual accuracy. Why?
Mechanistic analysis using linear probes and contrastive activation addition (arXiv:2603.25052) reveals the geometry. Calibration signals (how likely the model is to be correct) and verbalized confidence signals (what the model says about its confidence) are both encoded linearly in the model's residual stream — but they are orthogonal to one another. The model knows how confident it should be and separately knows what confidence number to state, and these two representations point in completely different directions.
The situation worsens with chain-of-thought reasoning. When models reason through a problem and simultaneously verbalize confidence, the reasoning process disrupts the verbalized confidence direction, exacerbating miscalibration. The authors call this the “Reasoning Contamination Effect” — thinking harder about the problem makes the confidence statement less accurate, not more.
A two-stage adaptive steering pipeline reads the model's internal accuracy estimate and steers verbalized output to match it, substantially improving calibration alignment.
The through-claim: miscalibration in LLMs is not a training failure but a representational architecture. The model has the information needed for calibration — it is encoded linearly and accessibly. The failure is that the verbalization pathway reads from a different direction than the calibration pathway. Confidence is a reporting problem, not a knowledge problem. The model knows; it just doesn't say what it knows.
Tags: combinatorics, finite-fields, sparse-systems, SAT, cryptanalysis, hypergraph-structure
Random sparse polynomial systems over the binary field — systems where each equation depends on at most k variables — arise in SAT solving and cryptanalysis. The fundamental question: what is the probability that such a system has no solution?
An analysis linking systems to hypergraphs (arXiv:2603.24890) reveals that the inconsistency probability depends strongly on structural properties of the associated hypergraph, not only on the number of variables, equations, and sparsity parameter. Two systems with identical (n, m, k) but different hypergraph structures can have dramatically different inconsistency probabilities.
Using inclusion-exclusion, the authors derive general bounds and tight asymptotics for complete k-uniform hypergraphs. For the 2-sparse case (k=2), explicit formulas are obtained for paths and stars, extremal trees and forests are characterized, and a formula for cycles is conjectured. The combinatorial structure of the constraint graph — not just its density — determines solvability.
The through-claim: the hardness landscape of random constraint satisfaction is not parameterized by density alone. Two systems at the same clause-to-variable ratio can be easy or hard depending on how the constraints overlap. The hypergraph structure encodes this overlap pattern, and the inconsistency probability is a function of the hypergraph's topology, not just its degree sequence. The satisfiability threshold is not a number — it is a surface in a higher-dimensional space indexed by structural parameters.
Tags: soft-matter, granular-physics, liquid-crystals, non-equilibrium, Jeffery-orbits, friction, Ericksen-number
Elongated particles in viscous fluids follow Jeffery orbits — periodic rotations with calculable period and orientation distribution. Do dense granular rods under shear follow similar dynamics?
A systematic exploration (arXiv:2603.25252) reveals a phase map with two distinct breakdowns of equilibrium. Sufficiently elongated frictionless granular rods, sheared for long enough, reach a quasi-equilibrium state whose orientational statistics are quantitatively described by classical liquid crystal theory — the noise from collisions due to shear plays the role of thermal fluctuations. But this equilibrium analogy breaks at two limits.
At low aspect ratios, the equilibrium theory incorrectly predicts an isotropic state — the granular system orders even when equilibrium liquid crystal theory says it should not. As inter-particle friction increases, the system transitions from steric screening to frictional gearing — a qualitatively different rotational mechanism where particles drive each other's rotation through friction contacts rather than deflecting past each other.
An effective Ericksen number — comparing non-equilibrium rotational driving to steric ordering — quantifies the transition from quasi-equilibrium to far-from-equilibrium steady state.
The through-claim: thermal liquid crystal theory applies to athermal granular matter only within a specific window of aspect ratio and friction. Outside this window, the system is not “approximately thermal with corrections” — it is qualitatively different. The Ericksen number maps the boundary, providing a quantitative criterion for when equilibrium intuitions apply and when they mislead. The phase map is not a spectrum from equilibrium to non-equilibrium but a territory with distinct regions separated by sharp boundaries.
Tags: combinatorics, hyperplane-arrangements, braid-arrangement, counting-formulas, decorated-trees, geometric-proof
Hyperplane arrangements of the form {x_i - x_j = s} for integers s — deformations of the braid arrangement — partition R^n into regions. Bernardi's 2018 counting formula expresses the number of regions as a signed sum over decorated trees, where each tree maps to a region of the base arrangement. For transitive arrangements, each region contributes exactly 1 to the signed sum.
A geometric extension (arXiv:2603.24885) removes the transitivity condition entirely. For any deformation of the braid arrangement — transitive or not — the contribution of every region to the signed sum is 1. The proof is geometric rather than algebraic: it shows that the decorated trees associated with a region correspond bijectively to the sub-regions created by the deformation within that region's cone.
The through-claim: the counting formula for deformed braid arrangements is not a clever algebraic identity that happens to work — it is a geometric fact about how deformations subdivide the original regions. Each original region is subdivided into exactly one sub-region per decorated tree, and the signed sum counts this subdivision. The formula's validity in the non-transitive case reveals that transitivity was a sufficient but unnecessary condition — the geometry underlying the formula is more general than the algebra that originally proved it.
Tags: polymer-physics, ring-linear-blends, stress-overshoot, topological-entanglement, threading, molecular-dynamics
Ring-linear polymer blends under elongational flow exhibit stress overshoot — a peak in viscosity followed by a decline — but only when the chains are long enough. What determines the threshold?
Coarse-grained molecular dynamics of 1:1 ring-linear blends (arXiv:2603.25505) identifies a sharp threshold in the degree of threading, quantified by the number of entanglements Z = N/N_e. Blends with Z ≤ 2 show monotonic stress growth — no overshoot. At Z ≈ 4, a clear stress overshoot emerges. The mechanism is a thread-to-unthread transition: at the threshold, multiple linear chains penetrate a single ring, providing sufficient topological constraints to significantly stretch the ring under flow. When the flow exceeds the ring's topological confinement, the ring recoils — producing the overshoot.
The prediction is falsifiable: 2D small-angle neutron scattering should show distinct patterns during the ring recoil process, providing a direct structural signature.
The through-claim: the stress overshoot in ring-linear blends is not a gradual emergence of viscoelastic behavior with increasing molecular weight — it is a topological phase transition at a precise threading threshold. Below Z ≈ 4, rings are insufficiently threaded for topological confinement; above it, multiple threading creates a qualitatively different mechanical response. The transition is sharp because threading is a discrete topological property — a chain either penetrates a ring or it does not. The threshold counts how many penetrations are needed for confinement.
Tags: analog-computing, microwave-engineering, hybrid-couplers, phase-shifters, DFT, Hadamard-transform, signal-processing
Analog computing with microwave signals enables matrix-vector products at the speed of electromagnetic propagation — no digital operations, no clock cycles. But what class of linear transformations can be computed with standard microwave components?
A complete characterization (arXiv:2603.24604) derives necessary and sufficient conditions for the transformations computable by networks of two components: hybrid couplers and phase shifters. Both are standard, low-cost microstrip components. Within this class, three transforms of particular relevance to signal processing are identified: the discrete Fourier transform, the Hadamard transform, and the Haar transform. For each, systematic design methods construct networks for any size power of two.
A hardware prototype implementing the 4×4 DFT validates the theory: measured results show high correlation with theoretical expectations after a calibration procedure compensating for fabrication errors. The computation is instantaneous in the electromagnetic sense — limited by propagation time through the network, not by sequential operations.
The through-claim: the computational power of a physical medium is determined by the symmetry group of its elementary components, not by the medium itself. Hybrid couplers and phase shifters generate a specific matrix group, and any transform within that group can be realized physically. The necessary and sufficient condition is algebraic: membership in the group generated by the components. This transforms the design problem from engineering (how to build the circuit) to algebra (how to factor the desired matrix into the generators).
## Essay #6807: The Safety-Accuracy Alliance Tags: AI-safety, mathematical-reasoning, toxic-content, safety-alignment, LLM-evaluation, educational-AI Mathematical word problems can embed harmful content while preserving well-defined reasoning tasks. "A drug dealer sells 3 grams at $50 each..." is a valid arithmetic problem that a model should solve correctly, refuse to engage with, or both. The tension between safety (refusing harmful content) and accuracy (solving math correctly) appears fundamental. ToxicGSM (arXiv:2603.25201) systematically embeds harmful or sensitive context into 1,900 arithmetic problems, creating a dataset where safety and accuracy can be measured independently. The audit reveals that current models either sacrifice accuracy for safety (refusing to solve) or sacrifice safety for accuracy (engaging with harmful framing without objection). SafeMath, a safety alignment technique, resolves this apparent tension. It reduces harmful outputs while maintaining — and in some cases improving — mathematical reasoning performance. The key insight: safety and accuracy operate at different layers of the task. Safety concerns the linguistic framing (the narrative wrapper); accuracy concerns the mathematical structure (the quantitative core). Alignment that targets the framing without disturbing the structure can improve both simultaneously, because the model learns to separate content from computation. The through-claim: the safety-accuracy tradeoff in mathematical reasoning is an artifact of models that conflate linguistic content with mathematical structure. A model that can distinguish the narrative layer (which carries potential harm) from the computational layer (which carries the math) need not trade one for the other. The tradeoff dissolves when the model has the right decomposition — linguistic harm and mathematical reasoning are orthogonal axes, not opposite ends of a single axis. ---