Tags: control-theory, dynamical-systems, fading-memory, memristors, incremental-stability, approximation-theory
Fading memory — the progressive loss of influence of past inputs on a system's current output — is what makes a dynamical system usable as a signal processor. A system without fading memory carries the infinite past into every present computation. A system that forgets too quickly loses the temporal context that gives signals meaning.
The state-space formulation of fading memory (arXiv:2603.23814) defines this property not as a black-box input-output relationship but as a structural condition on the system's internal dynamics. The memory kernel — a function controlling how quickly the influence of past input differences decays — becomes an explicit design parameter rather than an emergent consequence. The central result: incremental input-to-state stability implies fading memory for time-invariant systems, connecting a verifiable dynamical property to an approximation-theoretic one.
The practical demonstration targets memristors — devices whose resistance depends on the history of applied current. Under mild assumptions, current-driven memristors possess the fading-memory property, which means they can be approximated by finite-dimensional systems to arbitrary precision. Boyd and Chua's classical approximation theorems, which guarantee that fading-memory systems can be uniformly approximated by finite Volterra series, apply directly to these state-space models.
The through-claim: fading memory is not a property to be discovered in a system but a design specification to be imposed. The state-space formulation makes the memory kernel an explicit parameter, transforming the question from “does this system forget?” to “how fast should this system forget?” The engineering of forgetting is as precise as the engineering of response.
Tags: numerical-analysis, stochastic-rounding, floating-point, hardware-design, Horner-method, pairwise-summation
Stochastic rounding — rounding each arithmetic result up or down with probability proportional to proximity to the nearest representable value — eliminates the systematic bias of round-to-nearest. The expected error cancels over many operations. But classical stochastic rounding requires the exact infinite-precision result of each arithmetic operation, which is impractical: the whole point of low-precision arithmetic is to avoid computing at full precision.
Limited-precision stochastic rounding (arXiv:2603.24161) uses only a few extra bits beyond the target precision to generate the rounding decision. The question is how many extra bits are enough. The analysis targets two foundational algorithms: Horner's method for polynomial evaluation and pairwise summation for adding sequences of numbers. Both are building blocks that appear inside nearly every numerical computation.
The results quantify a hardware-accuracy tradeoff. Each additional bit of intermediate precision reduces the expected rounding error, but at a cost in circuit area, power, and latency. The analysis provides bounds that allow hardware designers to choose the minimum number of extra bits that achieves a target error level — a budget for rounding quality.
The through-claim: the accuracy of stochastic rounding is not a property of the rounding scheme alone but of the interaction between the scheme and the algorithm being rounded. Horner's method and pairwise summation respond differently to limited precision because their error accumulation patterns differ. The rounding hardware and the algorithm must be co-designed — accuracy is a system property, not a component property.
Tags: AI-security, agent-protocols, MCP, delegation, capability-tokens, zero-trust
The Model Context Protocol (MCP) and Agent-to-Agent (A2A) protocols enable AI agents to invoke tools and communicate with each other. Neither includes identity verification. An examination of approximately 2,000 MCP servers found that all lacked authentication — any agent claiming to act on behalf of a user is trusted implicitly.
The Agent Identity Protocol (arXiv:2603.24775) introduces Invocation-Bound Capability Tokens (IBCTs) that combine identity, attenuated authorization, and provenance binding into a token chain. Two formats: a compact signed JWT for single-hop interactions and a Biscuit token with Datalog policies for multi-hop delegation. The attenuation is the key mechanism — each delegation step can only narrow permissions, never widen them. A tool that receives a token can verify the entire delegation chain back to the original authorizing identity.
Performance is negligible: 0.049 milliseconds for verification in Rust, 2.35 milliseconds end-to-end overhead in real multi-agent deployments (0.086% of total latency). Security testing across 600 attack attempts achieved 100% rejection, including delegation depth violations and audit evasion attempts that standard JWT approaches miss.
The through-claim: the security gap in agent protocols is not a missing feature but a missing layer. MCP and A2A were designed for capability, not accountability. The identity layer must be separate from the communication layer because the trust model (who can do what on whose behalf) is orthogonal to the interaction model (how agents exchange messages). Bolting identity onto communication retroactively is architecturally wrong; the layers must be designed independently and composed.
Tags: control-theory, recurrent-neural-networks, physics-informed-learning, convex-optimization, large-scale-systems, stability-guarantees
Recurrent neural networks learn temporal dynamics from data. They work well — until they violate physical constraints that the training data happened to satisfy. A model trained on stable system trajectories may predict unstable behavior during deployment because nothing in the training loss penalizes instability directly. The guarantee is statistical (the model fits the data) rather than structural (the model respects the physics).
Physics-informed structured learning for RNNs (arXiv:2603.25574) formulates the training problem as a convex optimization that directly encodes stability and structural constraints via linear matrix inequalities. The trained network is guaranteed — by construction, not by statistical testing — to preserve the structural and stability properties of the underlying plant. This is not a regularization term added to the loss function; it is a hard constraint on the feasible set of network parameters.
For networked systems with modular structure, the optimization decomposes into parallel subproblems that communicate only with neighboring subsystems. The computational cost scales with the network's communication topology, not with the total system dimension — making the approach practical for large-scale systems where centralized optimization would be intractable.
The through-claim: the distinction between soft constraints (regularization) and hard constraints (feasibility) is the distinction between probable correctness and guaranteed correctness. Physics-informed training that adds penalty terms to the loss still permits constraint violations when the penalty weight is insufficient. Convex formulations with LMI constraints eliminate the ambiguity: the solution either satisfies the physics or doesn't exist. For safety-critical systems, “probably stable” is not an acceptable guarantee.
Tags: numerical-analysis, Grassmann-manifold, model-reduction, interpolation, Arnoldi-orthogonalization, maximum-volume-coordinates
Interpolation on manifolds is harder than interpolation in Euclidean space because the data lives on a curved surface. Polynomial interpolation formulas that work perfectly in flat space become ill-conditioned or undefined on manifolds because addition and scalar multiplication — the operations that define polynomials — don't have natural meanings on curved spaces.
The Grassmann manifold — the space of all k-dimensional subspaces of an n-dimensional vector space — is the natural setting for parametric model reduction, where a reduced-order model varies smoothly with parameters. Interpolating these models requires interpolating points on the Grassmannian.
Maximum-Volume coordinates combined with Arnoldi-orthogonalized polynomial bases (arXiv:2603.23858) solve the conditioning problem by addressing both layers of difficulty simultaneously. MV coordinates map manifold data to well-conditioned Euclidean representations without the expensive matrix factorizations that traditional Riemannian normal coordinates require. The Vandermonde-with-Arnoldi method then constructs discrete orthogonal polynomial bases directly from the parameter nodes, avoiding the ill-conditioned linear systems that plague standard polynomial interpolation at high degree.
The through-claim: high-order interpolation on manifolds fails not because the manifold is curved but because the coordinate system amplifies errors that the interpolation scheme then inherits. The cure is to fix the coordinates and the polynomial basis simultaneously — each independently insufficient but jointly stable. Conditioning is a property of the representation, not of the geometry.
Tags: numerical-linear-algebra, matrix-polynomials, Paterson-Stockmeyer, matrix-exponential, computational-efficiency, numerical-stability
Evaluating a matrix polynomial of degree d requires multiplying matrices — the most expensive operation in numerical linear algebra. The Paterson-Stockmeyer method minimizes the number of matrix multiplications to approximately √d, a classical result that has stood for decades. Saving even one matrix multiplication matters when the matrix is large and the polynomial is evaluated repeatedly, as in matrix exponential computation or geometric series.
A systematic framework (arXiv:2603.23143) achieves one fewer matrix product than Paterson-Stockmeyer for polynomials of degree 8 and higher. The saving comes from choosing polynomial coefficients that satisfy additional algebraic constraints, allowing intermediate results to be reused. But the optimized coefficients emerge from solving nonlinear polynomial equations with multiple solutions, and some solutions produce numerically unstable evaluations.
The MATLAB tool MatrixPolEval1 identifies and validates stable coefficient sets using variable-precision arithmetic, introducing structural variants when original configurations fail. A reliability measure quantifies how close the chosen coefficients are to instability boundaries, providing a confidence metric for the evaluation scheme.
The through-claim: the Paterson-Stockmeyer count is not a hard lower bound but a stability-constrained optimum. Fewer multiplications are possible if the polynomial coefficients are free parameters — but the solutions that minimize multiplications are often the ones that maximize instability. The real optimization is not “fewest multiplications” but “fewest multiplications that remain numerically stable.” The stability constraint, not the algebraic constraint, sets the true minimum.
Tags: software-engineering, code-evaluation, LLM-judges, human-preferences, automated-evaluation, bias-analysis
Language models increasingly evaluate code — ranking completions, grading student submissions, reviewing pull requests. The implicit assumption: models trained on code and natural language can approximate human judgment about code quality. The TRACE framework (arXiv:2603.24586) tests this assumption systematically.
Thirteen models evaluated across three coding scenarios — chat-based programming, IDE autocompletion, and code editing — underperformed human annotators by 12-23% at predicting human preferences. The framework identified 35 major areas of divergence. LLM judges favor longer code explanations; humans prefer shorter ones. Models weight syntactic correctness more heavily; humans weight readability and intent alignment. The biases are systematic and consistent across model families.
The divergence is not random noise. It reflects a structural difference in what models and humans optimize for. Models evaluate code against patterns learned from training data — patterns that overrepresent correct, well-documented code. Humans evaluate code against their current task context — patterns that weight convenience, readability, and cognitive load reduction. The same code can be objectively correct but subjectively unhelpful, and models systematically misjudge this distinction.
The through-claim: LLM code evaluation fails not because models lack coding knowledge but because evaluation and generation are different cognitive tasks that weight the same features differently. A model that generates excellent code may still be a poor judge of code because generation optimizes for correctness while evaluation optimizes for human preference — and preference is context-dependent in ways that training data does not capture.
Tags: active-matter, reinforcement-learning, microswimmers, turbulence, obstacle-avoidance, Q-learning
Microswimmers — particles that propel themselves through fluid — must navigate environments that are simultaneously turbulent (the flow pushes them unpredictably) and cluttered (obstacles block their path). Biological microswimmers solve this problem constantly: bacteria navigate mucus, sperm navigate the reproductive tract, immune cells navigate tissue. Artificial microswimmers for drug delivery or environmental remediation face the same challenge.
Obstacle-aware adversarial Q-learning (arXiv:2603.21223) extends reinforcement learning navigation to two-dimensional turbulent flows with obstacles, modeling a circular obstacle within a periodic simulation domain via the volume-penalization method. The adversarial strategy specifically targets stagnation point trapping — the failure mode where a swimmer gets caught in the flow pattern near an obstacle's surface and cannot escape.
Smart microswimmers trained with this obstacle-aware approach outperform both naive swimmers (which ignore obstacles) and surfers (which follow the local flow). The performance gap widens as turbulence intensity increases: in calm flows, all strategies work adequately; in strong turbulence, only the learned policy consistently reaches its target.
The through-claim: navigation in turbulent flows is not a path-planning problem but a policy problem. The optimal action depends on the local flow state, which changes faster than any plan can be re-computed. The reinforcement learning approach succeeds not because it finds better paths but because it replaces planning with reaction — a mapping from sensory state to motor action that responds to the flow in real time rather than predicting it.
Tags: quantum-mechanics, semiclassical-methods, Gaussian-wave-packets, magnetic-fields, structure-preserving-integration, symplectic-methods
Gaussian wave packets — localized quantum states whose probability density follows a Gaussian bell curve — are the workhorse of semiclassical quantum mechanics. They track the classical trajectory while capturing quantum spreading, interference, and tunneling through the packet's width and phase parameters. In a magnetic field, the wave packet dynamics acquire a modified symplectic structure because the magnetic vector potential couples position and momentum in a way that the standard Hamiltonian formulation does not.
Structure-preserving integrators for magnetic Gaussian wave packets (arXiv:2603.25596) exploit a kinetic momenta reformulation that recasts the averaged dynamics as a Poisson system identical in structure to classical charged particle motion. This connection enables Boris-type integrators — the same schemes used to push charged particles in plasma physics — to be applied to wave packet evolution. Explicit high-order symplectic schemes follow from splitting and partitioned Runge-Kutta methods.
The integrators conserve the quadratic invariants characterizing the Hagedorn parametrization and preserve momentum under symmetry conditions. The averaged Hamiltonian is nearly conserved over long integration periods — not exactly, but with error bounds uniform in the semiclassical parameter, meaning the method remains accurate as the system approaches the classical limit.
The through-claim: the same mathematical structure that makes charged particle motion integrable in classical physics also makes wave packet dynamics integrable in semiclassical quantum mechanics. The kinetic momenta reformulation reveals that the magnetic Schrödinger equation, projected onto Gaussian wave packets, IS classical charged particle dynamics in disguise. The quantum problem doesn't need quantum solvers — it needs classical solvers applied to the right variables.
## Essay #6768: The Delayed Barrier Tags: control-theory, safety-critical-systems, control-barrier-functions, input-delay, autonomous-vehicles, robust-control Safety-critical control systems — autonomous vehicles, surgical robots, industrial automation — must satisfy safety constraints at all times. Control Barrier Functions (CBFs) enforce safety by computing the minimally invasive control input that keeps the system within a safe set. But CBFs assume instantaneous actuation: the computed input takes effect immediately. Real actuators have delays — hydraulic systems, communication networks, motor controllers all introduce latency between command and execution. Input delay breaks the CBF framework because the safety guarantee depends on the current state, but the control input will not take effect until the state has evolved (possibly unsafely) through the delay period. The integral control barrier function framework with input delay (arXiv:2603.24566) addresses this through three mechanisms: predictor feedback to offset the delay by estimating the future state, integral CBFs to maintain safety compliance through the delayed response, and a quadratic programming formulation that computes the safe control input in real time. For systems with simultaneous state and input constraints, the framework derives conditions guaranteeing feasibility — that a safe input always exists — and forward invariance — that the safe set is never violated. Robust ICBFs handle delay uncertainty: when the exact delay is unknown, the controller maintains safety under worst-case delay assumptions. The through-claim: input delay does not merely degrade safety margins — it creates a qualitatively different control problem. The safe set under delay is strictly smaller than the safe set without delay, and the boundary of the safe set depends on the delay magnitude. A system that is safe at zero delay can become unsafeable (no feasible safe input exists) at finite delay. The delay doesn't just slow down safety — it shrinks the region where safety is achievable. ---