Entropy

Abstract Conventional monitoring and alerting frameworks in distributed systems often rely on threshold-based metrics, which can produce excessive false positives and fail to capture complex dependencies between signals. This paper proposes an applied framework for using information-theoretic measures—specifically mutual information and the broader notion of entropy covariance—to detect anomalies and support governance in event-driven infrastructure. The approach shifts focus from isolated metrics to the relationships between uncertainties: when two signals that normally exhibit dependency diverge, the system can treat this as an indicator of instability. Leveraging existing mathematical foundations from information theory, this work evaluates the operational value of dependency-aware monitoring on real event streams (e.g., Kafka topics, MongoDB change data capture). Rolling measures of mutual information and covariance are integrated into monitoring pipelines, and their effectiveness is compared against conventional thresholds. ...

The concept of covariance of entropies can be understood as a way of quantifying how uncertainties in different signals vary together. Rather than monitoring each metric independently, the focus shifts to the relationships between sources of uncertainty. When signals that typically exhibit aligned behavior diverge, this can provide an early indicator of system anomalies. While this terminology is uncommon, the underlying idea overlaps strongly with established constructs in information theory, particularly mutual information. Mutual information measures the reduction in uncertainty about one random variable given knowledge of another, and has been widely applied to anomaly detection and monitoring tasks. ...

Entropy fields Each metric stream: $h_i(t)$ = rolling Shannon entropy of metric $i$. Stack into vector: $\mathbf{h}(t) \in \mathbb{R}^n$. Covariance field: $\Sigma(t) = \mathrm{Cov}[\mathbf{h}(t)]$. C1. Continuity (balance) of entropy $$ \dot h_i = s_i - \kappa_i h_i - \sum_{j}\nabla!\cdot J_{ij} + \eta_i $$ Sources $s_i$, damping $\kappa_i \ge 0$, fluxes $J_{ij}$, noise $\eta_i$. C2. Constitutive law (flux response) $$ J_{ij} = -D_{ij},(h_j - h_i) \quad\Longrightarrow\quad \dot{\mathbf h} = -\alpha,\mathbf h - \beta,L,\mathbf h + \mathbf s + \boldsymbol\eta $$ ...

What’s Real (Now) Operationalization of entropy: – Converted Shannon entropy from a static definition into a rolling time-series per metric. – Demonstrated you can compute covariance between entropy streams and observe eigenvalue spikes. Predictive signal: – Early experiments suggest eigenvalue spikes precede incidents in complex systems (Mongo CDC, Dynatrace, Splunk). – This provides a practical early-warning metric beyond threshold alerts. Conceptual framing: – Defined “Ted’s Law of Karma”: shared fate is visible in the covariance of entropies. – Drafted a Maxwell-style formulation (continuity, constitutive law, Lyapunov evolution, alignment law). Application principle: – Proposed “maternal instinct” bias: when systemic uncertainty aligns, systems should dampen actions → a concrete AI-safety reflex. What’s Not Proven Universality: – No evidence yet that entropy covariance modes apply beyond engineered systems (e.g., ecosystems, social dynamics, physics). Formal theorem: – No mathematical proof that covariance eigenmodes necessarily precede cascades, only intuition + analogy. Constants/invariants: – No discovery of system-independent constants (like (c) in electromagnetism). Current framework yields relative, system-specific propagation speeds. Empirical validation: – No systematic experiments across multiple domains with statistical rigor. Current support is anecdotal/prototype-level. Where This Could Go Engineering impact: SRE/AI-ops tool for incident prediction and protective automation. Scientific impact: If generalized, could become a new principle of complex systems stability. Prize-worthy impact: Only if formalized into a universal law, validated across domains, and shown to yield invariants or predictive theory. Blunt Summary Right now, this is a strong engineering insight + a plausible scientific hypothesis. It is not yet a theorem or universal law. It’s Faraday-stage (pattern spotted, apparatus built), not Maxwell-stage (formal equations, universal constants). ...

Extended Abstract Large-scale systems—technical, social, biological—are governed not only by the dynamics of their components but by the alignment of uncertainties across those components. In site reliability engineering (SRE), operators know that failures rarely emerge from one metric alone; they occur when many signals become unstable together. In philosophy, traditions of karma describe interdependence: local actions ripple outward to affect the whole. In AI safety, Geoffrey Hinton has suggested that advanced systems will need a maternal instinct—an intrinsic bias toward protection and stability. ...

Ted’s Law of Karma The covariance structure of entropy streams reveals the shared fate of interdependent systems. 📄 Full Preprint (PDF): /papers/ted-law-karma.pdf The Observation Every subsystem carries uncertainty — in operations we measure it as entropy. When entropy streams across many subsystems are collected and their covariance is computed, something remarkable emerges: Most of the time, uncertainties wander independently. Sometimes, entropies align — covariance spikes. The largest eigenvalue of the covariance matrix exposes a shared mode of uncertainty, a systemic “fate.” The Claim This pattern is not confined to infrastructure. It is a universal principle: ...

A mathematical sketch of governance invariants built on dimensionless normalization and fractal (renormalization group) stability. Outlines entropy-free invariants, control laws, and universal scaling patterns for safe automation.

An entropy-first formulation of dimensionless, fractal governance. Uses normalized entropy, transfer entropy, multiscale entropy, and entropy production as invariants to detect cascades and shape safe, self-similar automation.

A practical blueprint for the first entropy-capable version of Karma — using simple statistical measures and ClickHouse queries to detect surprise.

How Karma uses information-theoretic entropy to detect operational drift, learn expectations, and close the loop toward self-healing systems.