Forrelation, Bent Functions, and a Spectral Criterion for Hardness

This note clarifies the two endpoint regimes of the Forrelation problem—Aaronson–Ambainis constant-gap hardness and the Girish–Servedio extremal bent frontier—and reframes them as anchors of a continuous “spectral hardness” picture. It proposes a conjectural criterion based on Fourier flatness and low-degree suppression, and adds a simple cubic Hermite spline baseline that smoothly interpolates the classical hardness exponent between the AA-scale and the extremal regime.

Identity Half-Life and Mutual Information Decay in Open Quantum Systems

A quantitative notion of sameness over time for a subsystem, defined not by narrative continuity but by correlation with a fixed reference. This note shows that an open subsystem cannot increase its informational connection to its past under ordinary physical evolution. The result is an identity half-life picture, supported by exact dynamical identities and concrete examples including beta decay and decoherence. The Ship of Theseus becomes operational: identity is measurable, and it decays.

Tail Risk and Compute Allocation in Heavy-Tailed ML Training Systems

Modern training pipelines often exhibit heavy-tailed runtimes where a small number of pathological jobs dominate deadlines and reliability. This note studies two queues with skewed service times and shows that optimizing for average performance and optimizing for tail performance are fundamentally different problems. The allocation boundary shifts in a precise way, revealing when throughput-optimal policies become fragile under rare event regimes and clarifying how to think about risk-sensitive compute.

Meaning as Symmetry Breaking: Why One-Way Messages Cannot Create Semantics

A one-way bitstring can be perfectly structured and still carry zero semantic content if the receiver lacks the decoder that turns symbols into meaning. This note reframes understanding as a symmetry-breaking process in which interaction supplies the work required to align an interpreter. It develops information-theoretic limits on decoder learning and introduces a conserved decoder potential that must be reduced through informative exchange. Bits are cheap. Interpreters are expensive.

The “Patch Droid” Alcubierre Drive

A speculative and playful exploration that reframes the Alcubierre warp bubble as a distributed control and logistics problem: when negative energy density, extreme pressures, horizon indicators, or curvature spikes “overload” a region of the bubble wall, a fleet of auxiliary patch modules reallocates stress–energy on the fly to keep the geometry within prescribed bounds. The note formalizes this idea as a constrained design/optimization system in the 3 + 1 (ADM) framework. It is less a propulsion proposal than a humorous, math-first way to ask what “plugging the holes” would even mean.

On the Smoothness Hierarchy of Time Differentiation in Classical Motion

A conceptual essay on regularity, causality, and time differentiation in analytical mechanics. This work interrogates the foundational idealization that acceleration may “switch on” instantaneously, showing that such assumptions are equivalent to admitting distributional impulses in higher-order dynamics. By reframing discontinuous acceleration as singular structure in jerk and beyond, and contrasting ballistic (wave-mediated) with diffusive (aggregate) response geometries, it demonstrates how Dirac and Gaussian kernels arise as forced limits of physical mediation, not mathematical conveniences. In this view, smoothness becomes a concrete statement about transport, locality, and scale.

Noether, Energy-Conserving Transitions, and Distance-to-Conservation

A geometric formalization of energy conservation as state-space filtration. This note discretizes physical evolution into candidate next states and quantifies the dimensional reduction imposed by Noether's theorem: conservation laws don't guide trajectories, they eliminate exponentially many futures. By introducing a distance-to-conservation metric and deriving a gradient-based correction formula, it bridges continuous symmetry with discrete computational geometry, making admissibility constraints explicit and measurable.

Bounds for Lognormal HVAC Perturbations

A theoretical investigation of seasonal energy shape distortion driven by lognormal HVAC multiplicative effects. This note introduces a smooth canonical seasonal envelope and proves a sharp bound linking physical HVAC parameters to probabilistic profile deformation. The analysis extends naturally to a rare-event tail regime, revealing how sub-exponential risk geometry governs the emergence of pathological summer load behavior across heterogeneous building populations.

Geodesics on the Permutohedron

Permutohedron Geometry & Distance Comparisons

Stochastic Redistribution vs. LP Surrogates for Circuit Balancing

A work-related exploration of optimal energy grid allocation framed through the lens of stochastic optimization. This investigation models service-point imbalances driven by lognormal load deviations and shows that, under exogenous dynamics, a seemingly discrete redistribution problem collapses to an exact convex program. The result bridges practical grid engineering with elegant stochastic theory, revealing when linear relaxations remain integral, and when real-world coupling demands MILP or MIQP extensions.

Decomposing Time Series into Marginal and Dependence Components

This note presents a compact factorization of a univariate time series into a marginal law of values and a dependence process of ranks, formalizing the informational split between “what occurs” and “how it is arranged in time.” Using Sklar’s theorem, it connects copula decomposition to information-theoretic and PAC-learnability frameworks, showing how modular learning of marginals and dependence composes into learning the full process.

Opposite-Skewness Symmetry & TV Bounds

Neural Net Dropout Viewed as Probability-Mass Dilution

On the Geometry of Fractal Boundaries in ReLU Networks

This paper develops a constructive and theoretical account of fractal geometry in neural network decision boundaries, introducing explicit ReLU modules that realize tent-map and Cantor-style refinements with provable dimensions. It proves that exact self-similarity arises only under measure-zero weight settings, while empirical probes reveal finite-range fractal mimicry and propose boundary fractal dimension as a diagnostic of overfitting.

Quantum-Accelerated Stabilization for Markov Chains

This note introduces a band–window stabilization criterion for Markov-chain averaging and demonstrates a quadratic quantum speedup via amplitude estimation. It contrasts classical and quantum complexities, presenting both immediate and anytime stabilization theorems with clear operational interpretations.

Misspecification, Quantile Mobility, and Arc Length

This note unifies three threads: distributional misspecification bounds for a lognormal truth fit by a moment-matched normal; quantile mobility difficulty under additive vs. multiplicative geometries; and a geometric relation between quantile shifts and the arc length of the PDF, clarifying why mobility differs across shapes.