Jonathan R. Landers

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

A derivation bounding the number of shortest paths between permutations in the permutohedron in terms of Kendall distance. The result makes the exponential branching structure of permutation space explicit.

This complements the geometric sorting framework in: my arXiv paper .

Permutohedron Geometry & Distance Comparisons

A short note analyzing three natural metrics on the permutohedron: the graph distance, geometric edge distance, and direct Euclidean distance. The construction gives clean comparison theorems showing how inversion count controls edge-based paths, and why straight-line Euclidean distance is always strictly smaller than any route restricted to edges. This clarifies how discrete order geometry interacts with embedded Euclidean structure.

This relates to the geometric sorting formulation in: my arXiv paper .

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.

Forrelation, Bent Functions, and a Spectral Criterion for Hardness

This note unifies the random-oracle and bent-function regimes of the Forrelation problem into a continuous spectral framework. By introducing a quantitative criterion based on Fourier flatness and resilience, it defines a smooth spectrum of classical hardness and quantum advantage. The discussion bridges Aaronson–Ambainis and Girish–Servedio, extending Forrelation theory through a geometric and information-theoretic lens.

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

A short, self-contained note showing how two scaled Beta distributions can share the same support, mean, and median while exhibiting opposite skewness via centered reflection. It gives a clean total-variation expression in terms of the density’s overlap with its mirror and a small-imbalance approximation that links TV directly to parameter differences. In ticket-pricing applications, this symmetry reflects opposite-skew pricing distributions for the same event snapshot, highlighting why full-shape geometry (not just mean/variance) matters for model robustness and divergence-based evaluation.

For related work and broader context, see my arXiv paper .

Neural Net Dropout Viewed as Probability-Mass Dilution

This short note extends the implicit regularization mechanism proved in my Random Forests work to neural networks. By viewing dropout as random thinning of active units, it derives a concise odds-compression inequality showing how dropout reduces dominance of high-scoring units. The note directly connects to Section 5 of my arXiv paper .

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.

Limits of Hawking-Induced Magnetism

This series of short notes explores the boundary between quantum evaporation effects and classical plasma dynamics around Kerr black holes. Each note tackles a different facet of why Hawking-induced magnetic fields remain negligible and structurally undetectable compared to accretion-disk–driven fields.

Magnetic Fields from Hawking Radiation vs. Accretion Disk Dynamics – Scaling arguments show Hawking-radiated charges cannot compete with disk dynamo fields across astrophysical regimes.
Unphysical Magnetic Field Parity – Proves impossibility in two directions: capped growth forbids parity in forward models, and inverse decompositions cannot recover Hawking modes.
Non-Identifiability of Subcomponents – A general theorem on information loss in aggregates, applied to Hawking fields, showing zero channel capacity for detection.

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.

Magnetic Reactor Clock

A minimalist physics clock driven by an electron orbiting a loop. The simulated current and Biot–Savart field evolve with real time, with a live readout and cumulative |B|. Blends clean WebGL shaders with an elegant EM formulation for a memorable, functional demo.

Mark Twain fMRI

A playful reading visualization: as a Twain passage scrolls and words highlight, simulated brain regions (frontal, visual, auditory, motor, emotional, spatial) light up in a 3D point cloud. Technically polished and human-centered, showcasing lateral thinking and real-time rendering.

Research Notes