Jonathan R. Landers – Research and Exploration Space

Noether, Energy-Conserving Transitions, and Distributed Repair

This cluster develops energy conservation as geometry: first as a Noether-style admissibility filter on possible futures, then as a repair-and-redistribution mechanism on a thickened energy shell. Together, the note, paper, and companion animation move from discrete constraint counting to a visual picture of how excess energy is corrected and spread across the system.

Conservation as filtration. Noether, Energy-Conserving Transitions, and Distance-to-Conservation - formalizes conservation laws as a dimensional reduction on candidate next states and introduces an explicit distance-to-conservation correction metric.
Distributed repair. Distributed Energy Borrowing - treats off-shell mismatch as excess that can be repaired by minimal coordinated adjustment across degrees of freedom on an admissible energy band.
Companion visualization. Distributed Energy Borrowing - Shell Ripples Animation - visualizes normal-to-shell repair together with ripple-like redistribution along the conserving shell.

A Continuous Information Landscape for Computational Problems

This paper models computational problems as binary objects in a shared Hamming space and measures relatedness through algorithmic mutual information. It derives a local entropy envelope for bit-flip perturbations, introduces a continuous similarity kernel on problem space, and proves that neighborhoods around low-complexity problems remain exponentially sparse below an explicit entropy threshold.

Constant-Time vs Constant-Velocity Transport: Anytime Geodesic Generation with Transfer to Flow Matching

A fixed-speed, arc-length reparameterization replaces fixed-time transport, making compute proportional to geodesic distance traveled. A metric-tensor view of warped geometry yields prefix/suffix optimality for minimizing geodesics, so truncation corresponds to calibrated geometric progress. In constant-speed flow matching, the same structure implies exact failure recovery and supports a segmentwise parallelization theorem.

Dynamics and Long-Horizon Stability

This four-note arc explains why models that look stable over short windows can fail under deployment-time drift at long horizons. It connects geometric error decomposition and directional compounding sensitivity to a practical robustness strategy: constrain tangent response specifically along empirically observed drift directions. To formalize this progression, the notes develop a sequence of theoretical results on horizon-scaled error bounds, directional sensitivity of long-run functionals, and variance control through drift-aligned tangent geometry.

Horizon geometry. Fan-of-Rays Geometry: Bounding Long-Horizon Prediction Error - decomposes terminal miss into initial offset, angle mismatch, and curvature mismatch with explicit horizon scaling.
Functional sensitivity. Long-Horizon Sensitivity of CAGR Under Infinitesimal Model Perturbations - converts geometric amplification into a concrete compounding metric and shows directional concentration along realized displacement.
Risk over time. Dynamics Shift and Geometric Robustness in Classification - moves from endpoint error to deployment-time risk volatility, controlled by drift speed and network tangent gain.
Capstone regularization. Drift-Tangent Regularization - turns the mechanism into a training and monitoring recipe by penalizing directional Jacobian gain along estimated drift subspaces.

Density-Based Clustering: A Geometry of Scale

In clustering, turning the density dial reveals a hidden continuous geometry underneath the discrete labels. A short arc: operational event discovery → DBSCAN knob dynamics → a continuous scale surrogate → level-set erosion calculus → equivalence with centroid-based clustering.

Grounding. Spatiotemporal Event Detection in Structured Grid Data — comments over incidents; delayed timestamps; clustering as inference.
Discrete behavior.
- Differential Effects of the Minimum Point Threshold on Topology and Motion in DBSCAN — matching splits motion (peeling) from birth/death events.
- A Continuous Sensitivity Theory for Density-Based Clustering Scale — smooth “effective cluster count” via diffusion; sensitivity scales like ε⁻².
Continuous geometry. Density-Level Clustering: Erosion, Centers, and Separation Across a Continuous Threshold — derivative laws for mass/centroid motion under density-threshold erosion.
Unification. Blob Topology Equals Centroids — after whitening + separation, DBSCAN components coincide with the k-means minimizer.

Creativity and Sorting: Rarity at the n log n Boundary

A concrete “creativity = rare compression” story for comparison sorting. It proves a boundary/rarity theorem for decision trees under an uninformed prior, lifts the same idea to the permutohedron where efficient sorts correspond to geodesic-like paths, and connects the discrete picture to the continuous contraction view of sorting as gradient flow.

This connects to the gradient-flow formulation in: my arXiv paper .

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

An operational notion of subsystem persistence defined by past-present mutual information with a fixed reference record. Using the data-processing inequality for open-system (CPTP) dynamics, it shows identity cannot increase and introduces an identity half-life via a stability threshold. Worked examples include beta decay and dephasing / decoherence, separating classical record persistence from coherent identity loss, with a concrete Ship of Theseus framing.

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.

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.

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.

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.

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.

Research Notes