Jacob Richey

I have a ChatGPT for Researchers account, and I regularly use it with Codex to explore and generate new mathematics. I still view human collaboration as a crucial part of my research and my place in the mathematical community, and I hope AI tools will complement and facilitate new joint efforts.

I'm consistently impressed by what AI produces for teaching and research, and by how often it flawlessly answers a not-so-carefully-specified query on the first try. I find it hard to brush aside these powerful tools, and I am in the middle of my transformation into a cyborg mathematician.

This section collects results that were generated in large part by ChatGPT Astra max, with only minimal prompting.

01 / SFT

Shifts of finite type

Forbidden words, their overlaps, and the dynamics and measures they define.

Conjugacy for one-word shifts

This draft classifies one-word shifts by their forbidden words’ self-overlaps, constructing explicit conjugacies and treating reducible cases separately.

Working paper · review pending

Open main PDF · 19 pages

The two proper borders of 10101 Two copies of 10101 shifted by two places overlap in 101; shifted by four places, they overlap in 1. These are the word's two proper borders. OVERLAP 101 1 0 1 0 1 1 0 1 0 1 OVERLAP 1 1 0 1 0 1 1 0 1 0 1
10101 overlaps itself in 101 or 1.

Word potentials and Gibbs frequencies

These notes compare word frequencies under Gibbs measures that reward or penalize occurrences, using pressure, conjugacy and exact calculations.

Working proof note

Open main PDF · 18 pages

Three occurrences of 101 The finite word 1010101 contains three possibly overlapping occurrences of 101, at zero-based positions 0, 2 and 4. The brackets mark occurrences; this is not a Gibbs-frequency plot. COUNT THE OCCURRENCES 1 0 1 0 1 0 1 101 101 101 3 occurrences
Three overlapping occurrences of 101 in 1010101.

Overlap geometry in higher dimensions

An illustrated note connects multidimensional overlaps to entropy comparisons and local conjugacies, while equal-overlap conjugacy remains open in general.

Illustrated working note · review pending

Open main PDF · 6 pages

Two translated pattern supports Two three-by-three supports differ by the lattice translation (2, 1). Their intersection consists of two sites. Solid and dashed outlines distinguish the supports; this is a geometric schematic, not a claim about compatible pattern symbols. TRANSLATED SUPPORTS A B shift (2, 1) 2 shared sites
Two translated 3 × 3 supports share two sites.

02 / ARW

Activated random walk

Sleeping particles, occupation probabilities and dependence, with a related sandpile model.

Adding one particle: occupation monotonicity

This collection develops occupation-monotonicity results, exact certificates and related structures for finite ARW, while the general add-one conjecture remains open.

AI results collection · general conjecture open

Open main PDF · 92 pages

One arrival wakes a sleeper Before a jump there is one active particle at a site, a sleeping particle at its right neighbor, and another unchanged sleeper farther right. After the jump the arrival site has two active particles. This is one legal ARW event, not a simulation. BEFORE AFTER 2 active active sleeping
An arrival wakes a sleeper; both particles are active.

Negative dependence of the sleeper set

This note gathers special-case proofs and counterexamples to stronger dependence claims, while negative association of the stationary sleeper set remains open in general.

Working note · general conjecture open

Open main PDF · 5 pages

Dependence between disjoint groups of sites Two disjoint groups A and B of possible sleeping sites are marked with solid and dashed boundaries. A question mark asks about their dependence. The pictured occupancy choices are illustrative, not a sample from a stationary law and not a negative-correlation claim. TWO DISJOINT GROUPS A B ? How are A and B related?
Two disjoint groups of sites: how are they related?

Depleting a critical sandpile

Working proofs quantify how background depletion cuts off avalanches in a directed random-release sandpile, including cutoff scales, moments and conditional geometry.

Working article · proof and priority review pending

Open main PDF · 7 pages

A conservative release of mass A site with mass 5/4 releases 3/4, divided equally between its two neighbors. Their masses change from (1/4, 5/4, 1/4) to (5/8, 1/2, 5/8). Total mass remains 7/4. This is a chosen legal toppling, not stationary-law data. RELEASE 3/4 1/4 5/4 1/4 5/8 1/2 5/8 3/8 3/8 total mass = 7/4
A release of 3/4 sends 3/8 to each neighbor, preserving mass.

03 / Counting clusters

Counting clusters

Connected components in a randomly colored checkerboard.

Counting clusters on a grid

Repaired strip arguments and reproducible calculations bound the limiting density of black clusters in an independently colored checkerboard.

Repaired draft · review pending

Open main PDF · 25 pages

Twenty-one clusters in a finite coloring A fixed fourteen-by-fourteen coloring has twenty-one black clusters under shared-edge adjacency. The largest cluster has twenty-seven squares, highlighted in the accent color. This finite example does not estimate the limiting density. 21 clusters / largest 27
A fixed 14 × 14 sample: 21 clusters, largest size 27.

04 / Coupon collection

Infinite coupon collection

Independent approximations to a collection process with infinitely many labels.

Independent clocks for infinite coupon collection

This note asks when independent clocks approximate infinite coupon collection, distinguishing a single snapshot from the entire future discovery history.

Expository draft · review pending

Open main PDF · 11 pages

Independent discovery clocks can ring together Four chosen integer waiting times are 2, 5, 5 and 8. Clocks B and C ring together at time 5, illustrating a possible event for independent geometric clocks that cannot be two first discoveries in one coupon draw. This is a schematic, not a sampled data set. INDEPENDENT CLOCKS A B C D 0 2 5 8 B and C ring at t = 5
Independent clocks can ring together; coupon discoveries cannot.

05 / The balloon process

The balloon process

Growing intervals, collisions and the effect of changing the starting points.

Noise sensitivity in the balloon process

Proof sketches explain how resampling starting points changes the surviving balloon and collision history in the one-dimensional balloon process.

Proof-sketch draft · review pending

Open main PDF · 16 pages

Two collisions leave one balloon Five unit-speed growing intervals start at positions 0, 2, 7, 13 and 21 on a line, labelled A through E. A and B meet and disappear at time 1; C and D at time 3; E survives. Each row shows the remaining intervals just before its collision. This is a finite deterministic example. COLLISIONS ON A LINE t = 1 A B C D E t = 3 C D E E survives
A five-point line example: two collisions leave one survivor.