The Standard Model census I built last time gave me the itch to do the same treatment for something I actually use, so here is a field guide to probability distributions.

It collects 24 named distributions, lets you bend their parameters and watch the curve respond, and shows how they all descend from a handful of mechanisms. You can open it here: A field guide to probability distributions.

It has three parts. The atlas gives every distribution its own plate with the probability mass or density, the cumulative function, and the moments (mean, variance, skew, kurtosis) computed in closed form rather than estimated from samples. The lines of descent is a map of how the distributions connect, because most of them are not separate inventions. One turns into another as a limit, a sum, a transform, or a conjugate prior, and you can follow any edge to read the relation. Drawing by hand is a sampling lab: pick a source, draw from it, and the histogram fills toward its curve. Turn on averaging and the central limit theorem pulls the result toward a normal, no matter where you started.

The maths runs in the page. Cumulative functions use the regularized incomplete gamma and beta functions with continued fractions, plus the error function, and sampling uses inverse-transform and the usual generators. No libraries, no network calls, and it runs offline once you save the file.

Enjoy!

I read this great Quanta piece and the part that stuck with me is that the question has no single answer. So I made a thing.

It is a scrolling census of the Standard Model where the particle count grows depending on how strictly you count. You can open it here: The Standard Model census.

You start at 17, the textbook poster of twelve matter particles, four force carriers and the Higgs. Then you scroll and the number climbs. Add the antiparticles and you get 30. The one gluon is really eight, which takes you to 37. Every quark comes in three colours, so 61. Split the fermions into left and right chiralities and give the bosons their polarizations and you reach 118, where the lopsided total comes from neutrinos only ever showing up left-handed. Keep going and you stop counting particles and start counting the underlying quantum fields, which lands on 995.5 degrees of freedom from a 2011 theorem. The same equations give very different answers.

There is also a four-forces matrix with gravity’s missing graviton, a “build a hadron” tool that only lets you assemble colour-neutral combinations, and a clickable dossier for every particle with its mass, lifetime, discovery and decay channel. The numbers follow Particle Data Group 2024 central values. No tracking, no network calls, and it runs offline once you save the file.

Enjoy!

Pokémon Go player scans are training military drone navigation.

This shouldn’t be surprising and somehow still is. Niantic was never a games studio that wandered into mapping. It spun out of Google’s Geo team, the Keyhole and Earth lineage, and Ingress then Pokémon Go were the friendliest data-collection apparatus ever shipped. Millions of people walking to exact coordinates and filming 360° video of doorways, statues and alleys, for free, for a Pokéball. You will not find a cleaner ground-truth dataset than humans voluntarily standing where you tell them and pointing a camera.

The part that actually lands is the consent bait-and-switch. “I was just playing a game” is the whole story. Nobody scanning a Pokéstop in 2021 thought they were seeding a vision system that lets a drone hold its position when GPS is jammed. And that is the real prize, not the game: navigating by matching a camera feed against a 3D model of the world instead of trusting satellites. GPS-denied navigation is what every defense program on the planet currently wants.

What I can’t get past is the shape of the denial. Vantor says it doesn’t use Pokémon Go data, but won’t say whether the models it already trained excluded it. Once 30 billion scans are baked into weights the question stops being answerable, because you can’t pull a person’s street corner back out of a model. The ethicist in the piece puts it well: proving your contribution is in there is nearly impossible. That isn’t a bug, it’s the useful property.

The thing I keep relearning: when the data is valuable and the collection is fun, the fun was the collection.

Dropped yesterday. 68 minutes covering C++ from Bell Labs to the modern renaissance, with interviews from Bjarne Stroustrup, Alexander Stepanov, Anders Hejlsberg, Chris Lattner, Brian Kernighan, and John Romero. That lineup alone is worth the watch.

I just read First public macOS kernel memory corruption exploit on Apple M5.

Stronger mitigations don’t end exploitation, they just raise the cost. If a team can still land a working kernel exploit in five days on hardware designed to make memory corruption dramatically harder, then the race is really about how quickly models and experts can turn bug classes into repeatable results.

I’m calling the winner here: Anthropic.

For reasons that made sense at the time, I built a small emergency video system.

The question behind it was pretty simple: if a connection gets bad enough, how little video can you still send before it stops being useful?

Repo is here: github.com/tgergo1/emergency-video.

The core of it is a custom C++ codec hooked up to a live webcam feed. Frames are shrunk to a tiny internal resolution, converted to grayscale, quantized to 4-bit luma, and encoded as periodic keyframes plus sparse interframes. If a block has not changed, it gets skipped, because bandwidth is what we are trying to save in the first place. If it has changed, it can be sent as coarse 4x4 cells, residuals against the previous reconstructed frame, or full raw 8x8 samples when the block is too messy for anything cheaper.

So yes, the result looks rough. Typical modes are 128x96 or 96x72 at only a few frames per second. That is on purpose. I was not trying to make nice-looking video. I was trying to keep enough scene information to answer basic questions: is somebody there, did something move, is that smoke, did the person just wave, is the situation getting worse?

What started as a codec experiment turned into a slightly larger communication prototype. There is a browser dashboard with live raw, sent, and received views, plus controls for the link, text messages, relay import/export, and general status. Underneath that is a simple packet layer with fragmentation, CRCs, ACK and retry handling, presence messages, queue management, and config beacons. Video is only one payload type. The same system can also send text, snapshots, and low-rate fallback traffic.

The transport side also got a bit out of hand. The app can move data over acoustic bursts through a speaker and microphone, serial, optical patterns, and a file-relay mode. When the link gets worse, a fallback controller steps down through lower FPS, lower resolution, snapshot-only mode, and finally text-only mode. There is also a low-rate text path with plain text, Morse, and DTMF-style encoding.

There are clear weak spots. This is not a standard video codec, and it is not trying to compete with one. The acoustic path depends a lot on the room, the hardware, and how noisy the environment is. The FEC is still fairly simple. Fast motion can still turn into stale or blocky output. The upside is that the whole thing is small enough to read, test, and package without it turning into a giant black box.

A few practical extras made it in: face-aware ROI prioritization, live bitrate and PSNR metrics, config hash checks before decode, browser-accessible HTTP endpoints, and tests for deterministic encoding, corruption handling, queue deduplication, communicator envelopes, and fallback logic. Somewhere along the way it stopped feeling like a codec toy and started feeling more like a small field tool.

If I had to describe it in one line, it is a video system that gives up slowly.

You can build and run it locally with:

cmake -S . -B build
cmake --build build -j
./build/emergency_video

Then open:

http://localhost:8080

These are basically my notes from working on this, but the math should still be right.

limitless numbers started from a simple thing: useful numbers don’t always fit into 32 or 64 bits, and rounding is not “good enough” when you actually need exact values.

Most of the numeric bugs I’ve run into were not really theory bugs. They were representation bugs:

• int overflows or wraps silently
• double cannot represent most decimal fractions exactly
• conversions between types lose information
• two values that look equal in source are not equal in memory

If you work on finance, symbolic transforms, exact ratios, small crypto exercises, protocol math, or generated test vectors, you bump into this fast.

So limitless had one job: exact arithmetic until you run out of memory, with a C API that is direct, portable, and small enough to embed without dragging in a lot of dependencies.

One number type, two exact forms

At runtime the type is a tagged union with two modes:

• integer: arbitrary-precision signed bigint
• rational: num/den, both bigints, always normalized

The mental model is three rules:

1. If a result is an integer, keep it integer.
2. If an operation requires a fraction, promote to rational.
3. If a rational simplifies back to denominator 1, collapse to integer again.

So 6 / 3 stays integer 2, but 7 / 3 becomes exact 7/3. No hidden float conversion, no binary rounding drift.

This is not “big float”. It is exact integer/rational arithmetic.

The real constraint is invariants, not operators

Adding + - * / is the easy part. Keeping the invariants intact after every operation is the harder one:

• denominator is always positive
• zero has one canonical representation
• rational is always reduced (gcd(num, den) = 1)
• outputs are failure-atomic: on error, the caller’s output is unchanged

If those invariants drift, everything built on top of them starts misbehaving.

So internally the library normalizes aggressively and uses temporary compute-and-swap patterns so error handling stays predictable.

Many C libraries keep allocator state in globals. Convenient for a demo, painful in real use.

limitless uses a per-context model:

• allocator hooks live in limitless_ctx
• no hidden global mutable state
• no mandatory libc dependency if you override alloc/realloc/free
• thread safety is scoped by context ownership

There is also a beginner path:

• limitless_ctx_init_default(&ctx) for quick start
• limitless_ctx_init(&ctx, &alloc) for custom environments

So you can start with the default and later switch to an arena/pool/failing allocator without changing how you call the API.

Big integer magnitude is a little-endian limb array.

• limbs are 32-bit by default
• optional 64-bit limb mode is available when supported
• sign is tracked separately from magnitude
• zero is represented canonically (sign=0, used=0)

The little-endian limb layout keeps carry/borrow loops simple and cache-friendly for schoolbook ops.

Rationals are just two bigints with strict normalization rules. It sounds trivial, but keeping that stable across parse/format/ops/conversions is where most of the actual work is.

Division is the part where many libraries do something I don’t like.

In a lot of libraries, integer division truncates unless you call a separate rational function. In limitless, limitless_number_div() returns exact math by default:

• integer result if divisible
• rational otherwise
• divide-by-zero is explicit LIMITLESS_EDIVZERO

So expressions stay safer because the default is mathematically correct.

Other operations are standard exact arithmetic with type promotion where needed:

• add/sub/mul across int/rat combinations
• unary neg/abs
• total compare via exact cross-multiply logic

Advanced integer operations

A second layer exposes integer-only helpers:

• gcd
• pow_u64 (exponentiation by squaring)
• modexp_u64 (modular exponentiation)

These are useful for small crypto things, number theory experiments, and parser/generator test fixtures.

The rules stay strict:

• integer-only APIs reject non-integer values with LIMITLESS_ETYPE
• modulus constraints are validated
• range and invalid states return status codes, not undefined behavior

Most numeric libraries do okay on arithmetic and then break on conversion edges.

I tried to spend real effort there, since conversion bugs are usually the ones that fail only in CI:

• parse from bases 2..36
• optional base autodetect (0x, 0b, etc.)
• rational text input as numerator/denominator
• formatting with caller buffer and explicit required-length reporting
• integer exports with range checks (to_i64, to_u64)

There are convenience aliases too:

• from_str() and to_str() as base-10 shortcuts

The formatting API returns LIMITLESS_EBUF when capacity is too small and reports the required length so callers can resize.

Float import is the part that is easy to skip and important in practice.

limitless_number_from_float_exact() and ..._from_double_exact() decode the IEEE-754 object representation and construct the exact rational/int value for that bit pattern.

• finite values become exact numbers
• NaN / Inf are rejected (LIMITLESS_EINVAL)
• 0.1 becomes its exact binary-rational equivalent, not a rounded decimal string

That is useful when debugging cross-language serialization and reproducible numeric tests.

C and C++ usability without splitting the core

The core stays strict C (limitless.h), with a C++ wrapper (limitless.hpp) on top for convenience.

C side:

• explicit lifecycle
• explicit context
• explicit status handling
• no operator overloading

C++ side:

• RAII wrapper class
• operator overloads for natural expressions
• primitive-left operators (int + limitless_number, etc.)
• parse/string helpers (parse, str)
• works in no-exception builds

So you can write:

limitless_number x = 33424234;
limitless_number y = (x + 3) / 2.3f;

C users keep full control over memory and error flow; C++ users get friendlier syntax without changing what runs underneath.

Testing

For exact arithmetic, you can’t really get confidence from a few unit tests.

Test coverage includes:

• generated deterministic vector matrices (large pairwise op sets)
• parse/format round-trips across multiple bases
• float/double exact import edge sets (normals, subnormals, signs)
• property-style randomized checks
• differential checks against Python fractions.Fraction
• allocator fault injection to force OOM at many allocation sites
• failure-atomic guarantees (out unchanged on failure)
• single-TU and multi-TU include model checks
• C++ wrapper equivalence behavior vs C core

So it is not just “worked on my machine”.

Packaging and release shape

A lot of small open-source libs stop at “here is a header”. I wanted this one to be a bit more usable, so it ships:

• CMake package target: limitless::limitless
• pkg-config metadata
• Conan recipe
• vcpkg overlay port
• GitHub Actions matrix across OS/compiler families
• coverage and release automation

Not the interesting part, but it is what makes a library actually usable by other people.

What this is good for and not good for

Good fit:

• exact fraction arithmetic
• educational math tooling
• parser/transformation pipelines that need deterministic exactness
• interoperability tests where numeric drift is unacceptable

Bad fit:

• performance-critical floating-point simulation
• vectorized numeric workloads
• cases where approximate real arithmetic is fine

It is precision-first, not throughput-first. That trade-off is on purpose.

Open-source, you can check it out here: limitless

Yet another thing:

The basic idea behind superblocks is simple: take a neighborhood, block through-traffic from cutting through it, and you end up with quiet streets where kids can play and people can walk without dodging cars. Barcelona did this and it actually works.

The question I had was: how do you systematically plan this for a whole city?

Most of the problem is really graph theory.

A city’s street network is a graph:

• Nodes are intersections
• Edges are streets
• Every edge has properties (road type, capacity, speed limit, one-way, etc.)

When you load OpenStreetMap data for a city, you basically get a big weighted graph with tens of thousands of nodes and edges.

The goal of superblock planning is to partition this graph into cells where:

• Through-traffic is impossible inside each cell
• Every address is still reachable from the outside

So this is a constrained graph partitioning problem.

The first step is to identify the arterial network. These are your primary, secondary, and tertiary roads, the ones that are designed to carry traffic. They form a subgraph that should stay fully connected and handle all the through-traffic.

Everything else (residential streets, service roads, living streets) becomes a candidate for the interior of a superblock.

The arterial network defines the boundaries of your superblocks. If you treat arterials as edges you are not allowed to cut, then superblocks are the faces of the planar graph formed by those arterials.

In graph theory terms, you find the minimal cycles in the arterial subgraph, and each cycle bounds a potential superblock.

But you cannot just say “everything inside this arterial boundary is a superblock”.

People still need to reach their homes by car. Deliveries need to happen. Emergency vehicles need access. So you need entry points.

The entry point problem is about finding nodes on the boundary where vehicles can enter the superblock interior. The main constraint is the no through-traffic property:

Once you are inside, you should not be able to exit from a different entry point without going back out and around on the arterial network.

Mathematically, this means the interior subgraph of each superblock has to be partitioned into sectors, where each sector is only reachable from its designated entry point. If you can drive from a north entry to a south entry through the interior, that is through-traffic, and it has to be eliminated.

To do that, you modify the interior graph:

• Modal filters: delete edges from the vehicle graph entirely (cyclists and pedestrians can still use them)
• One-way conversions: change undirected edges to directed edges
• Turn restrictions: keep the edges but forbid certain transitions at nodes

The algorithm tries to find the minimal set of modifications that achieves the no-through-traffic property while keeping all addresses reachable. This is a min-cut problem with reachability constraints.

For each superblock, you compute which nodes are reachable from which entry points after applying the modifications:

• If a node becomes unreachable from all entries, the modification set is too aggressive.
• If a node is reachable from multiple entry points in different sectors, the modification set is not aggressive enough (through-traffic is still possible).

The reachability check is just BFS from each entry point, respecting directed edges and modal filters. You mark which nodes each entry can reach, and then verify two things:

1. Every interior node is reached by at least one entry.
2. No interior node is reached by entries from different sectors (unless they share a boundary, which gets more complicated).

Sector assignment itself uses angular partitioning: compute the centroid of the superblock polygon, then assign each entry point to a sector based on its angle from the centroid.

Four sectors give you north/east/south/west. Eight sectors add the diagonals. The point is that entries on opposite sides of the superblock should be in different sectors, so traffic between them has to use the arterial network.

Validating the constraint globally is a graph connectivity problem. After all modifications, you build the “can I get from A to B” relation for all pairs of entry points:

• If entry A and entry B are in different sectors, there should be no path between them through the interior.
• If they are in the same sector, there should be a path (so addresses in that sector are reachable).

So you basically check that the interior graph has been decomposed into the right number of connected components.

The routing part is also pure graph theory. Given an origin and destination, find the shortest path that respects superblock constraints:

• If both points are on the arterial network, it is just Dijkstra or A*.
• If one or both are inside superblocks, you first route to the appropriate entry point, then across arterials, then from the destination’s entry point to the destination.

The cost function penalizes interior travel to encourage using arterials.

Traffic estimation without real data uses graph-theoretic heuristics:

• Edge betweenness centrality measures how many shortest paths pass through each edge (high betweenness means high traffic).
• You can also simulate flow by assuming trips are generated proportionally to residential density and attracted proportionally to commercial density, then computing shortest paths between all origin-destination pairs. The fraction of paths using each edge estimates its load.

The whole thing is basically a pipeline:

1. Load the OSM graph.
2. Extract the arterial subgraph.
3. Find the faces (superblock candidates).
4. Compute entry points for each face.
5. Find the minimal modifications that enforce the sector constraint.
6. Validate reachability.
7. Provide routing that respects the new structure.

What makes it tractable is that real street networks have useful properties:

• They are roughly planar (streets rarely cross without intersections).
• They have hierarchical structure (arterials form a coarse grid, local streets fill it in).
• Superblocks are spatially local, so you can process them independently.
• The modification search space is bounded: you only consider edges in the interior, and most interiors have maybe 20–50 edges.

The output is a transformed graph where through-traffic is structurally impossible, not just discouraged. You are not relying on signs or enforcement, the network topology itself prevents it. That is roughly the idea behind superblocks.

Open-source, contributions welcome: Superblocker

I’m pretty enthusiatic about world-models and especially JEPA (and V-JEPA), so I started a small prediction engine, that predicts the next few frames based on the imput stream’s current frame, visualizes the latent space (and its movement) and creates a guess about motion. I’m planning to use this in a car to build up a driving world-model.

The idea is that instead of just classifying images, the model should learn the dynamics: how things move, change, and evolve over time. That is what you actually need for robotics, autonomous vehicles, or any system that has to act in the real world. Before doing something, you want to ask: “what will happen if I do X?”.

I made a thing: It is a Riemann Zeta function convergence explorer with a 3D surface view. You can open it here: Riemann Zeta explorer. It lets you sweep the complex plane domain, pick presets (critical strip, first zeros, near the pole), and toggle layers like the critical line, known zeros, reference planes, and divergence region. There are also modes to compare analytic continuation against Dirichlet partial sums or prime zeta partials, plus error and relative error views.

Enjoy!

GTD with this.

So, this is getting better. I’ve always wanted to be able to just casually drop any thought here and however the markdown post are working fine, it is a bit hard to just e.g. post something from my phone. But with a custom GitHub action that accepts all the necessary data to create a new MD file and then pushes and updates the site: 🔥.

This post was sent from my phone via GitHub actions

ENKAIDU.

Currently listening to this episode:

A Threads post embedded right here:

Instagram embedding with Markdown! So, I promise this won’t be a meme site, I just try this out. I consider this progress.

I’m happy to announce that this site can handle two posts together. Can it handle three? Come back later to find out!

So, I decided to make a bloggish landingish page. A brutalist corner of the internet where I’ll share thoughts on research, random ideas, and whatever else crosses my mind. I tried some kind of weird fusion of OpenVibe, Threads etc., but I think it’ll be better if I just post things here and maybe share to other platforms.

I’m not very good at this, so improvements are on their way.

Follow via RSS if that’s your thing.