a trail of mini-lessons

Breadcrumbs into a paper

Short, interactive lessons that lay down one idea at a time — each a step on a trail that ends at the paper itself, now readable.

Trail 01  ·  The global workspace inside a language model
built from what you already know  ·  ~10 min per step  ·  interactive

1

the math object

What a Jacobian actually is

The ordinary derivative — “nudge the input, watch the output move” — grown up to handle vectors in and vectors out. With a live tangent-line you can drag.

calculus → linear algebrastart here →
2

the instrument

Why “transport a hidden activation into the final-layer basis via J”

Point the Jacobian from mid-stack to the model’s exit and it becomes a lens — a readout of what a buried thought is disposed to make the model say. Click down the layers and watch a thought resolve into words.

transformers · interpretabilitycontinue →
3

the claim

The four tests that make it a “workspace”

Verbal report, directed modulation, internal reasoning, flexible generalization — the criteria that promote “the lens finds verbalizable representations” into “these form a global workspace.” And then: could a beauty axis live there?

theory · consciousnessfinish the trail →

the destination — the paper

Verbalizable Representations Form a Global Workspace in Language Models

Gurnee, Sofroniew, Pearce, …, Batson, Lindsey (16 authors) · Anthropic / Transformer Circuits · July 2026

A new interpretability method (the Jacobian lens) surfaces a small, privileged set of “word-like” representations that the model can report on, steer, and reason with — structurally like the global workspace of consciousness theory. Once the trail is behind you, the paper reads like a home you already know.

What this is

Each paper worth understanding has a few load-bearing ideas that, once you hold them, make the whole thing legible. Paper Breadcrumbs pulls those out into a short trail — one idea per step, built up from something you already know, and interactive so you can poke at it rather than just read.

This first trail walks from the plain calculus derivative to the interpretability instrument at the heart of Anthropic’s 2026 global-workspace paper. More trails are coming, including material for fall coursework.