brightlabYear 11 · 90 minutes
When time becomes a calculation
Does a calendar-shaped representation mean a model calculates directly on a calendar?
You will learn to: Compare direct cyclic arithmetic with encode → ordinary sum → remap, use Fourier probes, and evaluate transfer and causal controls.
Your mathematical model
Follow the calculation
Before opening the researchers’ model below, build a prediction you can test. A circle keeps the remainder after complete turns.
Start 0 · add 0
Start 3 · add 1
Start 8 · add 6
Blue solid = start. Pink dashed = added number. Thick green = sum. Each angle is 2π × remainder ÷ period. These three views repeat together every 10: they cannot distinguish 7 from 17.
Can you break this number code?
The research question: August + 6 has ordinary sum 14, which maps to February. A final calendar circle alone cannot tell us whether an internal step used base-10 addition or direct mod-12 arithmetic. We need intermediate measurements and interventions.
Goodfire’s original research demonstration
Inspect Llama’s geometric calculator
Start with August + 6, then try 16 months after August. Follow the input and output representations on the period-2, period-5 and period-10 views. Which information survives a complete turn? Compare your mathematical prediction above with these recorded model activations.
The interactive below is hosted by Goodfire. If it does not fit your screen or your school blocks it, open the original in a full window ↗.
Source measurements and interface: Goodfire. Spark does not generate these recorded Llama results.
Goodfire’s original research demonstration
Change the internal representation
This experiment fixes the prompt at sixteen months after August. Change the available steering controls and compare the result with the baseline. This modifies internal features; it is stronger evidence than merely seeing a circle. It still does not establish every step of the model’s algorithm.
The interactive below is hosted by Goodfire. If it does not fit your screen or your school blocks it, open the original in a full window ↗.
Source measurements and interface: Goodfire. Spark does not generate these recorded Llama results.
Now investigate for yourself
Your Marimo research notebook
The browser calculator is a transparent mathematical model. A small language model may not learn or reuse Llama’s mechanism. Poor transfer is a valid finding; a circular plot does not prove the mechanism.
Teacher notes & evidence task
Starting knowledge: Modular arithmetic, sine/cosine and train/test splits; basic Python is helpful for extensions.
- Trace August + 6 months as 8 + 6 = 14, then February. Contrast mod-10 intermediate information with mod-12 output mapping.
- Compare the source’s weekday, month and hour examples. Include 16 months after August and a wraparound hour.
- Use the original causal steering demo; record its fixed prompt and the intervention.
- Fit Fourier probes to actual hidden states from addition prompts in Marimo, reserving evaluation examples.
- Transfer the probe to time prompts; compare next-token predictions, probe errors and a no-op-controlled intervention.
Evidence to collect: Submit a cross-task experiment with model revision, split, layer, projection, held-out error and a limited causal claim.
Research basis: The paper’s Llama analysis finds ordinary addition shared across months, weekdays and hours, followed by remapping to the requested cycle. Calendar geometry alone would not establish that algorithm.
Goodfire research article ↗ · Full research paper ↗ · Setup and teaching guidance