Year 12 · 55 minutes · Explicit geometry model
Compare chord and arc interventions on an explicit circle, inspect decoded angles and quantify off-manifold distance.
Student paper journal · Interactive notebook · Editable Python
Try the starting controls, save two runs and reopen a downloaded project. Print the student journal before class. If Python cannot load, use the supplied example below for a prediction and evidence critique; rerunning changed inputs requires the live or native notebook. Do not present paper discussion as a new model experiment. Keep an individual record even when you investigate in pairs. Record the exact settings, source and data split.
Start with 5–8 minutes of prediction, use about half the lesson for paired comparisons, then reserve at least 10 minutes for a counterexample and an individual explanation. Use the undefined centre as an explicit failure case. Connect the idea to cyclic concepts without claiming all model concepts form circles.
Settings: {'start': 0, 'angle': 180, 'path': 0}
Compare a straight chord with an arc on a known unit-circle representation. At the centre, the angle decoder is undefined.
| fraction | x | y | off_manifold_distance | decoded_degrees |
|---|---|---|---|---|
| 0.0 | 1.0 | 0.0 | 0.0 | 0.0 |
| 0.05 | 0.8999999999999999 | 6.123233995736766e-18 | 0.10000000000000009 | 3.8981718325193764e-16 |
| 0.1 | 0.8 | 1.2246467991473533e-17 | 0.19999999999999996 | 8.770886623168595e-16 |
| 0.15000000000000002 | 0.7 | 1.83697019872103e-17 | 0.30000000000000004 | 1.5035805639717594e-15 |
| 0.2 | 0.6000000000000001 | 2.4492935982947065e-17 | 0.3999999999999999 | 2.3389030995116253e-15 |
| 0.25 | 0.5 | 3.061616997868383e-17 | 0.5 | 3.508354649267438e-15 |
| 0.30000000000000004 | 0.3999999999999999 | 3.67394039744206e-17 | 0.6000000000000001 | 5.262531973901159e-15 |
| 0.35000000000000003 | 0.2999999999999999 | 4.2862637970157366e-17 | 0.7000000000000002 | 8.186160848290693e-15 |
The table shows up to eight rows. Inspect the notebook for all values, controls and denominators.
Method and scope: Arc(t) = [cos(θ₀+tΔ), sin(θ₀+tΔ)]; chord(t) = (1−t)a+tb. Distance to the unit circle = |‖x‖−1|. Geometry is hand-specified here; it is not extracted from a large language model.
Assess four criteria, each 0–2: testable prediction; comparison identifying what stayed fixed; accurate use of exact evidence; counterexample and limited conclusion. 0 means absent or contradicted by the record, 1 means partly supported, and 2 means clear and supported. A surprising result earns no penalty.
Beginning response: ‘The picture looks right.’ Ask for a particular case. Developing response: names one number without its setting. Ask what comparison supports it. Secure response: states the setting and measured change and separates the notebook's result from the source paper's claim. Extending response: authors or reserves a new case, tests an alternative explanation, and revises the claim if needed. These are marking examples, not pupil data.
Extension: Implement a noisy ellipse and test whether a wrongly fitted circle misleads the intervention.
System and data: Historical Llama steering demonstrations
Method: Changes selected internal feature strengths and inspects example outputs; demonstrations need independent task and side-effect checks.
Task: Compare activation interventions with prompting and weight updates. Record the source model/task, a baseline, the changed factor, a measurement and an alternative explanation. State exactly which part your notebook investigates and which part it does not reproduce.
Boundary: Steering is not guaranteed control or a source of new knowledge. Ember links are historical.
System and data: Llama 3.1 8B arithmetic and cyclic tasks
Method: Tracks layer/token representations, identifies a shared addition mechanism and checks it with causal interventions.
Task: Contrast an explanatory geometry with the arithmetic mechanism established by causal interventions. Record the source model/task, a baseline, the changed factor, a measurement and an alternative explanation. State exactly which part your notebook investigates and which part it does not reproduce.
Boundary: The paper does not claim that Llama simply adds directly around a circle. Our calendar is a teaching model.
System and data: Many-shot context and activation-steering experiments
Method: Fits a Bayesian account that distinguishes likelihood updates from changes to priors; tests how context strength changes steering effects.
Task: Compute posterior log-odds and test saturation under strong contextual evidence. Record the source model/task, a baseline, the changed factor, a measurement and an alternative explanation. State exactly which part your notebook investigates and which part it does not reproduce.
Boundary: A fitted account of model behaviour does not establish that a model has human beliefs.
System and data: Llama 3.1 8B weekday behaviour and activations
Method: Fits related behavioural and activation geometry, then compares movement along fitted manifolds with linear edits.
Task: Measure off-manifold distance and decoded behaviour under two steering paths. Record the source model/task, a baseline, the changed factor, a measurement and an alternative explanation. State exactly which part your notebook investigates and which part it does not reproduce.
Boundary: Only selected fitted manifolds and tasks were tested; not every concept is circular.
System and data: Research collection across several model domains
Method: Organises related studies of representation geometry; the collection is not an independent experiment beyond its linked papers.
Task: Use the series as a reading order, then inspect each paper's evidence separately. Record the source model/task, a baseline, the changed factor, a measurement and an alternative explanation. State exactly which part your notebook investigates and which part it does not reproduce.
Boundary: This is an index, not an additional independent experiment.
System and data: Image-patch representations and a BatchTopK SAE
Method: Decomposes image representations into sparse features and edits selected patches; visual examples and reconstruction are distinct from reliable semantic control.
Task: Consider fragmentation, spatial location and interactions between prompts and feature interventions. Record the source model/task, a baseline, the changed factor, a measurement and an alternative explanation. State exactly which part your notebook investigates and which part it does not reproduce.
Boundary: This is image generation, not an LLM experiment; the classroom intervention is a transferable analogy.
System and data: Evo 2 representations of cross-species DNA
Method: Constructs data to distinguish evolutionary relationship from simple sequence similarity and compares distances along learned geometry.
Task: Ask whether a fitted representation generalises to held-out clades and controls for sequence similarity. Record the source model/task, a baseline, the changed factor, a measurement and an alternative explanation. State exactly which part your notebook investigates and which part it does not reproduce.
Boundary: Biological geometry does not establish an equivalent map inside text LLMs.
System and data: Llama 3.1 8B and SimpleStories
Method: Collects sentence-end activations, fits trajectories and compares emotional organisation with behavioural readouts and human valence/arousal judgements.
Task: Compare decoded narrative variables, causal steering and entangled concepts. Record the source model/task, a baseline, the changed factor, a measurement and an alternative explanation. State exactly which part your notebook investigates and which part it does not reproduce.
Boundary: Representing a character's emotions does not mean the model has emotions. Classroom story probabilities are hand-specified.
System and data: Cross-domain neural-geometry perspective
Method: Connects structured data to learned representations and an unsupervised geometry-discovery pipeline; individual causal claims require their own experiments.
Task: Distinguish a geometric hypothesis from predictive and causal validation. Record the source model/task, a baseline, the changed factor, a measurement and an alternative explanation. State exactly which part your notebook investigates and which part it does not reproduce.
Boundary: A cross-domain research perspective does not prove every useful concept has an easily readable geometry.
Independent Brightlab adaptations; no Goodfire endorsement or full reproduction claim. Source methods reviewed 7 September 2026. Curriculum connections are selected planning links; confirm your school syllabus.