If you have read the primer, you already have the setup. This note only sharpens the question the paper asks. I am keeping the method and the findings out of it while the work is under double-blind review; I will expand this into a full write-up once it is public.
Where the memory goes
Recall the one tension from the primer. An analytic continual learner carries a small state between tasks, and a random expansion into more coordinates — more width — raises accuracy. But the part of that state which records how coordinates interact grows quadratically with width. When the state you carry has to stay within a fixed budget, that quadratic term is what limits how wide you can go.
The choice
So a fixed budget forces a choice, and it is the choice in the title:
Keep exact interactions among fewer coordinates — or keep more coordinates and store how they interact only approximately?
Why it is not obvious
You might guess width always wins, or that fidelity always wins. Neither is safe. More coordinates help only if a straight line can still use them; compressing interactions too hard can throw away exactly what separated two classes. And earlier comparisons in the literature mixed this trade-off in with differences in training recipe and state size, so they could not isolate it. Answering it cleanly means holding everything else fixed and changing only width against interaction fidelity — which is what the paper is built to do.
On the paper
The method, the experiments, and the answer — including where the trade-off does not pay, which I think is the most useful part — are in the submission, currently under double-blind review at ICLR 2027. Out of respect for that process I am not posting the architecture or the results here yet.
When it is public, this note will grow into the full walkthrough, the way the primer is written. For now, the question above is the part I most wanted to share.