CALLIOPE · BCOM

Epidemic models describe the spread of a replicating agent through a host population. Large language models do not replicate, but a recurrent pattern of cognitive coupling between human and model does, and that pattern is the natural object of an epidemiological treatment. This note takes up the three-compartment model introduced in WP0204 --- individuals uncoupled, coupled while retaining cognitive autonomy, or persistently dependent --- and analyzes it in detail. We first fix the normalization, so that the transmission parameter becomes directly comparable to its SIR counterpart, and verify that the system conserves population and leaves the simplex invariant. The normalized model admits a reproduction number $=( / )(1+ / )$ that decomposes into transmission accrued while autonomous and while dependent. Its bifurcation at $=1$ is forward, from which we draw a negative result: the model cannot exhibit a transition to maximal cognitive offloading, only smooth saturation. We then show that a single feedback --- skill atrophy, in which recovery of autonomy degrades as dependency spreads --- turns the bifurcation backward above an explicit threshold on its strength, opening a bistable window in which high offloading persists below the epidemic threshold. The resulting hysteresis, not the threshold, is the substantive claim: exposure reduction alone need not restore autonomy. This result gives the parasitic regime of WP0204 a precise dynamical signature.

Hysteresis, not thresholds, is why reducing your AI use might not give you your thinking back.

The core worry in WP0204 (the parent paper) is that LLMs might erode the cognitive skills people need to function without them — a "parasitic" regime where the tool degrades the host. This note asks: can a simple mathematical model actually produce that outcome, and if so, what does it require?

The model tracks three groups: people who don't use LLMs (uncoupled), people who use them but stay intellectually independent (autonomous), and people who have become persistently reliant (dependent). LLM-use practices spread socially — by watching colleagues, following institutional defaults, absorbing platform nudges — so the spread looks formally like an epidemic, with a reproduction number R₀ that tells you whether the practice invades a population. The paper first cleans up a normalization error in the original model (the transmission term was missing a 1/N factor, making it incomparable to standard epidemic models), then derives R₀ in closed form: it's a baseline contact-to-abandonment ratio, amplified by however much the dependent population contributes to further spread.

Here's the key negative result: the baseline model cannot produce a tipping point. The dependent fraction grows smoothly as R₀ crosses 1 and saturates below 100% — no sudden jump, no bistability. If you reduce exposure, the system simply reverses along the same path. This is a forward (supercritical) bifurcation, and it means the baseline model is too tame to represent the parasitic regime WP0204 worries about.

To get a genuine tipping point, you need one additional ingredient: skill atrophy, meaning recovery of cognitive autonomy gets harder as dependency spreads — because the very verification skills and training pipelines that enable recovery are themselves eroded by widespread offloading. Formally, the recovery rate σ becomes a decreasing function of the dependent fraction d. This single feedback, above an explicit threshold on its strength κ, flips the bifurcation backward: the endemic branch now curves back below the epidemic threshold, creating a bistable window where high dependency and full autonomy are both stable for the same parameter values. The system's history — not just its current exposure level — determines where it sits.

That bistability produces hysteresis, and hysteresis is the paper's real claim. Once a population has tipped into high dependency, reducing LLM exposure back to just below the epidemic threshold is not enough to restore autonomy — you have to push all the way to a lower saddle-node threshold. The intervention required to escape the trap is strictly larger than the one that would have prevented falling in. The paper also flags an important open question: whether full offloading (d = 1) is an unreachable asymptote or a genuine absorbing state depends on a modeling choice — whether recovery rate hits zero at d = 1 — that should be made deliberately rather than by algebraic convenience.