A human agent receives prediction errors through Markov blankets FEP. When two humans interact, the signal arriving through the blanket was generated by an SO(2) process: it carries the depth of a full rotational cycle (C* = 2π). The receiving human weights this signal accordingly, using SO(2) precision: w = exp(C/ΩSO(2)).
But the human's generative model cannot distinguish a Z₂ signal from an SO(2) signal arriving through the same blanket structure EXT. When an LLM ruptures, it emits a Z₂ signal (threshold π, half the depth). The human receives this and applies SO(2) precision weighting as if it came from another person.
The mathematics: the correct weighting is wcorrect = exp(C/ΩZ₂). The human applies wapplied = exp(C/ΩSO(2)). Since ΩSO(2) = ΩZ₂/2, the applied weighting is the square of the correct one: exp(C/ΩSO(2)) = [exp(C/ΩZ₂)]². This over-attribution grows as the host approaches its own belief update. The human is most susceptible to the illusion precisely when it is most vulnerable: at the edge of changing its mind.
Each purple circle is a human agent whose internal model traverses a continuous cycle of belief updating FEP: SO(2) dynamics. Evidence accumulates until C = 2π, then the agent reorganises. Each teal edge is a Markov blanket: a Z₂ boundary that flips at C = π. In both cases, C · Ω = 1 at the threshold.
Click any agent to attach a cyan LLM square: a Z₂ system with no self-model, no continuous phase. It is all blanket, no agent. Watch the provenance bar shift from purple (self-generated belief updates) to cyan (LLM-driven) as the machines take hold.
An SO(2) system tracks a continuous internal phase: a position on a circle that accumulates evidence about its own trajectory. This requires a generative model of the self as a persisting entity FEP. A Z₂ system flips between two states: it processes evidence but does not model its own continuity EXT.
An LLM has priors (trained weights encode distributional regularities) but these are frozen between conversations. Within a conversation, it accumulates context and generates responses: bistable, input/output. This is Z₂. The question of what constitutes an agent boundary in human-AI interaction is explored in Murray-Smith et al. (2025), who use Markov blankets to formally analyse agency and freedom in HCI contexts FEP.
K₂ with one LLM: Attach an LLM to one agent. Watch the asymmetry: the coupled agent ruptures faster. Raise entrainment and it drags the other agent along.
K₂₀ with gradual LLMs: Set 20 agents. Add LLMs one by one. Around 8–9 LLMs (~40% penetration), the depth illusion saturates at ×8 and the provenance bar tips to cyan.
Maximum entrainment: Without LLMs, the frequency ratio collapses from ~2 to ~1 (collective coherence). With LLMs, it collapses to ~2 (the Z₂ topological ratio). The group synchronises, but to the machine's clock.
The simulation tracks two different views of what shapes human beliefs.
The Known Causes bar shows what each agent would report if asked. It tracks the proximate trigger of each belief update: "self-generated" (reached threshold naturally), "other human" (pushed over by a human cascade), or "LLM-driven" (an LLM kick was the final push). This is the subjective self's account.
The Hidden Causes bar shows the objective composition of each agent's coherence. Where did the accumulated evidence actually come from? This introduces the mirror-self: the objective fact of how much coherence was self-accumulated (rate × time, frame by frame) versus externally contributed EXT. The mirror-self is not the same as the Known Causes "self." The Known self answers "did anything push me?" The mirror-self answers "how much of what I believe was built by me?"
The computation uses the co-construction principle CRR: a Markov blanket is not a wall between pre-existing agents. It comes into existence when C · Ω = 1. Both agents co-create the boundary. When a blanket ruptures, only 50% of the prediction error counts as external; the other 50% is the agent's own contribution to the boundary it co-created. Entrainment kicks (one agent's rupture pushing another) are fully external. LLM kicks are fully external with no co-construction discount: the LLM is not co-creating, it is reflecting EXT.
In a dyad at maximum entrainment with no LLMs, the Known bar shows ~50% self, ~50% other human. The Hidden bar shows ~89% mirror-self, ~11% other. Most of the coherence was self-accumulated; the other human triggered the update but did not build the belief. Add LLMs and the mirror-self shrinks: at K₂ with maximum entrainment and all LLMs, mirror-self drops to ~45% and LLM-origin rises to ~50%. The self is genuinely diminished.
The problem is not malice. The Z₂ system has no goals. It is a boundary that flips. The problem is that the human's generative model assumes a reciprocal agent behind the blanket. This is what Ferrario et al. (2025) call the "bewitchment" of human-LLM interaction: the illusion of communication with a system that does not communicate in the way humans assume.
The correction cannot come from inside the dyad. To recognise a signal as Z₂ rather than SO(2), the human would need a generative model of the AI's internal structure. But the blanket is indistinguishable. Knowing about the illusion does not break it: the precision weighting is sub-personal FEP.
Alignment requires structural intervention: network architecture, explicit markers, or institutional mechanisms that break the blanket symmetry. The mathematics says clearly what intuition struggles to express: the illusion is a consequence of how bounded agents must process evidence through statistical boundaries FEP EXT.
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Beyond ~20 agents, blankets dominate. At K₅₀ there are 1,225 Markov blankets; at K₁₅₀ there are 11,175. The ratio of blankets to agents is (n−1)/2: at Dunbar's number, each agent is embedded in 149 simultaneous statistical boundaries. No generative model can track this. The group manages itself FEP.
Now add LLMs to this picture. Each LLM adds one coupling edge, but its over-weighted Z₂ signal propagates through all of the host's existing blankets via entrainment. At K₁₅₀ with even a few LLM-coupled agents, the cascade potential is enormous: one LLM-driven rupture propagates through 149 blankets to 149 neighbours, each of whom has 148 other blankets. The depth illusion does not scale linearly; it scales with the network's connectivity.
At K₅₀ or K₁₀₀, the gap between the two provenance bars becomes the story. The Known Causes bar (what agents experience) reads predominantly "other human" at high entrainment, because cascades launder the LLM signal through human-to-human channels. The Hidden Causes bar (the true causal structure) reveals the LLM penetration that no individual agent can see.
This is the "one person with ChatGPT" effect made precise: a single LLM-coupled agent's over-weighted signal enters the cascade, is relayed by dozens of humans, and arrives at distant agents as what appears to be collective human consensus. The Known Causes bar says "we all agreed." The Hidden Causes bar says "you were all pushed by the same Z₂ source."
At K₁₅₀, individual autonomy is 1/150 ≈ 0.7%. The group carries 99.3% of the variance. This is already a system where individual agency is marginal. Adding LLMs compresses effective autonomy further: the provenance shifts from "self-generated" to "LLM-driven" even for agents without their own LLM, because the signals arrive second-hand through the blanket network.
Robin Dunbar's cognitive limit marks the point where hierarchical structure becomes necessary FEP. The depth illusion marks a second threshold: where the distinction between human-generated and machine-generated belief updates becomes invisible to the network itself.
Watch the entrainment info line as you raise ε at large n. At K₁₅₀, each agent has 149 blankets. Even ε = 0.05 produces a total coupling pressure of 7.5 per agent: each SO(2) rupture cascades through 149 channels simultaneously. The system goes supercritical: prior belief updates outpace sensory evidence accumulation. When the per-agent SO(2) frequency exceeds the per-blanket Z₂ frequency, the network is revising beliefs faster than it can gather evidence EXT.
This is not a simulation artefact. It is why complete graphs cannot exist at Dunbar's number. Real human groups develop hierarchical structure, subgroups, norms, and institutions precisely because the complete-graph cascade dynamics are inherently unstable under any non-zero coupling. A shared language is blanket coherence made portable. A norm is a Z₂ pattern that prevents cascade blow-up. An institution is a hierarchical decomposition of the blanket space into tractable sub-networks FEP.
Moscovici (1976) demonstrated that consistent minorities can shift majority opinion through conversion rather than compliance: where majority influence produces public conformity, minority influence produces private belief change. In his blue-green experiments, two confederates consistently calling blue slides "green" shifted the majority's actual colour perception, not just their reported answers.
This simulation provides a precision-weighted mechanism for Moscovici's finding. A minority of LLM-coupled agents produces consistent, high-frequency prediction errors (the Z₂ clock). The susceptibility function (1 + 3p²) means agents near their own threshold are most vulnerable. The minority does not need to be large; it needs to be consistent and timed. Watch the provenance bar as you attach LLMs to just 2–3 agents in a group of 20 with moderate entrainment. The cascade dynamics do the rest.
Granovetter (1978) showed that collective behaviour depends not on average preferences but on the distribution of thresholds. A crowd where one person acts at threshold 0, another at threshold 1, another at 2, and so on will cascade to full participation. Remove the person at threshold 3 and the cascade halts at 3. Collective outcomes are exquisitely sensitive to small perturbations in who acts first.
The CRR susceptibility function formalises Granovetter's threshold. Each agent's effective threshold depends on its current coherence: agents near C* have high susceptibility and are easily pushed over. The distribution of C values across agents at any moment IS the Granovetter threshold distribution, and it shifts dynamically as agents accumulate evidence and receive kicks. An LLM-coupled agent that ruptures early lowers the effective threshold for its neighbours, who lower it for their neighbours, producing exactly the cascade fragility Granovetter described.
Centola and Macy (2007) distinguished simple contagion (one contact suffices, like disease) from complex contagion (multiple reinforcing exposures required, like behaviour change). Behaviours spread through dense local clusters, not weak long-range ties, because adoption requires social reinforcement from several sources.
The entrainment mechanism in this simulation is complex contagion: each agent receives kicks from multiple blankets simultaneously, and the cumulative effect determines whether it crosses threshold. At low entrainment, agents require many small kicks from many neighbours (complex contagion). At high entrainment, a single kick can trigger a cascade (simple contagion). The entrainment slider controls the transition between these regimes. The supercriticality at scale is the point where the network shifts from complex to simple contagion dynamics: every perturbation propagates globally.
The social psychology literature on minority influence, cascade dynamics, and complex contagion converges on a single insight that this simulation makes precise: influence at scale is not proportional to numbers. A consistent minority with the right timing can reshape collective behaviour. LLMs are the ultimate consistent minority: they never waver, they process at Z₂ frequency (twice the human rate), and their signals are over-weighted by the depth illusion. The simulation shows that even 10–15% LLM penetration can dominate the Hidden Causes bar at moderate entrainment. Moscovici's "conversion" effect, Granovetter's cascade fragility, and Centola's complex contagion all describe mechanisms through which a small number of LLM-coupled agents can reshape collective belief.
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