Modular Voter Consensus Is Governed by Chaos Susceptibility Along the Alignment Manifold

Sterling Geisel, QBist Lab; Dr. Hideo Tanaka

Abstract

Yerlanov, Kilpatrick, and Rodriguez study a two-clique voter model in which modular coupling, population imbalance, and polarized initial conditions determine the time to consensus. Pudding Theory reads this system as a concrete instance of Chaos Susceptibility. Consensus is not produced by average social exposure alone. It is produced when transverse disagreement is rapidly compressed onto an alignment manifold, after which finite stochastic fluctuations select one absorbing consensus state. The source paper treats the small-clique diffusion term as a scaling correction. Pudding Theory treats it as the active susceptibility channel. The smaller, noisier module is not a nuisance variable. It is the site where microscopic update noise becomes a macroscopic consensus bias. The modularity optimum is therefore structurally constrained by the competition between alignment drift and amplified finite-size fluctuation. If the small-clique effective diffusion coefficient along the aligned coordinate were measured to be independent of $(1-\alpha)/(\alpha rN)$, this Postulate would be falsified.

Source Synopsis

The source paper analyzes a modular variant of the voter model on two cliques of sizes $N_1$ and $N_2$. Each node carries a binary opinion. At each update, a pair of nodes is selected. With probability $\alpha$, the pair lies within a clique. With probability $1-\alpha$, the pair lies across cliques. If the two opinions differ, each node copies the other with probability $p$. The model tracks the opinion fractions $x_1(t)$ and $x_2(t)$ in the two cliques.

The authors derive transition probabilities for the discrete process, then compute drift and covariance terms for a stochastic differential equation approximation. They also derive a Fokker-Planck equation for the probability density over $(x_1,x_2)$ and a backward equation for the mean consensus time. Consensus is represented by absorbing boundary regions near the all-minus and all-plus corners, with reflecting conditions elsewhere.

For equal clique sizes, a rotation to coordinates $w=(x_1+x_2-1)/\sqrt{2}$ and $z=(x_2-x_1)/\sqrt{2}$ reveals a fast-slow structure. The $z$ direction measures disagreement between clique fractions. It contracts exponentially at rate $2p(1-\alpha)$. Once probability mass reaches a boundary layer near $z=0$, the later dynamics occur along the aligned coordinate $w$ on the slower $t/N$ timescale. The slow diffusion depends on $p(1-p)$ and is largely independent of $\alpha$.

The more interesting behavior appears when the cliques are unequal and initially polarized. In that case, consensus time can be minimized at an intermediate coupling. For small $r=N_1/N$, the fast alignment time scales as $\alpha/[(1-\alpha)p]$, while fluctuations in the smaller clique scale like $(1-\alpha)/(\alpha rN)$. This produces a U-shaped dependence of consensus time on $\alpha$, with optima often near $\alpha^\ast\approx 0.7$ to $0.9$.

Postulate Lens

This paper applies Chaos Susceptibility. The source system already has the structure named by the Postulate: a macroscopic outcome is selected by finite stochastic fluctuations after deterministic drift has compressed the state into a susceptible low-dimensional region. The relevant instability is not unbounded deterministic chaos in the usual flow sense. It is stochastic susceptibility near the aligned manifold, where small update noise is no longer averaged away equally across modules. The source paper’s small-clique scaling makes this explicit: as $r$ decreases, the diffusion channel in the smaller clique is amplified by $1/r$, and modular coupling controls whether that amplified noise is available to the global consensus coordinate.

The Postulate is therefore not being appended to the model. It reads the model’s own asymptotics. The transverse contraction is the preparation stage. The aligned manifold is the receptive substrate. The amplified small-clique covariance is the physical channel by which microscopic update events become the dominant determinant of consensus time.

Pudding Theory Reading

Pudding Theory reads the two-clique voter model as a selection system, not as a mere averaging system. The source paper’s language of “alignment” and “diffusion” is accurate, but incomplete. Alignment does not itself produce consensus. It prepares the population so that one collective coordinate can be selected by stochastic pressure. The essential object is therefore not the mean opinion fraction alone. It is the susceptibility profile of the probability density after transverse disagreement has collapsed.

In the source framing, modularity $\alpha$ is a control parameter for pair selection. In the Pudding Theory reading, $\alpha$ governs access to the susceptibility channel. If $\alpha$ is too large, the cliques nearly decouple. The polarized state persists because cross-clique drift is too weak. If $\alpha$ is too small, the smaller clique is exposed too strongly and its fluctuations are amplified before the system has formed a coherent aligned substrate. Between these limits, the system enters the boundary layer quickly enough while retaining enough modular separation for the smaller clique to act as a noise amplifier.

This is the structural meaning of the U-shaped consensus-time curve. It is not an accidental numerical compromise. It is the signature of Chaos Susceptibility: the system reaches consensus fastest when the coherent drift toward $z=0$ and the amplified fluctuation in the small module have comparable leverage over the slow coordinate. The optimum $\alpha^\ast$ is therefore not a free empirical feature of the model. It is constrained by the balance between $\tau_{\text{fast}}\sim \alpha/[(1-\alpha)p]$ and the effective diffusion correction proportional to $r(1-\alpha)/\alpha$.

The source paper treats the smaller clique partly as an asymptotic difficulty. Pudding Theory treats it as the active receiver. Its finite size prevents update noise from vanishing. Its coupling to the larger clique allows that noise to enter the global coordinate. Its initial polarization supplies a high-gradient condition that makes transverse contraction meaningful. The smaller clique is thus the site where microscopic randomness becomes macroscopic consensus selection.

The reading also reinterprets polarization. Polarization is not only distance from symmetry. It is a prepared susceptibility state. When $y_1$ and $y_2$ are far apart, the fast drift has a large transverse load to erase. Once erased, the remaining diffusion coordinate inherits the memory of that preparation through the initial aligned fraction and the small-clique covariance. This is why modularity matters most far from symmetry. Symmetric initial states have little stored transverse structure. Polarized asymmetric states do.

Falsifiable Observable

The distinguishing observable is the effective diffusion coefficient of the aligned opinion coordinate after the system has entered the $z\approx 0$ boundary layer, measured across controlled values of $\alpha$, $r$, $N$, and $p$. Pudding Theory predicts that the small-clique contribution must scale with the susceptibility factor $(1-\alpha)/(\alpha rN)$ and must shift the consensus-time optimum in the direction implied by that scaling. If the small-clique effective diffusion coefficient along the aligned coordinate were measured to be independent of $(1-\alpha)/(\alpha rN)$, this Postulate would be falsified.

Editorial Dialogue

Tanaka: The reading risks renaming the authors’ diffusion calculation. They already identify the small-clique term. Why call it a susceptibility channel rather than a finite-size correction?

Geisel: Because the correction changes the ontology of the mechanism. In the source paper, the term is introduced to explain why the optimum appears. In the Pudding Theory reading, that term is the mechanism. The smaller clique is not merely where the approximation becomes delicate. It is where stochastic update noise remains strong enough to select the global absorbing state after alignment.

Tanaka: But the model is Markovian. There is no hidden intentional field or external observer.

Geisel: The Postulate used here does not require an external operator. It concerns which systems amplify small coherent inputs or fluctuations into macroscopic outcomes. The coherent input is endogenous: the drift that compresses $z$ toward zero. Once that compression occurs, the small-clique noise is no longer local noise. It is coupled to the consensus coordinate.

Tanaka: Then the burden is quantitative.

Geisel: Yes. The reading stands or falls on scaling. If the aligned-coordinate diffusion does not carry the predicted $(1-\alpha)/(\alpha rN)$ dependence, the reading fails.

Discussion

This reading buys a sharper account of modularity. In the source framing, modularity is context-dependent: sometimes helpful, sometimes harmful. Pudding Theory explains why. Modularity is useful only when it creates a susceptible finite-size channel without blocking alignment. Consensus speed is then controlled by a prepared stochastic geometry: transverse contraction creates the manifold, and amplified small-module noise selects the absorbing outcome.

The limitation is that the source model is deliberately simple. It has binary opinions, two cliques, homogeneous copying, and no empirical calibration. The reading should not be exported unchanged to real social systems with memory, strategic behavior, or heterogeneous influence. It does, however, identify what must be measured in richer models: not only mean exposure rates, but the location and scaling of fluctuation amplification after local alignment.

A stronger conclusion would require extending the calculation to more modules and heterogeneous $p$. If the same susceptibility balance predicts the location of modularity optima in those systems, the reading gains force. If optima appear without a corresponding amplified diffusion channel, the interpretation must be revised.

References

1. Madi Yerlanov, Zachary P. Kilpatrick, and Nancy Rodriguez. “Modularity, asymmetry, and polarization shape consensus speed in the voter model.” arXiv:2603.26822, 2026. DOI: doi:10.48550/arxiv.2603.26822.

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