CoCre-Sam (Kokkuri-san): Modeling Ouija Board as Collective Langevin Dynamics Sampling from Fused Language Models

Haruumi Omoto; Masatoshi Nagano; Tadahiro Taniguchi; Yoshiki Hayashi

REVIEW 3 major objections 6 minor 20 references

CoCre-Sam (Kokkuri-san): Modeling Ouija Board as Collective Langevin Dynamics Sampling from Fused Language Models

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The planchette's collective motion is provably equivalent to Langevin Monte Carlo sampling from the sum of the participants' language-model energy landscapes, so the emitted characters form a tempered product of experts.

desk verdict A novel interpretation of the Ouija board as Langevin sampling, with a genuinely suggestive model but over-stated convergence guarantees and a real gap between the analyzed equations and the implemented algorithm. read the letter →

arxiv 2507.11906 v1 pith:FANTO4OF submitted 2025-07-16 cs.MA cs.HC

Tadahiro Taniguchi , Masatoshi Nagano , Haruumi Omoto , Yoshiki Hayashi This is my paper

classification cs.MAcs.HC

keywords collectiveLangevindynamicsOuijaboardKokkuri-sanlanguagemodelfusionproductofexpertsenergy-basedmodelsMonteCarloemergentintelligence

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proposes a computational account of why Ouija boards and Kokkuri-san sessions sometimes emit coherent words that no single participant intended. In the model, each participant is an agent that pushes a shared pointer in the direction that reduces an energy function derived from the agent's internal language model, plus independent Gaussian noise. Summed across agents, these pushes are precisely the update step of the Unadjusted Langevin Algorithm on the fused energy equal to the sum of the agents' energies. The paper proves that the pointer's long-run position distribution is therefore a Gibbs distribution over that fused energy, which takes the form of a tempered product of experts of the individual language models. This turns a culturally mysterious phenomenon into an embodied sampling process.

What carries the argument

The load-bearing object is the fused energy landscape and the equivalence between the collective update and the Unadjusted Langevin Algorithm. Each agent's energy $E_i(x|c_{1:\tau}) = \sum_c P_i(c|c_{1:\tau}) E_i(x;g_c)$ maps next-character probabilities onto board positions, and the collective update $x(t+1) = x(t) - \eta \nabla_x E_{\mathrm{fused}}(x) + \sqrt{2D_{\mathrm{fused}}}\,\xi(t)$ is exactly a ULA step because independent Gaussian noises sum to a Gaussian with variance $2D_{\mathrm{fused}}$. This identity turns a decentralized sum of micro-forces into a centralized MCMC sampler, with the effective temperature $T_{\mathrm{fused}} = D_{\mathrm{fused}}/\eta$ controlling how much the group explores.

What would settle it

In a controlled repeated Ouija session with two participants whose next-letter probabilities are independently measured, record the planchette trajectory distribution; if the stationary density is not proportional to $\exp(-(E_1+E_2)/T)$ for a single temperature, the claimed ULA equivalence does not describe the actual phenomenon.

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Extended reading notes

Core claim

The central claim is that the collective planchette dynamics $x(t+1) = x(t) + \sum_i a_i(t)$ with $a_i(t) = -\eta \nabla_x E_i(x(t)) + \sqrt{2D_i}\xi_i(t)$ is an instance of the Unadjusted Langevin Algorithm targeting $E_{\mathrm{fused}} = \sum_i E_i$. Theorems 4.5 and 4.6 state that this dynamics converges to $P^{\mathrm{stat}}_{\mathrm{fused}}(x) \propto \exp(-E_{\mathrm{fused}}(x)/T_{\mathrm{fused}})$ with $T_{\mathrm{fused}} = D_{\mathrm{fused}}/\eta$, so the emitted characters approximate a tempered product of experts $\prod_i \{ \bar{P}_i \}^{T_i/T_{\mathrm{fused}}}$ of the agents' language models. Simulations with two n-gram agents biased toward different flower names generate words such as 'quaker' and 'jacobinia' that sit between the individual preferences, and perplexity scores confirm that the output is coherent under the fused model. The paper uses this result to argue that the Ouija board can be understood as a physical sampler of a collective prior.

Load-bearing premise

The load-bearing premise is Eq. (7): that each real participant's micro-movement is exactly a gradient step on the energy landscape of their own internal language model plus independent Gaussian noise, a premise the paper does not test with human data.

Editorial extensions

If this is right

If the claim is correct, any system that sums noisy gradient forces from several language-model agents samples from the product of their distributions, giving an embodied and decentralized route to language-model fusion.
Zero-noise dynamics reduce to gradient descent and collapse the output to a single sequence, so stochasticity is not a defect but a required component of the emergent sampling.
The character-level output inherits the same product-of-experts structure, yielding a quantitative prediction: the group's letter odds are the product of single-agent letter probabilities raised to temperature ratios.
The effective temperature $T_{\mathrm{fused}}$ directly controls the reliability-diversity trade-off, so a group's motor noise changes what the board 'says'.
The framework extends the ideomotor account of the Ouija board into a generative statistical model, connecting individual cognition to collective output.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If human micro-movements are not exactly Langevin, the core identity may still hold approximately, with fitted diffusion coefficients serving as measurable per-participant 'motor temperatures'.
The same sum-of-energies sampler could serve as a decentralized design principle for human-AI teams, where agents with incompatible priors jointly explore a fused decision space without explicit message passing.
The theorem predicts a quantitative signature for real Ouija experiments: position histograms should match $\exp(-(E_1+E_2)/T)$, which would distinguish this mechanism from a purely ideomotor response.
Extended from two agents to many heterogeneous language models, the framework suggests group output will systematically favor tokens that all members find plausible, a form of consensus that may explain the perceived coherence of emergent group speech.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

What should you know? The paper is the first to treat the Ouija board as Langevin sampling from a fused product of language models, and that interpretation is genuinely suggestive. But the formal claims overreach: Theorem 4.6's exact Gibbs stationarity for fixed-step ULA is simply false – fixed-step ULA leaves a discretization bias. And the analysis in Theorem 4.5 is for the averaged-energy update (Eq. 7), not for the goal-resampling process actually implemented in Algorithm 1. For DeltaT=1, each agent re-samples a temporary goal at every inner step; that adds state-dependent non-Gaussian noise. The invariant measure of that chain is not shown to equal the Gibbs measure of the averaged potential. So the central 'correct sampling' claim is unproven for the implemented dynamics.

What is new: the modeling frame, the hierarchical resampling idea (a temporary goal letter per agent), and the product-of-experts reading of collective agency. The simulations are honest, use simple n-gram models, and the code is on Zenodo/GitHub. Ablations on noise intensity show the expected reliability/diversity trade-off. The paper's own discussion flags the lack of human data as future work.

Soft spots, in proportion: the convergence theorem is wrong as stated; the resampling mismatch is a real gap, not a nitpick; the discrete PoE formula (Eq. 13) is asserted without derivation; the premise that micro-movements are Langevin over LM potentials is untested, which the authors acknowledge. None of this kills the conceptual contribution, but it means the current text claims more than it proves. Where the paper says 'we theoretically prove', it should say 'we show an algebraic equivalence for the averaged process, and leave the resampled chain's invariant measure as an open question.'

Who is this for? People working on collective intelligence, shared control, or cultural phenomena as sampling. It's worth a serious referee, and with revisions – correcting the convergence statement, analyzing or simulating the resampled chain, and softening the claims – it could be a solid contribution. I would not desk reject it.

Referee Report

3 major / 6 minor

Summary. The paper proposes CoCre-Sam, a framework that models an Ouija board/Kokkuri-san session as N agents performing Langevin steps on energy landscapes derived from their internal language models, with the shared planchette update equal to the sum of agent actions. It defines an averaged individual energy (Eq. 4) and a fused energy (Eq. 6), proves (Theorems 4.5 and 4.6) that the collective update corresponds to ULA on the fused energy and that the stationary distribution is the corresponding Gibbs/tempered product-of-experts distribution, and supports this with simulations using two n-gram agents generating flower names, perplexity comparisons, and noise ablations. The paper concludes that CoCre-Sam provides a computational mechanism for emergent collective linguistic behavior.

Significance. If the theoretical claims were established as stated, the paper would offer a fresh conceptual link among the ideomotor effect, collective predictive coding, and Langevin MCMC, and the released code would make the simulations reproducible. The PoE interpretation is attractive and the empirical ablations cleanly illustrate the role of stochasticity. However, the current manuscript is better viewed as a proposal of a plausible mechanism than as a verified account of the Ouija phenomenon: the human micro-force premise of Eq. (7) is untested, and the main theorems require restatement as approximate or algorithm-adjusted results. With those fixes, the framework could be a useful starting point for human experiments.

major comments (3)

[Algorithm 1, lines 8 and 11; §3.3, Eq. (7)] The theorem is proved for the average-energy action -η∇E_i(x|c_{1:τ}) + √(2D_i)ξ_i(t), but the implemented algorithm samples a goal c̄_i ∼ P_i(·|c_{1:τ}) at every ΔT step and evaluates E_i(x; c̄_i). For ΔT=1 these updates have the same conditional mean, but the resampled-goal process has a transition kernel that is a mixture of Gaussians rather than a single Gaussian with covariance 2D_i I, so the additional randomness is not absorbed by the √(2D_i)ξ_i(t) term. No argument is given that this mixture update chain has the Gibbs measure of E_fused as its invariant measure. Please either prove the invariance for the resampled-goal chain, change Algorithm 1 to use Eq. (7) directly, or state Theorem 4.5 for the average-energy process only.
[§4.3, Theorem 4.6] The conclusion that the fixed-step ULA chain has the exact stationary distribution P_stat(x) ∝ exp(-E_fused/T_fused) is not correct. With a fixed positive step size η, ULA is a biased discretization; its stationary distribution differs from the target by a bias that vanishes only as η → 0, and the cited results of Dalalyan (2017) and Roberts and Tweedie (1996) do not imply exact invariance for the discretized chain. The theorem and the surrounding text should be restated as an approximate-sampling guarantee with an explicit bias bound, or the analysis should be applied to a Metropolis-adjusted or otherwise exact sampler.
[Remark following Theorem 4.6, Eq. (13)] The step from the continuous stationary distribution P_stat_fused(x) to the discrete character probabilities P̄_fused(c) is asserted but not derived. Algorithm 1 selects a character by nearest-goal voting over the final 5% of the trajectory (lines 15–18), which introduces temporal correlations, burn-in effects, boundary clipping, and a voting aggregation rule; the probability of selecting a character is therefore not simply the integrated mass of P_stat over that character's Voronoi region, and Eq. (13)'s product-of-experts form does not follow. Since the paper's headline outputs are discrete character sequences, this is a load-bearing step; please provide a derivation or state Eq. (13) as an approximate relation with an explicit caveat.

minor comments (6)

[Section 3.1, Eq. (2)] The text says η is a step-size parameter, but η does not appear in Eq. (2); clarify that η enters through the agent actions in Eq. (7).
[Theorem 4.5] The theorem statement gives Teff = ηD_fused, while the proof derives T_fused = D_fused/η; these are inconsistent and the theorem should be corrected.
[Section 4.3] The text contains the typo 'Tf used' in the sentence defining the effective temperature; replace it with 'T_fused'.
[Algorithm 1, lines 15–20] The voting step is described only informally ('vote', 'considering vote results'); the precise rule (final 5% of the inner trajectory, nearest-goal, majority frequency) appears only in Section 5.1 and should be stated in the algorithm.
[References] The SpecFuse entry is dated 2023 but the cited arXiv identifier is 2412.07380, which is from 2024; please correct the year.
[Table 1] In the 'No Noise' row, the frequency 100 is reported alongside probability 0.010; please clarify that the probability column is the model likelihood rather than the empirical frequency, since the empirical frequency is 1.0.

Circularity Check

1 steps flagged · score 6.0 of 10

The fused-sampling theorem is a restatement of the model's definitions: because E_fused is defined as the sum of the same E_i that define each agent's Langevin action, Theorem 4.5 follows by linearity and the product-of-experts conclusion is built in rather than derived.

self definitional [Section 3.2 Eq. (5)-(6); Section 3.3 Eq. (7); Section 4.2 Theorem 4.5; Remark after Theorem 4.6]
"Efused(x | c1:τ ) = X i∈I Ei(x | c1:τ ); ai(t) = −η∇xEi(x(t) | c1:τ ) + p 2Diξi(t); Using the definition Efused = P i Ei, we have ∇xEfused = P i ∇xEi."

The claimed fused target is introduced as the sum of the same individual energy functions that define the agents' actions. Equation (7) makes each action a single ULA step on E_i, and Eq. (5)/(6) defines E_fused as the sum of those E_i. Theorem 4.5's proof does nothing beyond substituting Eq. (7) into Eq. (2) and invoking linearity of gradients and Gaussian summation. Therefore the stationary distribution exp(−E_fused/T_fused), and the tempered product-of-experts form in the Remark, are restatements of the model's definitions rather than consequences obtained from independent constraints or empirical evidence. The 'prediction' that the planchette samples from the fused distribution is forced by construction.

full rationale

The main circularity is a self-definitional step: the fused energy is defined as the sum of the individual energies, and each agent's micro-action is defined as a Langevin step on that same individual energy. Summing these actions yields ULA on the summed energy by elementary algebra, so Theorem 4.5 and the product-of-experts Remark do not add independent content beyond the definitions. This is partial circularity in the paper's claim to 'theoretically prove' that the collective motion samples from a fused distribution. The simulations and ablations are not circular in themselves; they are internal demonstrations that the constructed dynamics generate flower names, but they do not validate the fusion mechanism against external human data. I do not count the Algorithm 1 versus Eq. (7) discrepancy or the exact stationarity claim for fixed-step ULA as circularity; those are separate correctness gaps (the resampled-goal chain is not shown to have the averaged-energy Gibbs invariant measure, and standard ULA with fixed step size has an O(η)-bias). Self-citations such as Taniguchi (2024) and Matsui et al. (2025) appear in discussion of related ideas and are not load-bearing for the main theorem, so they do not raise the score further.

Assumptions & free parameters 5 free parameters · 4 assumptions · 1 invented entities

The ledger shows that the central theoretical result rests on the standard convergence theory of ULA and on definitions that bake in the fusion. The only genuinely ad hoc assumption is that human micro-movements are Langevin steps in an energy landscape derived from an internal language model, which is plausible but untested with human data. The algorithm-theory mismatch around high-level goal resampling adds an unmodeled source of stochasticity, and the character-level voting step is a heuristic that the theory does not cover.

free parameters (5)

potential radius r0 = 0.3
Hand-chosen width for the Cauchy potential; affects the shape of all energy landscapes and thus the selected character distribution.
step size eta = 0.1
Sets the mobility of the planchette and the effective temperature; hand-chosen for numerical stability.
individual noise intensity D_i = 0.01
Sets stochasticity, with T_fused = D_fused/eta = 0.2 for two agents; varied in the ablation.
n-gram order = 6
The internal language models are 6-gram character models built from 100,000 weighted samples per agent; chosen for tractability.
voting window = last 100 of 2000 steps
Only the final 5% of the inner trajectory votes for the next character; this quantization step affects character probabilities and is not part of the theory.

assumptions (4)

standard math Standard ULA convergence theory (Roberts-Tweedie 1996, Dalalyan 2017) applies to the discrete process.
The proof of Theorem 4.6 imports geometric ergodicity and convergence to the Gibbs stationary distribution; this is a background result, but ULA with fixed step size actually has a bias.
domain assumption The individual energy E_i(x|c) is the probability-weighted sum of potentials used directly as the potential in each agent's Langevin step (Eq. 4).
In Algorithm 1 the agent samples a single temporary goal and moves on that potential; the proof replaces this by the averaged energy without modeling the extra resampling variance.
ad hoc to paper Human micro-forces during Ouija sessions follow Langevin dynamics on an LM-derived energy landscape (Eq. 7).
No human experiments or psychophysical evidence support this equivalence; it is the core modelling assumption.
ad hoc to paper The nearest-goal voting over the final 5% of the trajectory yields character probabilities proportional to the integrated mass of the continuous stationary distribution (Remark after Theorem 4.6).
Stated as an approximation without proof; finite-time voting and discrete goals are not in the ULA convergence theorem.

invented entities (1)

Implicitly fused language model / Kokkuri-san (collective creature)
purpose: Explains the perceived external agent that produces the emergent output; in the model it is the product-of-experts distribution over characters.
The fused distribution is defined as the sum of individual energies from the outset, so it has no support outside the model's own construction. The paper offers no human data showing actual Ouija outputs follow the fused distribution.

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Cite this review

Pith. "Pith review of CoCre-Sam (Kokkuri-san): Modeling Ouija Board as Collective Langevin Dynamics Sampling from Fused Language Models." pith.science (2026). https://pith.science/paper/FANTO4OF

@misc{pith2026250711906,
  author       = {Pith},
  title        = {Pith review of: CoCre-Sam (Kokkuri-san): Modeling Ouija Board as Collective Langevin Dynamics Sampling from Fused Language Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FANTO4OF}},
  note         = {Machine review of arXiv:2507.11906}
}

read the original abstract

Collective human activities like using an Ouija board (or Kokkuri-san) often produce emergent, coherent linguistic outputs unintended by any single participant. While psychological explanations such as the ideomotor effect exist, a computational understanding of how decentralized, implicit linguistic knowledge fuses through shared physical interaction remains elusive. We introduce CoCre-Sam (Collective-Creature Sampling), a framework modeling this phenomenon as collective Langevin dynamics sampling from implicitly fused language models. Each participant is represented as an agent associated with an energy landscape derived from an internal language model reflecting linguistic priors, and agents exert stochastic forces based on local energy gradients. We theoretically prove that the collective motion of the shared pointer (planchette) corresponds to Langevin MCMC sampling from the sum of individual energy landscapes, representing fused collective knowledge. Simulations validate that CoCre-Sam dynamics effectively fuse different models and generate meaningful character sequences, while ablation studies confirm the essential roles of collective interaction and stochasticity. Altogether, CoCre-Sam provides a novel computational mechanism linking individual implicit knowledge, embodied collective action, and emergent linguistic phenomena, grounding these complex interactions in the principles of probabilistic sampling.

Figures

Figures reproduced from arXiv: 2507.11906 by the authors.

**Figure 2.** Flowchart of the hierarchical control in CoCre-Sam. The outer loop for character sequence [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Ouija board (left) and simulated player trajectories (right). Remark (Connection to the Product of Experts). Theorem 4.6 exposes an explicit link between CoCre-Sam and the Product of Experts (PoE) framework. Y i∈I {P stat i (x)} Ti/Tfused ∝ Y i∈I {exp (−Ei(x | c1:τ )/Ti)} Ti/Tfused (9) = exp X i∈I −Ei(x | c1:τ )/Tfused! (10) = exp (−Efused(x | c1:τ )/Tfused) (11) ∝ P stat fused(x) (12) where {P stat i } is the i-the… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Step-by-step visualization of the word generation process for the word [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

**Figure 6.** Figure 6: Ablation study on noise intensity (Tfused). The top panel shows the number of valid words generated, and the bottom panel shows the entropy of the word distribution. 6 Discussion The CoCre-Sam framework proposed in this work extends beyond a computational model for th…

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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arXiv 2024