Method / Results / Boundaries · v0.1.0

Ten witnesses.
How many observations?

Evidence status: executed, exact finite synthetic enumeration. No external language-model agents, learned policies, or human participants. Same-author verification; independent review has not occurred.

Question and intervention

At a fixed ten-report display budget, how do copied observations and a shared upstream fault change confidence and decisions? We hold each root witness’s marginal accuracy fixed while changing the source of its errors. This is not an equal-compute or equal-data-acquisition comparison: acquiring ten observations can cost more than copying one.

Supplied world

A lot is fresh (T=1) or stale (T=0), with equal prior probability. A shared binary fault C flips every root’s reading with probability q. Each root i also has its own independent error Eᵢ with probability e. All latent draws are mutually independent before these signals are constructed.

Sᵢ = T XOR C XOR Eᵢ
r = q + (1 − 2q)e
e = (r − q) / (1 − 2q)

We test marginal errors r ∈ {0.1, 0.2, 0.3}, and shared-fault probabilities q = αr for α ∈ {0, 0.25, 0.5, 0.75}. The 80%-accurate condition (r=0.2) is primary; 90% and 70% are stress panels. Adjusting e keeps every root’s marginal accuracy at 1−r. The shared-error allocation α is a model parameter, not a measured correlation coefficient.

Six layouts produce ten displayed reports: [10], [8,1,1], [5,5], [4,3,2,1], [2,2,2,2,2], and ten distinct sources. Copies are noiseless duplicates. Root IDs are truthful and supplied; forged provenance is outside this version.

Three rules, same observations

Headcount treats ten displayed reports as independent sensors of known marginal accuracy. Receipts counts each root once but still assumes roots are independent. Shared cause uses the exact dependence model supplied by the generator. The third rule has privileged structural information; it is not an agent that inferred the fault from agreement.

For s positive observations among n independent sensors with error x, define:

Fₓ(s,n) = [(1−x)ˢ xⁿ⁻ˢ] / [(1−x)ˢ xⁿ⁻ˢ + xˢ(1−x)ⁿ⁻ˢ]
Headcount: Fᵣ(report positives, 10)
Receipts: Fᵣ(root positives, root count)
Shared cause: q + (1−2q) Fₑ(root positives, root count)

Every rule buys at P(fresh) ≥ 0.9, rejects at P(fresh) ≤ 0.1, and otherwise holds. Correct decisive actions earn +1 toy token; wrong ones −9; holding 0. The threshold follows those supplied costs, with decisive actions chosen at equality. An all-hold rule has zero coverage, not perfect accuracy.

What was actually run?

72 parameter/layout cells; all 45,612 positive-probability latent histories; three rules per history, or 136,836 rule evaluations. These are not independent agents or equally likely trials. Every history receives its exact rational probability mass. q=0 histories with an impossible shared fault are excluded rather than padded into the count.

Outcomes are probability-weighted Brier score, decisive coverage, wrong-decision probability mass, accuracy conditional on deciding, and expected toy payoff. Since the declared finite distribution is exhausted, sampling confidence intervals would be inappropriate. This exactness says nothing about how well the model represents real agents.

Results, including the unfavorable controls

Copies inflate certainty. With one 80%-accurate observation copied ten times, headcount claims greater than 99.9999% confidence, decides every time, and is wrong 20% of the time. Receipt counting correctly remains at 80% confidence and holds; its zero errors come with zero decisions.

Independent evidence works. Ten independent 80%-accurate roots yield 99.3458% accuracy among decisions, with 97.3576% coverage. All three rules agree in this control. More agents or agreement are not automatically harmful.

Lineage is not full independence. With ten distinct roots at the same 80% marginal accuracy but a 15% shared-fault probability, headcount and receipt counting yield 84.9985% decisive accuracy with 99.9677% coverage. Their expected toy payoff is approximately −0.5000 per case. The model-aware rule cannot reach 90% confidence and holds throughout; it has not completed the classification task.

The model-aware reference has no worse Brier score and no lower expected toy payoff than the misspecified rules in all 72 cells. This is a consistency check for the specified Bayesian model, not evidence of universal superiority in an unknown real-world environment.

Verification and preserved evidence

The protocol was written and SHA-256 hashed locally before implementing and running the sweep. This is a local freeze, not independent preregistration. Raw histories, exact weights, predictions, actions, and aggregate metrics are retained. A second calculation groups joint history mass by observed roots to reconstruct the posterior, rather than using the closed-form expression.

Verification checks all 72 probability masses, unchanged root accuracy, every recorded posterior and metric, complement symmetry, copy invariance, the no-shared-fault controls, and the shared-fault confidence bound. Unit checks and actual source mutations are included. JavaScript and Python are compared over the full recorded support. Consult the final verification receipt for exact executed counts and browser checks.

Run it yourself

python -m unittest -v test_engine
python verify.py
python mutation_check.py
python reproduce.py --output reproduced

Python 3.10 or newer, standard library only. The browser exhibit runs locally without external dependencies. Optional browser tests use Playwright and Chromium; setup is documented in the README. A complete Python reproduction deterministically regenerates results.json, results-data.js and histories.csv.gz.

Play rounds are selected illustrations

The five market inspections are authored witnesses: copied consensus, a loud duplicated minority, a shared-fault failure, genuinely independent corroboration, and an incorrect dissenter. They are not randomly drawn, and their frequency says nothing about prevalence. The interface separates the model-supported expected value of a choice from the realized outcome. Truth is in the downloaded source; this is a learning tool, not a secure hidden benchmark.

Frontier context, not borrowed validation

Du et al., Improving Factuality and Reasoning in Language Models through Multiagent Debate (May 23, 2023) report benefits from debate. Batra et al., Who Convinced Whom? (October 18, 2026) examine peer influence, agreement, and shared errors. Those are external research contexts; we did not replicate their model runs, and neither validates this toy model.

What this does not establish

No claim about actual frontier-model accuracy, private reasoning, spontaneous coordination, motive, or adoption follows from these constructed signals. Receipts need not be complete or trustworthy in real systems. Errors need not be binary, symmetric, stationary, or governed by one known shared cause. The zero cost of delay is a simplifying choice. The experiment does not test whether a model can discover dependencies, price new evidence, or choose an independent tool. Those are separate empirical questions.

No third-party analytics, cookies, browser storage, external model calls, or hidden collection are added. Site-host traffic logging is separate from this exhibit’s code. No public license has been selected for the lab’s collected exhibits.