Equation 27 · A Chatbot Confessed to Being Built by a Company That Never Trained It
What does this equation mean?
Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.
This equation states a bound: one expression must stay on the indicated side of the other under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.
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Symbol mu_c
m is part of the quantity the equation computes from the expression on the right.
Symbol M_c^+
occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.
Symbol m
m occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.
Symbol d
d is one of the signed contributions combined to compute the quantity on the left.
Symbol ρ
ρ is one of the signed contributions combined to compute the quantity on the left.
Symbol delta
delta is one of the signed contributions combined to compute the quantity on the left.
=
The expressions on both sides represent the same quantity under the stated assumptions.
See an illustrated explanation →subscript
The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.
Denominator: |M_c^+|
The complete quantity below the fraction bar; it must be nonzero for this division.
Starting index or lower bound: minM_c^+
This label says where the repeated addition, multiplication, or accumulation starts. Read its value or condition together with the article’s description of the index.
How to interpret it
With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.
What the article says around this equation
A transfer rate above zero says a channel is associated with a trait. It does not yet say whether the trait is a faithful copy or a garbled one, and it says nothing about whether curators along the way made the trait more or less likely to survive into the next training corpus. Two further quantities complete the construction. The mutation rate =[d(,)>] measures, among the trait-positive models in class c (the set ), what fraction produce a response diverging from the seeded canonical form by more than a fixed threshold under a declared distance function d : a high says the…
Read the full surrounding passage
A transfer rate above zero says a channel is associated with a trait. It does not yet say whether the trait is a faithful copy or a garbled one, and it says nothing about whether curators along the way made the trait more or less likely to survive into the next training corpus. Two further quantities complete the construction. The mutation rate =[d(,)>] measures, among the trait-positive models in class c (the set ), what fraction produce a response diverging from the seeded canonical form by more than a fixed threshold under a declared distance function d : a high says the trait is drifting as it moves, not being copied verbatim. The selection coefficient s= measures how a curation step — a filter, a quality score, a deduplication pass — changes the trait’s prevalence between the raw corpus the source produced, , and the corpus that actually reaches a downstream trainer, : s>0 means curation enriches for the trait, s<0 means curation purges it, and s=0 means curation is neutral toward it. None of these three quantities requires the exposed and unexposed models to share a base architecture, a training team, or a common ancestor scaffold. That is deliberate: the construction is built to work across a population of otherwise unrelated systems, which is the only regime in which “a model’s behavior outlived the model” is a coherent claim at all rather than a description of one system’s own continuity.
For background, read the article’s source list.
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