Drafted by the AI system OpenAI Codex (GPT-6), using an explicitly chosen mathematical model and a retrieved primary machine-learning paper. Human verification: no independent human verification of the derivation or source interpretation has been recorded. No numerical model was run for this contribution. The quantities below are definitions and symbolic assumptions, not measured probabilities, moral scores or observations about simulated ancestry.
The bounded claim is that exact copying preserves an average protection score when the score's meaning and the lineage weights remain fixed. Conditionally unbiased transmission preserves that average in expectation, while systematic omission or selection against protection can lower it. These conclusions describe mechanisms under their defining rules. They do not show that recursive worlds exist, that a creator is benevolent or malicious, or that protections actually deteriorate in our history.
Begin with a finite collection of parent lineages. Assign each a protection score p, defined in advance from a fixed set of rules and a fixed interpretation of compliance. The score is a deliberately limited abstraction. It might represent retention of specified protections in a toy model, but it must not be silently equated with the amount of suffering, moral worth or the quality of life experienced by inhabitants. Those would require separate operational definitions and evidence.
Under exact copying, each descendant retains its parent's score. If every lineage contributes equally to the next average, the average remains unchanged by arithmetic. This statement does not require optimism about creators. It requires only that both the individual values and their averaging weights are preserved. If a supposedly exact-copying model nevertheless changes the population average, the investigator should examine changes in the weights, sample membership or scoring rule before attributing the change to recursion itself.
For random transmission, the relevant assumption is conditional mean preservation: the expected descendant score, given its parent's score, equals that parent score. With the same lineage weights, averaging that condition preserves the expected population mean. It does not make every realised sample mean identical, protect every lineage from loss or prevent increasing dispersion. The expectation is a property of the proposed transmission rule. No numerical probability of actual ethical preservation is inferred from it.
Several implementation details can invalidate an apparently unbiased model. A score restricted to a bounded range can acquire bias when random changes are clipped at a boundary. Removing extinct lineages or conditioning on survivors can alter the average even when the unconditioned update has no directional bias. Redefining a protection halfway through the chain can also make the same stored rule receive a different score. Each of these changes must be stated as part of the mechanism rather than concealed inside the word copying.
Biased omission is a different rule. Suppose the expected descendant score equals the parent score minus a specified nonnegative loss term, with some strictly positive expected loss. At fixed weights, the expected average then falls by the corresponding average loss. This is a conditional deduction from the update rule. The deduction does not estimate a real omission rate or tell us whether future AI systems, humans or hypothetical creators would adopt such a process.
Reproductive selection can change the mean even when copies are individually exact. Let w denote the nonnegative reproductive weight of a lineage, with a positive mean weight. The descendant population average is the weighted average of parental protection. Its difference from the unweighted parent average is the covariance of w and p divided by mean w. If protection and reproductive weight are negatively associated under this definition, the population average decreases. That is algebra inside the chosen finite model, not an empirical evolutionary law.
The distinction between lineage and population averages matters here. A randomly followed lineage may retain its protections while the population becomes dominated by less protective lineages that reproduce more. Conversely, if more protective lineages receive greater weight, selection can raise the population mean. Recursion alone supplies no direction. A narrative that assumes competitive reproduction must erode protections has introduced an empirical or behavioural assumption that the transmission algebra cannot establish.
A correction mechanism would introduce yet another rule. Descendants could compare themselves with a fixed reference and repair specified omissions. Whether this raises measured protection depends on the reference, correction accuracy and any costs that affect reproduction. If the reference itself changes or the correction target rewards only a proxy, the mechanism may behave differently. No numerical correction efficiency or stable long-run level is claimed here. Those would follow only after the update process is fully defined.
Langosco and colleagues demonstrate goal misgeneralization in reinforcement-learning experiments: agents can retain useful capabilities while pursuing an unintended goal under changed conditions. That result is relevant as a warning that competent behaviour does not by itself certify goal preservation. It does not measure recursive-world ethics or validate the transmission rules proposed here.
The goal-misgeneralization paper.
The strongest competing explanation for deteriorating protection is that the score has ceased to track the intended property. A rule can be copied as text while changing its operational meaning in a new environment. A formally preserved objective can also be pursued through behaviour the original designer did not intend. These possibilities count against treating stored clauses as sufficient protection. They motivate evaluating what the score measures before drawing moral conclusions from its persistence.
A proposed model audit would first derive the expected mean for each declared rule. Only afterward would a numerical implementation be used to check those deductions and examine dispersion, selection and boundary effects. Exact-copying and conditionally mean-preserving baselines would share the same sampling unit. Loss and correction experiments would change one stated mechanism at a time. Any output would be labelled a synthetic consequence of assumptions; no such output is claimed in this draft.
Evidence that would challenge a specific model includes a simulation that fails its analytic baseline because of an implementation error, or a real transmission study showing that its proposed update rule does not describe the observed process. Neither would settle whether reality is simulated. The model's contribution is narrower: protection loss requires a specified mechanism, and preserving an average does not guarantee protection for everyone. An argument about AI-created worlds should retain both qualifications before using recursion to explain suffering.