We study Bayesian inference of persistence from early observations of a heterogeneous susceptible--infected--susceptible (SIS) reaction--diffusion model with standard incidence. We show that the leading prevalence dynamics depend on the unknown coefficients only through the difference between the transmission and recovery rates and the infected diffusivity. The same pair determines the principal eigenvalue whose sign governs persistence, so early prevalence can inform the threshold even when the individual rates remain uncertain. We prove global weak well-posedness and quantitative parameter stability for the SIS system and use the resulting observation-map estimates to establish Bayesian well-posedness. For an initial infection of size $\varepsilon$, we derive a uniform second-order expansion that describes how the separate rates enter prevalence. Incidence depends explicitly on transmission at leading order. When the noise standard deviation is proportional to $\varepsilon$, we prove $O(\varepsilon)$ convergence in Hellinger distance of the nonlinear posteriors to their low-prevalence limits for both prevalence alone and joint prevalence and incidence data. We bound the principal eigenvalue using the growth rate of the infected $L^2$ norm between two times. This rate gives a lower bound for the full nonlinear system, and the upper bound separates spatial transients from nonlinear saturation. Uniform spectral estimates and bounds on infection size and measurement error give sufficient conditions for persistence and extinction. We also bound the error in posterior persistence probabilities under forward and eigenvalue approximation, accounting for posterior mass near the threshold. Numerical examples illustrate threshold inference and the predicted \(O(\varepsilon)\) convergence of the nonlinear posteriors to their low-prevalence limits in Hellinger distance.
Evolution can favour a trait for two reasons: an instructive cause makes selection favour the trait; a permissive one lets it persist once selection no longer removes it. The distinction matters most for phenotypic plasticity and its replacement by inherited behaviour, genetic assimilation, the Baldwin effect. Social learning is credited with favouring plasticity on evidence that cannot tell the two apart: how much plasticity remains at the end of a simulation. Here we show, analytically and computationally, which role it plays in a minimal model of learning and evolution. Writing expected fitness as a function of plasticity, we find that environmental change is the only source of benefit from plasticity, and the social channel enters only its cost. Simulation confirms this: in a static environment selection against plasticity never reverses sign at any channel strength; when the environment changes, it reverses at a threshold the channel lowers, and without the channel the population collapses before reaching it. Social learning is permissive, not instructive: it supplies the means to be plastic, never the reason. For the debate over whether plasticity leads evolution, this is one case settled by measurement: environmental change directs, socially inherited information permits, and neither suffices alone.
What structure on the rule space of a cellular-automaton family exposes which rules can support which behaviours? For the 262,144 life-like rules -- the two-dimensional outer-totalistic family containing Conway's Game of Life -- the standard coordinate, Langton's activity parameter, is long known to be too coarse. I organise the family by three symmetry-derived rule-table coordinates: an unweighted activity, a birth-survival asymmetry (odd under the swap of birth and survival), and the Vichniac-Binder Boolean-derivative sensitivity in closed form, whose centre-bit component vanishes exactly on the centre-blind (B=S) submanifold. The cardinality coordinates lambda, delta are necessary but never sufficient to certify behaviour; the centre-bit sensitivity labels the B=S submanifold on which the paper's one exact exclusion holds. Combined with an exhaustive enumeration of every rule's compact gliders, still lifes and oscillators (three-by-three seed, period at most sixteen), they expose a hierarchical factorisation: the birth condition is a narrow gate deciding whether anything can propagate, the survival condition tunes storage, and every compact-glider-supporting rule is birth-light and survival-rich. One exclusion is exact: no B=S rule supports a compact glider at that footprint, proven by exhaustive enumeration and closing only for compact-footprint patterns -- 36 B=S rules ship at four-by-four seed and 40 at five-by-five. Two dynamical results frame the whole: glider-capable rules that random sampling never expresses show that capability is certifiable from the table while its expression is only estimable from dynamics; and every uncatalogued rule producing a richly Life-like ecology turns out to be a Conway or HighLife variant on the neighbour counts its trajectories actually visit.
Generative models trained on the output of earlier models lose knowledge across generations, rare knowledge first. The remedies that work all bring in information from outside the loop, and the theory of model collapse treats that supply as given: a fixed archive, drawn from a fixed distribution. We argue that the supply is the problem, and that cultural evolution has the theory of it. We read model collapse as Rogers' paradox: a population that comes to rely on copying ends up no better off than one that never copied, because copies go stale unless someone keeps learning for themselves. We test the identification in three steps. First, Rogers' model, fitted only to fully synthetic training, predicts to within a few percentage points how much rare knowledge language models, from word trigrams to a pretrained transformer, retain under other mixtures of fresh data. Second, when the world itself changes, a lineage that copies its own past needs more fresh data to keep up the faster the world changes, as Rogers' model says it must and as analyses that assume a fixed target cannot; a lineage that restarts every generation from fixed pretrained weights is largely insulated, because those weights are knowledge that is never copied. Third, we close the loop: when the people who write a model's training data can ask the model instead of finding things out, the supply of fresh data becomes endogenous. At equilibrium, the model is exactly as reliable as it is costly for people to find out for themselves, the population is no better off than without the model, and every increase in the share of model-written text must be paid for by more people learning for themselves. Users who check what they are told and relearn when it is wrong break the paradox. Filters that select plausible text are conformist and accelerate collapse; only checks that carry new information act as critical learning.
Can AI contribute to formal demography? We report an experiment in which an AI model, Claude, was asked to explain and then simplify the proof of an earlier result: that vaccinating the oldest against COVID-19 saves both the most lives and the most years of life, because the years saved per person treated rise with age. The model proposed a new proof that is shorter, more intuitive, and somewhat more general than the original, built from a known life-table identity and the definition of convexity. It states the condition for the result as the convexity of the reciprocal of the hazard, which can be checked from the hazard alone and holds for most familiar parametric models, and it yields a lower bound on life expectancy whenever that condition is met. Prompted by a further question, the model also gave an exact form of the condition, showing how far convexity can be relaxed. The model traced these identities and related results in earlier literature. An edited record of the sessions is provided, and complete transcripts are available on request. The exercise suggests that AI can revisit established results, make them easier to understand, and prove valuable generalizations.