The Universal Inbreeding Law: Closure, Diversity Loss, Correlated Error, and Integrity Decay in Self-Organizing Systems

The Universal Inbreeding Law: Closure, Diversity Loss, Correlated Error, and Integrity Decay in Self-Organizing Systems Civilization Physics — Series: Boundary Laws of Self-Organizing Systems This paper proposes a Universal Inbreeding Law for self-organizing systems. Biological inbreeding is treated as the most visible instance of a broader structural principle: when a system generates future states mainly from its own prior outputs while external difference and corrective contact remain insufficient, diversity contracts, deviations become inherited across cycles, and integrity declines. The law is structural rather than substrate-specific. Genes, neural networks, developing minds, languages, and institutions do not obey identical local mechanisms. They nevertheless exhibit a recurrent topology of failure under closure. Rare states disappear, internal distortions become part of the next generative baseline, and the system increasingly validates itself through the same narrowed distribution that produced its outputs. The article develops this argument through several linked mechanisms: A canonical closed-resampling model: The paper represents minimal closed recursion as a finite stochastic process in which each generation is reconstructed from a sample of the current distribution. Without mutation, migration, independent real data, or external correction, the future state is generated entirely from the present internal state. Mathematical diversity contraction: Using the Gini–Simpson diversity index, the model shows that expected diversity contracts by the factor 1−1/n1-1/n1−1/n per generation under finite multinomial resampling. The contraction requires no explicit preference for homogeneity. Each finite generation captures only part of the variation present in the distribution from which it was produced. Rare-state extinction and absorption: A low-frequency state has a disproportionately high probability of disappearing from the next generation. Once its frequency reaches zero, the closed process cannot recreate it internally. Repeated finite resampling therefore moves almost surely toward an absorbing point-mass distribution in which one state remains and the others have disappeared. Inherited deviation and external integrity: Closed resampling remains unbiased relative to the current internal distribution, but it does not remain faithful to an original source or external reality. Sampling deviations become part of the reproductive baseline of future cycles. Relative to a fixed external distribution, expected error increases because the process adds new sampling variance without an independent corrective channel. A shared stochastic core across biology and AI: Wright–Fisher genetic drift and discrete recursive-model collapse instantiate the same finite closed-resampling structure. In finite populations, heterozygosity contracts and alleles reach fixation or extinction. In recursive generative systems, low-probability states disappear, support narrows, variance collapses, and the model moves away from the original data distribution. Broader structural extensions: Serial cloning, early language deprivation, public-language decay, echo chambers, and institutional self-validation are not claimed to obey the same multinomial transition equation. They extend the wider topology of closure, difference loss, inherited distortion, and declining fidelity through mechanisms native to their own domains. The paper retains two compact structural expressions: Quality=Integrity×Diversity Stability=Presence×Integrity Diversity preserves adaptive range, minority signals, and anomaly sensitivity. Integrity preserves reliable relation to an external reality or adaptive landscape. Presence names the continued entry of externally grounded information that the current internal distribution could not generate by itself. Openness therefore requires genuine difference as well as trustworthy input. The law does not treat every form of convergence as failure. Concentration may be functional when a system has reached a correct and stable optimum. Integrity decay begins when contraction occurs while the system still needs to represent a wider distribution, preserve adaptive capacity, track a changing environment, or retain distinctions that its own recursion has eliminated. The practical implication is that sustainable systems require structural reopening. Biology uses migration, outcrossing, and recombination. AI systems require continued real data, source integrity, and human correction. Developing intelligence requires timely exposure to language and social reality. Public language and institutions require dissent, provenance, external evidence, and retained minority signals. The Universal Inbreeding Law therefore identifies two levels of universality: an exact closed-resampling structure shared by finite-population genetics and recursive generative models, and a broader cross-domain morphology in which systems lose adaptive or epistemic integrity when they copy themselves too completely. Difference is not a luxury. It is the condition of correction. Keywords: Universal Inbreeding Law · Closed Recursion · Finite Resampling · Diversity Contraction · Rare-State Extinction · Absorbing States · Wright–Fisher Model · Genetic Drift · Model Collapse · Information Inbreeding · Correlated Error · Integrity Decay · Presence · External Correction · Self-Organizing Systems · Civilization Physics