Invariant reproduction as a fixed-point phenomenon
Abstract
Classically, biological reproduction is depicted as the building of a new organism from replicated genomic instructions, which then act as a representation of the organism. The corresponding theoretical model is von Neumann’s self-reproducing machine, which relies on a universal constructor that can build any machine (in particular itself) from instructions, shielded from the variable structure of the system. Biological reproduction does not work in this way, because a living organism cannot in fact build any other organism, and because the constructing processes are not isolated from the constructed structure. I propose instead that invariant reproduction is not, in fact, an inherent property of living systems. Rather, invariant reproduction must be understood as a fixed-point phenomenon. Generically, the parent produces a new system with a different structure. As this production process is iterated over generations, the sequence of produced systems may converge to a fixed point, at which point a system produces an identical system. I describe a number of biological observations in ciliates that support the idea that invariant reproduction is indeed not generic but rather a dynamical convergence phenomenon, among other possibilities: structural alterations may result in convergence to the typical form over a few generations, or to an alternative fixed point (a different self-reproducing organism with the same genome), or to a diverging sequence of descendants. In this view, the genome is a transmissible constraint on development, not a representation of the organism. I suggest that this dynamical view offers a more coherent perspective on biological reproduction than von Neumann’s computational framework.
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BibTeX
@misc{brette2026invariant,
title = {{Invariant reproduction as a fixed-point phenomenon}},
author = {Romain Brette},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05491732v4},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Invariant reproduction as a fixed-point phenomenon
AU - Romain Brette
PY - 2026
DA - 2026/09/29
PB - HAL
UR - https://hal.science/hal-05491732v4
AB - Classically, biological reproduction is depicted as the building of a new organism from replicated genomic instructions, which then act as a representation of the organism. The corresponding theoretical model is von Neumann’s self-reproducing machine, which relies on a universal constructor that can build any machine (in particular itself) from instructions, shielded from the variable structure of the system. Biological reproduction does not work in this way, because a living organism cannot in fact build any other organism, and because the constructing processes are not isolated from the constructed structure. I propose instead that invariant reproduction is not, in fact, an inherent property of living systems. Rather, invariant reproduction must be understood as a fixed-point phenomenon. Generically, the parent produces a new system with a different structure. As this production process is iterated over generations, the sequence of produced systems may converge to a fixed point, at which point a system produces an identical system. I describe a number of biological observations in ciliates that support the idea that invariant reproduction is indeed not generic but rather a dynamical convergence phenomenon, among other possibilities: structural alterations may result in convergence to the typical form over a few generations, or to an alternative fixed point (a different self-reproducing organism with the same genome), or to a diverging sequence of descendants. In this view, the genome is a transmissible constraint on development, not a representation of the organism. I suggest that this dynamical view offers a more coherent perspective on biological reproduction than von Neumann’s computational framework.
ER -