I plainly agree with Vincent Fleury that animal embryos are material objects and their development is according laws of condensed physics. But I wouldn’t call them “things” but rather objects and I would extend these considerations to all living organisms, not just animals, and at every phase of their lives, not just development.
He offers to describe an animal Ai as follows:
which grabs the spatio-temporal variation of genes expression specifying Ai. The morphogenetic field1 contains certainly all the physical parameter specified by the genome but not only. The developing organism isn’t a closed system, it continuously interacts with it’s environment and variations of physical (e.g. temperature, hygrometry) or chemical (e.g. O2, ethyl-alcool) factors influence it. Thus this equation needs the addition of at least one term describing them2. This will add in the complexity which seems already being overwhelming for Fleury.
I do understand that in simple situations it is of good advice to adopt a phenomenological approach, trying to “encapsulate this relationship into a constitutive equation for the said material“. That could be possible if the physical properties of the material was simple, which is not the case of the developing embryos; as many “physical phenotypes”3 as expression patterns arise, during a process chiefly known to be based on continuous differentiation of the tissues and phenotypic changes of the cells.
The attempt would be at least as much complex as the description of the molecular phenotypes observed within the same object, adding the complexity of the measures necessary to establish the “physical phenotype” and not giving access to data allowing to predict future phenotypic changes.
Following this preamble, the author describes the “genetic approach” as completely different, and presumably more complex and irreducible to a simple phenomenological description, failing to acknowledge that for the definition of the “physical phenotypes” necessary to feed his morphogenetic field not only genetic information but also gene expression data are necessary.
Then he describes the “paleontological approach”, rightly stating that except rare cases it can’t use genetic data. This is also true for embryology elements as the fossil record is relatively sparse and the fossilized embryos quite rare.
Before going on with the rest of the Introduction, a remark. One of the most relevant disciples for the paper‘s subject is simply omitted: evo-devo. Which is the contraction of evolution-development, the second component encompassing embryology. In an attempt to clarify embryogenesis (of tetrapods or other) with extensive discussion on evolution, not mentioning the discipline in charge of these studies is at least weird. Not even embryology is mentioned!
The word “embryology” appears just twice in the text, in a footnote and concerning embryology textbooks, which Fleury don’t seem to have consult thoroughly4.
D’Arcy Thompson’s allometric transformations are introduced to support the following idea:
[T]here is a possibility that animal formation may be understood on the basis of physical principles in the following way: genetic parameters, and their inductive relationships set a number of mechanical properties, such as material constants, instability parameters and volume forces, then the integral during development in each individual of the deformation rate tensor gives the animal shape at birth.
which is a reedition of the morphogenetic field and can certainly be used when comparing a restricted set of species presenting the same morphological traits which then can be morphed. This comes to define a class of species with a common set of attributes corresponding to the morphological features. You can call that an archetype if you want, in the sense of the ancient progenitor of the class, build on the pattern considered. The set being a construction on the basis of common morphological traits one should avoid to consider the method powerful.
Charles Darwin is cited5, badly, in support to this idea. The bad and fuzzy quotation wouldn’t be quite important if the author wasn’t complaining that “Out of context, the original sentence is quite ambiguous2.” And he adds to the ambiguity.
However, it is an extraordinary fact, unexplained by Darwin, that the limb of a mouse can extend directly, in a straightforward manner, by modifying a single gene promoter. When Darwin states that limbs extend by “conserving the scaﬀold”, he means that all the other erroneous plans that may have emerged were put oﬀ by selection.
Is it really an extraordinary fact that an organ developing following a proximo-distal axis? Not.
Is Darwin meaning that “that all the other erroneous plans that may have emerged were put oﬀ by selection.“? Not.
One must have a very poor understanding of the theory of evolution to reach Fleury’s exegesis. Before trying to explain what Darwin means it’s preferable to read what Darwin wrote. Let’s add context to the citation of the footnote 2 of p3.:
Nothing can be more hopeless than to attempt to explain this similarity of pattern in members of the same class, by utility or by the doctrine of final causes . The hopelessness of the attempt has been expressly admitted by Owen in his most interesting work on the ‘Nature of Limbs.’ On the ordinary view of the independent creation of each being, we can only say that so it is;—that it has so pleased the Creator to construct each animal and plant. 
The explanation is manifest on the theory of the natural selection of successive slight modifications,—each modification being profitable in some way to the modified form, but often affecting by correlation of growth other parts of the organisation . In changes of this nature, there will be little or no tendency to modify the original pattern, or to transpose parts. The bones of a limb might be shortened and widened to any extent, and become gradually enveloped in thick membrane, so as to serve as a fin; or a webbed foot might have all its bones, or certain bones, lengthened to any extent, and the membrane connecting them increased to any extent, so as to serve as a wing: yet in all this great amount of modification there will be no tendency to alter the framework of bones or the relative connexion of the several parts. If we suppose that the ancient progenitor, the archetype as it may be called, of all mammals, had its limbs constructed on the existing general pattern, for whatever purpose they served, we can at once perceive the plain signification of the homologous construction of the limbs throughout the whole class. So with the mouths of insects, we have only to suppose that their common progenitor had an upper lip, mandibles, and two pair of maxillæ, these parts being perhaps very simple in form; and then natural selection will account for the infinite diversity in structure and function of the mouths of insects. Nevertheless, it is conceivable that the general pattern of an organ might become so much obscured as to be finally lost, by the atrophy and ultimately by the complete abortion of certain parts, by the soldering together of other parts, and by the doubling or multiplication of others,—variations which we know to be within the limits of possibility. In the paddles of the extinct gigantic sea-lizards, and in the mouths of certain suctorial crustaceans, the general pattern seems to have been thus to a certain extent obscured.
Darwin address teleology  and creationism (the special creation flavor)  vs the theory of natural selection. And explain the persistence of particular patterns by common ancestry, the ancient progenitor qualified as the archetype of the class.
We are far from the presentation Vincent Fleury made. In general, Vincent Fleury is far from understanding that natural selection do not necessarily make disappear alternative phenotypes, which explain the variety of animal appendages today, or the fact that we dispose of a rather complete collection of transitional forms from fins to limbs.
The remark concerning the correlations between the dimensions of homologous bones in different species is also interesting. Let’s quote again the relevant part from the above quote of Darwin:
each modification being profitable in some way to the modified form, but often affecting by correlation of growth other parts of the organisation
Difficult to be more clear in acknowledging correlations of growth between different parts. Context matters.
Fleury attribute such correlations to morphogenetic constrains, meaning probably mechanical constrains (it would be nice of him to correct me if I’m wrong on that). Mechanical constrains are one class of the many natural selection factors and I really don’t see any reason to singularize them.
After reading the extended citation of Charles Darwin (relevant part underlined), one will find difficult to accept that “It comes as a new feature of animal morphogenesis, that the animal parts may be physical entities which may be elongated or shortened, by direct modiﬁcation of genes, without changing at all the pattern, in its dimensionless form“, with the exception of genes modification, genes being unknown by Darwin.
And the idea that new parts may emerge is older than the Origin.
Fleury’s hypothesis, is in the direct line of Biological Structuralism, a rather old approach of biological classification recently revived, rooting at Plato’s typology and much praised by theologists seeking alternative scientific paradigms.
It is obvious that animals are natural entities developing according to natural laws. It is probably obvious that animals do not have the same morphological characteristics (If it isn’t obvious for you, compare this to this, two animals). Limited classes, chosen for short range evolutionary history since their last common ancestor will tends to have more homologous morphologic characters and developmental processes. By considering tetrapods (a single taxon), one should expect that they share a lot, including parts of the embryogenetic process.
Vincent Fleury announce that he will try to defend the hypothesis that biological structuralism6 accounts for animal (and in particular tetrapods) morphogenesis, as it accounts for morphological evolutionary convergence.
And defend the existence of morphological archetypes, a term used by Darwin certainly not in the same sense as Fleury’s, but to design the “ancient progenitor”, what we call today the last common ancestor, going back to the history of life to the LUCA.
So, after the quote-mining of the “Origin“, the false display of the sense of “archetype” by Darwin, the denial of experimental data (see Abstract) and a non displayed connection to Biological Structuralism, the reader may expect a rather nasty exposé trying to look compatible with the theory of evolution.
[edited 09.04.26 I did forgotten the copy/paste of the last part of my notes, I’m sorry]
So, his personal point of view is a mathematical description of embryos movements in terms of a hyperbolic flow. And he heavily insist about the existence of a hyperbolic point, one of the necessary elements for his interpretation. I’ll comment on that more extensively at “5.4 Signiﬁcance of the hyperbolic point”.
About the concluding remarks of the introduction:
- His understanding of how Darwin describe the archetype of a class of animals is wrong, and so are his attempts to connect it to the sense he is using. The concerns of Fleury seem (at least by the title) limited to a particular taxon, tetrapods, which blind him about general morphological adaptations, which was the concern of Darwin. Taking in account their fundamentally different use of the term “archetype” that will be difficult to consider Fleury’s proposition as en extension of Darwin’s intuitions.
Let’s assume that7:
The transformation that forms an archetypic ﬁsh, from a round blastula is a vortex ﬂow, with a hyperbolic singularity, as has been demonstrated recently.
What are the determinants, not of the values of the parameters, of the terms of the describing equation?
My option is: genetic, inherited by the last common ancestor presenting one could describe using the said equation.
1. unwisely, the term being already used in embryology, thus adding confusion at the very first paragraph of his clarification.
2. I will not try myself to that.
3. I’m using the term in the sense presented by Damon, B.J., et al. at Limb bud and ﬂank mesoderm have distinct “physical phenotypes” that may contribute to limb budding, Dev. Biol. (2008 ), doi:10.1016/j.ydbio.2008.06.018
6. without citing the term, which isn’t fair to those who came before him
7. That was my initial approach, but as some elements are still invisible and thus can’t be reality checked, I’m waiting for a clear display.