I fear that I must once again disagree with your strong conclusions about the relations between mathematics and genetics. I would urge you to attempt to find exact correspondence relations between empirical evidence and your views of models based on numbers.
I find your post to be outside the scope of the standard theories of biochemistry and genetics.
Chemical information is grounded in the list of chemical elements and the relations among them.
The terms "DNA" and "RNA" etc, are chemical names of specific relationally rich bio-molecules.
The information content of chemical molecules must be expressed in terms of atomic numbers and relations among the electrical particles (graphs).
Biological information emerges as flows of changes of chemical relations - metabolic dynamics.
In general, chemical structures / information does support transitive relations among the atomic numbers organized into graphs.
Thus, if one wishes to develop a compelling argument about chemical numbers and structures and genetic information, one should start with relational algebras that keep track of changes of relations.
Bijective maps are not a suitable basis for describing change of chemical relations and hence the flow on biological information.
Finally, if one wishes to describe a mathematics of biological information, the suitable starting point is the fact that a single position in a DNA sequence can control the fate of the entire organism. A living system is a society of associative relations among atomic numbers.
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