Chapter 40
Behavioral Primes
Chapter 8 gave the definition of an interactive GSLT — a theory with a designated site \(\inter\) at which two terms meet, a base rule that fires there without contextual hypothesis, and context rules saying when the base rule may propagate — and argued that the symmetry or asymmetry of that site is a real ontological distinction rather than a notational one. The argument of this part needs one further thing from that structure, and this short chapter supplies it.
The question is whether an interactive theory has atoms: a supply of terms out of which everything else can be built by interaction alone. If it does, then an agent doing science on such a theory has something to find, and the population arguments of the chapters that follow have a target. If it does not, there is no bottom for the scientific method to converge to, and the whole question of what an agent can and cannot discover changes character.
40.1 Behavioral Primes
A combinator set for an interactive GSLT \(\GSLT\) is a set \(\comb \subseteq \terms(\GSLT)\) such that every term \(P \in \terms(\GSLT)\) is bisimilar to a finite interaction of elements of \(\comb\): \[\forall P \in \terms(\GSLT),\; \exists c_1, \ldots, c_n \in \comb \text{ such that } P \bisim c_1 \inter c_2 \inter \cdots \inter c_n.\]
An element \(c \in \comb\) is a behavioral prime if it is not bisimilar to any interaction \(c' \inter c''\) of smaller terms. A combinator set \(\comb\) is prime if every element is a behavioral prime and no proper subset of \(\comb\) is still a combinator set.
The elements of a prime combinator set \(\comb\) are the computational quarks of \(\GSLT\): irreducible behavioral units whose interactions generate all observable behavior. They are not the bisimulation equivalence classes, which may be equivalence classes of arbitrarily complex composite behavior. The primes are indivisible in the interaction-factorization sense: they cannot be split into independently interacting parts.
Every interactive GSLT with a finitely generated grammar admits a prime combinator set.
Unique factorization — the analogue of the fundamental theorem of arithmetic for behavioral primes — is a stronger condition that need not hold in general. Its failure in a given GSLT would be physically meaningful: it would correspond to composite objects that cannot be assigned a unique quark content, analogous to the situation in QCD where the quark content of hadrons is not always uniquely determined. Whether unique factorization holds is a structural question about the specific GSLT.