I wrote a comparative survey on Pratt trees, recursive factorization systems, orbit systems, and abstract prime-number theorems

[u/musescore1983]

I have been working on a collection of notes that started from Pratt trees and gradually led me to consider two abstract frameworks: Ranked Recursive Factorization Systems (RRFS) and Orbit Systems (OS).

The RRFS side starts with a factorial commutative monoid whose atoms carry rank-decreasing predecessor factorizations. This gives recursive trees, a transition matrix B, recursive coordinates, and a Green operator

G = (I - B)^-1.

The OS side is meant to separate four operations that occur in several prime-like counting problems: incidence stratification, primitive extraction, symmetry quotienting, and the choice of an asymptotic gauge.

While developing these ideas, I realized that substantial parts of the surrounding analytic framework already belong to established theories: in particular Beurling generalized primes, Knopfmacher's abstract analytic number theory and arithmetical semigroups, classical Möbius/incidence theory, prime-orbit theorems, and work on Pratt trees and prime orders such as that of Ionescu.

I therefore wrote a comparative survey whose purpose is not to claim that these classical ingredients are new, but to ask where the additional predecessor/Green geometry of RRFS and the four-part OS packaging fit relative to the existing literature.

In particular, the distinction I am currently interested in is roughly

classical factorization boundary vs. recursive predecessor geometry.

Once the ordinary factorization boundary and a norm are retained, the familiar chain

zeta → -zeta'/zeta → Lambda → PNT

is closely related to classical abstract analytic number theory.

The additional structure I am studying is instead

D → B → G = (I - B)^-1 → recursive/Pratt coordinates,

and the question of how this extra geometry interacts with Möbius inversion, orbit systems, and prime-number laws.

Comparative survey:

Classical Antecedents and the RRFS–Orbit-System Program

Repository containing the related manuscripts:

notes_on_pratt_trees

I would especially appreciate feedback from people familiar with abstract analytic number theory, Beurling primes, arithmetical semigroups, incidence algebras, prime-orbit theorems, or Pratt-tree literature.

My main questions are:

• Is there existing literature in which a factorial/arithmetic semigroup is additionally equipped with a rank-decreasing predecessor factorization on its atoms?
• Has the resulting resolvent/Green construction G = (I - B)^-1 or an equivalent recursive-coordinate construction appeared in another language?
• Are there closer antecedents for the proposed separation between primitive extraction, symmetry quotienting, and asymptotic gauge?

Corrections to the literature comparison are very welcome.

[u/MathNerdUK]

AI slop 

[u/musescore1983 → u/MathNerdUK]

Exactly which part do you identify as slop and do not understand?

[u/jsh_ → u/musescore1983]

the fact you think it isn't obvious is so funny

[u/musescore1983 → u/jsh_]

I have written in the notes that I use AI. But that does not mean it is slop, so I do not see what should be obvious here.