Recursive Definition of ρ and Expression Compression Complexity / ρ的递归定义与表达式压缩复杂度

Paper 6 of the ZFCρ series. Paper 5 (DOI: 10.5281/zenodo.18934515) provided structural induction as ρ-arithmetic's proof tool. This paper supplies the measurement tool.

Paper 3's conservation principle is transformed into a recursive definition of ρ on history terms. The compact history-term space Hist*(n) is introduced, excluding trivial operations (add-zero, multiply-by-one, zero-containing multiplication) to make each fiber finite. The first concrete measure ρ_w (weighted node count, α=1, β=2) is provided.

Core results: (1) The recurrence for ρ_E(n) = min ρ_w(h) over Hist*(n), simultaneously considering successor and multiplication paths. (2) The Addition Domination Lemma: addition paths never beat successor paths (when α=1), proved via subadditivity and the successor bound. (3) δ(n) = n − ρ_E(n) is monotonically non-decreasing. (4) The Prime Fixed-Point Theorem: if n is prime, then δ(n) = δ(n−1) — primes are fixed points of δ. This is ρ-arithmetic's first non-trivial number-theoretic theorem. δ measures expression compression complexity, not primality; primes are those numbers that do not increase compression efficiency. The structural correspondence between ρ_E and the Integer Complexity Problem is established. Bilingual CN-EN.

Resource type: Preprint

License: CC BY 4.0

Language: English, Chinese

Keywords: ρ-arithmetic, expression compression complexity, integer complexity, ρ-conservation, recursive definition, compact history terms, prime fixed-point theorem, addition domination, δ-function, number theory, foundations of mathematics

Related identifiers:

• Is supplement to: 10.5281/zenodo.18914682 (Paper 1)
• Is supplement to: 10.5281/zenodo.18927658 (Paper 2)
• Is supplement to: 10.5281/zenodo.18929819 (Paper 3)
• Is supplement to: 10.5281/zenodo.18930810 (Paper 4)
• Is supplement to: 10.5281/zenodo.18934515 (Paper 5)