Exact Combinatorics of History Fibers / 历史纤维的精确组合学

ZFCρ Series, Paper 9. Papers 1–8 completed the foundations of the multi-sorted first-order theory Tρ (ρ-arithmetic): syntax, axioms, term model, equiconsistency with PA, and conservative extension. This paper begins the internal structure theory of the term model.

We study the fiber counting problem: for each natural number n, how many compact history terms have extensional value n? The count h*(n) = |Hist*(n)| satisfies a three-branch recurrence (Theorem 1), decomposed by root constructor into successor, additive Cauchy convolution, and truncated multiplicative Dirichlet convolution. Removing the multiplicative branch yields a sequence identical to the large Schröder numbers (Theorem 2), with closed-form generating function and growth rate 3+2√2 ≈ 5.828 (Theorem 3). The full recurrence exhibits a mixed Cauchy–Dirichlet convolution structure without precedent in the combinatorics literature. Supermultiplicativity (from the additive branch) combined with an exponential upper bound (from a linear bound on compact tree node counts) and Fekete's lemma establish that the growth rate limit λ* = lim h*(n)^{1/n} exists and is finite (Theorem 5). Numerical computation to n = 500 suggests λ* ≈ 5.923; the exact value remains open.

Chinese and English versions under one DOI.

Keywords: ρ-arithmetic, term model, history fiber, compact history term, fiber counting, Schröder numbers, Cauchy convolution, Dirichlet convolution, supermultiplicativity, Fekete lemma, generating function, integer complexity, ZFCρ

Related identifiers (IsPartOf):

• 10.5281/zenodo.18914682 (Paper 1)
• 10.5281/zenodo.18927658 (Paper 2)
• 10.5281/zenodo.18929819 (Paper 3)
• 10.5281/zenodo.18930810 (Paper 4)
• 10.5281/zenodo.18934515 (Paper 5)
• 10.5281/zenodo.18934531 (Paper 6)
• 10.5281/zenodo.18943944 (Paper 7)
• 10.5281/zenodo.18952912 (Paper 8)