A Draft Term Model for ρ-Arithmetic: First Steps toward a History-Preserving Arithmetic Framework / ρ-算术的项模型草案:保留操作历史的算术框架初探

Paper 4 of the ZFCρ series. Papers 1–3 completed the diagnostic chain: remainder exists (Paper 1), remainder has quantitative identity (Paper 2), remainder is conserved (Paper 3). This paper takes the first constructive step: a draft term model for ρ-arithmetic.

Ordinary arithmetic (Peano Arithmetic) takes natural numbers as objects; operations output only results, discarding process. ρ-arithmetic takes history terms as objects — formal terms encoding the complete generative path from zero to the current value. The paper defines: (1) a recursive grammar for history terms; (2) an evaluation function val(h) = n; (3) two-level equality — history equality (term isomorphism) and extensional equality (value equality); (4) ρ-addition and ρ-multiplication with merge semantics; (5) ρ-conservation as a built-in constraint. Standard algebraic laws (commutativity, associativity, distributivity) hold under extensional equality but fail under history equality — this is not a defect but the core feature. A roadmap from draft to complete axiomatic system (M1 syntax → M2 axiom schema → M3 model theory → M4 relative consistency) is provided. Bilingual CN-EN.

Resource type: Preprint

License: CC BY 4.0

Language: English, Chinese

Keywords: ρ-arithmetic, history terms, term model, free term algebra, operative history, extensionalization, two-level equality, ρ-conservation, foundations of mathematics, philosophy of mathematics, Peano arithmetic, set theory, ZFC, primes, Goldbach conjecture, twin primes, structural induction, compression preorder

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)