Description: Paper 3 of the ZFCρ series. Paper 1 (DOI: 10.5281/zenodo.18914682) established the remainder thesis: ρ≠∅, remainder is ineliminable, only displaceable. Paper 2 (DOI: 10.5281/zenodo.18927658) provided the quantitative identity of the remainder: π as the first complete specimen, Euler's formula as "one act binding two remainders to produce closure," and the research program of self-referential generation unfolding across layers. This paper proposes the ρ-Conservation Principle: grounded in Paper 1's ρ-thesis and the premise "extensionalization does not create information," remainder is neither created nor destroyed under any extensionalization or layer transition — it is only redistributed between remainder (extensional product) and act (operative performance). Applications include: Gödel incompleteness re-read as ρ-overflow; proof complexity as ρ-conservation cost (illustrated by Fermat's Last Theorem and the Poincaré Conjecture); and a working ρ-classification and pairing of six major open problems (Goldbach, Twin Primes, Riemann, Collatz, Navier-Stokes, P≠NP). Bilingual CN-EN. Resource type: Preprint License: CC BY 4.0 Language: English, Chinese Keywords: remainder, conservation law, ρ-conservation, ZFC, extensionality, formalization, meta-theory, philosophy of mathematics, foundations of mathematics, Gödel incompleteness, proof complexity, Goldbach conjecture, twin primes, Riemann hypothesis, Collatz conjecture, Navier-Stokes, P≠NP, self-referential generation, operative content, ρ-arithmetic Related identifiers: Is supplement to: 10.5281/zenodo.18914682 (Paper 1) Is supplement to: 10.5281/zenodo.18927658 (Paper 2)
Han Qin (Mon,) studied this question.