Rel0 and Rel1 in NAND Flash Memory

This paper presents the first algebraic reinterpretation of threshold voltage distributions in NAND flash memory using the framework of Relational Algebra.

Three physically distinct states are identified in existing published measurement data: **Rel0** — the erased state, characterized by a broad, continuous threshold voltage distribution; **Rel1** — programmed states, characterized by narrow, discrete voltage peaks; and the **Rel0→Rel1 transition** — the programming operation, identified as a controlled symmetry break.

Five algebraic axioms (A1–A5) from the Relational Algebra framework [14, 15] are shown to be directly and quantitatively consistent with published characterization data spanning 2Y-nm MLC to current-generation 3D QLC NAND flash memory [1, 2, 3].

A quantitative metric — the **Rel0-Index R₀** — is introduced, defined as the ratio of erased-state to mean programmed-state distribution width ($\sigma$). R₀ is analytically derived to scale as $2^n$ across cell generations (SLC through QLC) and confirmed by published voltage margin data [4, 5, 6].

All findings are verified against existing peer-reviewed literature. No new experiments are required for the core claims. One measurement gap — direct characterization of the erased-state distribution — is identified as a target for future experimental work.

**Keywords:** NAND flash memory · threshold voltage distribution · Relational Algebra · Rel0-Index · erased state · multi-level cell · data retention · symmetry break · flash reliability · information theory