Register-Enrichment Correspondence

The Register-Enrichment Correspondence is the order-preserving mapping between the four registers of reason and the four enrichment layers of τ: Reg_P ↔ E_0 (pre-structural), Reg_E ↔ E_1 (data), Reg_D ↔ E_2 (formal proof), Reg_C ↔ E_3 (saturation). The correspondence identifies where each register *begins* — its native enrichment layer — not where it terminates. It reveals the structural unity of the diagrammatic and practical sectors as two traversals of the same enrichment hierarchy.

Categorical invariant. Order-preserving bijection between the indexed family {Reg_P, Reg_E, Reg_D, Reg_C} and the enrichment chain E_0 ⊆ E_1 ⊆ E_2 ⊆ E_3. Each enrichment layer presupposes the previous; each register builds on the structural results of the preceding registers in this ordering.

Primary registry anchor: VII.D75

Supporting items: VII.D01, VII.D02, VII.D03, VII.D04

1. I.K0 — Universe Postulate
2. VII.D01 — Empirical Register Reg_E (codomain Obs)
3. VII.D02 — Practical Register Reg_P (codomain Norm)
4. VII.D03 — Diagrammatic Register Reg_D (codomain Proof)
5. VII.D04 — Commitment Register Reg_C (codomain Stance)
6. VII.D75 — Register-Enrichment Correspondence: Reg_P↔E_0, Reg_E↔E_1, Reg_D↔E_2, Reg_C↔E_3

The correspondence is instantiated wherever a single ethical or epistemic content is traced through its four-layer ascent: from pre-structural intuition (E_0, Reg_P moral sense), through empirical observation of consequences (E_1, Reg_E data), through formal derivation (E_2, Reg_D proof), to saturated commitment (E_3, Reg_C postulate). The CI proof programme is the canonical worked example: it carries practical content through all four enrichment layers, identifying *where each register begins*, not where it terminates.

Examples:

• The CI proof programme: moral intuition (E_0, Reg_P) → empirical observation of consequences (E_1, Reg_E) → formal derivation of the CI as j-closed fixed point (E_2, Reg_D) → commitment to Kantian postulates (E_3, Reg_C)
• The diagrammatic synthesis: pre-symbolic pattern (E_0) → symbolic encoding (E_1) → formal proof (E_2) → saturated completion (E_3) — the structural unity-mate of the practical traversal
• Why Reg_P maps to E_0, not E_1: moral intuition ('lying is wrong') is pre-structural — immediate and pre-formal, prior to articulation of reasons — exactly the character that the zeroth enrichment captures

Register codomain: Meta-statement linking each register's codomain (Obs, Norm, Proof, Stance) to its native τ-enrichment level (E_1, E_0, E_2, E_3 respectively)

Manuscript reference: manuscript-sources/book-07/part07/ch93.tex

Status: Planned

Related glossary entries

• MG-R01-empirical-register Empirical Register (Reg_E)