Register Completeness (Mediator Uniqueness)

Register completeness is the structural fact that the four canonical registers (Reg_E, Reg_P, Reg_D, Reg_C) exhaust the modes of engagement available to a self-modelling observer at E_3, and that the Logos sector S_L is the unique mixed sector. The Saturation Theorem (E_4 = E_3) closes the register count at four; mediator uniqueness (VII.P30) shows that S_L is the only sector where two registers coincide. There is no fifth register, and no second mixed sector.

Categorical invariant. Two-part completeness statement: (i) exhaustiveness of {Reg_E, Reg_P, Reg_D, Reg_C} at E_3, certified by the Sector Decomposition Theorem (VII.T03); (ii) uniqueness of S_L as the sole mixed sector, certified by the Mediator Uniqueness remark (VII.P30) — the only register pair admitting a natural transformation between codomains is (Proof, Stance).

Primary registry anchor: VII.P30

Supporting items: VII.D01, VII.D02, VII.D03, VII.D04, VII.D11

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.D11 — Logos sector S_L — the unique pair (Proof, Stance) admitting a natural transformation
7. VII.P30 — Mediator uniqueness — S_L is the only mixed sector at E_3

Register completeness is instantiated whenever a candidate fifth mode of engagement is examined and shown to reduce to one of the canonical four — or to a mixed-sector intersection that already exists. Every act of self-modelling reasoning falls under exactly one of: observe (E), do (P), prove (D), live-as-true (C), or — uniquely — prove-and-live-as-true together (L). The completeness is not stipulated; it is a structural theorem.

Examples:

• A claim that initially appears to unify ethics and observation ('this measurement obliges that action') resolves into a Reg_E-readout plus a Reg_P-readout — not a new fifth register
• A purported 'aesthetic' register reduces to Reg_D content (pattern-as-proof) by the diagrammatic-sector synthesis (Book VII, ch. 92)
• A purported 'religious' register reduces to S_L content — proof-and-commitment-coincident — already accounted for by the Logos sector

Register codomain: Meta-statement about the four codomains (Obs, Norm, Proof, Stance) and the unique mixed locus where two of them are naturally transformable into each other

Manuscript reference: manuscript-sources/book-07/part01/ch08.tex

Status: Planned

Related glossary entries

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