Logos Sector (S_L)

The Logos Sector S_L is the unique mixed sector at E_3 where the diagrammatic register and the commitment register coincide. Formally, S_L = { c ∈ Adm_{E_3} | Coh_D(c) = Coh_C(c) }. A content c belongs to S_L iff its proof-validity and stance-stability are mutually witnessing — c is provable iff it is stably committable. S_L is named by its universal property; the name 'Logos' describes the structure, nothing more. It is explicitly not a theological claim.

Categorical invariant. Equaliser of Coh_D and Coh_C inside Adm_{E_3}. The unique mixed sector in the 4+1 decomposition; the only register pair admitting a natural transformation between codomains is (Proof, Stance), and that coincidence defines S_L.

Primary registry anchor: VII.D11

Supporting items: VII.D03, VII.D04, VII.D09, VII.D10, VII.P30

1. I.K0 — Universe Postulate
2. VII.D03 — Diagrammatic Register Reg_D with codomain Proof
3. VII.D04 — Commitment Register Reg_C with codomain Stance
4. VII.D11 — Logos sector S_L — equaliser of Coh_D and Coh_C in Adm_{E_3}
5. VII.P30 — Mediator uniqueness — S_L is the only mixed sector at E_3

S_L is instantiated wherever proof-validity and stance-stability are mutually witnessing — where 'I can prove it' and 'I am willing to live as true' converge on the same content. This is the structural location, not a theological claim. The Logos sector is the meeting point of reasoned demonstration and existential commitment; its dedicated treatment is Book VII, Part X.

Examples:

• A foundational theorem one is willing to stake one's research programme on — content whose Coh_D-witness and Coh_C-witness coincide
• The dignity axiom in its mature form: provable as label-independence (Reg_D) and stably committable as the meta-ethical ground (Reg_C) — S_L content where the two readouts witness each other
• A categorical universal property held both as theorem and as orientation — S_L content where 'provable' and 'lived-as-true' are inseparable

Register codomain: Mixed: the equaliser of Reg_D : K_τ → Proof and Reg_C : K_τ → Stance under their natural transformation. S_L lives weakly in both S_D and S_C.

Manuscript reference: manuscript-sources/book-07/part01/ch05.tex

Status: Planned

Related glossary entries

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