Logos Sector Closure
Logos Sector Closure (VII.D86) is the architectural property that the Logos sector S_L = S_D ∩ S_C is closed under mutual witnessing: a content φ is S_L-admissible iff (i) φ is Reg_D-derivable from the seven axioms and five generators, (ii) φ is Reg_C-stable under the commitment register's admissibility rules, and (iii) the Reg_D-proof and the Reg_C-stance are mutually witnessing — neither derives from the other; they are the same structural datum read through two registers.
τ-Definition
Logos Sector Closure (VII.D86) is the architectural property that the Logos sector S_L = S_D ∩ S_C is closed under mutual witnessing: a content φ is S_L-admissible iff (i) φ is Reg_D-derivable from the seven axioms and five generators, (ii) φ is Reg_C-stable under the commitment register's admissibility rules, and (iii) the Reg_D-proof and the Reg_C-stance are mutually witnessing — neither derives from the other; they are the same structural datum read through two registers.
Categorical invariant. S_L = S_D ∩ S_C with coincidence property: φ ∈ S_L ⟺ (i) Reg_D-validity (derivable from seven axioms + five generators within Reg_D's logical architecture), (ii) Reg_C-stability (admissible under commitment-register rules), (iii) mutual witnessing (Reg_D-proof = Reg_C-ground, no priority either way). The name 'Logos' is a universal-property assignment: it describes structure, nothing more.
Primary registry anchor:
VII.D86
τ-Derivation Chain
-
I.K0— Universe Postulate -
VII.D11— Logos Sector S_L — Coh_D = Coh_C mixed sector at E₃ -
VII.T45— Logos Sector Uniqueness — S_L unique up to natural isomorphism -
VII.D86— Logos Sector (Extended) — closure under mutual witnessing -
VII.T46— Bridge Equivalence at S_L — Reg_D and Reg_C agree there
Lean modules referenced:
TauLib.BookVII.Logos.Sector
Phenomenological Correlate
Logos Sector Closure is instantiated wherever a content's proof and its commitment-stance are mutually witnessing — the proof IS the ground for stance, and the stance IS the proof, with neither derived from the other. Examples: a mathematical theorem one is willing to stake one's professional reputation on (proof and stance coincide); a moral truth that survives both deductive scrutiny and lived commitment.
Examples:
- Mathematical conviction: 'I have proved this and I stand by it' — proof-validity and stance-stability coincide on the same structural datum
- Moral truth lived: 'I cannot consistently prove and yet refuse to stand by what I prove' — Reg_D-proof = Reg_C-ground
- Scientific integrity: a result the researcher both derives and is willing to defend in public discourse — S_L admissibility test
- Failure case: a content that derives in Reg_D but is not stably committable (technically true, ethically unholdable) — outside S_L
Register codomain: S_L (the Logos sector) — the unique closed intersection of Reg_D's proof-content and Reg_C's stance-content; closure holds by mutual witnessing rather than priority of one register over the other
Manuscript reference: manuscript-sources/book-07/part10/ch119.tex
Lean Coverage
Status: Formalized
Module: TauLib.BookVII.Logos.Sector
Lean kind: structure
Lean symbol: Tau.BookVII.Logos.Sector.LogosSectorExtended