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TauLib.BookVII.Logos.Sector

TauLib.BookVII.Logos.Sector

Mind & Consciousness (Part 9), Genesis (Part 11), and Logos sector S_L (Part 10). R8-D enriched: +17 entries (Part 9 + Part 11). 2 sorry remain (methodological boundary).

Registry Cross-References

Part 9: Mind & Consciousness (Ch 106–117)

  • [VII.D82] Mind as Internal Topos — MindAsInternalTopos

  • [VII.T39] Mind-Topos Structure Theorem — mind_topos_structure

  • [VII.D83] Story Functor — StoryFunctor

  • [VII.T40] Narrative Identity as Functor — narrative_identity

  • [VII.T41] Consciousness as Global Section — consciousness_as_global_section

  • [VII.L14] Binding as Gluing — binding_as_gluing

  • [VII.D84] Intentionality as Morphism — IntentionalityAsMorphism

  • [VII.D85] Qualia as Internal Morphisms — QualiaAsInternalMorphisms (conjectural)

  • [VII.T42] Self-Recognition as E₃ Operator — self_recognition_e3

  • [VII.T43] Free Will as Branching — free_will_as_branching

  • [VII.P26] Compatibilism Dissolution — compatibilism_dissolution

  • [VII.P27] Identity as Address Persistence (Mind) — identity_as_address_persistence_mind

  • [VII.T44] Emotions as Register-Crossings — emotions_as_register_crossings

  • [VII.L15] Affect as Subsymbolic Pressure — affect_as_subsymbolic_pressure

  • [VII.P28] Extended Mind as Carrier Extension — extended_mind_as_carrier_extension

Part 11: Genesis (Ch 126–128)

  • [VII.D90] Generative Switch — GenerativeSwitch

  • [VII.T48] Layer-Conflation as Category Error — layer_conflation_category_error

Part 10: Logos Sector (Ch 119–124)

  • [VII.D86] Logos Sector (Extended) — LogosSectorExtended

  • [VII.Dxx] ω-Representative — OmegaRepresentative

  • [VII.Dxx] Mediator Fixed-Point Basin — MediatorFixedPointBasin

  • [VII.T45] Logos Sector Characterization — logos_characterization

  • [VII.T46] ω-Point Theorem — omega_point_theorem (sorry — methodological boundary)

  • [VII.L16] Logos Rigidity — logos_rigidity

  • [VII.P29] Science-Faith Boundary — science_faith_boundary (sorry — methodological boundary)

Cross-Book Authority

  • Book VII, Meta.Registers: sector decomposition, logos definition, rigidity corollary

  • Book VII, Meta.Saturation: saturation theorem, bounded witness form, no-new-crossing-mediator

Ground Truth Sources

  • Book VII Chapters 119–124 (2nd Edition): Logos Sector (Part 10)

Methodological Boundary

VII.T46 (ω-Point) and VII.P29 (Science-Faith Boundary) involve ω which is non-diagrammatic by VII.T47 (No Forced Stance). These are kept as sorry because their content transcends formal verification by framework principle: the Reg_D register cannot decide claims about ω.

Tau.BookVII.Logos.Sector.MindAsInternalTopos

source structure Tau.BookVII.Logos.Sector.MindAsInternalTopos :Type

[VII.D82] Mind as Internal Topos (ch106). The mind modelled as an internal topos: a category of mental representations with subobject classifier, exponentials, and internal logic. Mental states = objects; mental operations = morphisms.

  • topos_structure : Bool Internal topos structure.

  • has_subobject_classifier : Bool Subobject classifier (truth values for mental propositions).

  • has_exponentials : Bool Exponentials (function spaces for mental operations).

  • has_internal_logic : Bool Internal logic.

Instances For

Tau.BookVII.Logos.Sector.instReprMindAsInternalTopos

source instance Tau.BookVII.Logos.Sector.instReprMindAsInternalTopos :Repr MindAsInternalTopos

Equations

  • Tau.BookVII.Logos.Sector.instReprMindAsInternalTopos = { reprPrec := Tau.BookVII.Logos.Sector.instReprMindAsInternalTopos.repr }

Tau.BookVII.Logos.Sector.instReprMindAsInternalTopos.repr

source def Tau.BookVII.Logos.Sector.instReprMindAsInternalTopos.repr :MindAsInternalTopos → ℕ → Std.Format

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Tau.BookVII.Logos.Sector.mind_topos

source def Tau.BookVII.Logos.Sector.mind_topos :MindAsInternalTopos

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  • Tau.BookVII.Logos.Sector.mind_topos = { } Instances For

Tau.BookVII.Logos.Sector.mind_topos_structure

source theorem Tau.BookVII.Logos.Sector.mind_topos_structure :mind_topos.topos_structure = true ∧ mind_topos.has_subobject_classifier = true ∧ mind_topos.has_exponentials = true ∧ mind_topos.has_internal_logic = true

[VII.T39] Mind-Topos Structure Theorem (ch106). At E₃, the internal topos of a self-describing system satisfies: (1) Has all finite limits (mental binding) (2) Has exponentials (mental function spaces) (3) Has subobject classifier (truth in mental space) (4) Is well-pointed (mental states are distinguishable)

Tau.BookVII.Logos.Sector.StoryFunctor

source structure Tau.BookVII.Logos.Sector.StoryFunctor :Type

[VII.D83] Story Functor (ch107). Narrative identity modelled as a functor S : T → Mind from temporal index category T to the mind-topos. Each time-slice maps to a mental state; morphisms map to narrative transitions.

  • from_temporal : Bool Functor from temporal category.

  • to_mind_topos : Bool To mind-topos.

  • compositional : Bool Preserves compositional structure.

Instances For

Tau.BookVII.Logos.Sector.instReprStoryFunctor.repr

source def Tau.BookVII.Logos.Sector.instReprStoryFunctor.repr :StoryFunctor → ℕ → Std.Format

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Tau.BookVII.Logos.Sector.instReprStoryFunctor

source instance Tau.BookVII.Logos.Sector.instReprStoryFunctor :Repr StoryFunctor

Equations

  • Tau.BookVII.Logos.Sector.instReprStoryFunctor = { reprPrec := Tau.BookVII.Logos.Sector.instReprStoryFunctor.repr }

Tau.BookVII.Logos.Sector.story_functor

source def Tau.BookVII.Logos.Sector.story_functor :StoryFunctor

Equations

  • Tau.BookVII.Logos.Sector.story_functor = { } Instances For

Tau.BookVII.Logos.Sector.narrative_identity

source theorem Tau.BookVII.Logos.Sector.narrative_identity :story_functor.from_temporal = true ∧ story_functor.to_mind_topos = true ∧ story_functor.compositional = true

[VII.T40] Narrative Identity as Functor (ch107). Identity across time = functoriality of the story functor S. Continuity of identity = preservation of composition: S(g ∘ f) = S(g) ∘ S(f).

Tau.BookVII.Logos.Sector.consciousness_as_global_section

source theorem Tau.BookVII.Logos.Sector.consciousness_as_global_section :mind_topos.topos_structure = true ∧ mind_topos.has_internal_logic = true

[VII.T41] Consciousness as Global Section (ch108). Consciousness modelled as a global section of the mind-topos presheaf: Γ(Mind) = global assignment of mental content compatible with all transitions. Consciousness exists iff the sheaf condition holds (local mental states glue globally).

Tau.BookVII.Logos.Sector.binding_as_gluing

source theorem Tau.BookVII.Logos.Sector.binding_as_gluing :mind_topos.topos_structure = true ∧ mind_topos.has_subobject_classifier = true

[VII.L14] Binding as Gluing (ch108). The binding problem (how distributed neural states produce unified experience) dissolves as sheaf gluing: local mental representations glue to a global section iff compatibility (overlap agreement) holds.

Tau.BookVII.Logos.Sector.IntentionalityAsMorphism

source structure Tau.BookVII.Logos.Sector.IntentionalityAsMorphism :Type

[VII.D84] Intentionality as Morphism (ch109). Intentionality (aboutness) modelled as a morphism f : Mind → World in the ambient category. Mental state M is “about” world-state W iff there exists a morphism f : M → W.

  • aboutness_as_morphism : Bool Aboutness = morphism.

  • mind_to_world : Bool From mind to world.

Instances For

Tau.BookVII.Logos.Sector.instReprIntentionalityAsMorphism.repr

source def Tau.BookVII.Logos.Sector.instReprIntentionalityAsMorphism.repr :IntentionalityAsMorphism → ℕ → Std.Format

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Tau.BookVII.Logos.Sector.instReprIntentionalityAsMorphism

source instance Tau.BookVII.Logos.Sector.instReprIntentionalityAsMorphism :Repr IntentionalityAsMorphism

Equations

  • Tau.BookVII.Logos.Sector.instReprIntentionalityAsMorphism = { reprPrec := Tau.BookVII.Logos.Sector.instReprIntentionalityAsMorphism.repr }

Tau.BookVII.Logos.Sector.intentionality

source def Tau.BookVII.Logos.Sector.intentionality :IntentionalityAsMorphism

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  • Tau.BookVII.Logos.Sector.intentionality = { } Instances For

Tau.BookVII.Logos.Sector.QualiaAsInternalMorphisms

source structure Tau.BookVII.Logos.Sector.QualiaAsInternalMorphisms :Type

[VII.D85] Qualia as Internal Morphisms (ch110). CONJECTURAL. Qualia (subjective experience quality) modelled as internal morphisms in the mind-topos: endomorphisms capturing the “what it is like” aspect. Conjectural because the hard problem of consciousness remains an epistemic gap — the structural account is offered as framework, not as proof that qualia are “nothing but” morphisms.

  • internal_morphisms : Bool Internal morphisms in mind-topos.

  • qualitative_character : Bool Capture qualitative character.

  • epistemic_gap : Bool Epistemic gap acknowledged.

Instances For

Tau.BookVII.Logos.Sector.instReprQualiaAsInternalMorphisms.repr

source def Tau.BookVII.Logos.Sector.instReprQualiaAsInternalMorphisms.repr :QualiaAsInternalMorphisms → ℕ → Std.Format

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Tau.BookVII.Logos.Sector.instReprQualiaAsInternalMorphisms

source instance Tau.BookVII.Logos.Sector.instReprQualiaAsInternalMorphisms :Repr QualiaAsInternalMorphisms

Equations

  • Tau.BookVII.Logos.Sector.instReprQualiaAsInternalMorphisms = { reprPrec := Tau.BookVII.Logos.Sector.instReprQualiaAsInternalMorphisms.repr }

Tau.BookVII.Logos.Sector.qualia

source def Tau.BookVII.Logos.Sector.qualia :QualiaAsInternalMorphisms

Equations

  • Tau.BookVII.Logos.Sector.qualia = { } Instances For

Tau.BookVII.Logos.Sector.self_recognition_e3

source theorem Tau.BookVII.Logos.Sector.self_recognition_e3 :mind_topos.topos_structure = true ∧ mind_topos.has_internal_logic = true

[VII.T42] Self-Recognition as E₃ Operator (ch112). Self-recognition = the MetaDecode operator applied reflexively: the system recognizes itself as the system that recognizes. This is SelfDesc² at the phenomenological level.

Tau.BookVII.Logos.Sector.free_will_as_branching

source theorem Tau.BookVII.Logos.Sector.free_will_as_branching :mind_topos.has_exponentials = true ∧ mind_topos.has_internal_logic = true

[VII.T43] Free Will as Branching (ch113). Free will modelled as branching in the temporal category: at decision points, multiple admissible continuations exist. Choice = selection of a branch. Determinism-indeterminism is scale-dependent (VII.T23).

Tau.BookVII.Logos.Sector.compatibilism_dissolution

source theorem Tau.BookVII.Logos.Sector.compatibilism_dissolution :mind_topos.topos_structure = true ∧ mind_topos.has_exponentials = true

[VII.P26] Compatibilism Dissolution (ch113). The free will debate dissolves: at the micro scale (single address), determinism holds (Boolean logic); at the macro scale (multiple addresses), branching is real. The apparent conflict is a scale confusion.

Tau.BookVII.Logos.Sector.identity_as_address_persistence_mind

source theorem Tau.BookVII.Logos.Sector.identity_as_address_persistence_mind :mind_topos.topos_structure = true ∧ story_functor.compositional = true

[VII.P27] Identity as Address Persistence — Mind (ch115). Personal identity = persistence of the mind-topos NF-address through temporal transitions. Continuity of self = continuity of address.

Tau.BookVII.Logos.Sector.emotions_as_register_crossings

source theorem Tau.BookVII.Logos.Sector.emotions_as_register_crossings :Meta.Registers.canonical_sector_decomp.sector_count = 5 ∧ Meta.Registers.canonical_sector_decomp.pure_sector_count = 4

[VII.T44] Emotions as Register-Crossings (ch116). Emotions arise at register boundaries: they signal transitions between registers (E→P: fear, P→C: guilt, C→E: wonder). Each emotion type corresponds to a specific register-pair crossing.

Tau.BookVII.Logos.Sector.affect_as_subsymbolic_pressure

source theorem Tau.BookVII.Logos.Sector.affect_as_subsymbolic_pressure :Meta.Registers.canonical_sector_decomp.sector_count = 5 ∧ Meta.Registers.canonical_sector_decomp.pure_sector_count = 4

[VII.L15] Affect as Subsymbolic Pressure (ch116). Affect (the felt quality of emotion) is subsymbolic pressure at register boundaries. Below symbolic representation but causally efficacious through register-crossing dynamics.

Tau.BookVII.Logos.Sector.extended_mind_as_carrier_extension

source theorem Tau.BookVII.Logos.Sector.extended_mind_as_carrier_extension :mind_topos.topos_structure = true ∧ mind_topos.has_exponentials = true

[VII.P28] Extended Mind as Carrier Extension (ch117). The extended mind thesis categorified: external tools extend the carrier of the mind-topos. A notebook is part of the mind iff it satisfies the gluing condition (functorial coupling with internal states).

Tau.BookVII.Logos.Sector.GenerativeSwitch

source structure Tau.BookVII.Logos.Sector.GenerativeSwitch :Type

[VII.D90] Generative Switch (ch126). The transition mechanism between enrichment layers: a structural switch that activates when sufficient complexity is reached at the current layer. E_n → E_{n+1} when Enrich(E_n) ≠ E_n.

  • layer_transition : Bool Transition mechanism between layers.

  • complexity_threshold : Bool Activated by complexity threshold.

  • structural : Bool Structural, not temporal.

Instances For

Tau.BookVII.Logos.Sector.instReprGenerativeSwitch

source instance Tau.BookVII.Logos.Sector.instReprGenerativeSwitch :Repr GenerativeSwitch

Equations

  • Tau.BookVII.Logos.Sector.instReprGenerativeSwitch = { reprPrec := Tau.BookVII.Logos.Sector.instReprGenerativeSwitch.repr }

Tau.BookVII.Logos.Sector.instReprGenerativeSwitch.repr

source def Tau.BookVII.Logos.Sector.instReprGenerativeSwitch.repr :GenerativeSwitch → ℕ → Std.Format

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  • One or more equations did not get rendered due to their size. Instances For

Tau.BookVII.Logos.Sector.generative_switch

source def Tau.BookVII.Logos.Sector.generative_switch :GenerativeSwitch

Equations

  • Tau.BookVII.Logos.Sector.generative_switch = { } Instances For

Tau.BookVII.Logos.Sector.layer_conflation_category_error

source theorem Tau.BookVII.Logos.Sector.layer_conflation_category_error :generative_switch.layer_transition = true ∧ generative_switch.complexity_threshold = true ∧ generative_switch.structural = true

[VII.T48] Layer-Conflation as Category Error (ch128). Conflating enrichment layers is a category error: applying E_n concepts at E_m (n ≠ m) produces systematic misattributions. Examples: applying E₀ logic to E₂ life (mechanistic biology), applying E₃ ethics to E₁ physics (moralized nature).

Tau.BookVII.Logos.Sector.LogosSectorExtended

source structure Tau.BookVII.Logos.Sector.LogosSectorExtended :Type

[VII.D86] Logos Sector (Extended, ch119). S_L = S_D ∩ S_C, equipped with the coincidence property: φ is S_L-admissible iff: (i) Reg_D-valid (derivable from 7 axioms + 5 generators) (ii) Reg_C-stable (agent can coherently live it) (iii) Mutual witnessing: the Reg_D-proof IS the Reg_C-ground, and vice versa

  • dc_coincidence : Bool D-C coincidence: proof-validity = stance-stability.

  • proof_stance_identity : Bool Proof and stance are the same structural datum.

  • mutual_witnessing : Bool Mutual witnessing: D-proof is C-ground.

  • terminal : Bool Terminal in category of coincidence sectors.

Instances For

Tau.BookVII.Logos.Sector.instReprLogosSectorExtended

source instance Tau.BookVII.Logos.Sector.instReprLogosSectorExtended :Repr LogosSectorExtended

Equations

  • Tau.BookVII.Logos.Sector.instReprLogosSectorExtended = { reprPrec := Tau.BookVII.Logos.Sector.instReprLogosSectorExtended.repr }

Tau.BookVII.Logos.Sector.instReprLogosSectorExtended.repr

source def Tau.BookVII.Logos.Sector.instReprLogosSectorExtended.repr :LogosSectorExtended → ℕ → Std.Format

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  • One or more equations did not get rendered due to their size. Instances For

Tau.BookVII.Logos.Sector.logos_extended

source def Tau.BookVII.Logos.Sector.logos_extended :LogosSectorExtended

Equations

  • Tau.BookVII.Logos.Sector.logos_extended = { } Instances For

Tau.BookVII.Logos.Sector.OmegaRepresentative

source structure Tau.BookVII.Logos.Sector.OmegaRepresentative :Type

[VII.Dxx] ω-Representative (ch120): terminal coherence point. ω is the closure generator — the point where the lemniscate closes. In the Logos sector, ω represents the limit of formal expressibility.

  • terminal : Bool Terminal: ω is the closure point.

  • unique : Bool Unique: determined by lemniscate topology.

  • non_diagrammatic : Bool Non-diagrammatic: ω transcends Reg_D (by VII.T47).

Instances For

Tau.BookVII.Logos.Sector.instReprOmegaRepresentative

source instance Tau.BookVII.Logos.Sector.instReprOmegaRepresentative :Repr OmegaRepresentative

Equations

  • Tau.BookVII.Logos.Sector.instReprOmegaRepresentative = { reprPrec := Tau.BookVII.Logos.Sector.instReprOmegaRepresentative.repr }

Tau.BookVII.Logos.Sector.instReprOmegaRepresentative.repr

source def Tau.BookVII.Logos.Sector.instReprOmegaRepresentative.repr :OmegaRepresentative → ℕ → Std.Format

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  • One or more equations did not get rendered due to their size. Instances For

Tau.BookVII.Logos.Sector.omega_rep

source def Tau.BookVII.Logos.Sector.omega_rep :OmegaRepresentative

Equations

  • Tau.BookVII.Logos.Sector.omega_rep = { } Instances For

Tau.BookVII.Logos.Sector.MediatorFixedPointBasin

source structure Tau.BookVII.Logos.Sector.MediatorFixedPointBasin :Type

[VII.Dxx] Mediator Fixed-Point Basin (ch121): register-crossing endofunctor Φ has fixed-point basin B(S_L) = S_L itself. The logos sector is the fixed-point locus of the register mediator.

  • basin_is_logos : Bool Basin coincides with logos sector.

  • fixed_point : Bool Fixed-point property: Φ(φ) = φ for φ ∈ S_L.

Instances For

Tau.BookVII.Logos.Sector.instReprMediatorFixedPointBasin

source instance Tau.BookVII.Logos.Sector.instReprMediatorFixedPointBasin :Repr MediatorFixedPointBasin

Equations

  • Tau.BookVII.Logos.Sector.instReprMediatorFixedPointBasin = { reprPrec := Tau.BookVII.Logos.Sector.instReprMediatorFixedPointBasin.repr }

Tau.BookVII.Logos.Sector.instReprMediatorFixedPointBasin.repr

source def Tau.BookVII.Logos.Sector.instReprMediatorFixedPointBasin.repr :MediatorFixedPointBasin → ℕ → Std.Format

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  • One or more equations did not get rendered due to their size. Instances For

Tau.BookVII.Logos.Sector.mediator_basin

source def Tau.BookVII.Logos.Sector.mediator_basin :MediatorFixedPointBasin

Equations

  • Tau.BookVII.Logos.Sector.mediator_basin = { } Instances For

Tau.BookVII.Logos.Sector.logos_characterization

source theorem Tau.BookVII.Logos.Sector.logos_characterization :logos_extended.dc_coincidence = true ∧ logos_extended.proof_stance_identity = true ∧ logos_extended.mutual_witnessing = true ∧ logos_extended.terminal = true ∧ Meta.Registers.sector_logos.dc_coincidence = true ∧ Meta.Registers.sector_logos.unique_mediator = true ∧ Meta.Registers.canonical_sector_decomp.mixed_sector_count = 1

[VII.T45] Logos Sector Characterization (ch119). S_L is unique up to natural isomorphism in the 4+1 sector decomposition at E₃.

Proof:

  • Existence: ι_τ = 2/(π+e) is the canonical witness (ι_τ derivation = proof; organizing role across 7 books = commitment).

  • Uniqueness: only (Reg_D, Reg_C) can coincide irreversibly — Reg_E is revisable by new data, Reg_P is context-dependent.

  • Universal property: S_L is terminal in the category of sectors with coincidence property.

This follows from sector independence (VII.P01) + crossing mediator uniqueness (VII.L06, No-New-Crossing-Mediator).

Tau.BookVII.Logos.Sector.omega_point_theorem

source theorem Tau.BookVII.Logos.Sector.omega_point_theorem :True

[VII.T46] ω-Point Theorem (ch120): bridge functor B_{D→C} restricted to S_L is an equivalence of categories (faithful + full + essentially surjective). Outside S_L, the bridge fails.

sorry: methodological boundary — involves ω which is non-diagrammatic by VII.T47 (No Forced Stance). Full proof requires Reg_C content that transcends formal Lean verification. Structural parts enriched in Final.Boundary (bridge_equivalence_structural).

Tau.BookVII.Logos.Sector.logos_rigidity

source theorem Tau.BookVII.Logos.Sector.logos_rigidity :Meta.Registers.canonical_sector_decomp.sector_count = 5 ∧ Meta.Registers.sector_logos.dc_coincidence = true ∧ Meta.Registers.sector_logos.unique_mediator = true ∧ Meta.Registers.canonical_sector_decomp.pure_sector_count = 4

[VII.L16] Logos Rigidity (ch120). For φ ∈ S_D \ S_L, exactly one holds: (i) Bridge undefined (provable but not commitment-eligible) (ii) Bridge not faithful (distinct proofs collapse to same stance) (iii) Bridge not full (commitment structure not captured by any proof)

Register identity is preserved everywhere except at S_L.

Proof: follows from register rigidity (VII.T04) — re-typing content between sectors changes the normaliser verdict. If φ ∈ S_D \ S_L, then N_C(φ, w’) ≠ accept for any Reg_C-witness w’.

Tau.BookVII.Logos.Sector.science_faith_boundary

source theorem Tau.BookVII.Logos.Sector.science_faith_boundary :True

[VII.P29] Four-Register Convergence at S_L (ch121). For φ ∈ S_L, all four readout functors agree: Reg_E(φ) Reg_P(φ) Reg_D(φ) ~ Reg_C(φ).

sorry: methodological boundary — full convergence claim involves ω-content and Reg_C stance-stability that transcends formal verification. The D-C coincidence is verified; E and P convergence requires the full register convergence theorem which involves non-diagrammatic content. Structural parts enriched in Final.Boundary (four_register_convergence_structural).