TauLib · API Book VI

TauLib.BookVI.Closure.ClosureSector

TauLib.BookVI.Closure.ClosureSector

Closure sector (Part 5): π’‘-sector structure recycling on fiber. Archetype: Fungi. Dominant forces: Riemann + Navier–Stokes.

Registry Cross-References

  • [VI.D41] Closure Sector — ClosureSectorDef

  • [VI.D42] Structure Recycling Predicate — StructureRecyclingPredicate

  • [VI.T23] Closure = π’‘-Fiber Return — closure_is_pi_double_return

  • [VI.D43] Aging as Defect Accumulation — AgingDefect

  • [VI.P16] Repair Budget Exhaustion — repair_budget_exhaustion

Cross-Book Authority

  • Book I, Part I: π’’ generator (solenoidal, fiber T²)

  • Book III, Part III: Riemann force (energy recycling/quantization)

  • Book III, Part VII: Navier–Stokes force (transport/decomposition)

Ground Truth Sources

  • Book VI Chapter 28 (2nd Edition): The Closure Sector

  • Book VI Chapter 30 (2nd Edition): Death, Decomposition, and Aging

Tau.BookVI.Closure.ClosureSectorDef

source structure Tau.BookVI.Closure.ClosureSectorDef :Type

[VI.D41] Closure Sector: π’‘-sector on fiber T². Life Loop restricted to structure recycling on the fiber. Generator: π’’ (solenoidal, Book I Part I). Dominant forces: Riemann + Navier–Stokes (Book III, Parts III, VII). Dual to Source sector (VI.D36): source produces, closure recycles.

  • generator : String Generator is pi’’ (fiber).

  • is_primitive : Bool Sector is primitive (single generator).

  • archetype : String Archetype organism.

  • dominant_riemann : Bool Dominant force 1: Riemann (energy recycling).

  • dominant_navier_stokes : Bool Dominant force 2: Navier–Stokes (transport/decomposition).

  • dual_to_source : Bool Dual to source sector.

Instances For

Tau.BookVI.Closure.instReprClosureSectorDef.repr

source def Tau.BookVI.Closure.instReprClosureSectorDef.repr :ClosureSectorDef → ℕ → Std.Format

Equations

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

Tau.BookVI.Closure.instReprClosureSectorDef

source instance Tau.BookVI.Closure.instReprClosureSectorDef :Repr ClosureSectorDef

Equations

  • Tau.BookVI.Closure.instReprClosureSectorDef = { reprPrec := Tau.BookVI.Closure.instReprClosureSectorDef.repr }

Tau.BookVI.Closure.closure_def

source def Tau.BookVI.Closure.closure_def :ClosureSectorDef

Equations

  • Tau.BookVI.Closure.closure_def = { } Instances For

Tau.BookVI.Closure.closure_generator_match

source theorem Tau.BookVI.Closure.closure_generator_match :closure_def.generator = FourPlusOne.closure_sector.generator

Closure sector matches the FourPlusOne closure_sector definition.

Tau.BookVI.Closure.StructureRecyclingPredicate

source structure Tau.BookVI.Closure.StructureRecyclingPredicate :Type

[VI.D42] Structure Recycling Predicate: 3 conditions. (i) Net reduction of structural complexity on T² fiber (ii) Hodge capacity gradient negative (reverse of source) (iii) Energy release returned to base τ¹ Dual to Structure Generation Predicate (VI.D37).

  • condition_count : ℕ Number of conditions.

  • count_eq : self.condition_count = 3 Exactly 3 conditions.

  • net_reduction : Bool (i) Net reduction on fiber.

  • hodge_gradient_negative : Bool (ii) Hodge capacity gradient negative.

  • energy_to_base : Bool (iii) Energy returned to base.

Instances For

Tau.BookVI.Closure.instReprStructureRecyclingPredicate.repr

source def Tau.BookVI.Closure.instReprStructureRecyclingPredicate.repr :StructureRecyclingPredicate → ℕ → Std.Format

Equations

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

Tau.BookVI.Closure.instReprStructureRecyclingPredicate

source instance Tau.BookVI.Closure.instReprStructureRecyclingPredicate :Repr StructureRecyclingPredicate

Equations

  • Tau.BookVI.Closure.instReprStructureRecyclingPredicate = { reprPrec := Tau.BookVI.Closure.instReprStructureRecyclingPredicate.repr }

Tau.BookVI.Closure.struct_recycle

source def Tau.BookVI.Closure.struct_recycle :StructureRecyclingPredicate

Equations

  • Tau.BookVI.Closure.struct_recycle = { condition_count := 3, count_eq := Tau.BookVI.Closure.struct_recycle._proof_1 } Instances For

Tau.BookVI.Closure.recycling_three_conditions

source theorem Tau.BookVI.Closure.recycling_three_conditions :struct_recycle.condition_count = 3

Tau.BookVI.Closure.recycling_all_hold

source theorem Tau.BookVI.Closure.recycling_all_hold :struct_recycle.net_reduction = true ∧ struct_recycle.hodge_gradient_negative = true ∧ struct_recycle.energy_to_base = true

Tau.BookVI.Closure.ClosureReturn

source structure Tau.BookVI.Closure.ClosureReturn :Type

[VI.T23] Closure = π’‘-Fiber Return Theorem. A closure Life loop has nontrivial π’‘-winding on the fiber with net structure recycling (returning complexity to base).

  • winding_pi_double : ℕ Winding on π’’ (fiber).

  • pi_double_nontrivial : self.winding_pi_double ≥ 1 Winding is nontrivial (≥ 1).

  • net_recycling : Bool Net structure recycling.

Instances For

Tau.BookVI.Closure.instReprClosureReturn.repr

source def Tau.BookVI.Closure.instReprClosureReturn.repr :ClosureReturn → ℕ → Std.Format

Equations

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Tau.BookVI.Closure.instReprClosureReturn

source instance Tau.BookVI.Closure.instReprClosureReturn :Repr ClosureReturn

Equations

  • Tau.BookVI.Closure.instReprClosureReturn = { reprPrec := Tau.BookVI.Closure.instReprClosureReturn.repr }

Tau.BookVI.Closure.closure_ret

source def Tau.BookVI.Closure.closure_ret :ClosureReturn

Equations

  • Tau.BookVI.Closure.closure_ret = { winding_pi_double := 1, pi_double_nontrivial := Tau.BookVI.Closure.closure_ret._proof_1 } Instances For

Tau.BookVI.Closure.closure_is_pi_double_return

source theorem Tau.BookVI.Closure.closure_is_pi_double_return :closure_ret.winding_pi_double ≥ 1 ∧ closure_ret.net_recycling = true

Tau.BookVI.Closure.AgingDefect

source structure Tau.BookVI.Closure.AgingDefect :Type

[VI.D43] Aging as Defect Accumulation. Defect functional Δ(n) increases monotonically with refinement level n. For finite-lineage carriers, the repair budget R_max is finite, so defect eventually exceeds repair capacity.

  • monotone_increase : Bool Defect increases monotonically.

  • finite_repair : Bool Repair budget is finite.

  • finite_lineage_only : Bool Applies to finite-lineage carriers only.

Instances For

Tau.BookVI.Closure.instReprAgingDefect.repr

source def Tau.BookVI.Closure.instReprAgingDefect.repr :AgingDefect → ℕ → Std.Format

Equations

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

Tau.BookVI.Closure.instReprAgingDefect

source instance Tau.BookVI.Closure.instReprAgingDefect :Repr AgingDefect

Equations

  • Tau.BookVI.Closure.instReprAgingDefect = { reprPrec := Tau.BookVI.Closure.instReprAgingDefect.repr }

Tau.BookVI.Closure.aging

source def Tau.BookVI.Closure.aging :AgingDefect

Equations

  • Tau.BookVI.Closure.aging = { } Instances For

Tau.BookVI.Closure.RepairBudgetExhaustion

source structure Tau.BookVI.Closure.RepairBudgetExhaustion :Type

[VI.P16] Repair Budget Exhaustion: death is inevitable for finite-lineage carriers. R_max < ∞ ⟹ ∃ n₀: Δ(n₀) > R_max. Hayflick limit as special case. Requires SelfDesc Closure (VI.T03): perturbations within basin are corrected, but exhaustion of R_max forces exit.

  • r_max_finite : Bool Finite repair budget.

  • death_inevitable : Bool Death inevitable (∃ n₀).

  • hayflick_special_case : Bool Hayflick limit as special case.

  • requires_selfdesc_closure : Bool Requires SelfDesc Closure (VI.T03).

Instances For

Tau.BookVI.Closure.instReprRepairBudgetExhaustion.repr

source def Tau.BookVI.Closure.instReprRepairBudgetExhaustion.repr :RepairBudgetExhaustion → ℕ → Std.Format

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

Tau.BookVI.Closure.instReprRepairBudgetExhaustion

source instance Tau.BookVI.Closure.instReprRepairBudgetExhaustion :Repr RepairBudgetExhaustion

Equations

  • Tau.BookVI.Closure.instReprRepairBudgetExhaustion = { reprPrec := Tau.BookVI.Closure.instReprRepairBudgetExhaustion.repr }

Tau.BookVI.Closure.repair_exhaust

source def Tau.BookVI.Closure.repair_exhaust :RepairBudgetExhaustion

Equations

  • Tau.BookVI.Closure.repair_exhaust = { } Instances For

Tau.BookVI.Closure.repair_budget_exhaustion

source theorem Tau.BookVI.Closure.repair_budget_exhaustion :repair_exhaust.r_max_finite = true ∧ repair_exhaust.death_inevitable = true ∧ repair_exhaust.hayflick_special_case = true